Document [original]

Towards ab initio assisted materials design:

DFT based thermodynamics up to the melting point

Dissertation

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften (Dr. rer. nat.)

vorgelegt dem

Department Physik der Fakult¨at f¨ur Naturwissenschaften

an der Universit¨at Paderborn

Blazej Grabowski

Promotionskommission

Vorsitzender Prof. Dr. phil. Klaus Lischka

Gutachter Prof. Dr. rer. nat. Wolf Gero Schmidt

Gutachter Prof. Dr. rer. nat. J¨org Neugebauer

Tag der Einreichung: 20. Oktober, 2009

Tag der m¨undlichen Pr¨ufung: 08. Dezember, 2009

To Anna, Maximilian, Christoph, Barbara, and Stanislaw.

Acknowledgments

I would like to express my gratitude to Prof. Dr. J¨org Neugebauer. It is a pleasure to know that one

of the leading experts has given you the chance to benefit from his knowledge about such a highly

complex field of physics. His influence goes however beyond this specific field. He taught me to

identify general concepts, to efficiently approach physical problems, and, in particular, that physics

is not merely the research itself but the way of how you ”bring” it to other people. I am also deeply

indebted to my supervisor Tilmann Hickel for his continuous support and encouragement. We

had numerous discussions which stimulated my research and which are the basis for the major part

of this work. Tilmann also helped me in extensive detail in improving the written part of the thesis.

I would like to thank Sixten Boeck who spent much time in introducing me into the field of

computational physics. His knowledge was the methodological basis allowing me to implement the

various concepts and approaches I have developed. Without the help of Lars Ismer, I could have

not accomplished the very intricate molecular dynamics simulations of the classical anharmonic

contribution. In particular, the methods he developed in a previous study were of fundamental

importance. Fritz K¨ormann contributed a great part of work and knowledge to the quantum me-

chanical anharmonicity investigations. I thank him also for proof reading of my thesis and helping

me in optimizing my figures. I am especially grateful to Alexander Udyansky. Alexander has not

only supported me from a scientific point of view, but he also strongly and continuously encouraged

me from a rather personal side.

My wife is an irreplaceable part of my life. Writing this thesis has brought us even closer

together, since in tough times you really recognize how much you can rely on your partner. She

did not only take over any other work and responsibility during this time, moreover she helped me

with loving detail in correcting and proof reading the thesis. I am absolutely obliged to say ”Thank

you for everything” to my family, in particular, to my parents. They have always put the needs of

their children above their own needs. This allowed us to receive a good education. My parents are

therefore the real basis of this thesis.

Abstract

Metals are ubiquitous in numerous aspects of our lives, due to the possibility of adjusting their

properties to a wide range of technological needs (light-weight, high-strength, durability, etc.). The

ability to provide such a large variability in metallic properties is based on a profound knowledge

gathered over centuries of research. One might, therefore, be tempted to consider the task of metal

research as being completed in general. This is however by far not the case. The reason is the

tremendous complexity of such materials at microscopic scales, which is actually the origin of their

versatility. This complexity is still poorly understood and most traditional approaches are facing

fundamental difficulties in providing further progress.

A very recent approach to enable further progress is so called ab initio methods. Their basic

idea is to start the materials description directly at the electronic scale, in contrast to many

traditional approaches which focus on the meso-/macroscopic scale. The key advantage of ab initio

methods is their derivation from universal quantum mechanical laws which allow, in principle, to

fully incorporate the complexity inherent to metallic materials. The actual application of these

methods faces serious challenges: (i) A direct quantum mechanical solution is not feasible and

approximations are unavoidable. For instance, the density functional theory (DFT), a particularly

successful ab initio approach, relies in practical applications on the so called exchange-correlation

functional, which cannot be systematically improved. (ii) Despite various approximations, ab initio

calculations are computationally highly demanding and the development of advanced simulation

techniques is needed. (iii) Typically, only T=0 K conditions are considered and the extension to

finite temperatures – crucial for metals – is related to even larger CPU requirements.

The general objective of the present work is to address various aspects related to these chal-

lenges. For that purpose, a systematic and with respect to numerical accuracy fully controlled DFT

study of thermodynamic properties for an extensive set of metals is provided. A special focus is

on the assessment of the predictive power of present day’s exchange-correlation functionals and on

the influence of temperature in the full temperature window from zero Kelvin up to the melting

point. We study in detail the central thermodynamic quantity, the free energy surface, and show

that a high quality prediction of its temperature and volume dependence is crucial to guarantee an

unbiased description of derived materials properties. This turns out to be particularly challenging

at high temperatures due to the fact that the numerical/controllable errors propagate in a strongly

increasing fashion with temperature. We therefore developed and applied a set of novel approaches

going significantly beyond previous studies: 1) A method to efficiently assess the controllable errors

in all relevant free energy contributions and to reduce them to a few meV/atom even at the highest

temperatures. 2) A hierarchical coarse graining scheme to efficiently determine the anharmonic

free energy contribution, which accounts for the atomic interaction beyond the simple analytic

harmonic description and which therefore usually represents a formidable computational challenge.

3) A general and intuitive treatment of the free energy contribution due to point defects from which

the standard approaches can be easily derived as approximations.

Our methods can be applied to resolve long standing uncertainties about physical mechanisms

such as, e.g., the evaluation of the effects eventually leading to the transition from the solid to the

liquid phase. One of these decisive problems, which remained unresolved for over 90 years, is the

detailed balance of contributions to the heat capacity of a metal before melting. Investigating the

example of aluminum in detail, our approach allowed for the first time an accurate quantification

of all relevant excitation mechanisms and thus to settle a long standing debate. These findings

indicate that the methods developed and applied in this study represent an important step towards

the general goal of a materials design solely on the computer.

Contents

1 Motivation: Ab initio assisted metals design 1

1.1 General ideas and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Goals of the present work: Achievable accuracy and efficiency . . . . . . . . . . . . . 2

2 Theoretical background 5

2.1 From quantum mechanics to density functional theory . . . . . . . . . . . . . . . . . 5

2.2 Capturing electronic bonding effectively: The EAM approach . . . . . . . . . . . . . 31

2.3 Exploring the nuclei phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 From the free energy to materials properties . . . . . . . . . . . . . . . . . . . . . . . 59

3 Methodological developments 63

3.1 An intuitive description/treatment of point defect formation . . . . . . . . . . . . . . 63

3.2 Accelerating DFT calculations of the anharmonic free energy: The UP-TILD method . 71

3.3 Integrated approach to thermodynamic quantities . . . . . . . . . . . . . . . . . . . . 77

3.4 Towards highly accurate DFT free energies . . . . . . . . . . . . . . . . . . . . . . . 84

4 Results: Selected topics 100

4.1 Assessing DFT accuracy in predicting thermodynamic properties of metals . . . . . 100

4.2 Beyond the conventional scheme: Temperature dependent dynamical matrix . . . . . 117

4.3 Beyond the quasiharmonic approximation: Anharmonicity and vacancies in Al . . . 120

4.4 Quantum mechanical treatment of the anharmonic contribution . . . . . . . . . . . . 131

4.5 Achievable accuracy with empirical approaches: EAM vs. DFT . . . . . . . . . . . . 134

5 Conclusions 140

A Supplement 143

A.1 Technical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

A.2 Frequently used notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Bibliography 152

List of Publications 161

TEXT IN WHITE in order to keep this page empty

Chapter 1

Motivation: Ab initio assisted metals

design

1.1 General ideas and philosophy

The ever ongoing demand for increasing performance, quality, and price reduction of steel and other

metals has lead to a change in the philosophy of designing such materials. Traditionally, metals were

designed based on physical understanding derived from the mesoscopic and macroscopic scale (top-

down approaches). A wide range of elaborated methods has been developed (e.g. Refs. [1, 2]) over

the past decades allowing to perform efficient and accurate macroscopic simulations, for instance

to simulate a car crash. Despite their advantages and successes, there are however fundamental

difficulties in using top-down approaches: Their application often requires expensive or partly

even inaccessible experimental input and they need to be re-derived for each material class anew.

Rather recently, an alternative approach has therefore attracted a lot of attention in metals design.

The key idea is to start from the same fundamental principles as nature does which amounts to

describing the interaction between electrons and nuclei, the basic constituents of any material,

using quantum mechanics (bottom-up approach). The basis for these bottom-up approaches are

so called ab initio methods, the Latin term for from the beginning, and their crucial advantage is

that they, in principle, allow reliable predictions without the need of any experimental input.

In various branches of the metals science community, the ab initio approach is expected to be

the future key to a large variety of metals science issues as the following selected expert statements

may illustrate: ”Conventional materials design hits upon its limits for the newly developed high

manganese steels. We need to understand the atomistic interactions among the various alloying

elements to predict ordered and defect structures which affect the mechanical properties.” (Prof.

W. Bleck [3]) – ”Mesoscopic models need energetic and kinetic data which is practically not ac-

cessible by experiment. Examples are interface energies of metastable phases or mobilities in

multicomponent systems. Ab initio is the unrivaled approach capable of providing such data.”

(Prof. G. Gottstein [4]) – ”In designing and optimizing new steels, it is often helpful to decom-

pose the complex processes into sub processes and to understand each independent of the others.

For such purposes, ab initio methods become increasingly important, since they allow to perform

’virtual’ experiments. A prominent example is the study of hydrogen behavior in metals.” (Dr.

H.-P. Schmitz [5]) – ”We need modeling approaches taking advantage of the new ab initio meth-

ods, in order to support specific material data acquisition for an understanding of physically based

models. This will facilitate a faster development of such models and make them applicable to

conventional fabrication routes of semi-finished high quality aluminum products made from more

recycling friendly alloys.” (Prof. J. Hirsch [6]).

2 1.2. Goals of the present work: Achievable accuracy and efficiency

Let us discuss one specific example in more detail: the application of ab initio methods as a tool

to provide input parameters to meso/macroscopic approaches. We focus hereby in particular on

the so called calphad approach [7]. The calphad approach is a widely used method in materials

design for predicting phase diagrams as a function of temperature and concentration for multi-

component materials. Such phase diagrams are fundamental tools for engineers when determining

processing routes (road maps [8]) to design materials. An important difficulty related to the typical

calphad approach is however the fact that it requires experimental input. The reason is an often

challenging and expensive sample preparation and the necessity of high precision measurements.

Further, from a principle point of view, some necessary input (e.g., energetics of metastable phases)

is missing due to the lack of corresponding samples. The ab initio approach constitutes therefore

a very promising possibility to provide the missing experimental data [9]. Further, it would also

provide a reference to evaluate the physical concepts that enter the interpolation formulas used in

calphad (e.g., with respect to stacking fault energies or magnetism).

The quantum mechanical and electrostatical concepts, upon which ab initio approaches are

based, are well established and have been verified in numerous experimental studies (see e.g. the

collection in Ref. [10]). Therefore, at energy scales relevant for materials science, any such problem

could be in principle solved by applying the ab initio concept. However, a direct solution of the

underlying quantum mechanical equation – the Schr¨odinger equation – would be restricted to a

few electrons. This is a serious limitation since realistic materials science problems require the

description of a huge number of nuclei (≈1023 in one cubic centimeter) with each having up to a

hundred electrons. In order to make an application of ab initio techniques nonetheless feasible, a

combination of numerically highly efficient methods and various physically justified approximations

is indispensable. The development and falsification of these methods and approximations cannot

be performed by tackling directly such complex problems as mentioned above (multicomponent

systems, non-periodic structures, metastable phases) where ab initio is expected to predict prop-

erties. It is mandatory to address these challenges first for simple materials systems and problems

for which sufficient experimental data are available which can be used to falsify the developments

and approximations. The general goal of the present study is to address the latter issue.

1.2 Goals of the present work: Achievable accuracy and efficiency

There are various ab initio approaches available,1out of which one is exceptionally well suited for

the application to metals: the density functional theory (DFT [12]; introduced in detail in Sec. 2.1).

For that reason, the focus of the present study is on DFT based ab initio calculations. In general,

DFT assisted metals design faces the following challenges:

A. Extension to finite temperatures.

Originally, DFT has been designed and applied as a ground state theory and therefore the

directly accessible materials properties refer to T= 0 K. For realistic simulations, in particular

for metals, the extension to finite temperatures is necessary. Calculating materials properties

at finite temperatures results however in a large increase in computational effort. This is due

to the dramatically increased number of accessible microscopic configurations as compared

to T= 0 K. In fact, calculating finite temperature properties is only feasible using statistical

approaches. Therefore, the development and improvement of the combination of DFT and

statistical approaches is crucial.

1The existing ab initio approaches can be roughly grouped into density functional theory based, Hartree-Fock

based, and quantum Monte Carlo based [11].

1.2. Goals of the present work: Achievable accuracy and efficiency 3

B. Assessment of the systematic error.

Formally, DFT is an exact theory. In practice however, the so called exchange-correlation

(xc) functional (see Sec. 2.1.6) needs to be approximated, since the exact functional, mathe-

matically proven to exist, is not known. This approximation causes an unknown systematic

error, because there exists no practical approach to systematically improve the accuracy of

the xc functional. It is therefore necessary to systematically assess the achievable accuracy

of calculated materials properties.

C. Assessment of controllable errors.

Any practical realization of DFT is based on a set of specific parameters. These parame-

ters control for instance how formally infinite sums or continuous integrals are mapped onto

practical finite sums and discrete integrals. There is no principle difficulty arising from this

mapping since the introduced errors can always be controlled by scanning the correspond-

ing parameter space, even though at an increased computational cost. In actual calculations

however, the assessment of these controllable errors is rarely done when it comes to finite tem-

perature calculations (challenge A) which themselves require highly increased computational

resources. A challenge is therefore to develop a detailed understanding of the dependence of

the controllable errors on the various parameters for finite temperature properties. This will

allow to devise optimum parameters, i.e., the ones yielding the desired accuracy with lowest

computational cost.

The present thesis aims at tackling several aspects of these challenges. For that purpose and

according to the philosophy introduced in Sec. 1.1, we focus on the thermodynamics of simple – i.e.,

elementary and non-magnetic – metals, which are experimentally well investigated. In particular,

we employ a hierarchical scheme, in order to study the various excitation mechanisms contributing

to the equilibrium thermodynamic properties. This scheme is illustrated in Fig. 1.1 and its key

steps (levels) are:

I. The basis of the present thesis is a systematic study of the two dominating excitation mech-

anisms (challenge A): the electronic and the quasiharmonic one. A key point distinguishing

these investigations from previous ones is the large and comparable set of investigated metals

and the wide range of studied thermodynamic properties. Moreover, we perform all calcu-

lations using two complementary xc functionals (challenge B) and put special emphasis on

numerically highly converged thermodynamic properties with an assessment of the remaining

numerical error (challenge C). As a further special feature, we provide a systematic compar-

ison of DFT results with data obtained using the calphad method discussed in Sec. 1.1.

We focus hereby on the key quantity of the calphad approach: the free energy at constant

pressure.

II. Based on the results from LEVEL I, two representative elements are identified, for which the

quasiharmonic calculations are extended beyond the conventionally applied scheme by includ-

ing the explicit dependence on the electronic temperature. Typically, also the corresponding

theoretical background is not treated and we therefore present a discussion of the necessary

formalism (Sec. 2.1.3) and its implementation (Sec. 3.3.4). These developments eventually

allow to calculate the phonon shift due to electronic temperature and further to investigate

its influence on thermodynamic properties.

III. On LEVEL III of the hierarchy, the excitation spectrum is further extended such as to in-

clude all relevant mechanisms contributing to thermodynamic properties of the studied ele-

ments. The additionally included contributions, the explicitly anharmonic and the vacancy

4 1.2. Goals of the present work: Achievable accuracy and efficiency

el,qh

el, qh, qhel

el,qh, ah,vac

qhel

,expected

per ele-

ment

contribution to resources

thermodynamics Al,Rh,Pd,Pb,

Au,Pt,Ir,Cu,Ag

Al,Rh

computational

LEVEL I

LEVEL II

LEVEL III

Figure 1.1: Schematic illustration of the hierarchical approach to thermodynamic properties employed in the

present thesis. As shown by the first (from left) triangle the number of considered excitation mechanisms

increases with the level. The abbreviations stand for: el=electronic, qh=quasiharmonic, qhel=qh + the

influence of electronic temperature (cf. Sec. 3.3.4), ah=explicitly anharmonic, and vac=vacancies. At each

level the newly included contributions are emphasized by a larger fontsize. The second triangle indicates the

expected contribution of the newly included excitations to thermodynamic properties and the third triangle

the corresponding CPU requirements. The last triangle shows the elements considered at each level.

one, constitute a tremendous theoretical challenge: For instance, a molecular dynamics based

calculation of the anharmonic part of the free energy would necessitate computational time

in the range of several CPU years when using standard methods even on high performance

computers (challenge A). In order to make such a study nonetheless feasible, we have devel-

oped a fully DFT based coarse graining technique in configuration space and succeeded in

reducing the number of configurations from 106to a few hundreds at the highest and thus

computationally most expensive level. Using this approach we were able to guarantee a nu-

merical accuracy of better than 1 meV/atom (challenge C) - remaining errors can thus be

exclusively related to the xc functional (challenge B). As will be discussed in Sec. 2.3.5, it

is useful to reconsider the anharmonicity calculations using complementary methods, which

include quantum mechanical effects. In order to estimate the resulting corrections, we have

calculated the anharmonic phonon shift in aluminum using quantum mechanical perturbation

theory up to fourth order.

The methods developed and applied within this hierarchical scheme provide a complete de-

scription of all physically relevant contributions to the free energy of a non-magnetic elementary

metal. Based on this knowledge we can computationally address key questions of materials design.

For example, the highly accurate approach will allow us to tackle a long standing debate about

the dominating physical mechanisms determining the isobaric heat capacity of aluminum close to

the melting point. Furthermore, the obtained DFT results provide a sound basis for evaluating

theoretical approaches for larger scale applications. For example, inter-atomic potentials such as

the embedded atom method (EAM) are widely applied for finite temperature properties of several

million atoms, although they are designed by fitting to a limited range of usually temperature

independent data. Based on the complete set of DFT calculated excitation mechanisms for alu-

minum, we will therefore perform a detailed evaluation of the quality of three state-of-the-art EAM

parametrizations. The examples chosen in this study shall demonstrate the capability of DFT

based thermodynamics to open new and exciting routes towards an ab initio assisted materials

design.

Chapter 2

Theoretical background

A key objective of this thesis is to identify and develop efficient ab initio approaches to predict mate-

rials properties at finite temperatures. For that purpose, concepts from thermodynamics, statistics,

and quantum mechanics need to be combined. One of the central quantities in thermodynamics

is the free energy surface F(V, T ) as a function of volume Vand temperature T. It is a thermo-

dynamic potential and therefore fully determines all thermodynamic properties, for example the

isobaric heat capacity CP(T) (P=pressure) or the adiabatic bulk modulus BS(T) (S=entropy):

CP(T) = −T∂2F(V, T )

∂T 2V,P

, BS(T) = V∂2F(V, T )

∂V 2T,S

.(2.1)

According to statistical physics, Fis fully determined by the partition function Zaccording to

F(V, T ) = −kBTln Z(V, T ),(2.2)

where kBis the Boltzmann constant, and

Z(V, T ) = X

e−βEξ(V),(2.3)

with β= (kBT)−1. In Eq. (2.3), Eξis the volume dependent but temperature independent ξth

energy level of the considered system. The sum runs over all such energy levels including possible

degeneracies. These energies can be fully determined using the concepts of quantum mechanics.

The detailed determination of the Eξand subsequently of Ffor realistic solid state materials

uses an involved apparatus of numerous methods developed over the past decades. A schematic

overview illustrating the task to be accomplished and the relation of various key concepts is given

in Fig. 2.1. The remainder of this chapter and Chap. 3 address the various parts of Fig. 2.1 in

detail.

2.1 From quantum mechanics to density functional theory

2.1.1 Motivation

The basic quantum mechanical principles are: The motion and interaction of microscopic particles

(electrons and nuclei in the present case) cannot be described/predicted on an individual basis.

Their behavior can be only captured in a statistical fashion, i.e., based on probabilities for ensembles

of particles. The probabilities can be described by wave mechanics, which accounts for the observed

6 2.1. From quantum mechanics to density functional theory

Coulombs law

Embedded

atom method

Quantum mechanics

Schrödinger equation

Born-Oppenheimer

approximation

Nuclei motion

Vibrations of perfect crystal

DFT = density functional theory

MD = molecular dynamics

BE = Bose-Einstein statistics

= 17.6 Å ,

xc = exchange correlation

FD = Fermi-Dirac statistics

= 17.1 Å ,

= 16.6 Å

Quasiharmonic

Independent phonons

kvector qvector 0 300 600 (K)T

0 300 600 (K)T

0 300 600 (K)T0 300 600 (K)T0 100 200 (K)T

FD( )T

BE( )T

wq,s( )

Dwah ( , )V T

F V T

f( , )

F V T

el( , )

F V T

qh( , )

F V T

vib( , )

F V T( , )

F V T

ah( , )

F V T

vac( , )

( )

n,kV

w(meV)

C k

P( )

a(10 K )

-5 -1

S/B

S(T=0K)

C T F T

P V,P

= - ( / )¶ ¶

2 2 BS= ( / )V F V¶ ¶

2 2

T,S

a ¶ ¶=( / )/3V T V

eq eq

F V F V( ) = min ( )

w(meV)

F(eV)

e(eV)

0.97

0.96

0.6

0.3

0.0

0.8

0.6

0.4

-5

-10

-15

Interacting phonons

Formation

free energy

Anharmonic

Vacancy formation

Electronic free energy surface

Electron motion

(many particle

wave function)

Electronic density

Effective one

electron equation

DFT

Harmonic

oscillator

DFT

Experiment

Isobaric heat capacity

Equilibrium thermo-

dynamic properties

Expansion coefficient Relative bulk modulus

MD, thermodynamic

integration,

perturbation theory

Dilute limit,

Stirling’s

approximation

Kohn-Sham

(xc functional)

Figure 2.1: Schematic visualization of the ab initio approach that is developed in this thesis and employed

to derive equilibrium thermodynamic properties of non-magnetic elementary metals. The plotted data are

for aluminum and highlight some of the results of this work. Experimental values are from Refs. [13], [14],

and [15], for CP,α, and BS, respectively. The various expressions and symbols will be defined in this and

the following chapter, except for Fvib which denotes the vibrational free energy and which is defined only

here for the purpose of a convenient representation.

2.1. From quantum mechanics to density functional theory 7

interference phenomena and the impossibility of localization. These principles are embodied in the

Schr¨odinger equation, which additionally incorporates Coulomb’s law. We are interested only in

equilibrium properties here, for which it is sufficient to consider the time independent Schr¨odinger

equation: ˆ

HΨξ=EξΨξ.(2.4)

Here, the Eξare the energy levels entering Eq. (2.3). Further, Ψξ= Ψξ({ri},{RI}) is the many-

body wave function of the ξth energy depending on the set of electronic coordinates {ri}, where

iruns over all electrons in the system, and on the set of nuclei coordinates {RI}with Irunning

over all nuclei. The Ψξcontain the full information about the system and they must obey the

fundamental requirement of antisymmetry with respect to the interchange of electrons which are

identical particles (fermions)

Ψξ(r1,...,ri,...,rj,...,rNe,{RI}) = −Ψξ(r1,...,rj,...,ri,...,rNe,{RI}),(2.5)

where Neis the number of electrons. In principle, also nuclei of the same species are identical

particles and would need to obey either a symmetry or antisymmetry condition depending on the

number of fermions they are composed of [16]. In crystals however, the motion of the nuclei is

fixed to the vicinity of their crystallographic positions and this renders the nuclei distinguishable.

Further in Eq. (2.4), ˆ

His the Hamilton operator given by:

H=ˆ

Tel +ˆ

Tnuc +ˆ

Vel +ˆ

Vnuc +ˆ

Ve-n.(2.6)

The operators on the right are the kinetic energy for the electrons (nuclei) ˆ

Tel (ˆ

Tnuc), the electron-

electron repulsion ˆ

Vel, the nucleus-nucleus repulsion ˆ

Vnuc, and the electron-nucleus attraction ˆ

Ve-n

given by:

Tel({ri}) = −

2me∇2

i,(2.7)

Tnuc({RI}) = −

2MI∇2

I,(2.8)

Vel({ri}) = 1

j6=i

4πǫ0|ri−rj|,(2.9)

Vnuc({RI}) = 1

J6=I

ZIZJe2

4πǫ0|RI−RJ|,(2.10)

Ve-n({ri},{RI}) = −

ZIe2

4πǫ0|ri−RI|.(2.11)

Here, ~is the reduced Planck constant, methe electron mass, ethe elementary charge, ǫ0the

electric constant, Nnthe number of nuclei, and MI(ZI) the mass (proton number) of nucleus I.

At this point, the ab initio problem is fully defined: Only known fundamental constants enter the

various formulas and thus a unique solution of Eq. (2.4) could in principle be obtained. However, the

numerical effort to directly solve Eq. (2.4) is not practical even for systems with a few electrons. In

order to illustrate this in a bit more detail, we show how one could proceed in solving it numerically.

Let us sketch the procedure for a simple system consisting of one nucleus at R, containing

one proton, and of one electron at r, i.e., the hydrogen atom: First, we need to construct a box

8 2.1. From quantum mechanics to density functional theory

around the atom, realized by energetic barriers, to fix the atom in space. The box needs to be

large enough to capture the relevant physics, which is technically ensured by converging its size. A

reasonable box size for the present problem is 20 a0[17] (a0:= 1 Bohr radius ≈0.05 nm). Next,

we need to discretize the box introducing another convergence parameter, the corresponding mesh

which we assume here for simplicity to be equidistant. A reasonable mesh distance is 0.1a0=: δ

[17] yielding a mesh of 2003= 8 ·106=: N3

mesh. The possible positions of the electron and the

nucleus are thus r=δ(x, y, z) and R=δ(x′, y′, z′) with x,y,z,x′,y′, and z′integers each running

over 1 . . . Nmesh. The wave function becomes in the discrete case a vector indexed by all possible

configurations the system can assume: Ψx,y,z,x′,y′,z′, i.e., a vector of size (N3

mesh)2(for convenience,

we drop the index ξon Ψ in this example). The Hamilton operator becomes a quadratic matrix, H,

of size (N3

mesh)2×(N3

mesh)2with rows and columns indexed accordingly. The attractive electron-

nucleus term, the only Coulomb term for the hydrogen atom, affects only the diagonal elements

of the matrix contributing a term proportional to 1/|(x, y, z)−(x′, y′, z′)|. To render the infinite

value occurring for (x, y, z) = (x′, y′, z′) finite, an infinitesimal parameter can be added to the

denominator. The kinetic energy operators can be, for the present purpose, approximated by the

central finite difference

∇2

xΨx,y,z,x′,y′,z′≈1

δ2Ψx+1,y,z,x′,y′,z′−2Ψx,y,z,x′,y′,z′+ Ψx−1,y,z,x′,y′,z′,

∇2

yΨx,y,z,x′,y′,z′≈1

δ2Ψx,y+1,z,x′,y′,z′−2Ψx,y,z,x′,y′,z′+ Ψx,y−1,z,x′,y′,z′,(2.12)

and so on for z,x′,y′z′. Equation (2.12) makes clear that the kinetic energy operator is non-local

in the chosen real space representation. In this approximation, it contributes a finite value to

all neighboring sites at x−1, x+ 1, y−1, and so on, and thus makes the Hamilton matrix H

non-diagonal, e.g.,

Hx,y,z,x′,y′,z′;x−1,y,z,x′,y′,z′=−~2/(2meδ2),

Hx,y,z,x′,y′,z′;x,y,z,x′−1,y′,z′=−~2/(2Mδ2),(2.13)

(M: proton mass). Having constructed H, the solution of the discretized Schr¨odinger equation

is obtained by diagonalizing H, i.e., by solving its eigenvalue equation. In principle, this is a

standard algebraic problem for a sparse matrix, however, for a matrix of extremely huge size:

(N3

mesh)2×(N3

mesh)2= 64 ·1012 ×64 ·1012. The currently largest manageable eigenvalue problems1

are performed on matrix sizes in the range of 1012 ×1012. Thus, the matrix of the electron-nucleus

problem is in the range of technically diagonalizable matrices provided one employs today’s world’s

highest performance algorithms and computer hardware.

For a general problem, the Hamilton matrix will have the schematic form given in Fig. 2.2 and

its size swill scale as:

s(N3

mesh, Ne, Nn) = (N3

mesh)Ne+Nn×(N3

mesh)Ne+Nn.(2.14)

It is clear that, for any problem larger than the hydrogen atom, a direct solution is not practical

in the next decades even if we assume that Moore’s law will continue to hold. For instance, if we

consider only a single aluminum atom, the atom with the smallest number of electrons (Ne= 13)

out of the studied elements, the matrix size (for the same box size) is s(Nmesh =200, Ne=13, Nn=

1) ≈1096 ×1096.

1Such eigenvalue problems are performed by Internet search engines [18]. The matrix size is determined by the

number of web pages available in the Internet, which was ≈1012 in 2008 [19]. However, strictly speaking, in these

algorithms only the lowest eigenvalue is determined (using the so called power method) rather than the full eigenvalue

problem [20].

2.1. From quantum mechanics to density functional theory 9

{ }ri1

{ }ri2

{ }RI1{ }RI2

T +

el ˆ

T +

nuc ˆ

V +

el ˆ

V +

nuc ˆ

Ve-n ˆ

T +

el ˆ

T +

nuc ˆ

V +

el ˆ

V +

nuc ˆ

Ve-n ˆ

T +

el ˆ

T +

nuc ˆ

V +

el ˆ

V +

nuc ˆ

Ve-n

{ }ri( )

NNe

mesh = { , , ... , }r r r

1 2 Ne

N N

mesh 3

mesh N3

mesh { }RI( )

NNn

mesh = { , , ... , }R R R

1 2 Nn

N N

mesh 3

mesh N3

mesh

{ }ri( )

NNe

mesh -1 = { , , ... , }r r r

1 2 Ne

N N

mesh 3

mesh 1{ }RI( )

NNn

mesh -1 = { , , ... , }R R R

1 2 Nn

N N

mesh 3

mesh 1

{ }ri N 3

mesh = { , , ... , }r r r

1 2 Ne

mesh 11{ }RI N 3

mesh = { , , ... , }R R R

1 2 Nn

mesh 11

{ }ri( )

mesh 2= { , , ... , }r r r

1 2 Ne

N N

mesh 3

mesh 1{ }RI( )

mesh 2= { , , ... , }R R R

1 2 Nn

N N

mesh 3

mesh 1

{ }ri( )

NNe

mesh -1+1 = { , , ... , }r r r

1 2 Ne

1 1 2{ }RI( )

NNn

mesh -1+1 = { , , ... , }R R R

1 2 Nn

1 1 2

{ }ri( )

mesh +1 = { , , ... , }r r r

1 2 Ne

1 2 1{ }RI( )

mesh +1 = { , , ... , }R R R

1 2 Nn

1 2 1

{ }ri( )

mesh 2+1 = { , , ... , }r r r

1 2 Ne

1 1 1{ }RI( )

mesh 2+1 = { , , ... , }R R R

1 2 Nn

1 1 1

{ }ri2= { , , ... , }r r r

1 2 Ne

2 1 1{ }RI2= { , , ... , }R R R

1 2 Nn

2 1 1

{ }ri1= { , , ... , }r r r

1 2 Ne

1 1 1{ }RI1= { , , ... , }R R R

1 2 Nn

1 1 1

{ }ri( )N3

mesh Ne

{ }ri( )N3

mesh Ne

{ }ri( )N3

mesh Ne{ }ri( )N3

mesh Ne

{ }ri( )N3

mesh Ne

{ }ri( )N3

mesh Ne

{ }ri( )N3

mesh Ne

{ }ri( )N3

mesh Ne

{ }ri( )N3

mesh Ne

{ }RI( )N3

mesh Nn

{ }RI( )N3

mesh Nn

{ }RI2

{ }RI1

Tel

Tnuc

b) c)

…

………

……

…

…………

………

……

…

Figure 2.2: a) Schematic visualization of the Hamiltonian matrix for a system of Neelectrons, with coordi-

nates {ri}, and Nnnuclei, with coordinates {RI}, discretized onto a Nmesh ×Nmesh ×Nmesh mesh. The rows

and columns are indexed by the possible microscopic configurations of the system enumerated as shown in

b) and c). On the right hand side of the equations in b) and c) the black index runs over the electrons and

nuclei, respectively, while the red index runs over the mesh. In a), the dashed lines indicate that the nuclei

configuration is the same for all electronic configurations captured by the line. Further, the placement of

the operators indicates which part of the matrix is affected. For instance, ˆ

Tel +ˆ

Tnuc +ˆ

Vel +ˆ

Vnuc +ˆ

Ve-n

affects only the dark shaded diagonal of the matrix.

10 2.1. From quantum mechanics to density functional theory

Table 2.1: Electron mass, me, to nucleus mass, MI, ratio for the studied elements (I= Al, Cu, ...). Addi-

tionally, the actual values for the off diagonal elements within the central finite difference method are shown.

Note that in the first column we set MI=mefor comparison. The mesh distance is δ= 0.1a0.

MImeMAl MCu MRh MPd MAg MIr MPt MAu MPb

me/MI(10−6) — 20.3 8.6 5.3 5.2 5.1 2.9 2.8 2.8 2.6

−~2

2MIδ2(meV) −1.3·106−27.7−11.7−7.3−7.0−6.9−3.9−3.8−3.8−3.6

A complete solution is, however, not needed to obtain accurate results. Numerous approxima-

tions have been developed which allow a numerically efficient treatment capturing the important

physics. A major role is played hereby by the Born-Oppenheimer approximation, which decouples

the electron and nucleus motion, and by density functional theory which removes the dependence

of son Ne. These concepts are introduced in the following.

2.1.2 Born-Oppenheimer approximation

Inspection of Eqs. (2.7) and (2.8) reveals that the kinetic energy operators for the electrons and

nuclei differ by a factor of me/MI. Tab. 2.1 shows this factor and actual values using the central

finite difference method for the studied elements. For the lightest element, aluminum, the off

diagonal elements [Eq. (2.13)] produced by the nucleus kinetic energy are five orders of magnitude

smaller than the off diagonal elements produced by the electron kinetic energy. Physically, this

means that the electrons can follow the nuclei almost instantaneously and can therefore relax

virtually to their ground state for each nuclei configuration on the time scale of atomic motion

as given e.g. by atomic vibrations. This observation suggests that decoupling the motion of

electrons and nuclei completely, i.e., accounting only for the electronic ground state at each nuclei

configuration, will be a reasonable approximation. This is the essence of the Born-Oppenheimer

approximation [21].

The formal introduction of the Born-Oppenheimer approximation is a three step procedure. In

a first step, we define an auxiliary Hamilton operator by:

Hel := ˆ

H−ˆ

Tnuc =ˆ

Tel +ˆ

Vel +ˆ

Vnuc +ˆ

Ve-n.(2.15)

We call ˆ

Hel an electronic Hamilton operator, because it contains only non-local terms in the

electronic coordinates after subtracting ˆ

Tnuc. The corresponding eigenvalue equation, the electronic

Schr¨odinger equation, reads ˆ

Hel ψ˜ν=Eel

˜νψ˜ν,(2.16)

where ψ˜ν=ψ˜ν({ri},{RI}) and Eel

˜νare the corresponding ˜ν’th eigenfunction and eigenvalue. An

important observation is that a complete solution to Eq. (2.16) can be obtained by solving an

eigenvalue equation for each nuclei configuration separately, which is due to the fact that ˆ

Hel

corresponds to a block diagonal matrix. As a consequence, we can relabel the eigenvalues as

Eel

˜ν=Eel

ν({RI}) and the eigenfunctions as ψ˜ν({ri},{RI}) = ψν({ri};{RI}), with νrunning only

over the eigenfunctions for a fixed nuclei configuration and with the semicolon indicating that the

set {RI}is treated as a parameter.

In a second step, we reintroduce the kinetic energy operator of the nuclei by defining a nuclei

2.1. From quantum mechanics to density functional theory 11

operator ˆ

Hnuc

νas ˆ

Hnuc

ν({RI}) = ˆ

Tnuc({RI}) + ˆ

1Eel

ν({RI}),(2.17)

where ˆ

1 is the identity operator. Similarly as ˆ

Hel,ˆ

Hnuc

νis block diagonal, having only coupling

terms in {RI}but none in ν, and therefore its eigenfunctions Λ and eigenvalues Enuc can be labeled

by νand an index µrunning over all eigenfunctions for a fixed ν, i.e.:

Hnuc

ν({RI}) Λν,µ({RI}) =

hˆ

Tnuc({RI}) + ˆ

1Eel

ν({RI})iΛν,µ({RI}) = Enuc

ν,µ Λν,µ({RI}).(2.18)

The key point of Eq. (2.18), called the nuclei Schr¨odinger equation, is that each of the Eel

νdescribes

a potential energy surface (PES) in which the nuclei move. Particularly, once νhas been fixed to

some value, say ν=ν′, the nuclei move only in the potential energy surface given by Eel

ν′without

changing to another Eel

ν.

The final step of the Born-Oppenheimer approximation is then to approximate the eigenvalues

Eξof the original Schr¨odinger equation, Eq. (2.4), by the eigenvalues Enuc

ν,µ , i.e.:

Eξ≈Enuc

ν,µ .(2.19)

As shown in Ref. [22], this approximation is equivalent to neglecting terms which couple the elec-

tronic wave functions ψνto the nuclei wave functions Λν,µ. These terms will be small and the

approximation therefore reasonable when me/MI≪1 as for the studied elements. Application of

Eq. (2.19) decouples the original Schr¨odinger equation, Eq. (2.4), into two block diagonal eigenvalue

equations, Eqs. (2.16) and (2.18), and the full problem scales therefore as:

s(Nmesh, Ne, Nn) = (N3

mesh)Nn·(N3

mesh)Ne×(N3

mesh)Ne

| {z }

Electronic Schr¨odinger eq.

for fixed nuclei coordinates

+ (N3

mesh)Ne·(N3

mesh)Nn×(N3

mesh)Nn

| {z }

Nuclei Schr¨odinger eq.

for fixed electronic PES

(2.20)

The symbolic notation, used here and frequently in the following to indicate the dimensions of a

problem, is explained in App. A.1.1. Equation (2.20) means that the original problem represented

by one large matrix, Eq. (2.14), has been transformed into various smaller matrix equations. The

remaining smaller matrix equations are however still too large to be solved directly. The actual

advantage of applying the Born-Oppenheimer approximation is that further approximations and

reformulations can be performed for the electrons and nuclei separately.

In typical applications of the Born-Oppenheimer approximation a further approximation is

implicitly performed. Out of the various potential energy surfaces, Eel

ν, only the energetically

lowest one (denoted as Eel

g), i.e., the one corresponding to the electronic ground state at T= 0 K,

is considered. Temperature dependence enters subsequently by determining the nuclei motion on

Eel

g. Such an approach is however not fully correct since at finite temperatures the nuclei motion

will be determined by the thermodynamic electronic ground state which corresponds to the minimal

electronic free energy rather than by the T= 0 K electronic ground state Eel

g[23]. Put differently,

the motion of the nuclei at some finite temperature will be determined by several Eel

νand the

specific contribution of each Eel

νwill be weighted according to the temperature. It is important to

stress that this coupling of different Eel

νis not contained in Eq. (2.18), it only enters when the Eel

are inserted into the partition function, Eq. (2.3). This fact is exploited in the next section.

12 2.1. From quantum mechanics to density functional theory

2.1.3 Free energy Born-Oppenheimer approximation

In order to derive the dependence of the nuclei motion on the electronic free energy, we pursue the

following strategy. We apply the Born-Oppenheimer approximation to the partition function and

transform the latter such as to separate a sum over the electronic eigenvalues Eel

νwhich we identify

with the electronic free energy. Let us thus apply Eq. (2.19) and insert the Enuc

ν,µ into the partition

function, Eq. (2.3),

Z=X

ν,µ

e−βEnuc

ν,µ =X

ν X

e−βEnuc

ν,µ !,(2.21)

where the sum has been adjusted to correspond to the new indices. We can use Eq. (2.18) to

transform the partition function further. For that purpose, we multiply Eq. (2.18) from left with

Λ∗

ν,µ and integrate over the nuclei coordinates to obtain (using Dirac’s notation; see App. A.1.3):

hΛν,µ|ˆ

Tnuc +ˆ

1Eel

ν|Λν,µi=Enuc

ν,µ .(2.22)

Inserting into Eq. (2.21) yields

Z=X

ν X

e−βhΛν,µ|(ˆ

Tnuc+ˆ

1Eel

ν)|Λν,µi!=X

ν X

µhΛν,µ|e−β(ˆ

Tnuc+ˆ

1Eel

ν)|Λν,µi!,(2.23)

where we have used Eqs. (A.4) and (A.5), which is possible since the Λν,µ are eigenfunctions

of ˆ

Tnuc +ˆ

1Eel

ν. It would be desirable to factorize the exponential to separate the Eel

ν. This

factorization needs however to be performed with caution since ˆ

Tnuc and ˆ

1Eel

νare non-commuting

operators (both act on {RI}). Therefore, we have to apply the so called Zassenhaus formula [24]:

e−β(ˆ

Tnuc+ˆ

1Eel

ν)=e−βˆ

Tnuc e−βˆ

1Eel

νeβ2/2 [ ˆ

Tnuc,ˆ

1Eel

ν]e−β3/6(2[ˆ

1Eel

ν,[ˆ

Tnuc,ˆ

1Eel

ν]]+[ ˆ

Tnuc,[ˆ

Tnuc,ˆ

1Eel

ν]])···.

(2.24)

Here, we have used the commutator as given in Eq. (A.3) and the dots denote exponentials corre-

sponding to higher orders in βand with increasingly nested commutators. An explicit formula for

the higher order terms is given in Ref. [24]. In Ref. [25], it is shown that exponentials corresponding

to orders β2and higher will be small if me≪MI, i.e., under the same condition as needed for the

Born-Oppenheimer approximation. It is therefore justified to approximate

e−β(ˆ

Tnuc+ˆ

1Eel

ν)≈e−βˆ

Tnuc e−βˆ

1Eel

ν,(2.25)

in any case in which the Born-Oppenheimer approximation is justified. Using Eq. (2.25), the

invariance property of the trace, Eq. (A.7), and the fact that (e−βˆ

Tnuc e−βˆ

1Eel

ν) corresponds to a

block diagonal matrix (which allows to choose the same basis for different ν, e.g., ν=ν′with a

fixed ν′) yields

Z=X

ν X

µhΛν,µ|e−βˆ

Tnuc e−βˆ

1Eel

ν|Λν,µi!

ν X

µhΛν′,µ|e−βˆ

Tnuc e−βˆ

1Eel

ν|Λν′,µi!

µhΛν′,µ|e−βˆ

Tnuc X

e−βˆ

1Eel

ν!|Λν′,µi=X

µhΛν′,µ|e−βˆ

Tnuc e−βˆ

1Fel |Λν′,µi,(2.26)

2.1. From quantum mechanics to density functional theory 13

with the electronic free energy defined by:

Fel({RI}) := −kBTln X

e−βEel

ν({RI}).(2.27)

In order to recombine the exponentials again, we need to apply the Baker-Campbell-Hausdorff

formula which reads [24]:

e−βˆ

Tnuc e−βˆ

1Fel =e−β(ˆ

Tnuc+ˆ

1Fel)−β2/2[ ˆ

Tnuc,ˆ

1Fel]]−β3/6[[ ˆ

Tnuc,ˆ

1Fel],ˆ

1Fel−ˆ

Tnuc]···.(2.28)

However, the terms in the exponential of order β2and higher correspond again to terms which are

small if me≪MIand we approximate therefore:

e−βˆ

Tnuc e−βˆ

1Fel ≈e−β(ˆ

Tnuc+ˆ

1Fel).(2.29)

In fact, it is shown in Ref. [25] that the terms neglected in Eq. (2.29) lead to contributions which

are of the same order as the contributions from the neglected terms in Eq. (2.25) having however

the opposite sign. Therefore, the approximations performed in Eqs. (2.25) and (2.29) partially

compensate each other. Inserting Eq. (2.29) into Eq. (2.26) yields for the partition function:

Z=X

µhΛµ|e−β(ˆ

Tnuc+ˆ

1Fel)|Λµi.(2.30)

Next we define the eigenvalue equation

ˆ

Tnuc +ˆ

1Fel˜

Λµ=˜

Enuc

µ˜

Λµ,(2.31)

with the eigenfunctions ˜

Λµand eigenvalues ˜

Enuc

µ, and we call it effective nuclei Schr¨odinger equation.

Equation (2.31) describes the nuclei motion in the electronic free energy surface (FES) correspond-

ing to the thermodynamic ground state. Using Eqs. (2.31) and (A.7), we can conveniently rewrite

Zas:

Z=X

e−β˜

Enuc

µ.(2.32)

In summary, we have therefore the following chain of key equations and flow of key quantities:

Hel ψν=Eel

ν({RI})ψν(2.33)

Fel({RI})=−kBTln X

e−βEel

ν({RI})(2.34)

nˆ

Tnuc +ˆ

1Fel({RI})o˜

Λµ=˜

Enuc

µ˜

Λµ(2.35)

F=−kBTln X

e−β˜

Enuc

µ.(2.36)

14 2.1. From quantum mechanics to density functional theory

Performing both approximations, Eqs. (2.25) and (2.29), is referred to as the free energy Born-

Oppenheimer (FEBO) approximation. The FEBO approximation has been introduced by Cao

and Berne in 1993 [25]. We stress that it is an additional approximation to the original Born-

Oppenheimer one, requiring however the same condition (me/MI≪1) to be applicable. The

scaling within the FEBO approximation changes to

s(Nmesh, Ne, Nn) = (N3

mesh)Nn·{sel := (N3

mesh)Ne×(N3

mesh)Ne}

| {z }

Electronic Schr¨odinger eq.

for fixed nuclei coordinates

+(N3

mesh)Nn×(N3

mesh)Nn

| {z }

Effective nuclei Schr¨odinger eq.

for electronic ground state FES

(2.37)

where we have defined for future reference sel =sel(Nmesh, Ne), the scaling of the electronic

Schr¨odinger equation for fixed nuclei coordinates. In comparison with the scaling within the Born-

Oppenheimer approximation, Eq. (2.20), the prefactor (N3

mesh)Nehas been removed rendering the

second term on the right hand site of Eq. (2.37) to be independent of Ne.

From this point on, we can discuss the electronic Schr¨odinger equation, Eq. (2.16), and the

effective nuclei Schr¨odinger equation, Eq. (2.31), separately.

2.1.4 Density functional theory

For the remainder of Sec. 2.1, we concentrate on the electronic Schr¨odinger equation for a fixed

nuclei configuration: ˆ

Hel ψν({ri};{RI}) = Eel

ν({RI})ψν({ri};{RI}).(2.38)

A direct solution for more than two electrons is still an infeasible task. In fact, the estimations

performed in Sec. 2.1.1 for the system of one electron and one nucleus can be taken directly over

for two electrons.

In 1964, Hohenberg and Kohn [12] showed that the ground state Eel

gof Eq. (2.38) is uniquely

determined by a functional of the electron density ρwhich is given in terms of the electronic wave

function ψν({ri}) = ψν({ri};{RI}) by:

ρ(r) := ρ(r1) = NeZ...Z|ψν({ri})|2dr2dr3. . . drNe.(2.39)

The crucial observation is that ρ(r) is only a function of the three spatial variables, in contrast to

ψν({ri}) which is a function of 3Nevariables, and that a functional is the central equation instead

of an eigenvalue equation. Thus, the scaling reduces as

sel(Nmesh, Ne) = (N3

mesh)Ne×(N3

mesh)Ne→sdft(Nmesh) = N3

mesh,(2.40)

with the DFT scaling sdft. The corresponding theory is called the density functional theory

(DFT) and it is important to stress that, despite the tremendous reduction of variables, DFT is in

principle exact. It is only the practical realization which makes an approximation necessary. The

derivation by Hohenberg and Kohn [12] is the most often presented formulation of DFT. However,

Eel

gcorresponds to a single electronic PES, the T= 0 K one, and as discussed in Sec. 2.1.2, it is not

sufficient for our purposes to have only a single PES available. In 1965, Mermin [26] showed that

there exists also a functional of the electronic density which uniquely determines the thermodynamic

ground state FES, i.e., Fel. We therefore consider here Mermin’s extended formulation, the finite

temperature DFT. Further, we rely on the Levy constrained search formulation [27, 28], which is

more generally applicable than the original approach by Hohenberg and Kohn/Mermin [12, 26].

2.1. From quantum mechanics to density functional theory 15

In order to introduce finite temperature DFT formally, we need to generalize the notion of the

free energy. Specifically, we define a free energy functional e

Fel constructed from an entropy and an

inner energy contribution by:

Fel[{pν},{ψν}] = T kBX

pνln pν

| {z }

(negative) Entropy term

pνhψν|ˆ

Hel|ψνi

| {z }

Inner energy contribution

.(2.41)

Here, the pνare weights which determine the contribution of the corresponding ψνto the free

energy and which must satisfy:

pν≥0 and X

pν= 1.(2.42)

The free energy functional, Eq. (2.41), is constructed such that its minimum with respect to {pν}

and {ψν}will always correspond to the free energy defined in Eq. (2.27) [29], i.e.:

Fel =−kBTln X

e−βEel

ν= min

{pν},{ψν}e

Fel[{pν},{ψν}].(2.43)

We can now perform the key step of DFT which is to rewrite the minimization as:

Fel = min

ρρ

min

{pν},{ψν}e

Fel[{pν},{ψν}].(2.44)

Here, the inner minimization is constrained in the sense that only sets of ({pν},{ψν}) are allowed

which yield ρ, using a generalized version of Eq. (2.39):

ρ(r) := ρ(r1) = NeX

pνZ...Z|ψν({ri})|2dr2dr3. . . drNe.(2.45)

The outer minimization relaxes then this condition by allowing to search for the minimum among

all electronic densities ρwhich satisfy [29]:

ρ(r)⩾0,Zρ(r)dr=Ne,and Z|∇ρ(r)1/2|2dr<∞.(2.46)

These conditions guarantee that the electronic density corresponds to an antisymmetric wave func-

tion describing Neelectrons [30]. Using Eqs. (2.15) and (2.41), we can rewrite the inner minimiza-

tion from Eq. (2.44) as:

min

{pν},{ψν}e

Fel[{pν},{ψν}] = ρ

min

{pν},{ψν}"X

pνkBTln pν+hψν|ˆ

Tel +ˆ

Vel +ˆ

Vnuc +ˆ

Ve-n|ψνi#

=F[ρ] + Zρ(r)v(r)dr+Vnuc.(2.47)

Here, we have defined the functional F[ρ], the external potential v(r), and the scalar quantity Vnuc

F[ρ] := ρ

min

{pν},{ψν}"X

pνkBTln pν+hψν|ˆ

Tel +ˆ

Vel|ψνi#,(2.48)

16 2.1. From quantum mechanics to density functional theory

Ve-n =−

ZIe2

4πǫ0|ri−RI|=:

v(ri),(2.49)

pνhψν|ˆ

Vnuc|ψνi=1

J6=I

ZIZJe2

4πǫ0|RI−RJ|=: Vnuc,(2.50)

and finally used the fact that:

pνhψν|ˆ

Ve-n|ψνi=X

pνZNe

v(ri)|ψν({ri})|2dri

=Zv(r)dr"NeX

pνZ...Z|ψν({ri})|2dr2. . . drNe#

Eq. (2.45)

=Zv(r)ρ(r)dr.(2.51)

To obtain the second equality in Eq. (2.51), the symmetry of |ψν({ri})|2with respect to an inter-

change of particle variables riand r1was used [cf. Eq. (2.5)]. Inserting Eq. (2.47) into Eq. (2.44)

gives:

Fel = min

ρF[ρ] + Zρ(r)v(r)dr+Vnuc.(2.52)

This is the final equation of finite temperature DFT. It states that the electronic free energy at

thermal equilibrium, Fel, can be determined by minimizing the functional F[ρ] + Rρ(r)v(r)drwith

respect to the electronic density ρ, while Vnuc is only a scalar shift. An important observation is

that F[ρ] does not depend on the external potential v(r), i.e., it is a universal functional of ρonly,

likewise applicable to atoms, molecules, or crystals. However, the exact functional form of F[ρ]

is not known and, in practice, one therefore relies on approximations which are discussed in the

following. Before, let us briefly comment on the presented derivation of Eq. (2.52).

The steps leading to Eq. (2.52) seem rather straightforward. In fact, the key step is the sepa-

ration of the minimization as given in Eq. (2.44) which allows the introduction of the functional

F[ρ] in Eq. (2.47). It is however important to stress that an indispensable requirement for such a

separation and for the correct minimization of F[ρ] is the knowledge of the domain of electronic

densities which are admissible. Specifically, only those densities are admissible which are derivable

from an antisymmetric wave function [Eq. (2.5)]. The determination of the conditions identifying

such densities is rather involved [30] and the conditions are therefore given here, in Eq. (2.46),

without proof.

2.1.5 Kohn-Sham approach

Equation (2.52) asserts only that a functional F[ρ] must exist, but it gives no prescription of

how to construct it. In 1965, Kohn and Sham [31] developed an indirect approach to handle this

problem. The basic idea is to introduce an auxiliary system of non-interacting electrons in addition

to the actual system of interest containing electron interactions. The auxiliary system is solved

exactly (this is straightforward since no interactions are present) and additionally the classical

electron interaction is included. As it turns out empirically, this accounts for the major part of the

physics contained in the actual interacting system and the remaining part can be approximated very

efficiently with a ”simple” functional dependence. To introduce the Kohn-Sham approach formally,

we need first to formulate the minimization of Eq. (2.52) in terms of a functional derivative.

2.1. From quantum mechanics to density functional theory 17

The minimum search in Eq. (2.52) is equivalent [29] to performing a variation of

δF[ρ] + Zρ(r)v(r)dr−µZρ(r)dr−Ne= 0,(2.53)

where µRρ(r)dr−Ne, with the Lagrange multiplier µ, is included in order to guarantee particle

conservation, i.e., that ρintegrates to Ne[cf. Eq. (2.46)]. Carrying out the variation yields the

Euler-Lagrange equation [29]:

µ=v(r) + δF[ρ]

δρ(r).(2.54)

We can now proceed with the Kohn-Sham (KS) approach. DFT and thus Eq. (2.54) is applicable

to any electronic system, in particular to a system of non-interacting electrons, i.e., one without

Vel, for which:

Fks[ρ] := F[ρ] = ρ

min

{pν},{ψν}"X

pνkBTln pν+hψν|ˆ

Tel|ψνi#.(2.55)

We hence introduce such a system with Neelectrons and require that its electronic density equals

the density of our interacting system. This can be achieved by varying the external potential

of the non-interacting system, which we call the effective potential, veff (r). We thus have two

Euler-Lagrange equations providing the same density and thus the same µ:

1) Interacting system:

µ=v(r) + δF[ρ]

δρ .(2.56)

2) Non-interacting system:

µ=veff (r) + δFks[ρ]

δρ .(2.57)

Combining Eqs. (2.56) and (2.57) gives for the effective potential:

veff (r) = v(r) + δF[ρ]

δρ −δFks[ρ]

δρ .(2.58)

To proceed further, we now introduce the first key step of the Kohn-Sham approach, namely the

separation of the interacting system into:

F[ρ] = Fks[ρ] + Eh[ρ] + Exc[ρ].(2.59)

Here, Eh[ρ] is the so called Hartree energy functional which accounts for the classical part of the

electron-electron interaction,

Eh[ρ] = 1

2Zρ(r)ρ(r′)

|r−r′|drdr′,(2.60)

and Exc[ρ] is the exchange-correlation functional which formally accounts for the unknown differ-

ence between Fks[ρ]+Eh[ρ] and the fully interacting system. The functional dependence of Exc[ρ]

is not known as is the case for F[ρ]. However, and this is the actual merit of this approach, the

functional dependence of Exc[ρ] can be approximated to good accuracy using simple formulas as

opposed to F[ρ]. This is a consequence of the fact that Fks[ρ]+Eh[ρ] contains already a major part

of the physics of the interacting system. The exchange-correlation functional is discussed further

18 2.1. From quantum mechanics to density functional theory

in the following section. Inserting Eq. (2.59) into Eq. (2.58) yields:

veff(r) = v(r) + δEh[ρ]

δρ +δExc[ρ]

δρ =v(r) + 1

2Zρ(r′)

|r−r′|dr′+δExc[ρ]

δρ .(2.61)

This is the final equation for the effective potential of the non-interacting system, which can be

evaluated assuming we have approximated Exc[ρ]. Due to the second and third term on the

right hand side this potential is in fact a functional of the density, i.e., veff (r) = veff [ρ](r). This

dependence causes however a difficulty since, while we have a prescription of how to calculate veff

from ρ, we have no prescription of how to obtain ρ. The solution to this issue is given by the

second key step of the Kohn-Sham approach: Up to this point, we have treated the non-interacting

system using DFT, but we can as well use a standard quantum mechanical approach based on an

eigenvalue equation. This is straightforward due to the missing interactions. The full solution of the

non-interacting system can be therefore obtained by solving the Kohn-Sham eigenvalue equation

Hks[ρ]ϕi(r) = ǫiϕi(r),(2.62)

where ǫiand ϕiare the ith eigenvalue and eigenfunction, and where the Kohn-Sham Hamiltonian

Hks assumes an effective one particle form:

Hks[ρ] = −1

2∇2+veff [ρ](r).(2.63)

The eigenfunctions ϕican be used to obtain the electronic density of the non-interacting system

by ρ(r) = X

fi|ϕi(r)|2,(2.64)

where fiis the Fermi-Dirac function:

fi=n1 + e(ǫi−µ)/(kBT)o−1.(2.65)

The origin of the Fermi-Dirac function is the fact that even non-interacting electrons need to obey

antisymmetry with respect to particle interchange. Its maximal value is 1 which means that a

particular energy level can be occupied by at most one electron (Pauli principle). An example

Fermi-Dirac function is plotted in Fig. 2.1 [denoted as FD(T)]. With Eqs. (2.63) to (2.65), we have

obtained a prescription for calculating ρfrom veff. If we finally combine both approaches, the

standard quantum mechanical and the DFT approach, we arrive at a set of fully coupled equations:

veff[ρ](r) = v(r) + δExc[ρ]/δρ +1

2Rdr′ρ(r′)/(|r−r′|)

−∇2+veff (r)ϕi(r) = ǫiϕi(r)

ρ(r) = Pifi|ϕi(r)|2

ϕi(r) (2.66)

ρ(r)veff(r)ϕstart

i(r)

2.1. From quantum mechanics to density functional theory 19

This cycle of equations is called the self consistency cycle. At start, one needs initial wave functions

ϕstart

iwhich can be constructed for instance from random numbers. The equations are then calcu-

lated self consistently until convergence has been achieved, i.e., until ϕi,ρ, and veff do not change

significantly anymore. The converged ϕiand the fican be used to calculate Fks, which contains a

kinetic energy contribution [this is the only contribution to the inner energy for a non-interacting

system; cf Eq. (2.41)] and an entropy contribution [29]:

Fks [{ϕi[ρ]},{fi[ρ]}] = −1

fiZϕi(r)∇2ϕi(r)dr

| {z }

Kinetic energy contribution

−T kBX

[filn fi+ (1 −fi) ln(1 −fi)]

| {z }

Entropy contribution

(2.67)

Here, we have indicated that Fks depends implicitly on ρthrough the self consistency cycle.

Equation (2.67) completes the solution of the non-interacting system. The equilibrium electronic

free energy of the system of interest including interactions can be finally obtained using Eqs. (2.52)

and (2.59):

Fel =Fks[ρ] + Eh[ρ] + Exc[ρ] + Zρ(r)v(r)dr+Vnuc.(2.68)

All terms are now explicitly given except for the exchange-correlation functional Exc[ρ] which is

discussed in the next section.

Let us summarize DFT and the Kohn-Sham approach with the key equations and their scaling

properties:

1) Many electron eigenvalue equation:

Helψν=Eel

νψν

Fel =−kBTln X

e−βEel

ν







with sel(Nmesh, Ne) = (N3

mesh)Ne×(N3

mesh)Ne.

(2.69)

↓DFT

2) Electronic density functional:

Fel = min

ρF[ρ] + Zρ(r)v(r)dr+Vnuc with sdft(Nmesh) = N3

mesh.(2.70)

↓Kohn-Sham

3) Self consistent effective one electron eigenvalue equation:

−1

2∇2+veff(r)ϕi(r) = ǫiϕi(r), ρ =X

fi|ϕi|2

Fel =Fks[ρ] + Eh[ρ] + Exc[ρ] + Zρ(r)v(r)dr+Vnuc











with sks(Nmesh, Nit) =

=Nit ·(N3

mesh ×N3

mesh).

(2.71)

20 2.1. From quantum mechanics to density functional theory

In Eq. (2.71), we have introduced sks the scaling function of the Kohn-Sham approach. It represents

a one particle matrix equation, which is a consequence of the introduction of the effective one

electron eigenvalue equation. We need, however, to solve an eigenvalue equation at each iteration

step of the self consistency cycle, Eq. (2.66), which results in Nit eigenvalue equations, with Nit

the number of iterations steps required for self consistency.

We finally remark that the usual Kohn-Sham approach, as presented here, is not applicable in

every case. Levy [32] and Lieb [28] have shown that there are physically reasonable interacting

systems which produce electronic densities that cannot be obtained from a pure ground state of

a non-interacting system. This means that if we are to use the occupation numbers as given by

Eq. (2.65), no effective potential for the non-interacting system can be constructed such as to yield

the density of the interacting system. Situations of this kind often arise for degenerate systems

and it is then possible to derive a more general Kohn-Sham formalism [33] which uses ensembles

of the degenerate ground states to describe the density. In this general approach, the occupation

numbers in Eq. (2.64) do not decrease monotonically with energy, in fact some of the occupation

numbers below µfrom Eq. (2.65) can go to zero leaving holes and some well above µcan become

1. Further, Englisch and Englisch [34] have shown that in rare cases there are even systems which

cannot be described by this generalized Kohn-Sham procedure. However, these issues are of no

concern for the present study: As remarked in Refs. [35] and [36], if Eqs. (2.62) and (2.64) are not

applicable the self consistency cycle, Eq. (2.66), will not converge. Such a situation did not occur

in the presented calculations.

2.1.6 Exchange correlation functional

We now turn to the discussion of the exchange-correlation energy functional Exc[ρ] which is for-

mally defined by Eq. (2.59). This is the remaining component of the Kohn-Sham approach to be

determined.

The exact functional dependence of Exc[ρ] is not known. There are however certain formal

properties and exact relations which must be fulfilled, like for instance the norm conservation for the

so called exchange-correlation hole [36]. Those can be used as guidelines in constructing/improving

functionals and, by now, a large set is available in literature. Two popular functionals representing

two important families of Exc[ρ] functionals are the local density approximation (LDA) and the

generalized gradient approximation by Perdew-Burke-Ernzerhof (GGA-PBE) [37]. LDA and GGA-

PBE have been widely applied to T= 0 K properties and have proven to be very reliable.2

The idea behind the LDA is to assume that the exchange-correlation energy is local, i.e., that

it depends only on the value of the electronic density at a single space point [29]. The actual

dependence is derived from the homogeneous electron gas and reads:

Exc[ρ] = Zρ(r)ǫxc

heg(ρ(r)) dr.(2.72)

Here, ǫxc

heg(ρ) is the exchange-correlation energy per particle of a homogeneous electron gas (heg)

with the density ρ. Note that ǫxc

heg(ρ) is merely a function and not a functional, since it depends on

a single density value. It is convenient to split ǫxc

heg into an exchange part, ǫx

heg, and a correlation

part, ǫc

heg, as: ǫxc

heg =ǫx

heg +ǫc

heg. The exchange part can be derived exactly analytically and is given

2One of many examples is the study performed in Ref. [38], where the T= 0 K energetics for a wide range

of intermetallics was computed with LDA and GGA, showing good agreement with experiment. In Ref. [38], the

PW91 [39] parametrization of GGA was employed. However, the PW91 and the PBE parametrization typically yield

very similar results [37].

2.1. From quantum mechanics to density functional theory 21

by [29]:

ǫx

heg(ρ) = −3

43

π1/3

ρ4/3.(2.73)

In contrast, the analytical dependence of ǫc

heg is not known. However, numerical results (based

on quantum Monte Carlo calculations) have been obtained for a wide range of densities [40] and

interpolations allow to obtain ǫc

heg for any density. In principle, it is also possible to derive an

exchange-correlation energy for the homogeneous electron gas which contains an explicit temper-

ature dependence [29]. The resulting corrections to the free energy are however small compared

to the error introduced by LDA in general [23]. We will not consider such temperature dependent

exchange-correlation functionals.

Due to the assumptions made, the LDA yields exact results only for the homogeneous electron

gas. Despite this restriction and its simplicity, the LDA has turned out to provide accuracies

sufficient to address a wide variety of material science issues and has been used successfully for

many years [41]. The enormous success of LDA is attributed to the fact that its derivation is based

on a physical system (homogeneous electron gas). This ensures that LDA reproduces the above

mentioned norm conservation and further important relations for the exchange-correlation hole (see

e.g. Ref. [29] for details).

The exact exchange-correlation energy depends on electron density values at all space points.

Therefore, a natural extension of the LDA seems to be a systematic increase of the space points

which contribute to Exc, for instance by including more and more neighboring points. The latter is

equivalent to performing an expansion in terms of the gradient of the electronic density. Exchange

correlation energy functionals of this kind are termed gradient expansion approximation (GEA) and

belong to the set of semi-local functionals. It turns out however that such a systematic increase

of gradient orders is not applicable [42]: The analytical and computational complexity becomes

higher with each order while the achievable accuracy drops as compared to LDA. This unintuitive

loss in accuracy is due to the fact that, in contrast to LDA, the GEA is not derived from a physical

system and it therefore does not fulfill the exact relations. In order to remove this deficiency of the

GEA, a generalized approach, i.e., the generalized gradient approach (GGA), has been developed.

Within GGA the exchange-correlation energy is given by:

Exc[ρ] = Zfxc(ρ(r),∇ρ(r)) dr.(2.74)

Here, fxc is a general function of the density and of its gradient at a single space point. The

important difference to a GEA functional is that within GGA an arbitrary dependence on ρand

∇ρcan be implemented into fxc while for a GEA functional a systematic power-series-like gradient

expansion is performed. The general fxc function gives GGA a greater flexibility which can be

used to construct functionals that fulfill the exact relations.3One of the most reliable and therefore

prominent functionals of this type is the GGA-PBE functional.

Further improvements of the exchange-correlation energy functionals proceed along the lines of

incorporating more non-locality. One such group of functionals are metaGGAs [42] and another

are explicitly orbital dependent functionals [42]. Both groups are however still in their infancy and

3There exists also another philosophy to constructing GGA functionals. The fxc function is constructed such as

to reproduce a certain set of experimental values rather than fulfill the exact relations. This approach is referred to

as semi-empirical and it is used predominantly in quantum chemistry calculations [36]. A prominent functional of

this type is BLYP [36]. To distinguish the semi-empirical and the approach presented in the main text, the latter

is referred to as ab initio or non-empirical. The non-empirical approach is predominantly used in condensed matter

theory and a prominent functional of this type, besides the PBE functional, is the PW91 [39] functional.

22 2.1. From quantum mechanics to density functional theory

the complexity and computational requirements of these functionals are significantly increased as

compared to LDA and GGA. We will not consider them in this study.

We finally comment on the philosophy with respect to the exchange-correlation functional taken

in this work. Since there exists no feasible way to systematically improve the exchange-correlation

functional as shown by the GEA functionals, we regard the errors introduced by LDA and GGA-

PBE as being uncontrollable. In contrast, all other approximations introduced subsequently will

be controllable in the sense that some convergence parameter can always be used to reduce the

corresponding error.

2.1.7 Periodic approach

The matrix corresponding to the Kohn-Sham eigenvalue equation scales as sks(Nmesh) = (N3

mesh ×

mesh) (with Nit = 1). Let us roughly estimate the size of a crystal that would be accessible

based on this equation. For that purpose, we consider however a more realistic situation than the

one sketched in Sec. 2.1.1 where we relied on the world’s most powerful computer machines. We

assume now a single node with, say, 16 GB of memory.4If we further represent our matrix by

single precision numbers (i.e., 4 bytes), we can store ≈63,000 ×63,000 elements and thus increase

Nmesh to ≈40. Using the mesh distance from Sec. 2.1.1, δ= 0.1a0, we are then able to describe

a cell of 4 ×4×4a3

0(= 43a3

0). A typical unit cell of the considered elements is however ≈83a3

(4 atoms) large, and thus too large to be fully captured by the mesh. In fact, due to the cubic

scaling, we would need already 1,000 GB (i.e., 1 TB) of memory to discretize a 83a3

0unit cell.

In Sec. 3.4.3, we will show that cell sizes of at least 163a3

0(32 atoms) are necessary to describe

thermodynamic properties, which is clearly not manageable if we tackle the Kohn-Sham equation

directly as described. In order to proceed, we need to further exploit the underlying physics and

symmetry of the system under study.

An inherent property of crystals is their periodicity. A more suitable basis, as compared to the

real space basis used above, is therefore a plane wave, i.e., periodic or reciprocal, basis. Using a

plane wave basis has in fact two advantages: 1) Only a small number of basis elements needs to

be considered explicitly in the calculation, since they describe already accurately the properties

of the system. 2) Surface effects are removed which allows to describe thermodynamic properties

accurately using smaller cell sizes. Strictly speaking, 1) is only valid if the strong oscillations of the

electronic wave function close to the ionic cores are removed. This can be achieved by employing

the PAW method and is described in Sec. 2.1.8. In the following, we implicitly assume that the

PAW method is employed and discuss the periodic ansatz with the general aim of rewriting the

Kohn-Sham equations in reciprocal space. We put special emphasis on revealing the role played by

two important parameters that are crucial to obtain the desired convergence: the ksampling and

the plane wave cutoff.

Every crystal structure can be classified into one of 14 Bravais lattices (see e.g. Ref. [43]). A

Bravais lattice consists of all points with position vectors Rof the form

R=n1a1+n2a2+n3a3,with n1, n2, n3∈Z,(2.75)

and with lattice vectors a1,a2, and a3. Specifically, the investigated elements belong to the face-

centered-cubic (fcc) Bravais lattice. The fcc lattice vectors can be specified in two different but

equivalent ways corresponding to the conventional unit cell or the primitive unit cell. The first one,

emphasizing the cubic symmetry, will be employed in Sec. 2.3.2. Here, we utilize the primitive cell

4Currently available RAM sizes in the high performance computing sector are in the range of 2 to 64 GB.

2.1. From quantum mechanics to density functional theory 23

with lattice vectors

a1=alat

2(0,1,1),a2=alat

2(1,0,1),a3=alat

2(1,1,0),(2.76)

where alat denotes the lattice constant. The real space representation, Eq. (2.75), has a correspond-

ing representation in reciprocal space, which is defined as the lattice of vectors Gfulfilling:

eiG·R=ei2πn = 1,with n∈Z.(2.77)

Equation (2.77) is satisfied if

G=k1b1+k2b2+k3b3,with k1, k2, k3∈Z,(2.78)

and with the reciprocal lattice vectors uniquely determined by

b1=2π

Va2×a3,b2=2π

Va3×a1,b3=2π

Va1×a2,(2.79)

where

V=a1·(a2×a3) = (alat)3/4 (2.80)

is the volume of the (real space) fcc primitive unit cell. With these definitions and considering the

fact that the periodicity of the lattice implies veff(r) = veff (r+R), we can expand the effective

potential by

veff(r) = 1

veff

GeiG·r,(2.81)

where veff

Gare reciprocal expansion coefficients. The sum in Eq. (2.81) runs only over the reciprocal

lattice vectors Grather than over all possible plane waves, to ensure the periodicity of the potential

in real space. The wave function, which does not need to have this symmetry, can be expanded

using Bloch’s theorem [44, 45] as

ϕi(r) = ϕν,k(r) = eik·rX

cν,k+GeiG·r,(2.82)

with expansion coefficients cν,k+Gand with the wave vector klying inside the so called first Brillouin

zone which is spanned by the vectors b1,b2, and b3from Eq. (2.79). We note two important

observations about Eq. (2.82): 1) A given eigenfunction ϕi(r) of the periodic real space Kohn-

Sham equation can be expanded using again the reciprocal lattice vectors Gwhich now however,

in contrast to Eq. (2.81), depend on a specific wave vector kand which are modulated by a plane

wave eik·rof this wave vector. 2) Wave functions of different kdo not couple to each other and

can be therefore treated separately. The latter point allows also to split the original index iinto an

index kand an index νwhich denotes the solutions for a fixed k. Inserting Eqs. (2.81) and (2.82)

into Eq. (2.62) yields

2meX

G|k+G|2cν,k+Gei(k+G)·r+1

G,G′

veff

G′cν,k+Gei(k+G+G′)·r=ǫν,kX

cν,k+Gei(k+G)·r,(2.83)

where the indices of the eigenvalues, ǫi=ǫν,k, have been adjusted to correspond to Eq. (2.82).

Since the set of Gvectors is infinite, we can shift the summation in the second term on the left

24 2.1. From quantum mechanics to density functional theory

side to ˜

G=G+G′and then rename the index ˜

G→G:

G,G′

veff

G′cν,k+Gei(k+G+G′)·r=1

G,G′

veff

G′cν,k+˜

G−G′ei(k+˜

G)·r

G,G′

veff

G′cν,k+G−G′ei(k+G)·r.(2.84)

Inserting Eq. (2.84) into Eq. (2.83) gives

G"~2

2me|k+G|2−ǫν,kcν,k+G+1

G′

veff

G′cν,k+G−G′#ei(k+G)·r= 0,(2.85)

and finally, since Eq. (2.85) needs to be satisfied for all r, we have:

~2

2me|k+G|2−ǫν,kcν,k+G+1

G′

veff

G′cν,k+G−G′= 0.(2.86)

Equation (2.86) is the final expression for the reciprocal Kohn-Sham equation. Similarly as

Eq. (2.82), it couples only expansion coefficients cν,k+Gfor the same vector k, i.e., it is block

diagonal (αi=kand βj=G). This allows to treat kas a parameter and the corresponding

eigenvalue equation independently of the equations for all the other kvectors. From a technical

viewpoint, this fact allows to parallelize computations in a straightforward manner. The density of

the kvectors, which we call ksampling in the following, becomes only important when calculating

for instance the kinetic energy contribution to Fks from Eq. (2.67):

−~2

2meX

ν,k

fν,kZϕν,k(r)∇2ϕν,k(r)dr=~2

2meX

ν,k

fν,kX

G|k+G|2|cν,k+G|2.(2.87)

Here, the summation, Pi=Pν,k, and the occupation numbers, fi=fν,k, have been adjusted to

correspond to Eq. (2.82) and the orthonormality of plane waves has been used. In principle, the

ksampling needs to be infinitely dense (within the first Brillouin zone) for a macroscopic crystal.

It turns out however that the dependence of Fks (and other properties) on kis smooth and a

rather small mesh can be used to accurately describe materials properties. [An example for the

dependence of ǫν,kon kalong special high symmetry directions (introduced in Sec. 2.4.1) including

additionally the volume dependence is given in Fig. 2.1 for the case of aluminum.] At this point

it is however important to distinguish metallic systems as investigated in the present study from

semiconductors and insulators. For the latter, rather small ksamplings are often sufficient. In

contrast, for metals, denser meshes are required as a consequence of the sharp interface at the

Fermi surface. In calculating thermodynamic properties of metals the convergence aspect of the

ksampling becomes therefore particularly important. This will be investigated in Sec. 3.4.2. In

performing ksampling tests, one should consider that a larger crystal cell in real space leads to a

smaller Brillouin zone in reciprocal space so that less kvectors suffice for the sums in Eq. (2.87).

For a convenient comparison between the ksamplings of different supercells, we will use the unit

kp·atom, which is obtained by multiplying the number of kpoints with the number of atoms in the

supercell, and which takes the inverse relation between real and reciprocal spaces into account.

Equation (2.86) can be considered as a matrix equation in Gand G′and its solution (with

the aim of determining the cν,k+G) scales therefore as s(g) = g3×g3where gis the number of

2.1. From quantum mechanics to density functional theory 25

Gvectors along 1 dimension. In principle, gwould need to go to infinity. Similarly as for the

ksampling, it turns out however that (if the PAW method is employed) only small Gvectors

contribute significantly to the expansion in Eq. (2.82). We therefore introduce a plane wave cutoff,

Ecut, such that only Gvectors are considered which fulfill

2me|G+k|2< Ecut ⇒ |G+k|<r2me

~2Ecut,(2.88)

or using the reduced quantities explained in App. A.1.2:

|Ga0+ka0|<q2Ecut

hand |Ga0+ka0|<qEcut

r.(2.89)

Thus, the allowed vectors lie within a sphere of radius q2meEcut/~2. Let us now reconsider the

estimation from the beginning of this section, i.e., 16 GB of memory allowing to store a matrix of

size 63,000 ×63,000. For the purpose of an estimation, we can choose k= 0 and further assume

that the number of plane waves, Npw, inside the sphere can be obtained from the volume of the

sphere, 4π/3(2meEcut/~2)3/2, divided by the volume of the Brillouin zone, b1·(b2×b3) = 8π3/V ,

i.e.:

Npw ≈V

6π22me

~2Ecut3/2

.(2.90)

Choosing aluminum as an example, we can take Ecut to be 14 Ry (see App. A.1.2 for atomic

units) which is a very well converged cutoff (cf. Sec. 3.4.4.3). Solving Eq. (2.90) for the real space

volume Vand using the maximum allowed Npw = 63,000 yields V≈403a3

0, i.e., significantly larger

than the V≈43a3

0achieved within the real space approach. Thus, by rewriting the Kohn-Sham

equation in reciprocal space and introducing a plane wave cutoff we are able to address cell sizes

which are large enough to calculate converged thermodynamic properties (cf. the discussion at the

beginning of the section). We stress however that the presented estimation has primarily relied

on the information about the converged plane wave cutoff obtained from the detailed convergence

tests performed within this work. Such convergence issues become particularly important when

calculating finite temperature properties.

The transition from the real to the reciprocal space changes the scaling of the Kohn-Sham

approach as follows:

sks =Nit ·(N3

mesh ×N3

mesh)

| {z }

Real space Kohn-Sham equation

→

sks =Nit Nk·Npw ×Npw

=Nit Nk·h(alat)3(Ecut)3/2×(alat)3(Ecut)3/2i.

| {z }

Reciprocal Kohn-Sham equation

(2.91)

Here, we have introduced Nk, the number of kvectors included in the ksampling, to account for

the fact that the matrix constructed from the Npw Gand Npw G′vectors needs to be solved for

each such kvector. We have also used Eqs. (2.80) and (2.90) to express Npw in terms of alat and

Ecut. In fact, in actual implementations, the solution of the Kohn-Sham equation does not scale as

given in Eq. (2.91). Highly optimized routines allow to improve the scaling considerably, however,

the actual realization and implementation is far more complex (and can differ among different

codes) than the presented derivation, which aims at introducing the key convergence parameters, k

sampling and plane wave cutoff. We briefly comment on four important ideas which further reduce

the involved computational effort, while details can be found for example in Refs. [46] or [47].

26 2.1. From quantum mechanics to density functional theory

The first idea is to calculate only the energetically lowest ǫν,k, in particular those that will

be actually occupied by the electrons at thermal equilibrium. For such a purpose, very efficient

techniques have been developed [46] which do not need to diagonalize the Hamiltonian matrix as a

whole. They rely instead on self consistent minimization techniques which use computationally far

less expensive operations as compared to a direct diagonalization. The scaling is then approximately

linear with N3

e(the cubic dependence is due to the necessary orthogonalization of the electronic

orbitals). The second idea is to use specific mixing schemes for the electronic density within the

self consistency cycle, Eq. (2.66). The density as obtained from the Kohn-Sham equation is not

taken directly to calculate the effective potential but it is mixed with previous densities. This

procedure introduces a certain memory effect which reduces Nit the number of self consistency

cycles. The third idea is to calculate not all of the contributions to the Kohn-Sham Hamiltonian

in reciprocal space, but to take the most convenient representation (reciprocal or real space) for

each contribution. For instance, a good representation for the kinetic energy is the reciprocal

space in which it is diagonal. In contrast, the exchange-correlation energy is diagonal in real space

in which it is therefore calculated. Such a dual space approach is only possible because there

are methods available (Fast Fourier Transforms [48]) which transform quantities from one to the

other space very efficiently. Finally, the fourth idea is to employ point symmetries to reduce the

ksampling to include only the inequivalent or irreducible kvectors, i.e., Nk→Nirr

k, with Nirr

the number of irreducible kvectors. Symmetries will be considered in more detail in the context

of the quasiharmonic approximation used to describe nuclei motion (Sec. 2.3.4). Based on these

optimizations, the solution of the Kohn-Sham equation scales as (cf. App. A.1.1 for the notation

convention)

sks =Nit Nk·h(alat)3(Ecut)3/2×(alat)3(Ecut)3/2i→sks(alat, Ecut, Nirr

k, Ne, Nit) =

= (alat)3(Ecut)3/2Nirr

kN3

eNit ,

| {z }

Optimized Kohn-Sham equation

(2.92)

which applies to both CPU time and memory.

2.1.8 LAPW+lo, PAW, and pseudopotentials

In the vicinity of the nuclei positions RI, the effective Kohn-Sham potential, veff(r), is dominated

by the strong Coulomb repulsion produced by the nuclei, v(r) = PNn

IZIe2/(4πǫ0|r−RI|) [cf.

Eqs. (2.49) and (2.51)]. This causes the Kohn-Sham wave functions ϕν,k(r) to oscillate strongly in

this region which can be understood as follows: Classically, if an electron approaches the nucleus

its potential energy decreases while the kinetic energy increases such as to keep the total energy

constant. Quantum mechanically, an increase in kinetic energy corresponds to an increase in the

oscillations of the wave function.

A plane wave description of these strong oscillations would require a large cutoff in Eq. (2.88)

obviating a treatment in reciprocal space. In order to circumvent this difficulty, the following

decomposition of the full crystal wave function might seem reasonable:

ϕν,k(r) = eϕν,k(r/∈ΩI)

| {z }

Smooth wave function,

suited for plane waves

+ϕΩ

ν,k(r∈ΩI)

| {z }

Strong oscillations at nucleus,

suited for atomic basis

.(2.93)

Here, the real space has been divided into two distinct regions: One region consists of non-

overlapping spheres which are centered at the ionic positions. The sphere corresponding to ion

2.1. From quantum mechanics to density functional theory 27

Ihas a volume of ΩI. In this region, ϕΩ

ν,k(r∈ΩI) describes the strongly oscillating crystal wave

function and it is well suited to be described by an atomic like basis. The other region consists

of all the remaining space not covered by the spheres. Here, eϕν,k(r/∈ΩI) describes the smooth

crystal wave function and it is well suited for a plane wave description.

It turns out however that a major problem of Eq. (2.93) is the matching of the two wave functions

from the distinct regions. This problem has two aspects: 1) The wave functions need to match

in real space at the interfaces of the spheres. This can be accomplished efficiently. 2) The wave

functions need to match also in energy space. This requires an additional self consistency loop which

is computationally highly expensive [49]. One possible approach to overcome 2) is to perform the

energy matching only up to the first order of a Taylor expansion. Since this so called linearization

procedure is an approximation, additional orbitals inside ΩI(referred to as local orbitals) are

introduced to reduce the corresponding error. Methods based on this approach are called linearized

augmented plane waves plus local orbitals (LAPW+lo) and despite the approximation introduced

by the linearization, they are considered as the most accurate methods presently available for

solving the Kohn-Sham equation [50]. They are however also the computationally most expensive

methods and, for the system sizes required in this study (up to 500 atoms), we had to apply

a numerically more efficient method, the projector augmented wave (PAW) approach introduced

next. Nonetheless, we will use the LAPW+lo method as a reference for validating the so called

PAW potentials using smaller system sizes.

The PAW method [51] approaches the time consuming energy matching problem (and in fact

the real space matching at the same time) from a conceptually different direction. Within PAW

the decomposition in Eq. (2.93) is reformulated to:

ϕν,k(r) = eϕν,k(r)

| {z }

Smooth wave function

over the full crystal

−eϕΩ

ν,k(r∈ΩI)

| {z }

Smooth wave function

inside spheres

+ϕΩ

ν,k(r∈ΩI)

| {z }

Strongly oscillating wave

function inside spheres

.(2.94)

Here, in contrast to Eq. (2.93), the smooth wave function eϕν,k(r) is allowed to extend over the

full crystal, i.e., also in the spheres. To compensate for the contribution of eϕν,k(r) inside ΩI, a

new smooth function eϕΩ

ν,k(r∈ΩI) is introduced which equals eϕν,k(r) in ΩIand is zero otherwise.

Therefore, Eq. (2.94) is effectively identical to Eq. (2.93). A key point of the PAW method is now

to treat

∆ϕΩ

ν,k(r∈ΩI) = ϕΩ

ν,k(r∈ΩI)−eϕΩ

ν,k(r∈ΩI) (2.95)

as a single object. In particular, the aim is to construct a convenient basis to expand it, i.e., a basis

from which only a few elements suffice to describe ∆ϕΩ

ν,k(r∈ΩI) accurately. This approach has

the following major advantage. The matching in real space and energy space is fully incorporated

into the construction of the basis. This construction is however performed once and can be then

applied to any PAW calculation. In a sense, the matching, in particular the time consuming self

consistency cycle for the energy matching, is outsourced from the actual calculation into the PAW

basis of ∆ϕΩ

ν,k.

The actual construction of a convenient basis for ∆ϕΩ

ν,kis an involved matter and in fact based

on empiricism and experience. We mention briefly only the key aspects. The expansion reads:

∆ϕΩ

ν,k(r∈Ω) = X

I,µ

cI,µ∆φI,µ(r−RI) = X

I,µ

cI,µ hφI,µ(r−RI)−e

φI,µ(r−RI)i.(2.96)

Here, cI,µ are the expansion coefficients and ∆φI,µ the basis elements with µrunning over basis

28 2.1. From quantum mechanics to density functional theory

functions for a single ion Iat RI. The basis elements ∆φI,µ are constructed from two parts

φI,µ and e

φI,µ. The first part φI,µ is called the atomic partial basis and is readily accessible. It

corresponds to a basis of a single isolated ion, i.e., to the eigenfunctions of a corresponding Kohn-

Sham equation. Constructed in this way, it turns out empirically that only a few elements φI,µ

suffice to accurately describe ϕΩ

ν,k. Therefore, the second part e

φI,µ, which is called auxiliary partial

basis, needs to be adjusted such that eϕΩ

ν,kis removed from ϕΩ

ν,kin order to satisfy Eq. (2.95).

Further, each element e

φI,µ needs to approach its dual part φI,µ smoothly at the interface of ΩI

in order to guarantee that ∆ϕΩ

ν,kvanishes outside ΩI. In practice, an auxiliary partial basis is

obtained by solving a Kohn-Sham equation of an artificial isolated ion with the strong Coulomb

potential replaced by an artificially smooth potential. This potential is fine tuned such that the

corresponding eigenfunctions have the desired properties. Finally, also the coefficients cI,µ need to

be determined. For that purpose, note that ∆ϕΩ

ν,kneeds to depend on eϕν,k(r) due to eϕΩ

ν,k. Imposing

additionally (for simplicity) a linearity constraint one obtains [51]:

cI,µ =hepI,µ|eϕν,ki,with hepI,µ|e

φJ,ξi=δI,J δµ,ξ.(2.97)

Here, epI,µ =epI,µ(r) are the so called projector functions. They are typically constructed from the

φI,µ with some optimized analytical dependence [51, 52].

In summary, there are three sets of functions {φI,µ},{e

φI,µ}, and {epI,µ}within the PAW method

which need to be determined for each species and also for each xc functional separately beforehand.

Their dependence is then stored in so called PAW potentials and imported in actual PAW calcu-

lations. In fact, since these functions are solutions to isolated ions, only their radial part needs to

be stored explicitly while the angular part can be treated efficiently analytically.

In principle, the PAW method is a formally exact way to transform the original wave function.

In practice, the expansion in Eq. (2.96) is finite and the quality of the basis must be checked,

i.e., the quality of the PAW potential. We use here PAW potentials [53] as provided with the

vasp code [46]. In addition to the finite basis approximation, these potentials treat only the

valence electrons explicitly while the core electrons are treated within the frozen core approximation.

This approximation exploits the fact that for most materials properties the strongly localized core

electrons do not participate in the chemical bonding and their wave functions retain the form found

for isolated atoms, i.e., they are frozen [54]. To validate if both approximations, the finite basis

and the frozen core, are justifiable, the reliability of the used PAW potentials will be cross checked

against the LAPW+lo method (Sec. 4.1.2). This comparison has in fact a further advantage which

concerns the heavier elements in our study (Ir, Pt, Au). For these elements, the core electrons are

expected to behave relativistically, since they attain high kinetic energies. Such effects (within the

scalar relativistic approximation) are in principle included during the construction of the frozen

core for the PAW potentials. Since the employed LAPW+lo method explicitly contains scalar

relativistic effects for the core electrons, the cross checks will implicitly include also this evaluation.

We need to comment on an important consequence of the wave function decomposition as used

in the PAW method. This issue will lead to the introduction of a new convergence parameter. The

Kohn-Sham equation within the PAW method is formulated in terms of the smooth wave functions

eϕν,kand reads [53, 55]: ˆ

Hpaw eϕν,k=ǫν,kˆ

Oeϕν,k.(2.98)

Here, ˆ

Ois the overlap matrix which is given by

O=ˆ

1 + X

I,J,µ,ξ |epI,µihhφI,µ|φJ,ξi−he

φI,µ|e

φJ,ξiihepJ,ξ|,(2.99)

2.1. From quantum mechanics to density functional theory 29

and which renders Eq. (2.98) a generalized eigenvalue problem. Further in Eq. (2.98), ˆ

Hpaw is

the PAW Hamiltonian which is more complex than the original Kohn-Sham Hamiltonian ˆ

Hks,

Eq. (2.63), due to non-local terms. It reads5[53]

Hpaw =ˆ

Hpaw[eρ, ρΩ,eρΩ, ρaug]

=ˆ

Hrks[eρ+ρaug]

| {z }

Local;

plane waves

I,J,µ,ξ |epI,µihhφI,µ|ˆ

Hrks[ρΩ]|φJ,ξi−he

φI,µ|ˆ

Hrks[eρΩ+ρaug]|e

φJ,ξiihepJ,ξ|

| {z }

Non local; spherical basis on radial grid

(2.100)

where ˆ

Hrks is a reduced Kohn-Sham Hamiltonian given by

Hrks =ˆ

Hks −v(r),(2.101)

and where the electronic charge densities eρ,ρΩ,eρΩ, and ρaug read:

Plane wave grid →eρ(r) = X

ν,k

fν,k|eϕν,k(r)|2,

Radial grid →









ρΩ(r) = X

I,J,µ,ξ

φ∗

I,µ(r−RI)φJ,ξ(r−RI)X

ν,k

fν,kheϕν,k|epI,µihepJ,ξ|eϕν,ki,

eρΩ(r) = X

I,J,µ,ξ e

φ∗

I,µ(r−RI)e

φJ,ξ(r−RI)X

ν,k

fν,kheϕν,k|epI,µihepJ,ξ|eϕν,ki,

Both grids →ρaug(r) = augmentation (or compensation) charges. (2.102)

We have indicated in Eqs. (2.100) and (2.102) in which basis, i.e., on which grid, the various terms

are calculated. Similarly as for the wave function, Eq. (2.94), the PAW Hamiltonian and the PAW

charge densities separate into a smooth part which can be conveniently calculated on a plane wave

grid and an oscillating part inside the spheres which can be efficiently treated on a radial grid.

In this respect, a note on the terms heϕν,k|epI,µiwhich enter ρΩand eρΩmight be useful. These

terms must be first calculated on a plane wave grid and thus the projector functions need to be

transformed from their radial representation onto plane waves. However, the latter transformation

needs to be performed only once at the beginning of the calculation and further heϕν,k|epI,µiincludes

a sum over the reciprocal vectors which renders these quantities mere numbers multiplying the

radial part [φ∗

I,µ(r−RI)φJ,ξ(r−RI)]. Also, possibly large reciprocal vectors (i.e., beyond Ecut)

in the plane wave representation of the projectors are of no concern, since they are simply cut of

when multiplied with the corresponding zero components from eϕν,k.

A complete separation of the PAW Hamiltonian into plane wave and radial components, how-

ever, cannot be achieved. There is one quantity the augmentation charge ρaug which enters both

Hamiltonian parts and which must be therefore calculated on both grids. This fact is indicated

in Eq. (2.100) by the blue color. The origin of the augmentation charge is the need to cancel a

5For the sake of clarity, we have omitted here the contribution of the core charges (corresponding to the quantities

nc,enc,nZc, and enZc in Ref. [53]) and of two further terms connected to the augmentation charge (corresponding to

the terms PLReveff (r)ˆ

ij(r)drand PLRΩrev1

eff (r)ˆ

ij (r)drin Ref. [53]). The omitted terms however do not change

the conclusions drawn in the main text.

30 2.1. From quantum mechanics to density functional theory

spurious electrostatic interaction between the charges in different spheres. An actual construction

scheme for ρaug is given for instance in Ref. [53]. The point of concern here is that its plane

wave representation needs to contain components significantly beyond Ecut. In order to avoid an

increase in Ecut, a dual grid technique is frequently employed [53, 55]. Within this approach, a

second, dense real space grid (augmentation grid; corresponding to high reciprocal components) is

introduced in addition to the usual plane wave grid, in order to treat the augmentation charges

efficiently, while the ”usual” plane wave grid is used for the time consuming operations of the

Kohn-Sham equation [53]. The density of the augmentation grid however corresponds to a new,

third convergence parameter besides the ksampling and plane wave cutoff. We will investigate in

Sec. 3.4.1 the influence of the augmentation grid on thermodynamic properties.

Finally in this section, we discuss the pseudopotential method which is an approximation to the

frozen core PAW method. In particular, we focus on the norm conserving pseudopotential approach

[56] in the separable Kleinman-Bylander form [57] which has the following Kohn-Sham equation

Hpp eϕν,k=ǫν,keϕν,k,(2.103)

with the pseudopotential (PP) Hamiltonian given by:

Hpp[eρ] = ˆ

Hrks[eρ] + ˆ

Vpp

I,µ′+X

I,µ |∆ˆ

VI,µ e

φpp

I,µihhe

φpp

I,µ|∆ˆ

VI,µ |e

φpp

I,µii−1he

φpp

I,µ∆ˆ

VI,µ|

| {z }

PP contribution; independent of density, i.e., fixed during calculation

.(2.104)

Here, ∆ ˆ

VI,µ =ˆ

Vpp

I,µ −ˆ

Vpp

I,µ′and the e

φpp

I,µ are obtained from solving a Kohn-Sham equation of

an artificial isolated ion with the strong Coulomb potential replaced by ˆ

Vpp

I,µ. This procedure is

analogous to determining the auxiliary partial basis e

φI,µ for the PAW potentials. In fact, the

solutions φI,µ to a Kohn-Sham calculation of an isolated atom (including the Coulomb potential)

are also here used to determine the shape of e

φpp

I,µ. In particular, the artificial potential ˆ

Vpp

I,µ is fine

tuned such that:

1) the eigenvalues corresponding to e

φpp

I,µ equal the eigenvalues corresponding to φI,µ,

2) e

φpp

I,µ and φI,µ match outside ΩI,

3) the norms R|e

φpp

I,µ|2drand R|φI,µ|2drare equal inside ΩI(norm conservation),

4) the radial part of e

φpp

I,µ has no nodes.

Further in Eq. (2.104), ˆ

Vpp

I,µ′is the so called local potential and corresponds to ˆ

Vpp

I,µ for a fixed

µ=µ′. For a sufficiently large basis e

φpp

I,µ (with respect to the atomic quantum number µ), the

choice of ˆ

Vpp

I,µ′is arbitrary. Since however the summation in Eq. (2.104) is typically truncated for

lower µ, the local potential needs to be chosen such that it adequately reproduces the scattering

properties in the higher angular momentum channels [56]. As in the case of the PAW potentials,

φpp

I,µ,ˆ

Vpp

I,µ′, and e

φpp

I,µ are determined beforehand, stored on radial grids in so called pseudopotentials,

and then imported during actual calculations.

The reason for introducing the norm conserving pseudopotential method are the following ad-

vantages as compared to the PAW method:

2.2. Capturing electronic bonding effectively: The EAM approach 31

1) The overlap matrix ˆ

O, Eq. (2.99), reduces to ˆ

1 due to norm conservation. This reduces the

pseudopotential Kohn-Sham equation, Eq. (2.103), to an ordinary eigenvalue problem which

is computationally less expensive than a generalized one.

2) Another consequence of norm conservation is that the spurious electrostatic interaction be-

tween different spheres is removed. Therefore, the augmentation charge needs not to be

introduced and the only convergence parameters remain the ksampling and the plane wave

cutoff.

3) The PP contribution to Eq. (2.104) is independent of the electronic density. Therefore, it

needs to be calculated once at the beginning of a calculation and does not have to be self

consistently updated. In fact, the PP contribution can be systematically derived from a

Taylor expansion of the non-local contribution to the PAW Hamiltonian, Eq. (2.100) [58].

The local potential ˆ

Vpp

I,µ′corresponds then to the first order term and the non-local term of

the PP contribution, i.e., the third term in Eq. (2.104), corresponds to the second order term.

In this respect, the PAW potential can be viewed as a dynamic pseudopotential which adapts

to the actual environment.

4) For some elements (aluminum in the present case), it is possible to generate pseudopotentials

which yield accurate results based on smoother wave functions as compared to the PAW.

Thus, the employed Ecut can be chosen smaller reducing the computational effort.

We will apply the pseudopotential approach for the time consuming calculations of anharmonic

contributions in aluminum. For the other elements in this study the pseudopotential method is not

well suited. It would produce ”harder” wave functions than the PAW method, i.e., wave functions

with larger reciprocal components, especially for the transition metals [59].

2.2 Capturing electronic bonding effectively: The EAM approach

DFT, the Kohn-Sham ansatz, the periodic representation, and the PAW method allow to solve quan-

tum mechanical electronic systems that are not addressable using the original electronic Schr¨odinger

equation, Eq. (2.38). At present, structures of up to ≈1000 atoms can be investigated based on

the combination of these approaches. As shown in Sec. 3.4.3, these system sizes are sufficient to

calculate accurate thermodynamic properties of elementary crystalline metals as intended here. In

general however, a variety of materials science issues will not be describable with this size restric-

tion. An example is the study of grain boundary migration which makes system sizes of several

10,000 atoms necessary [60]. Additionally, there is often also a time restriction when considering

dynamical phenomena. To address such issues nonetheless, the application of empirical approaches

is unavoidable. Such approaches capture the electronic bonding only effectively and are therefore

orders of magnitude faster than DFT but consequently significantly less accurate. For metals,

one particularly convenient empirical approach is the embedded atom method (EAM) [61]. So far

however, a well founded evaluation of the capability of the EAM approach to yield sound thermo-

dynamic properties up to the melting point, was hampered by the lack of reference data. Based

on our DFT calculations we will be able to fill this gap. In particular, we will investigate three

different EAM parametrizations for aluminum introduced in Sec. 4.5. In the following we discuss

briefly the theoretical background of the EAM approach, while details can be found for instance in

Ref. [61].

32 2.2. Capturing electronic bonding effectively: The EAM approach

3 4 5 6

|RI − RJ| (Å)

-0.3

-0.2

-0.1

Energy (eV)

0.0 0.5 1.0 1.5

Charge density (e)

-2

-1

Energy (eV)

Mei-Davenport

Zope-Mishin

Ercolessi-Adams

3 4 5 6

|RI − RJ| (Å)

0.03

0.06

0.09

Charge density (e)

a) b) c)

Figure 2.3: The three EAM parametrizations for aluminum (Mei-Davenport [62], Zope-Mishin [63], and

Ercolessi-Adams [64]) employed in this work. a) The pair potential vpair as a function of the distance

between two atoms. The vertical dashed (dotted) line marks the distance to the first (second) nearest

neighbor in fcc aluminum. Both distances are similar for all parametrizations. b) The embedding energy

femb as a function of the local density ρI, which is the sum of charge densities of all neighboring atoms

inside a certain cutoff radius. The charge density is given in units of the electronic charge e. The vertical

dot-dashed line marks ρIfor fcc aluminum at the equilibrium lattice constant. c) Charge density ρat as a

function of the distance between two atoms. The meaning of the vertical lines is as in a).

The total energy Eeam of an elementary material composed of Nnatoms reads within the EAM

approach [61]

Eeam({RI}) = 1

J6=I

vpair(|RI−RJ|) +

femb(ρI),(2.105)

with the local electron density ρIgiven by:

ρI=

J6=I

ρat

J(|RI−RJ|).(2.106)

Equation (2.105) provides a means to calculate the energy for any given atomic configuration {RI}

using only the three one dimensional and beforehand parametrized functions vpair,femb, and ρat.

Examples for these functions are given in Fig. 2.3. The pair potential vpair can be interpreted as

an electrostatic interaction between two atoms [61]. If vpair would be the only term contributing to

Eeam, the bond strength would be coordination independent, i.e., the bond energy of two atoms

Iand Jwould always be a constant regardless of how many further atoms are placed near Iand

J. This coordination independence strikingly contradicts the true physical behavior observed in

metallic systems [61]. The key idea of the EAM approach is to introduce an embedding energy

femb which accounts exactly for this missing dependence. It is a function of the local electron

density ρIwhich is produced by the spherically symmetric atomic densities ρat

Jof all atoms inside

a certain cutoff radius, except for I.

The EAM approach can be also motivated by starting with the Kohn-Sham expression Eq. (2.68)

[in fact the T= 0 K version of Eq. (2.68) is used] and performing two crude approximations which

2.3. Exploring the nuclei phase space 33

lead directly to Eq. (2.105) [61]. Despite this formal connection, however, no unique way exists to

determine vpair,femb, and ρemb from an underlying theory [65]. Consequently, various schemes have

been developed and we discuss three possibilities in Sec. 4.5. Assuming that we have determined

these functions, Eq. (2.105) allows to treat system sizes in the range of 107atoms on present day

high performance computers [66].

In order to show how the EAM approach is incorporated into the so far discussed methodology,

let us reconsider Eqs. (2.33) to (2.36) and combine them with Eq. (2.105):

Eeam({RI})=1

J6=I

vpair(|RI−RJ|) +

femb(ρI) (2.107)

nˆ

Tnuc +ˆ

1Eeam({RI})oΛeam

µ=Enuc,eam

µΛeam

µ(2.108)

F≈Feam =−kBTln X

e−βEnuc,eam

µ.(2.109)

Thus, Eeam({RI}) replaces the electronic free energy surface Fel({RI}) and determines now fully

the nuclei motion. We have indicated this fact by introducing corresponding eigenfunctions Λeam

and eigenvalues Enuc,eam

µand the subsequently resulting free energy Feam. The methods developed

in Sec. 2.3 for solving the nuclei equations, Eqs. (2.108) and (2.109), can be fully applied regardless

of which of the surfaces, Fel({RI}) or Eeam({RI}), is used. We stress however that if we employ

the EAM approach, the electron system is treated effectively at T= 0 K and we have no access to

the contribution of the electrons to the free energy.

2.3 Exploring the nuclei phase space

2.3.1 Motivation

In Secs. 2.3 and 3.1, we will be concerned with solving Eq. (2.35), i.e.,

nˆ

Tnuc({RI}) + ˆ

1Fel({RI})o˜

Λµ({RI}) = ˜

Enuc

µ˜

Λµ({RI}) (2.110)

and subsequently Eq. (2.36):

F=−kBTln X

e−β˜

Enuc

µ.(2.111)

Using Eqs. (2.37), (2.69) to (2.71), (2.91), and (2.92), we have the scaling behavior of the full

problem, i.e., involving the calculation of Fel({RI}), given by:

s=(N3

mesh)Nn·sks(alat, Ecut, Nirr

k, Ne, Nit)

| {z }

Optimized Kohn-Sham eq.

for fixed nuclei coordinates

+(N3

mesh)Nn×(N3

mesh)Nn

| {z }

Effective nuclei Schr¨odinger eq.

for electronic ground state FES

.(2.112)

We have hereby written in black the part of the problem that has been transformed into a solvable

expression by the methods introduced in Secs. 2.1.4 to 2.1.8, whereas red marks the two parts

34 2.3. Exploring the nuclei phase space

that render the full solution at this stage infeasible: 1) We need to solve the Kohn-Sham equation

(N3

mesh)Nntimes, i.e., for each nuclei configuration. If we take Nmesh = 200 from the example in

Sec. 2.1.1, Nn= 2, and 1 CPU hour for one Kohn-Sham equation, we would need a calculation

time of 1010 years. 2) As for the solution of the effective nuclei Schr¨odinger equation, we can apply

the estimations performed in Sec. 2.1.1 replacing the electron and nucleus system by two nuclei.

Inspection of the nuclei Schr¨odinger equation reveals that it scales exactly as the electronic

Schr¨odinger equation, Eqs. (2.38) and (2.40), but with Nereplaced by Nn. An intuitive way to

proceed might therefore be the application of DFT and the Kohn-Sham ansatz to remove the

dependence on Nnand to render issue 2) feasible. Indeed, such approaches are developed in the

field of nuclear physics [67]. For the case of thermodynamic properties of crystals however, a

conceptually different approach needs to be applied. The reason is the different physical nature of

the electronic and nuclei systems:

A) Focusing on metals, the valence electrons are free to move across the full crystal. In quantum

mechanical terms, an electron wave function extends over the full crystal and the overlap with

the wave functions of other electrons is significant. Due to this overlap, quantum mechanical

many-body effects need to be accounted for as it is done in DFT and the Kohn-Sham ansatz.

B) In contrast, the nuclei are closely bound to their T= 0 K equilibrium positions. This holds

true even if we consider thermal vibrations. The different one nucleus wave functions therefore

practically do not overlap and quantum mechanical many-body effects are negligible. Instead,

there are two other very important aspects that need to be accounted for: First, the nuclei

motion is strongly coupled in a classical fashion due to the presence of the valence electrons,

i.e., due to Fel({RI}). This is the origin of thermal vibrations and the major contribution to

all thermodynamic properties. Second, even though quantum mechanical many-body effects

are not present, each of the nuclei must be treated quantum mechanically. This means that the

uncertainty principle, i.e., the impossibility to localize a particle in both real and reciprocal

space, must be obeyed. The consideration of this principle is crucial for thermodynamic

properties at low temperatures.

An adequate approach to efficiently cover the issues in B) is the quasiharmonic approximation.

The quasiharmonic approximation is a perturbative approach respecting the special role played by

the T= 0 K equilibrium positions of the nuclei, {R0

I}, by expanding Fel({RI}) in a Taylor series

up to second order at {R0

I}. Considering only the second order term allows computationally an

extremely efficient treatment. Nonetheless, it describes a major part of the classical coupling of

the nuclei, while the rather small effects beyond the quasiharmonic approximation can be taken

into account by two distinct approaches which will be discussed in Sec. 2.3.5. The quasiharmonic

approach neglects by definition quantum mechanical many-body effects, which enables to solve

the resulting Hamiltonian using a standard procedure, which is the correspondence principle. This

means that, in a first step, we can define a purely classical variable U(R0

I), which denotes the

displacement of nucleus Iout of R0

I, i.e.,

U(R0

I) = RI−R0

I,(2.113)

and a corresponding classical momentum P(R0

I). This allows to solve the classical Hamilton func-

tion and by requiring that U(R0

I) and P(R0

I) obey the standard canonical commutation relations,

we obtain the quantum mechanical result. This procedure fully takes into account the quantum

mechanical nature of each of the nuclei discussed above in B), i.e., quantum mechanical one-body

effects. A further advantage of the quasiharmonic approximation is the fact that it tackles both of

2.3. Exploring the nuclei phase space 35

the difficulties with Eq. (2.112), issues 1) and 2), at the same time. Before discussing these issues

in detail and introducing the quasiharmonic approximation formally, we need to briefly comment

on defect formation and to introduce the supercell concept in the next section.

As schematically shown in Fig. 2.1, the step preceding the quasiharmonic approximation, which

separates the vibrations of the perfect crystal into quasiharmonic and anharmonic vibrations, is

the separation of the nuclei motion into vacancy formation and perfect crystal vibrations. Within

the present work, an intuitive model was developed to perform the latter separation. We therefore

postpone the discussion of this step to the methodological part, Chap. 3. We note here that in

calculating vacancy properties, we will need to apply the quasiharmonic approach to a perfect

crystal and to a crystal containing a vacancy. Due to the break of translational symmetry, some of

the equations given below need to be modified for the vacancy crystal. We will address this issue

in Sec. 3.3.3.

2.3.2 Supercell approach

So far, for the discussion of the electronic free energy, it was sufficient to consider a crystal lattice as

being defined by the primitive lattice vectors, Eqs. (2.75) and (2.76). These definitions correspond,

however, to the equilibrium positions {R0

I}of the nuclei at T= 0 K. In order to be able to describe

nuclei vibrations at finite temperatures within a periodic ansatz, we introduce the so called supercell

approach which will lead to a new convergence parameter, the supercell size. This approach is also

needed to describe defect structures, such as vacancies or self interstitials.

A supercell is defined to be a multiple of either the primitive unit cell, Eq. (2.76), or of an

equivalent unit cell. Specifically, we will use cubic supercells (sc)

s1=nsc a′

1,s2=nsc a′

2,s3=nsc a′

3,(2.114)

built up from the conventional fcc unit cell, which is given by the lattice vectors

a′

1=alat(1,0,0),a′

2=alat(0,1,0),a′

3=alat(0,0,1),(2.115)

and by four atoms placed at:

1=alat(0,0,0),R0

2=alat

2(0,1,1),R0

3=alat

2(1,0,1),R0

4=alat

2(1,1,0).(2.116)

The employed supercell sizes in the present study and the corresponding numbers Nnof atoms per

supercell will be:

nsc ∈ {1,2,3,4,5} ⇒ Nn∈ {4,32,108,256,500}.(2.117)

The reason for constructing cubic supercells are symmetry compatibility and efficient defect-defect

screening. After setting up the supercell, the atoms within are allowed to freely vibrate around

Iaccording to the given temperature. Beyond the supercell, we have however periodic boundary

conditions, which are implicit due to solving the Kohn-Sham equation in reciprocal space, and

hence, to each atom there is an infinite number of image atoms, i.e., atoms that move identically

to the original atom. In terms of the displacement U(R0

I), we have

U(R0

I+n1s1+n2s2+n3s3) = U(R0

I),(2.118)

for any integers n1,n2, and n3. The consequence of this implicit periodicity is best discussed in

reciprocal space. Let us therefore calculate Uq, the Fourier transform of all real space displacements,

36 2.3. Exploring the nuclei phase space

i.e., also those beyond the supercell, which is given by:6

Uq=N−1/2

nN−3

U(R0

I+n1s1+n2s2+n3s3)e−iq·(R0

I+n1s1+n2s2+n3s3).(2.119)

Here, qis a plane wave vector corresponding to the nuclei displacements. Further, the first sum runs

over all Nnnuclei within the supercell and the corresponding normalization factor N−1/2

nis split

symmetrically to this Fourier transform and its inverse given below in Eq. (2.128). The remaining

sums run each over Nsintegers, where Nsneeds to go in principle to infinity to account for the

periodicity. The resulting value of the sums however does not diverge, which becomes apparent

when Eq. (2.119) is rewritten using Eq. (2.118):

Uq=N−1/2

U(R0

I)e−iq·R0

I×

×"N−1

e−in1q·s1#"N−1

e−in2q·s2#"N−1

e−in3q·s3#

| {z }

1 for Ns→ ∞ and q=Gsc

0 for Ns→ ∞ and q6=Gsc

(2.120)

In general, a sum 1/N PN

ne−ixn for N→ ∞ is 1 if xis a multiple of 2πand 0 otherwise. Equa-

tion (2.120) will therefore be identical to 0 unless q·s1= 2πn,n∈Z, and similarly for q·s2and

q·s3. This is an analogous requirement to Eq. (2.77) however with Rreplaced by s1,s2, and s3

and therefore qvectors satisfying these conditions will lie on a supercell reciprocal lattice defined

Gsc =k1bsc

1+k2bsc

2+k3bsc

3with k1, k2, k3∈Z,(2.121)

and with the supercell reciprocal lattice vectors uniquely determined by

bsc

1=2π

Ωs2×s3,bsc

2=2π

Ωs3×s1,bsc

3=2π

Ωs1×s2,(2.122)

where Ω = s1·(s2×s3) is the volume of the supercell in real space. The Fourier transform Uqis

therefore fully determined by the Gsc vectors

Uq=UGsc =N−1/2

UIe−iGsc·R0

I,(2.123)

where we have now written UI=U(R0

I) for short. In essence, Eq. (2.123) is equivalent to the

intuitive statement that only plane waves are allowed which have periodicity according to the

supercell geometry. A consequence of this fact is the restriction that not all qvectors will contribute

to thermodynamic properties but only the set of Gsc vectors. Due to the inverse relation of the real

6Note that, since the investigations in this work focus only on elementary crystals with one basis atom per primitive

cell, we write all formulas regarding Fourier transforms in the corresponding reduced version, i.e., we do not include

sums over basis atoms. Similar arguments will apply (in Sec. 2.3.3) to the mass scaling factors. General expressions

can be found for instance in Ref. [68]. Note further that for the purpose of a convenient reading, we do not include the

mass scaling factor and the eigenvectors of the dynamical matrix into the Fourier transforms in the present section.

Both will be needed and thus included in Sec. 2.3.3. The important point is that the relations derived in this section

still hold true after this inclusion (the mass is a mere scalar and the eigenvectors obey the necessary properties).

2.3. Exploring the nuclei phase space 37

and reciprocal space, a larger supercell will produce a denser grid of Gsc vectors. In experiment,

the supercell corresponds in principle to the macroscopic crystal.7The plane waves contributing

to experimental thermodynamic properties are therefore lying on a significantly denser grid as

compared to the grids achievable theoretically [cf. Eq. (2.117)]. An interpolation scheme (based on

a Fourier transform) can be applied to produce also theoretically a dense plane wave grid, i.e., to

facilitate access to qvectors in between the Gsc vectors. We will introduce this scheme in Sec. 2.4.1.

It is nonetheless very important to carefully check the convergence of thermodynamic properties

with the supercell size. The reason is that frequently significant deviations (see Sec. 3.4.3) can

occur between the predicted interpolation and the true frequencies, if too few sampling points are

used. The number of sampling points, however, correlates with the supercell size and we thus have

to consider a new convergence parameter. The set of all convergence parameters increases therefore

to: ksampling, plane wave cutoff Ecut, augmentation grid, and supercell size nsc.

The reciprocal displacements UGsc and the corresponding Gsc vectors have three important

exact properties:

1) The UGsc obey a symmetry property which can be readily used to simplify the calculation of

thermodynamic quantities. Let us consider

UGsc+G=N−1/2

UIe−i(Gsc+G)·R0

=N−1/2

UIe−iGsc·R0

Ie−iG·R0

| {z }

= 1 due to

Eq. (2.77)

=UGsc ,(2.124)

where Gis a reciprocal lattice vector, Eq. (2.78), corresponding to the primitive cell, Eq. (2.76).

Due to the construction of the supercell, Eqs. (2.114) to (2.116), the equilibrium positions

Icorrespond to the lattice vectors Rdefined in Eq. (2.75) and therefore Eq. (2.77) can

be applied to e−iG·R0

I. Equation (2.124) states that the displacement UGsc has periodicity

corresponding to the reciprocal lattice of the primitive cell. We therefore need to consider

only displacements within the first Brillouin zone of the primitive lattice (prBZ). This applies

likewise to all thermodynamic quantities. In particular, as shown in Sec. 2.3.3 the calculation

of thermodynamic quantities will require a sum over Gsc vectors and we can confine this sum

to the prBZ. An example for the employed Gsc meshes is given in Fig. 2.4.

2) For the case of elementary crystals with one atom per primitive cell as considered here,

the number NGsc of Gsc vectors inside the prBZ equals the number Nnof atoms in the

corresponding supercell. One way to see this is to calculate the ratio of the volume of the

prBZ and the volume that each of the Gsc vectors occupies:

NGsc =b1·(b2×b3)

bsc

1·(bsc

2×bsc

3)= 4 (nsc)3=Nn.(2.125)

To obtain the second equality, we have used Eqs. (2.76), (2.79), (2.114), (2.115), and (2.122).

The last equality follows from comparison with Eq. (2.117). In practical applications, one has

to be cautious with Gsc vectors which lie on the boundary of the prBZ. Such vectors should

be taken only once into account as shown in Fig. 2.4.

7Strictly speaking, this is not fully correct since the boundary conditions are different. Those will be however

negligible for the relevant scales.

38 2.3. Exploring the nuclei phase space

4p/alat

Figure 2.4: The mesh of Gsc vectors (black dots) as defined by Eqs. (2.121) and (2.122) for a 2 ×2×2

supercell, i.e., nsc = 2 in Eq. (2.114). Only Gsc vectors lying inside the first primitive lattice Brillouin

zone (prBZ; heavy red lines), Eq. (2.76), are shown. The supercell Brillouin zone (scBZ) is indicated by the

heavy dashed lines. The reciprocal lattice vector is q= (qx, qy, qz). The thin black lines are a guide to the

eye. – Note that the scBZ defines a simple cubic mesh due to the cubic supercell, while the prBZ defines a

base-centered-cubic lattice (the reciprocal lattice of fcc).

3) In general, the UGsc are complex variables. Taking however the complex conjugate of

Eq. (2.123) and considering the fact that the real space displacements UIare always real, we

obtain

U∗

Gsc =N−1/2

UIeiGsc·R0

I=U−Gsc ,(2.126)

which means that the displacements at Gsc and at −Gsc are coupled to each other. This

coupling can be understood also by considering a degrees-of-freedom argument: In real and

in reciprocal space the number of degrees-of-freedom needs to be the same. If however the

reciprocal displacements are complex, they have two degrees-of-freedom each while the real

space displacements have only one. Therefore the coupling reduces the number of degrees-

of-freedom in reciprocal space and ensures that it is equal to the number given in real space,

which is the physically relevant one.

As a final step of the discussion of the supercell concept, we need to adapt the electronic system

to the supercell geometry. We consider for that purpose the expansion of the electronic wave

function in reciprocal space, Eq. (2.82), which now changes to:

ϕν,k(r) = eik·r

Ecut

Gsc

cν,k+Gsc eiGsc·rwith k∈scBZ.(2.127)

The sum here runs over the supercell reciprocal vectors Gsc, which fulfill the condition Eq. (2.88),

and kis now confined to the first Brillouin zone of the supercell lattice (scBZ), which is spanned by

the vectors bsc

1,bsc

2, and bsc

3from Eq. (2.122). It is useful to compare Eq. (2.127) with the inverse

2.3. Exploring the nuclei phase space 39

Fourier transform of the nuclei displacement given by:

UI=N−1/2

prBZ

Gsc

UGsc eiGsc·R0

I.(2.128)

Here, according to Eq. (2.124) and the following discussion, the sum runs only over the prBZ.

The distinction between the two Brillouin zones, prBZ and scBZ, is illustrated in Fig. 2.4. The

comparison between Eqs. (2.127) and (2.128) should be performed with caution, since, as discussed

in Sec. 2.3.1, the electronic and nuclei systems are treated on a different footing. Nonetheless, the

comparison helps to reveal the role played by the kand Gsc vectors:

1) The electronic wave function ϕν,k(r) depends on k, i.e., on wave vectors from inside the scBZ,

whereas UIdoes not. The origin of the kdependence is that (Sec. 2.3.1) the electrons are

free to move in the full crystal, i.e., also beyond the supercell. To describe this motion also

plane waves are needed, which have wave vectors incommensurable with the supercell. The

kvectors describe exactly such plane waves. In contrast, the nuclei are confined to their

equilibrium positions and do not require a kdependence.

2) The sum over Gsc vectors extends to Ecut in Eq. (2.127) and would need to go formally to

infinity, whereas the sum in Eq. (2.128) is rigorously confined to the prBZ. In a sense, the

prBZ defines a natural cutoff for the expansion of UI. The reason for this natural cutoff is

that, again due to the confinement of the nuclei, the real space displacement of the nuclei is a

function of the discrete equilibrium positions R0

I. Taking Gsc vectors beyond the prBZ into

account would only describe plane waves oscillating faster than the next nearest neighbor

distance and would thus contain no new information. This fact was implicitly used above in

Eq. (2.124). On the contrary, ϕν,k(r) depends on the continuous variable rand therefore

all possible Gsc vectors contain new information which could in principle be relevant.

The remaining equations from Sec. 2.1.7 need to be adapted to the supercell approach accordingly

to Eq. (2.127) by changing G→Gsc and by performing the ksampling over the scBZ. Additionally,

we have to adjust the dependence of the electronic free energy as

Fel({RI})→Fel ({RI},Ω ),(2.129)

with Irunning now only over the atoms inside the supercell. The dependence of Fel on the supercell

volume Ω will turn out to be a crucial ingredient of the quasiharmonic approach. It will give rise to,

e.g., the thermal expansion. With these preliminary considerations regarding the supercell concept,

we can proceed with the formal introduction of the quasiharmonic approximation.

2.3.3 Quasiharmonic approximation

The central equation of the quasiharmonic approach8is a Taylor expansion of the electronic free

energy Fel around the T= 0 K equilibrium nuclei positions {R0

I}:

Fel({RI},Ω, T) = Fel

0(Ω, T) + 1

I,J

α,β

UI,αUJ,β ∂2Fel({RI},Ω, T)

∂RI,α∂RJ,β {R0

+O(U3).(2.130)

8Note that the basic ideas underlying the quasiharmonic approximation are well known since already the beginning

of the last century. A main contributor to the field of the theory of lattice dynamics was M. Born [69, 70]. A more

recent discussion can be found for instance in Ref. [68].

40 2.3. Exploring the nuclei phase space

Here, the zeroth order term is abbreviated as Fel

0(Ω, T) := Fel({R0

I},Ω, T), αand βrun over the

three real space components of the vector UI, and O(U3) denotes terms of third and all higher

orders in UI. We have further explicitly indicated the dependence of Fel on T, which enters through

the Fermi-Dirac function, Eq. (2.65). The expansion of Fel does not contain the first order term

I,α

UI,α ∂Fel({RI},Ω, T )

∂RI,α {R0

,(2.131)

since, by definition of the equilibrium positions, it is identical to zero. We next define a key quantity

of the quasiharmonic approximation, the dynamical matrix D, with the components

DIα,Jβ(Ω, T) := 1

MI∂2Fel({RI},Ω, T)

∂RI,α∂RJ,β {R0

,(2.132)

where the mass MIof the nucleus Iis included for later convenience. The quasiharmonic approxi-

mation is now introduced by neglecting the terms O(U3)

Fel({RI},Ω, T)≈Fel

0(Ω, T) + 1

I,J

α,β

UI,αUJ,βDIα,Jβ(Ω, T )

| {z }

=: UD(Ω, T)U/(2M)

,(2.133)

where we have defined a short hand notation for future reference. The displacement vectors U

and mass vector Mcomprise hereby the displacements and masses of all nuclei {UI}and {MI}

respectively. ”Harmonic” refers to the inclusion of the second order term which will be subsequently

transformed into a form similar to that of a harmonic oscillator. The term ”quasi” emphasizes that

the dynamical matrix is not constant, but depends on the supercell volume Ω. The next step is

now to solve the effective nuclei Schr¨odinger equation, Eq. (2.110), based on the approximation

Eq. (2.133), i.e., to solve the eigenvalue equation of the quasiharmonic Hamiltonian ˆ

Hqh

Hqh ˜

Λµ=˜

Eqh

µ˜

Λµ,(2.134)

with eigenfunctions ˜

Λµand eigenvalues ˜

Eqh

µand with ˆ

Hqh given by

Hqh(Ω, T) = −

2MI∇2

I+ˆ

1Fel

0(Ω, T) + 1

I,J

α,β

UI,α ˆ

UJ,βDIα,Jβ(Ω, T ),(2.135)

where we have used the explicit form of the nuclei kinetic energy operator ˆ

Tnuc({RI}), Eq. (2.8).

It should be noted that the displacements UItransform to operators ˆ

UIwhen inserted into ˆ

Hqh

since they are functions of the nuclei positions. Further, due to the dependence of ˆ

Hqh on Ω and

T, the eigenfunctions and eigenvalues in Eq. (2.134) also require this dependence. As discussed in

Sec. 2.3.1, we can use the correspondence principle to solve Eq. (2.134), since we by definition ex-

clude quantum mechanical many-body effects (and since we have a well defined classical analogon

to the quantum mechanical operators). We therefore transform the quantum mechanical quasi-

harmonic Hamilton operator, Eq. (2.135), into its classical version, the quasiharmonic Hamilton

function Hqh, by the transformation rules

−i~∇I→PIand ˆ

UI→UI,(2.136)

2.3. Exploring the nuclei phase space 41

where PI=P(R0

I) is the classical momentum of nucleus I. Using Eq. (2.136), we obtain:

Hqh(Ω, T) =

2MI

I,α +Fel

0(Ω, T) + 1

I,J

α,β

UI,α UJ,β DIα,Jβ(Ω, T ).(2.137)

The reason for changing into the classical description is that we can now Fourier expand PIanalo-

gously to Eq. (2.128) for UI, which will allow to transform also Hqh into reciprocal space. The latter

will turn out to be significantly more convenient than the real space representation Eq. (2.137).

We need however to use for both a modified form:

UI=N−1/2

nM−1/2

prBZ

Gsc

UGsc

, s eiGsc·R0

IwGsc

, s,(2.138)

PI=N−1/2

nM1/2

prBZ

Gsc

PGsc

, s eiGsc·R0

IwGsc

, s.(2.139)

Here, the PGsc are expansion coefficient corresponding to the momentum of a plane wave with

vector Gsc and sruns over the three reciprocal components of the vectors UGsc and PGsc which

are commonly called branches. As compared to the Fourier transforms utilized in Sec. 2.3.2, in

particular Eq. (2.128), we have two changes: 1) The mass factors MIhave been included, which

correspond however merely to scalars introduced for convenience. 2) The vectors wGsc

, s, which

correspond to eigenvectors of the reciprocal dynamical matrix [defined in Eq. (2.141)], have been

introduced. This change is important and will be discussed below. First, using Eqs. (2.138) and

(2.139) and anticipating Eqs. (2.144) and (2.145), we can rewrite Eq. (2.137) as [68]:

Hqh(Ω, T ) = 1

prBZ

Gsc

PGsc

, s P−Gsc

, s +Fel

0(Ω, T) +

+N−1

prBZ

Gsc

UGsc

, s U−Gsc

, s

α,β

I,J

DIα,Jβ(Ω, T)eiGsc·(R0

I−R0

J).(2.140)

We can now define the reciprocal dynamical matrix DGsc as the Fourier transform of the real space

dynamical matrix Dwith the following components:

DGsc

, αβ(Ω, T ) = N−1

I,J

DIα,Jβ(Ω, T)eiGsc·(R0

I−R0

J).(2.141)

For each Gsc,DGsc is (in our study) a 3 ×3 matrix and has the important property of being

Hermitian, which can be seen by writing:

(DGsc

, αβ)∗=N−1

I,J

DIα,Jβ e−iGsc·(R0

I−R0

=N−1

I,J

DIα,Jβ e−iGsc·(R0

J−R0

I)=DGsc

, αβ.(2.142)

42 2.3. Exploring the nuclei phase space

To obtain the second line, we have interchanged the summation indices Iand Jand used the

fact that DIα,Jβ =DJβ,Iα, since changing the order of differentiation in the defining equation,

Eq. (2.132), does not affect the dynamical matrix. Any Hermitian matrix is diagonalizable and we

can therefore write an eigenvalue equation for DGsc :

DGsc (Ω, T)wGsc

, s = [ωGsc

, s(Ω, T)]2wGsc

, s.(2.143)

Here, wGsc

, s are the already introduced eigenvectors, which can be orthonormalized,

wGsc

, s ·wGsc

, s′=δs,s′,(2.144)

and which obey

w−Gsc

, s = (wGsc

, s)∗,(2.145)

due to D−Gsc = (DGsc )∗. Further in Eq. (2.143), (ωGsc

, s)2are the eigenvalues with the square

included in order to make an identification of the ωGsc

, s with phonon frequencies (introduced below)

possible. Using Eq. (2.143), we have P3

α,β DGsc

, αβ =P3

s(ωGsc

, s)2and we thus obtain:

Hqh(Ω, T ) = Fel

0(Ω, T) + 1

prBZ

Gsc

s|PGsc

, s|2+ [ωGsc

, s(Ω, T )]2|UGsc

, s|2.(2.146)

Here, we have also used the property Eq. (2.126) of the UGsc , which translates accordingly to

the PGsc . Equation (2.146) is the reciprocal representation of Hqh we were intending to derive.

The important difference to Eq. (2.137) is that the contributions from the different Gsc vectors

and different branches sdo not couple to each other. In contrast, the real space representation,

Eq. (2.137), has coupling contributions in the last term due to the real space dynamical matrix. At

this point, it is appropriate to discuss the necessity to use, in Eqs. (2.138) and (2.139), a modified

(and rather complicated) Fourier transform including the eigenvectors wGsc

, s. The reason is that, if

we used the ”simple” Fourier transform Eq. (2.128), we would in general not achieve a decoupling

of the different branches at a single Gsc vector. In contrast, including wGsc

, s into the Fourier

transform leads to a rotation of the reciprocal coordinates, UGsc

, s and PGsc

, s, which in turn leads

to their decoupling. In performing this rotation, one needs however to reconsider the relations

derived in Sec. 2.3.2 and to check if they also apply to the ”new” Fourier transforms Eqs. (2.138)

and (2.139). This is indeed the case since, with Eqs. (2.144) and (2.145), the eigenvectors wGsc

, s

obey the necessary conditions which were used in Sec. 2.3.2 for deriving these relations.

Equation (2.146) has a further important advantage: It has a form well known from the study

of the harmonic oscillator, which is characterized by the eigenvalue equation:

2ˆp2+ω2ˆu2|ni=En|ni.(2.147)

Here, ˆp, ˆu, and ωare respectively the momentum and displacement operator and the frequency

of the oscillator. The eigenfunctions are denoted by |ni. Their exact form does not concern us

here and nserves merely as an index for enumerating them and identifying the corresponding

eigenvalue En. We are only interested in the detailed form of the Enwhich can be derived from

the commutation relation

[ˆu, ˆp] = i~(2.148)

2.3. Exploring the nuclei phase space 43

and is given by (e.g. Ref. [71])

En=~ωn+1

2with n= 0,1,2, . . . , ∞,(2.149)

i.e., the eigenvalues are discrete, not degenerate, bound from below, and extend to infinity in

equidistant steps. In order to apply Eq. (2.147) to solving Eq. (2.146), we first transform UGsc and

PGsc again to operators

UGsc →ˆ

UGsc ,PGsc →ˆ

PGsc ,(2.150)

which are now required to obey the commutation relations:

[ˆ

UGsc

, s,ˆ

PGsc′

, s′] = −i~δGsc

,Gsc′δs,s′,(2.151)

[ˆ

UGsc

, s,ˆ

UGsc′

, s′] = [ ˆ

PGsc

, s,ˆ

PGsc′

, s′] = 0.(2.152)

The quantum mechanical quasiharmonic Hamiltonian in its reciprocal representation then reads

Hqh(Ω, T ) = ˆ

1Fel

0(Ω, T) + 1

prBZ

Gsc

sn|ˆ

PGsc

, s|2+ [ωGsc

, s(Ω, T)]2|ˆ

UGsc

, s|2o

=ˆ

1Fel

0(Ω, T) + 1

3Nn

in|ˆ

Pi|2+ [ωi(Ω, T)]2|ˆ

Ui|2o(2.153)

and its eigenvalue equation, Eq. (2.134), changes to a generalized version of Eq. (2.147)

Hqh |{ni}i =˜

Eqh

{ni}|{ni}i,(2.154)

with eigenfunctions |{ni}i and eigenvalues ˜

Eqh

{ni}given by:

Eqh

{ni}(Ω, T) = Fel

0(Ω, T) +

3Nn

~ωi(Ω, T)ni+1

2with ni= 0,1,2,...,∞.(2.155)

Here, the index iis a short hand notation for both the Gsc vector and the branch index s. As

shown in Eq. (2.125), there are NnGsc vectors in the prBZ and since to each correspond three

branches, iruns over 3Nnvalues. The eigenfunctions |{ni}i and eigenvalues ˜

Eqh

{ni}require a more

complex notation than the ones in Eq. (2.147), since we have now an oscillator for each iand

we therefore need the set {ni}of 3Nnindices for characterization. At this point, we can identify

ωi=ωGsc

, s with a frequency corresponding to the oscillator at Gsc and s. These oscillators are

reciprocal oscillators and the corresponding motion of the nuclei in real space is given by plane

wave lattice vibrations. A crucial observation is that the amplitude of these lattice vibrations is

quantized as a consequence of the discreteness of the eigenvalues ˜

Eqh

{ni}, which by itself is originally

rooted in the fundamental commutation relations, Eqs. (2.151) and (2.152). Such quantized plane

wave lattice vibrations are referred to as phonons. The corresponding frequency is a measure of

their quanta and is therefore called phonon frequency. Phonons can be viewed as a kind of particle,

the so called quasi particle, since we can assign a definite momentum (PGsc

, s) and a definite position

(UGsc

, s) to each phonon. Within the quasiharmonic approximation, phonons do not interact with

each other with the consequence that if a phonon is once excited, it will always be present. Taking

44 2.3. Exploring the nuclei phase space

higher order terms in the expansion in Eq. (2.130) into account (Sec. 2.3.5) switches on the phonon

interaction, which can lead to destruction and creation of phonons.

As the final step of the quasiharmonic approach, we approximate the eigenvalues of the orig-

inal effective nuclei Schr¨odinger equation, Eq. (2.110), with the eigenvalues of the quasiharmonic

Hamiltonian ˜

Enuc

µ≈˜

Eqh

{ni},(2.156)

and apply Eq. (2.156) to Eq. (2.111) in order to evaluate an explicit expression for the free energy

in terms of the frequencies ωGsc

, s:

F≈ −kBTln X

{ni}

e−β˜

Eqh

{ni}

=Fel

0(Ω, T)−kBTln

=: Zqh

z }| {

{ni}

exp "−β

3Nn

~ωi(Ω, T)ni+1

2#.(2.157)

The outer sum runs here over all possible sets {ni}which can be constructed when each nican

take the values 0,1,2,...,∞. We have also defined the quasiharmonic partition function Zqh which

can be rewritten as:

Zqh =X

{ni}

3Nn

e−β~ωi(ni+1/2) =

3Nn

e−β~ωi

∞

n=0

e−n β~ωi=

3Nn

e−β~ωi

/2h1−e−β~ωii−1

.(2.158)

To obtain the second equality, note that it is merely a generalization of

∞

n1,n2

n1n2= ∞

n1! ∞

n2!,(2.159)

whereas the last equality in Eq. (2.158) is obtained from the geometric series. Resubstituting Zqh

into Eq. (2.157), we finally obtain

F(Ω, T)≈Fel

0(Ω, T) + Fqh(Ω, T ),(2.160)

where we have defined the quasiharmonic free energy Fqh by

Fqh(Ω, T ) :=

prBZ′

Gsc

2~ωGsc

, s(Ω, T )

| {z }

=: Ezp(Ω; T) = zero point

energy

+T kB

prBZ′

Gsc

ln [1 −exp {−β~ωGsc

,s(Ω, T)}]

| {z }

=: Sqh(Ω, T) = entropic contribution

(2.161)

using again explicitly Gsc and sinstead of i. We have also changed the sum over the prBZ to a sum

over prBZ′which denotes that the Gsc ={0,0,0}vector needs to excluded. This vector corresponds

to a rigid shift of the full crystal and is unphysical since in a corresponding experimental setup

the crystal is fixed in space. Note that including this degree of freedom would lead to a diverging

Fqh since the corresponding frequencies are zero. The quasiharmonic free energy consists of a term

2.3. Exploring the nuclei phase space 45

which does not explicitly depend on the temperature and which is referred to as the zero point

energy, Ezp. Note however that Ezp(Ω; T) depends implicitly on Tthrough ωGsc

, s(T) (indicated

by the semicolon). The second term in Fqh depends implicitly and explicitly on Tand is referred

to as the entropic contribution, TSqh. Equations (2.160) and (2.161) are the key equations of the

quasiharmonic approach. In order to show their computational requirements and compare their

scaling behavior with the one of the original effective nuclei Schr¨odinger equation, Eqs. (2.110) and

(2.112), we need to discuss the properties of the dynamical matrix and to show how to obtain it

from a Kohn-Sham calculation. Before, let us add a brief comment on the statistical nature of the

phonons.

Based on a similar derivation as the one applied above to obtain Fqh, one can derive an expres-

sion for the average occupation number fniof the various phonon energy levels. It reads [68]

fni=heβ~ωi−1i−1(2.162)

and it is called the Bose-Einstein function. In contrast to the Fermi-Dirac function, Eq. (2.65), it

allows to occupy an energy level by an arbitrary number of phonons, i.e., phonons are bosons. An

example Bose-Einstein function is plotted in Fig. 2.1.

2.3.4 Dynamical matrix

There are basically two distinct approaches to calculate the dynamical matrix. The first one is

called the direct force constant method and it utilizes finite differences of forces to calculate the

real space dynamical matrix D. A part of the methodological work performed in the present study

consisted of implementing the necessary routines allowing to employ this method and the results

discussed in Chap. 4 are mainly based on the direct force constant method. We will therefore

discuss it in detail in this section along with symmetry properties of the dynamical matrix. The

second approach is the linear response method and it utilizes perturbation theory to calculate

directly the reciprocal matrix DGsc . It is discussed and compared with the direct force constant

method in App. A.1.4.

The direct force constant approach [72] is based on an expression for the real space dynamical

matrix in terms of forces acting on the nuclei which reads

DIα,Jβ(Ω, T) = 1

MI∂2Fel({RI},Ω, T)

∂RI,α∂RJ,β {R0

=−1

MI"∂Fhf

I,α({RI},Ω, T)

∂RJ,β #{R0

,(2.163)

where we have defined the force Fhf

I,α on nucleus Iin direction αby:

Fhf

I,α({RI},Ω, T) = −∂Fel({RI},Ω, T)

∂RI,α

.(2.164)

The ”HF” in the superscript is explained below. To show the advantage of formulating the dynam-

ical matrix in terms of forces, note that the only quantities in the defining equation of Fel({RI}),

Eq. (2.68), which explicitly depend on the nuclei coordinates {RI}are v(r) and Vnuc as can be

seen from their definitions in Eqs. (2.49) and (2.50), respectively. The other quantities depend

implicitly on the {RI}since the charge density ρdepends on them through the self consistency

cycle, Eq. (2.66). However, Fel is constructed such as to correspond to the minimum with respect

to ρand therefore this implicit dependence will not contribute to the derivative in Eq. (2.164). We

46 2.3. Exploring the nuclei phase space

thus can write:

Fhf

I,α({RI},Ω, T) = Zρ(r,Ω, T)∂v(r,{RI},Ω)

∂RI,α

dr+∂V nuc({RI},Ω)

∂RI,α

=Zρ(r,Ω, T)ZIe2(rα−RI,α)

4πǫ0|r−RI|3dr+

J6=I

ZIZJe2(RI,α −RJ,α)

4πǫ0|RI−RJ|3.(2.165)

Here, we have adjusted the dependence of ρ,v(r), and Vnuc to correspond to the supercell ap-

proach (Ω dependence). Further, we have indicated the temperature dependence of ρwhich enters

through the electron occupation numbers fiin its defining equation, Eq. (2.64). Equation (2.165)

is a version of the Hellmann-Feynman theorem [73–75] and the forces are therefore referred to as

Hellmann-Feynman (HF) forces. The key point of Eq. (2.165) is the second equality stating that

the Hellmann-Feynman forces can be expressed analytically. This approach to obtain the forces

should be contrasted with the other possibility which is a numerical finite difference approach

Fhf

I,α({RI},Ω, T) = 1

∆RhFel(R1,1,...,RI,α + ∆R,...,RNn,3,Ω, T)−Fel({RI},Ω, T)i,(2.166)

with ∆Ra suitably chosen displacement (typically in the range of 0.01a0). Equation (2.166) requires

two separate Kohn-Sham calculations of Fel, with and without the displacement, each consisting

of a full self consistency cycle Eq. (2.66). To calculate the forces on all Nnnuclei in all three spatial

directions, we need 3Nn+1 Kohn-Sham calculations. In contrast, utilizing the Hellmann-Feynman

theorem, we need only 1 Kohn-Sham calculation at {RI}. The corresponding converged charge

density ρcan then be employed in Eq. (2.165) to efficiently calculate the forces on all nuclei.

The Hellmann-Feynman theorem has been presented here in real space. It is however also fully

applicable when the Kohn-Sham equations are given in terms of plane waves in reciprocal space,

Eq. (2.86). The corresponding equations for the forces can be found for instance in Ref. [47]. In

fact, the possibility to efficiently compute forces is a further advantage, additionally to the two

mentioned at the beginning of Sec. 2.1.7, of the plane wave basis as compared for instance to an

atomic like basis. The latter requires additional terms (so called Pulay forces [76]) which arise due

to the explicit dependence of the basis on the nuclei coordinates.

For the calculation of the derivative of the forces in Eq. (2.163), we resort to the finite difference

method:

DIα,Jβ(Ω, T) = −(MI∆R)−1Fhf

I,α(R0

1,1,...,R0

J,β + ∆R,...,R0

Nn,3,Ω, T).(2.167)

We do not need to explicitly subtract Fhf

I,α({R0

I}) in Eq. (2.167), since it is identical to zero due to

the equilibrium nuclei positions. We hence have to perform 3Nnforce calculations, i.e., according

to the above discussion Kohn-Sham calculations, to obtain the full dynamical matrix Dfor a single

supercell volume Ω. We will show in the following that this number can be reduced to one for the

systems studied in the present work, if we employ translational and point symmetry arguments.

We first discuss translational symmetry. As a preliminary consideration, we point out the

translational property of a supercell with all nuclei at their T= 0 K equilibrium positions {R0

I}.

Due to the construction of the supercell, Eqs. (2.114) to (2.116), based on the conventional fcc unit

cell, it is possible to translate the nuclei by a difference vector ∆R0

I,J =R0

I−R0

Jwhere Iand

Jcan be any two nuclei out of the supercell. The resulting structure will match the original one

exactly. We only need to take special care of the nuclei that move out of the supercell by mapping

them back using the supercell vectors s1,s2, and s3from Eq. (2.114). We write the translation of

2.3. Exploring the nuclei phase space 47

nucleus Kincluding the mapping as:

mod[R0

K+ ∆R0

I,J,{si}], i = 1,2,3.(2.168)

Let us now consider a calculation in which we displace a particular nucleus Iinto direction α:

{R0

1,1,...,R0

I,α + ∆R,...,R0

Nn,3} ⇒ {Fhf(R1),...,Fhf(RI),...,Fhf(RNn)}.(2.169)

We have indicted here that the displacement will create a force field, i.e., a force on each atom in

the supercell. From the force field Eq. (2.169), we can construct the force field to the displacement

of any other nucleus Jin the same direction α,

{R0

1,1,...,R0

J,α + ∆R,...,R0

Nn,3}(2.170)

without additional Kohn-Sham calculations by applying:

nFhf mod[R0

1+ ∆R0

I,J,{si}],...,Fhf mod[R0

I+ ∆R0

I,J ,{si}], . . . ,

. . . , Fhf mod[R0

J+ ∆R0

I,J ,{si}],...,Fhf mod[R0

Nn+ ∆R0

I,J ,{si}]o.(2.171)

Translational symmetry thus allows to reduce the computational effort from 3Nnto 3 calculations.

Let us now turn to point symmetry and consider first the symmetry properties of the perfect

fcc crystal. There are 48 point symmetry operations (including mirror symmetries) which map the

fcc crystal into itself. They can be represented by 3 ×3 matrices acting on Cartesian coordinates.

Among them are

Sxy =



0 1 0

1 0 0

0 0 1 

and Sxz =



0 0 1

0 1 0

1 0 0 

,(2.172)

which map the xaxis onto the yaxis and the xaxis onto the zaxis, respectively. We can use Sxy

and Sxz to construct the force field corresponding to the displacement of nucleus Iinto direction

β=yand β=zfrom a single Kohn-Sham calculation for Iand α=x. To do so, we have

however to ensure that our supercell is compatible with both symmetry operations. To illustrate

this, consider a supercell with:

s1= (sx,0,0),s2= (0, sy,0),s3= (0,0, sz).(2.173)

If we displace nucleus Iby ∆Ralong x, we obtain a pattern of displacements due to the image

atoms. According to Eq. (2.118) [with U= (∆R, 0,0)] and using Eq. (2.173), this pattern reads

U(R0

I+n1s1+n2s2+n3s3) = 



∆R

0



R0

I+



n1sx

n2sy

n3sz



,(2.174)

with n1,n2, and n3integers. Similarly, if we displace along y, we obtain:

U(R0

I+n1s1+n2s2+n3s3) = 



∆R

0



R0

I+



n1sx

n2sy

n3sz



.(2.175)

The displacement patterns, Eqs. (2.174) and (2.175), can only be mapped onto each other using

48 2.3. Exploring the nuclei phase space

Sxy, if sx=sy. This holds also true for the displacement along zand Sxz and we therefore require

sx=sy=sz, i.e., a cubic supercell. In general, our supercell needs to be compatible with the

symmetry operations that are to be used to map one displacement pattern onto the other. Based

on appropriate supercells, we require only one Kohn-Sham calculation for

{R0

1,1,...,R0

I,1+ ∆R,...,R0

Nn,3} ⇒ {Fhf(R1),...,Fhf(RI),...,Fhf(RNn)},(2.176)

from which we can construct the force field corresponding to

{R0

1,1,...,R0

I,2+ ∆R,...,R0

Nn,3}(2.177)

by using

nSxy·Fhf mod[Sxy ·R0

1,{si}],...,Sxy ·Fhf mod[Sxy ·R0

I,{si}], . . . ,

. . . , Sxy ·Fhf mod[Sxy ·R0

Nn,{si}]o (2.178)

and similarly for zand Sxz. Including both, translational and point symmetry operations, we can,

for the elementary fcc crystals studied here, reduce the number of necessary Kohn-Sham calculations

from 3Nnto 1 for each supercell volume Ω.

Finally in the discussion of the dynamical matrix, we need to comment on three important

issues. First, the value for the displacement ∆Rneeded for the finite difference of the forces was

introduced rather ad hoc. An appropriate displacement needs however to be derived from test

calculations scanning various displacements. In general, if the displacement is too small, the forces

will contain a significant amount of noise, while too large displacements will cause anharmonic

contributions to be present. Both situations need to be avoided. We will address this issue in

Sec. 3.4.4.1. Second, as described above, utilizing Hellmann-Feynman forces allows a significant

reduction of the number of necessary Kohn-Sham calculations. However, the calculation of forces is

rather sensitive with respect to convergence parameters (ksampling, Ecut, augmentation grid). It

is therefore very important to carefully check the resulting thermodynamic properties with respect

to the controllable errors. This issue will be addressed in Secs. 3.4.1 and 3.4.2. Third, to reduce

the computational effort, the dynamical matrix Dis typically approximated by:

DIα,Jβ(Ω, T) = 1

MI∂2Fel({RI},Ω, T)

∂RI,α∂RJ,β {R0

=−1

MI"∂Fhf

I,α({RI},Ω, T)

∂RJ,β #{R0

I}≈(2.179)

≈1

MI"∂2Eel

g({RI},Ω)

∂RI,α∂RJ,β #{R0

=: −1

MI"∂Fhf,0k

I,α ({RI},Ω)

∂RJ,β #{R0

=: D0k

Iα,Jβ(Ω).

Here, Eel

gis the T= 0 K ground state of the electronic Schr¨odinger equation, Eq. (2.16). The

corresponding Hellmann-Feynman forces Fhf,0kand the dynamical matrix D0khave been therefore

marked with the ”0K” superscript. Using D0k, the full formalism leading to phonon frequencies and

eventually to the quasiharmonic free energy can be applied as presented. It is however convenient

to adopt also for the other quantities the notation:

DGsc →D0k

Gsc , ωGsc

,s →ω0k

Gsc

,s, Ezp →Ezp,0k, Sqh →Sqh,0k,and Fqh →Fqh,0k.

(2.180)

2.3. Exploring the nuclei phase space 49

The D0kapproximation is performed in most applications of the quasiharmonic approach and its

influence has not been assessed yet. In the present study, we will investigate the validity of this

approximation in Sec. 4.2.

The discussion of the dynamical matrix and thus of the quasiharmonic approximation is now

complete and we summarize the key equations and the flow of the main quantities. In this summary

we use the D0kapproximation, while the full formalism based on Dis developed in Sec. 3.3.4:

HF forces from Kohn-Sham

calculations for different Ω

| {z }

Kohn-Sham calculations

for different Ω and T

| {z }

D0k

Iα,Jβ(Ω) =−(MI∆R)−1Fhf,0k

I,α (R0

1,1,...,R0

J,β + ∆R, . . . , R0

Nn,3,Ω) (2.181)

D0k

Gsc

, αβ(Ω) =N−1

I,J

D0k

Iα,Jβ(Ω) eiGsc·(R0

I−R0

J)(2.182)

D0k

Gsc (Ω) wGsc

, s =(ω0k

Gsc

, s)2(Ω) wGsc

, s (2.183)

F(Ω, T)≈Fel({R0

I},Ω, T) +

prBZ′

Gsc

s1

2~ω0k

Gsc

, s(Ω) +kBTln 1−exp −β~ω0k

Gsc

,s(Ω) 

| {z }

=Fqh,0k(Ω, T) = Ezp,0k(Ω) + TSqh,0k(Ω) (2.184)

F(Ω, T)/Nn=F(V, T).(2.185)

The free energy F(Ω, T ) corresponds to a supercell with Nnatoms. Scaling by Nntherefore gives

a per atom quantity F(V, T) which corresponds to the free energy from Eq. (2.2). In our case,

F(V, T ) is equivalent to the free energy per primitive unit cell with volume Vfrom Eq. (2.80).

We note that for reasons of a convenient discussion, we will hereafter also frequently change the

notation for the forces, the dynamical matrix, the phonon frequencies, and the quasiharmonic free

energy as

Fhf,0k

I(Ω) →Fhf,0k

I(V),D0k(Ω) →D0k(V),D0k

Gsc (Ω) →D0k

Gsc (V), ω0k

Gsc

, s(Ω) →ω0k

Gsc

, s(V)

Ezp,0k(Ω)/Nn→Ezp,0k(V), Sqh,0k(Ω)/Nn→Sqh,0k(V),and Fqh,0k(Ω)/Nn→Fqh,0k(V),

(2.186)

and analogously for the quantities without the ”0K” superscript. In using the replacements in

Eq. (2.186), it should be however remembered that to calculate these quantities (within the direct

force constant method) a supercell is needed. In Eq. (2.184), we have changed Fel

0back to its original

definition Fel({R0

I}) and explicitly written its dependence on the temperature, which enters through

the Fermi function, Eq. (2.65). To satisfy this dependence, we need to explicitly perform Kohn-

Sham calculations at different temperatures. The practical approach will be presented in Sec. 3.3

and we note here only that ≈10 temperatures are sufficient. We also need to consider ≈10 different

volumes for each temperature. For Fel({R0

I}), we can however use the fact that all atoms are in

their T= 0 K equilibrium positions which allows to use directly the one atomic primitive unit cell

50 2.3. Exploring the nuclei phase space

with volume V. As noted above, the volume dependence of the Hellmann-Feynman forces Fhf,0k

I,α

needs to be computed using the supercell with volume Ω. Within the T= 0 K approximation

of the dynamical matrix, Eq. (2.179), the Fhf,0k

I,α are however temperature independent and one

calculation at T= 0 K for each Ω suffices. The scaling sqh,0kof the quasiharmonic approach within

the D0kapproximation therefore reads:

sqh,0k=NVNT·sks(alat, Ecut, Nirr,pr

k, Npr

e, Nit)

| {z }

Kohn-Sham eq. for primitive cell

+NV·sks(asc, Ecut, Nirr

k, Ne, Nit)

| {z }

Kohn-Sham eq. for supercell

.(2.187)

Here, NVand NTare the number of volume and temperature points respectively. We have further

introduced Nirr,pr

k≈NnNirr,pr

kthe number of irreducible kvectors inside the prBZ, Npr

e=Ne/Nn

the number of electrons in the prBZ, and asc =nscalat. Equation (2.187) describes a readily

accessible problem for present day computational resources. Its performance will be addressed

in Sec. 4.1. In the following sections, we will present approaches that allow to go beyond the

quasiharmonic approximation.

2.3.5 Beyond the quasiharmonic approximation

Two conceptually distinct routes exist to take the terms O(U3) in the expansion Eq. (2.130) into

account:

1) Classical statistical approaches: The basic idea is to start with the general expression for the

quantum mechanical free energy Fand to replace it by its classical counterpart Fclas

F=−kBTln X

µh˜

Λµ|exp (−β"−

2MI∇2

I+ˆ

1Fel({RI})#)|˜

Λµi(2.188)

⇒Fclas =−kBTlnZdPIZdRIexp (−β"Nn

2MI|PI|2+Fel({RI})#)/(h3NnNn!),

(2.189)

with the Planck constant h= 2π~. The expression for Fis obtained by multiplying Eq. (2.110)

from left with ˜

Λ∗

µand integrating over the nuclei coordinates, by inserting the result and

Eq. (2.8) into Eq. (2.111), and by using Eq. (A.5). The expression for Fclas is obtained using

the transformation rule for the momentum Eq. (2.136), by replacing the trace, Eq. (A.6),

with an integral over the nuclei momenta and coordinates defined as in Eq. (A.8), and by

scaling for consistency with 1/(h3NnNn!) (see e.g. Ref. [77]). An advantage of Eq. (2.189)

over Eq. (2.188) is that the term inside the exponential is diagonal in the nuclei coordinates,

since PIis a local operator in contrast to ∇I, i.e., at each nucleus position RI,PIhas a

definite value. Therefore, Eq. (2.189) needs not to be treated as a matrix equation and hence

it scales as s=(N3

mesh)Nn·sks(alat, Ecut, Nirr

k, Ne, Nit), i.e., the second term is removed as

compared to the scaling of Eq. (2.188) given in Eq. (2.112). However, the prefactor (N3

mesh)Nn

corresponding to the sampling of the nuclei phase space (i.e., the integral RdRI) is still present

and difficulty 1) from the discussion at the beginning of Sec. 2.3.1 still applies. [In fact, the

prefactor doubles due to RdPI; see however Eq. (2.190) below.] There are two complementary

classes of methods to tackle this difficulty:

2.3. Exploring the nuclei phase space 51



Monte Carlo methods [78] are based on a stochastic/random sampling of the phase space.



Molecular dynamics simulations [78] use deterministic equations of motion (e.g., New-

ton’s equation of motion) to sample the phase space.

The advantages and disadvantages of both approaches are compared for instance in Ref. [79].

For our present purposes, we will employ a scheme based on a combination of both: The so

called Langevin dynamics which extends the original molecular dynamics by a random ele-

ment. This coupled approach has been implemented into the s/phi/nx9code and extensively

tested in a previous study [81]. It turns out to be a very efficient way to sample the phase

space. We will discuss it in Sec. 2.3.6.

The discussion of the classical statistical approaches so far has been based on the free energy

expression Eq. (2.189) which corresponds to a general definition of the free energy. In partic-

ular, evaluating Eq. (2.189) directly corresponds to calculating all terms in the expansion in

Eq. (2.130). We would however like to benefit from the results obtained within the quasihar-

monic approximation, which is, in comparison to the statistical methods, very efficient and

contains already quantum mechanical effects, i.e., we would like to calculate only the O(U3)

terms. The method which allows this is the so called thermodynamic integration discussed

likewise in Sec. 2.3.6.

2) Quantum mechanical perturbation theory: The idea here is to explicitly include the O(U3)

terms in the expansion Eq. (2.130) successively order by order. The solution of the quasihar-

monic approximation, i.e., the system of non-interacting phonons, is fundamentally included

as the unperturbed system which is subjected to quantum mechanical perturbation theory.

The O(U3) terms correspond to interactions between phonons and can also lead to their

creation or destruction. We will discuss this approach in Sec. 2.3.7.

Let us briefly contrast the two approaches with each other. Within the classical statistical approach

all, i.e., infinitely many, orders of the expansion in Eq. (2.130) are taken into account, since the

phase space is explicitly (numerically) sampled. Using quantum mechanical perturbation theory

only the lowest and in practice typically only the third and fourth order terms are taken into

account, since the corresponding calculations become rapidly infeasible. The quantum mechanical

character of the system is however fully included as opposed to the classical approach. Another

advantage is that with perturbation theory one obtains analytical formulas for thermodynamic

properties which allows to efficiently study their temperature dependence. In contrast, within the

statistical approach each temperature needs to be recalculated explicitly. It is difficult to a priori

decide which approach is better suited, a combination being certainly desirable.

2.3.6 Thermodynamic integration based Langevin dynamics

The central idea of the thermodynamic integration10 is to couple the full classical system, i.e.,

Eq. (2.189), to the classical quasiharmonic system, Eq. (2.146). The free energy that needs then

to be sampled is effectively the difference free energy between these two systems, i.e., the terms

O(U3) in Eq. (2.130). Since this difference is rather small, the corresponding sampling converges in

9The s/phi/nx code [80] is developed at the Max-Planck-Institut f¨ur Eisenforschung, D¨usseldorf, at which also the

main part of the present work was performed. s/phi/nx is an ab initio based multiscale library including, e.g., many

features necessary for anharmonic calculations. The results presented in Sec. 4.3 were obtained using this package.

10In fact, thermodynamic integration is a more general scheme applicable to any two systems, one which is not

directly solvable and one reference system which can be treated efficiently. The coupling of the two systems allows

to solve also the original system.

52 2.3. Exploring the nuclei phase space

reasonable numbers of sampling steps. This classical free energy for the O(U3) terms is eventually

added to the quantum mechanical free energy of the quasiharmonic approximation, Eq. (2.160),

resulting in the desired full free energy.

To introduce thermodynamic integration formally, we note first that in Eq. (2.189) the integral

over the nuclei momenta can be carried out analytically giving [77, p. 63]

Fclas =−kBTln ZdRIe−βF el({RI})/(Λ3NnNn!),(2.190)

where Λ = h/√2πMIkBTis the de Broglie wavelength. Let us next define a coupled system with

an electronic free energy Fel

λgiven by

Fel

λ({RI}, V ) = (1 −λ)hFel

0(V) + UD(V)U/(2M)i+λFel({RI}, V ),(2.191)

where we have explicitly written the dependence on the volume V, used the shorthand notation

from Eq. (2.133), and introduced a coupling parameter λwhich can take the values 0 ...1. Equa-

tion (2.191) describes a linear coupling between the actual system of interest, given by Fel, and a

system fully described by the (real space) dynamical matrix D, i.e., by the second order approxi-

mation to Eq. (2.130). Next, we differentiate the classical free energy corresponding to Fel

λ,

Fclas

λ=−kBTln ZdRIe−βF el

λ({RI})/(Λ3NnNn!) =: −kBTln Zλ,(2.192)

where we have defined the classical partition function Zλof the coupled system, with respect to λ

[the reason for this becoming apparent in Eq. (2.195) below]:

∂Fclas

∂λ=−kBTZ−1

λZdRIe−βF el

λ({RI})−β∂Fel

∂λ/(Λ3NnNn!)

=: ∂F el

∂λT,λ

=hFel({RI}, V )−Fel

0(V)−UD(V)U/(2M)iT,λ.(2.193)

Here, we have defined the thermodynamic average h.iT,λat temperature Twith respect to the λ

coupled system, Eq. (2.191). An important point is that, under the ergodicity hypothesis [77],

thermodynamic averages are equivalent to time averages, and the latter can be obtained directly

from a molecular dynamics or in particular from a Langevin dynamics simulation. We will discuss

this in more detail below. Assuming we know how to calculate h.iT,λ, we can obtain straightfor-

wardly the difference ∆Fclas between the classical free energy of the system of interest, Fclas, and

the classical quasiharmonic free energy Fclas,qh,

Fclas,qh(V, T ) = −kBTlnZdRIe−β[Fel

0(V) + UD(V)U/(2M)]/(Λ3NnNn!),(2.194)

by integrating over λfrom 0 to 1:

Fclas,ah(V, T ) := ∆Fclas(V, T) = Fclas(V, T )−Fclas,qh(V, T) = (2.195)

=Z1

dλ∂Fel

∂λT,λ

=Z1

dλhFel({RI}, V )−Fel

0(V)−UD(V)U/(2M)iT,λ.

2.3. Exploring the nuclei phase space 53

We have here defined Fclas,ah which is referred to as the explicitly anharmonic free energy. The

full free energy Fof the system is finally obtained by adding the classical Fclas,ah to the quantum

mechanical quasiharmonic result, Eq. (2.160):

F(V, T )≈Fel

0(V, T ) + Fqh(V, T ) + Fclas,ah(V, T).(2.196)

Equation (2.196) constitutes the final expression for the free energy of an elementary, non-magnetic

perfect crystal. The extension to crystals with vacancies is discussed in Sec. 3.1 with the resulting

free energy expression given in Eq. (3.35).

Let us now discuss how to calculate h.iT,λusing molecular dynamics in the special form of

Langevin dynamics. The Langevin scheme is discussed in detail in Ref. [81] and we therefore focus

only on the main ideas and relevant parameters. The central equation of Langevin dynamics is

an extended version of Newton’s equation of motion, which we adapt here directly to the coupled

system, Eq. (2.191),

∂2RI,α(t)

∂t2=−∂Fel

λ({RI(t)})

∂RI,α

| {z }

Newton’s equation of motion

−ζ∂RI,α(t)

∂t +Frand

I,α (t)

| {z }

Additional Langevin term

,(2.197)

with α∈ {x, y, z}. In Eq. (2.197), the nuclei positions {RI}are treated as functions of time

t. Further, the additional Langevin term consists of a friction term ζ ∂RI/∂t, with the friction

parameter ζ, and of a randomly generated force Frand

I. Integrating Eq. (2.197) twice will generate

a trajectory {RI}tthrough the phase space of the nuclei. Let us consider the time average h.it,λof

our quantity of interest, ∂Fel

λ/∂λ, over this trajectory given by

∂Fel

∂λt,λ

=Nld−1NLD

∂

∂λFel

λ({RI}ti),(2.198)

with irunning over the Nld elements (also steps, structures, or configurations) of the Langevin dy-

namics (LD) phase space trajectory. Further in Eq. (2.198), {RI}tidenotes the nuclei configuration

at time tiout of {RI}t. The key point is now that this time average will equal the corresponding

thermal average, i.e., ∂Fel

∂λT,λ

=∂Fel

∂λt,λ

,(2.199)

provided that the random force satisfies [81]

hFrand

I,α (t)it,λ≡0 and h(Frand

I,α )2(t)it,λ≡6ζNnkBT(2.200)

and Nld is sufficiently large, which needs to be checked by convergence. Using Eq. (2.199) in

Eq. (2.195), we can fully determine the anharmonic free energy Fclas,ah. We will refer to the

presented method, including thermodynamic integration and Langevin dynamics, as the thermo-

dynamic integration based Langevin dynamics (TILD) method. We will discuss the scaling per-

formance of the TILD method in Sec. 3.2.1. The discussion will serve as a direct motivation for

the UP-TILD method, which will be derived later (Sec. 3.2.2) in order to significantly improve the

performance of the TILD method. Before closing this section, we need however to comment on

several important technical details of the TILD approach:

54 2.3. Exploring the nuclei phase space

a) The correct choice of the friction parameter ζis important for a Langevin dynamics simula-

tion. If chosen inappropriately, h∂Fel

λ/∂λit,λmay converge to a wrong value. Moreover, the

computational effort depends on the choice of ζ. (See Ref. [81] for further details.)

b) A method for integrating Eq. (2.197) needs to be chosen. A possible choice (also used in this

study) is the van-Gunsteren-Berendsen scheme [82] which has been previously implemented

into s/phi/nx [80] and extensively tested [81]. This scheme is an advanced version of the

Verlet algorithm [83] which is a very popular integration scheme in MD simulations [79]. The

main idea of the Verlet algorithm is to expand the nuclei trajectory twice in time up to third

order at each MD step. One expansion is performed forward in time and one backward. The

reason for these two expansions is that adding them together removes the first and third

order terms. The remaining terms correspond to the positions and accelerations of the nuclei,

where the latter can be obtained from a finite difference of the forces, so that a closed set of

equations is available. An advantage of the Verlet algorithm is that it is symplectic, which

means in essence that it is stable over long time periods. The van-Gunsteren-Berendsen

scheme has the additional feature that the finite time step used for the integration is only

limited by the fastest oscillation of the system and not by stochastic fluctuations. A detailed

description of this scheme can be found in Ref. [82].

c) Since Eq. (2.197) is integrated twice, we need to fix two integration constants, which are

typically chosen to be the T= 0 K equilibrium nuclei positions {R0

I}and additionally random

start velocities. Such a choice is however far from the actual thermodynamic equilibrium and,

to reach it, a significant number of molecular dynamics steps is needed. A possible solution

(also employed in this study) is to use a pre-equilibration on the quasiharmonic reference

potential (likewise implemented in s/phi/nx). Since the quasiharmonic potential is given

analytically this procedure requires negligible CPU time, but it brings the system very close to

its actual equilibrium. The further equilibration requiring expensive Kohn-Sham calculations

is then reduced to ≈20 equilibration steps. The molecular dynamics steps used to achieve

equilibration have to be taken out when calculating the average Eq. (2.198).

d) A critical issue in using statistical methods is the correct estimate of the unavoidable statistical

error, in order to provide a trustful interval for the calculated data. The estimation is a three

step procedure:

1) We need to consider the fact that adjacent molecular dynamics structures {RI}tiwill be

strongly correlated as a consequence of the purely deterministic nature of the original

Newton equation of motion. The contribution of correlated structures to the time average

Eq. (2.198) is however effectively less than the one of uncorrelated/random structures.

As a preliminary consideration to estimating the statistical error, we therefore need

in a first place to determine the correlation time. For that purpose, we calculate the

correlation coefficient [79]

Ccor(t′) =

Nld

[∂Fel

λ({RI}ti)/∂λ− h∂Fel

λ/∂λit,λ][∂F el

λ({RI}ti+t′)/∂λ− h∂Fel

λ/∂λit,λ]

Nld

[∂Fel

λ({RI}ti)/∂λ− h∂Fel

λ/∂λit,λ]2

(2.201)

2.3. Exploring the nuclei phase space 55

which shows a steep change in its time dependence at the point when the molecular

dynamics steps become uncorrelated [79].

2) In a next step, we divide the full trajectory {RI}tinto Nunc segments each of the size

of the correlation time. Out of each segment we take systematically a single structure

and add it to a new set of uncorrelated (unc) structures. This procedure corresponds to

the stratified systematic sampling [79].

3) Finally, we calculate the statistical error σerr (cf. e.g. Fig. 4.13) from

σerr =pσvar/Nunc,(2.202)

where σvar is the variance (var) given by

σvar = (Nunc −1)−1

Nunc

uh∂Fel

λ({RI}tu)/∂λ−h∂Fel

λ/∂λiunc

t,λi2,(2.203)

with the uncorrelated time average

∂Fel

∂λunc

t,λ

= (Nunc)−1

Nunc

∂

∂λFel

λ({RI}tu),(2.204)

and with the uncorrelated structures {RI}tu.

e) Due to their correlation, adjacent molecular dynamics configurations {RI}tiare very similar.

The similarity translates to the corresponding Kohn-Sham wave functions and it is therefore

advantageous to use the converged Kohn-Sham wave function of the previous step as the initial

wave function ϕstart

iof the next self consistency cycle, Eq. (2.66). This reduces the number

of iteration steps Nit needed to reach self consistency. In fact, an even better performance

can be achieved by a wave function extrapolation. This feature is available in s/phi/nx and

was applied in the present work. The reduction (red) was Nit ≈10 →Nit,red ≈5.

f) An efficient scheme to perform the λintegration in Eq. (2.195) is the generalized Simpson’s

rule [84].

2.3.7 Quantum mechanical perturbation theory

The quantum mechanical perturbative treatment of the higher order terms O(U3) in Eq. (2.130)

proceeds along the following general lines. At first, we need to carefully decide which terms ac-

tually to include. The reason is that the contribution to materials properties does not increase

monotonically order by order, but different orders can yield similar contributions. In fact, this

situation occurs already for the third and fourth order term, which are known [68] to be equally

important and which therefore need to be treated simultaneously. (This will become also apparent

from our results in Sec. 4.4, Fig. 4.18.) Yet higher order terms yield a smaller contribution and are,

in practice, typically not considered due to the large analytical and computational requirements.

(We likewise follow this route.) Thus, the electronic free energy is expanded up to fourth order

and then each term is transformed into a representation based on phonon coordinates, which are

fully known/available from the diagonalization of the dynamical matrix (Sec. 2.3.3). Up to this

stage, the procedure is rather straightforward and corresponds merely to a change of the basis.

56 2.3. Exploring the nuclei phase space

This change is however needed, in order to provide a description based on a non-interacting second

order term. For such a description, various highly involved methods and techniques have been de-

veloped (Green’s functions methods [85], quantum field theoretical techniques [85], renormalization

methods [68]), which eventually allow to derive expressions for the desired materials properties. For

the particular case of nuclei vibrations, the operator renormalization method has been developed

in detail in Ref. [68].

The basic idea of the operator renormalization method is to ”design” modified/renormalized

phonon operators which do not interact if higher order terms are included into the Hamiltonian.

Note the important difference to the usual/original phonon operators (Sec. 2.3.3): The latter

are non-interacting if the Hamiltonian consists solely of the second order term, whereas they start

interacting if higher order terms are included. In contrast, the renormalized phonon operators would

interact if the Hamiltonian consisted of the second term only, but they do not interact if the higher

order terms are included. The actual construction procedure consists of setting the renormalized

phonon operator equal to the original phonon operator plus a remainder. The remainder is then

determined from a set of commutator relations for the higher order terms. Having calculated

the renormalized phonon operator, the corresponding renormalized frequencies can be calculated

straightforwardly since the new phonons do not interact. Eventually, phonon shifts (given below),

anharmonic free energies, and the full set of thermodynamic properties can be calculated.

In the following, we give the key equations regarding the above discussed change of the basis,

adjusted to the here investigated elementary materials. Concerning the operator renormalization

method, we present only the final equations for the phonon shifts, which were applied for the

calculations presented in Sec. 4.4, while details can be found in Ref. [68]. Let us thus expand Fel

up to fourth order [using Fel(Ω)/Nn→Fel(V) similarly as in Eq. (2.186)]

Fel({RI}, V ) = ... +1

I,J,K

α,β,γ

UI,αUJ,βUK,γ Φ3ord

Iα,Jβ,Kγ(V)

I,J,K,L

α,β,γ,δ

UI,αUJ,βUK,γUL,δ Φ4ord

Iα,Jβ,Kγ,Lδ(V) + O(U5),(2.205)

with the dots denoting the lower order terms given in Eq. (2.130) and with the real space third

order and fourth order tensors, Φ3ord and Φ4ord, with elements given by:

Φ3ord

Iα,Jβ,Kγ(V) = ∂3Fel({RI}, V )

∂RI,α∂RJ,β∂RK,γ {R0

,(2.206)

Φ4ord

Iα,Jβ,Kγ,Lδ(V) = ∂4Fel({RI}, V )

∂RI,α∂RJ,β∂RK,γ∂RL,δ {R0

.(2.207)

Using the Fourier transform for the UI, Eq. (2.138), and the quasiharmonic result for the lower

order terms, Eq. (2.153), we can rewrite Eq. (2.205) such that the full nuclei Hamiltonian from

Eq. (2.110) can be transformed to

nˆ

Tnuc +ˆ

1Felo=ˆ

Hqh(V) + ˆ

H3ord(V) + ˆ

H4ord(V) + O(U5),(2.208)

2.3. Exploring the nuclei phase space 57

with

H3ord(V) =

prBZ

Gsc

,Gsc′,

Gsc′′′

s,s′,s′′

Φ3ord

Gscs, Gsc′s′,Gsc′′s′′ (V)ˆ

UGsc

, s ˆ

UGsc′

, s′ˆ

UGsc′′

, s′′ ,(2.209)

H4ord(V) =

prBZ

Gsc

,Gsc′

Gsc′′,Gsc′′′′

s,s′

s′′,s′′′

Φ4ord

Gscs, Gsc′s′,Gsc′′s′′,Gsc′′′s′′′ (V)ˆ

UGsc

, s ˆ

UGsc′

, s′ˆ

UGsc′′

, s′′ ˆ

UGsc′′′

, s′′′ ,

(2.210)

where the Φ3ord

Gscs, Gsc′s′,Gsc′′s′′ correspond to the elements of the Fourier transform of Φ3ord,

Φ3ord

Gscs, Gsc′s′,Gsc′′s′′ (V) = 1

3! ~

2Nn3/2

(MI)−3/2ωGsc

, s(V)ωGsc′

, s′(V)ωGsc′′

, s′′ (V)−1/2+

I,J,K

α,β,γ

Φ3ord

Iα,Jβ,Kγ(V) (wGsc

, s)αwGsc′

, s′βwGsc′′

, s′′ γeiGsc·R0

IeiGsc·R0

JeiGsc·R0

K,(2.211)

and where the Φ4ord

Gscs, Gsc′s′,Gsc′′s′′,Gsc′′′s′′′ correspond to the elements of the Fourier transform of

Φ4ord:

Φ4ord

Gscs, Gsc′s′,Gsc′′s′′,Gsc′′′s′′′ (V) = 1

4! ~

2Nn21

IωGsc

, s(V)ωGsc′

, s′(V)ωGsc′′

, s′′ (V)ωGsc′′′

, s′′′ (V)−1/2

I,J,K,L

α,β,γ,δ

Φ4ord

Iα,Jβ,Kγ,Lδ(V) (wGsc

, s)αwGsc′

, s′βwGsc′′

, s′′ γwGsc′′′

, s′′′ δ×

×eiGsc·R0

IeiGsc·R0

JeiGsc·R0

KeiGsc·R0

.(2.212)

In Eqs. (2.211) and (2.212), we have used the notation (wGsc

, s)αfor the real space component α

of wGsc

, s. We stress again that, despite looking rather awkward, the preceding equations represent

merely a change of the real space Hamiltonian into the reciprocal phonon basis. As described above,

the operator renormalization method can be now applied to the reciprocal Hamiltonian Eq. (2.208)

and, eventually, materials properties can be derived. We will be in particular interested in the

quantum mechanical anharmonic phonon shift ∆ωqm,ah

Gsc

, s (V, T ) (an example temperature dependence

shown in Fig. 2.1),

∆ωqm,ah

Gsc

, s (V, T ) = ∆ω3ord

Gsc

, s(V, T ) + ∆ω4ord

Gsc

, s(V, T),(2.213)

consisting of the third order, ∆ω3ord

Gsc

, s, and fourth order, ∆ω4ord

Gsc

, s, shifts given by [68]

∆ω3ord

Gsc

, s(V, T) =−18 ~−2

prBZ′

Gsc′,Gsc′′

s′,s′′ |Φ3ord

Gscs, Gsc′s′,Gsc′′s′′,Gsc′′′s′′′ (V)|2f3ord

Gscs, Gsc′s′,Gsc′′s′′,Gsc′′′s′′′ (V, T ),

(2.214)

∆ω4ord

Gsc

, s(V, T) = 12 ~−1

prBZ′

Gsc′

s′

Φ4ord

Gscs, −Gscs, Gsc′s′,−Gsc′s′(V)f4ord

Gsc′s′(V, T),(2.215)

58 2.3. Exploring the nuclei phase space

with

f3ord

Gscs, Gsc′s′,Gsc′′s′′,Gsc′′′s′′′ (V, T ) = −fGsc′

, s′+fGsc′′

, s′′

ωGsc

, s +ωGsc′

, s′−ωGsc′′

, s′′ p

+fGsc′

, s′+fGsc′′

, s′′ + 1

ωGsc

, s +ωGsc′

, s′+ωGsc′′

, s′′

+fGsc′

, s′−fGsc′′

, s′′

ωGsc

, s −ωGsc′

, s′+ωGsc′′

, s′′ p−fGsc′

, s′+fGsc′′

, s′′ + 1

ωGsc

, s −ωGsc′

, s′−ωGsc′′

, s′′ p

(2.216)

f4ord

Gsc′s′(V, T) = 2fGsc′

, s′+ 1,(2.217)

where fGsc′

, s′is the Bose-Einstein function [using a slightly different notation than in Eq. (2.162)]:

fGsc′

, s′=fGsc′

, s′(V, T ) = exp β~ωGsc′

, s′(V)−1−1, β = (kBT)−1.(2.218)

Note that in Eqs. (2.216) and (2.217) all frequencies are volume dependent (but temperature

independent). Further, prBZ’ indicates the exclusion of the zero frequencies at Γ and (.)pdenotes

the principle value, which can be implemented using

(ω)p

=ω

ω2+γ2,(2.219)

and which is simply a device to avoid divergences. For that purpose, the generic infinitesimal

parameter γneeds to be finite but small [86].

In order to provide a better idea of the rather formal development, we sketch how one would

proceed in practice in calculating the phonon shift ∆ωqm,ah

Gsc

, s . We first fix the volume Vand the

supercell size. For the latter let us take a 23supercell with 32 atoms. The major task is then to

calculate the real space tensors Φ3ord and Φ4ord. For that purpose, we can apply the finite difference

method similarly as in the case of the dynamical matrix. Due to the increased order, the equations

become however more complex. For instance, to calculate a third order tensor element, we use:

Φ3ord

Iα,Jβ,Kγ =1

2∆R"Fhf

I,α(... R0

J,β +∆R ... R0

K,γ +∆R ... )−Fhf

I,α(... R0

J,β −∆R ... R0

K,γ +∆R ... )

2∆R

−

Fhf

I,α(... R0

J,β +∆R ... R0

K,γ −∆R ... )−Fhf

I,α(... R0

J,β −∆R ... R0

K,γ −∆R ... )

2∆R#.

(2.220)

The dots denote all the remaining (3 ·32 −2) nuclei coordinates which correspond to the T= 0 K

equilibrium positions. To calculate all Φ3ord tensor elements for that supercell, we have to solve

(3 ·32)2equations of a type as Eq. (2.220). For each such equation, we need however four Kohn-

Sham calculations so that we have in total (4 ·3·32)2≈105calculations. For the full fourth order

tensor, the number is (8·3·32)3≈108and we stress that each calculation is a complete Kohn-Sham

calculation in a 23supercell (≈1 CPU hour for aluminum for instance). In order to make a calcula-

tion of ∆ωqm,ah

Gsc

, s nonetheless feasible, it is crucial to employ symmetry arguments: There are generic

symmetries available in any tensor and related to the interchange of its indices, as a consequence

of the arbitrariness of the order of differentiation in Eqs. (2.206) and (2.207). Other symmetries

are of an intrinsic type and are related to the specific atomic structure. For the fcc structure, we

can significantly reduce the number of calculations to ≈103(3. order) and ≈104(4. order) using

2.4. From the free energy to materials properties 59

these arguments. Note however that the symmetry search routines become more complicated than

for the dynamical matrix, since we have to compare pairs and triplets of displacements. Having

computed Φ3ord and Φ4ord, we perform the Fourier transforms Eqs. (2.211) and (2.212) which cor-

respond merely to sums over the tensor elements dressed by the (quasi)harmonic quantities (ωGsc

, s,

wGsc

, s) determined in Sec. 2.3.3 and by structural quantities (Gsc,R). The Fourier transforms

along with the (quasi)harmonic frequencies ωGsc

, s can then be used directly in Eqs. (2.214) and

(2.215) to calculate the shift. The application of Eq. (2.214) is straightforward due to the simple

dependence of f4ord

Gsc′s′, whereas the treatment of f3ord

Gscs, Gsc′s′,Gsc′′s′′,Gsc′′′s′′′ in Eq. (2.214) is compli-

cated by additional checks of the γdependence. For our calculations, we found however that γ= 0

was always possible, since the available frequencies in the used supercells never fulfilled any of the

conditions needed to yield a diverging f3ord

Gscs, Gsc′s′,Gsc′′s′′,Gsc′′′s′′′ .

We sketch also a different approach to obtain the phonon shift, which is particularly convenient

if one is interested in ∆ωqm,ah

Gsc

, s only for a single Gsc and s. Note for that purpose that, if we

use the approach described above, we always need to calculate the full real space tensors before we

compute the Fourier transforms, since the latter depend on every real space tensor element. If it was

possible to calculate the Φ3ord

Gscs, Gsc′s′,Gsc′′s′′ and Φ4ord

Gscs, Gsc′s′,Gsc′′s′′,Gsc′′′s′′′ directly, we could confine

the calculations to the elements actually needed for a specific Gsc and sin the sums Eqs. (2.214)

and (2.214). This is particularly interesting for the fourth order shift which depends only on a

small set of diagonal elements for Gsc,s,Gsc′, and s′. (This rather simple dependence arises

due to various symmetry and conservation arguments; see Ref. [68].) A direct calculation of the

Fourier elements is indeed possible. Instead of performing the finite difference in real space, we

can employ it directly reciprocal space. This means that, rather displacing pairs and triplets of

atoms, we now displace pairs and triplets of phonons, i.e., collective atomic displacements. The

corresponding real space coordinates (needed for the actual calculation) can be obtained using the

Fourier transform Eq. (2.138). There is however a complication occurring in performing the finite

difference in reciprocal space: The Hellmann-Feynman theorem (Sec. 2.3.4) provides only forces

on individual atoms. We therefore need to calculate the forces on the phonons numerically which

increases the number of calculations by a factor of 2 as compared to the real space method.

2.4 From the free energy to materials properties

In the present section, we introduce the Fourier interpolation scheme allowing to obtain a con-

tinuous phonon dispersion and we collect the various formulas needed to obtain the investigated

thermodynamic properties from the free energy surface.

2.4.1 Phonon dispersion

A phonon dispersion denotes the dependence of the phonon frequencies on the phonon wave vector

qinside the prBZ. An example including also the volume dependence of the frequencies is shown in

Fig. 2.1 for aluminum. Further results will be discussed extensively in Chap. 4. Typically, certain

high symmetry directions are chosen for a convenient representation. For the fcc Brillouin zone, in

particular, the following directions are used

Γ→X,X′→K→Γ→L,(2.221)

with the coordinates (in units of 2π/alat):

Γ = (0,0,0),X = (1,0,0),X′= 2K/3,K = (0.75,0.75,0),L = (0.5,0.5,0.5).(2.222)

60 2.4. From the free energy to materials properties

Note that, due to point symmetry, the frequencies at X and X’ are equivalent so that we can join

the two points in plotting the dispersion. A difficulty in calculating phonon dispersions is the fact

that the wave vectors Gsc, at which phonon frequencies are available according to Eq. (2.143), are

rather sparse along the high symmetry directions for reasonable supercell sizes. In order to allow

nonetheless a reasonable comparison to an experimental phonon dispersion, we apply a Fourier

interpolation by replacing the Fourier transform of the dynamical matrix, Eq. (2.141), with

Dq,αβ(V) = N−1

I,J

k,l,m=0

AIJ,k1AIJ,l2AIJ,m3

P3

s=1 P1

k=0 AIJ,ksDIα,Jβ(V)eiq·[(R0

I−R0

J)−ks1−ls2−ms3],(2.223)

where the coefficients AIJ,ks are given by:

AIJ,ks =(1 for [k= 0 ∧(RI−RJ)·ss≤1/2)] ∨[k= 1 ∧(RI−RJ)·ss≥1/2)]

0 for [k= 0 ∧(RI−RJ)·ss>1/2)] ∨[k= 1 ∧(RI−RJ)·ss<1/2)] .(2.224)

In comparison to Eq. (2.141), the wave vector qin Eq. (2.223) is not limited to the Gsc vectors

corresponding to the considered supercell. It is rather allowed to take any value inside the prBZ.

To guarantee that Eq. (2.223) yields a Hermitian matrix Dqif qdoes not fall on a Gsc vector,

the coefficients AIJ,ks and the modified exponential function had to be introduced. Effectively, the

modification has two effects: 1) It ”centers” the dynamical matrix. To explain this, it is sufficient

to consider a fixed dynamical matrix element DIα,Jβ with RI= (x, 0,0) and RJ= (0,0,0). For

the supercell vectors used in this study [Eq. (2.114)], we then find that the phase factor of DIα,Jβ

is unchanged, i.e., as in Eq. (2.141), if x≤ |s1|. For the other case (x≥ |s1|), the exponent of the

phase factor is modified to [iq·(x−|s1|,0,0)], i.e., the corresponding atom is mapped such that it

is always closest to the origin. This procedure corresponds in essence to centering the dynamical

matrix. 2) Further, the modification maps atoms which lie on some border of the new centered

cell onto each border and scales the corresponding dynamical matrix entries appropriately. The

underlying idea to Eq. (2.223) has been given in Ref. [72], whereas the explicit expressions were

derived in the present study. The phonon frequencies for arbitrary qcan be now obtained by

solving:

Dq,s(V)wq,s = (ωq,s)2(V)wq,s.(2.225)

Besides allowing to construct a phonon dispersion, the dense sampling of the frequencies can be also

employed for the calculation of the quasiharmonic free energy. If we construct a mesh of m×m×m

points inside the prBZ the quasiharmonic free energy Fqh,mfrom this mesh is given by:

Fqh,m(V, T ) = 1

s1

2~ωq,s(V) + kBTln [1 −exp {−β~ωq,s(V)}].(2.226)

We have included here the scaling factor 1/m3which makes Fqh,ma per atom quantity justifying

the dependence on the atomic volume V.

2.4.2 Thermodynamic properties

In order to arrive at the thermodynamic quantities which can be compared directly to experiment,

it is convenient to first derive an entropy surface S(V, T ), a pressure surface P(V, T), an isochoric

[isobaric] heat capacity surface CV(V, T ) [CP(V, T )], and an adiabatic [isothermal] bulk modulus

2.4. From the free energy to materials properties 61

surface BS(V, T ) [BT(V, T)] from the free energy surface:

F(V, T ) (2.227)

S(V, T )= −∂F(V, T )

∂T V

P(V, T )= −∂F(V, T )

∂V T

(2.228)

CV(V, T )=T∂S(V, T )

∂T V

CP(V, T )=T∂S(V, T )

∂T P

BS(V, T )=−V∂P (V, T )

∂V S

BT(V, T )=−V∂P (V, T )

∂V T

(2.229)

The dashed lines indicate that the corresponding upper quantity enters the respective lower quantity

by determining the direction of differentiation (cf. the subscript on the square brackets of the lower

quantity). We stress that all quantities in Eqs. (2.227) to (2.229) are full V-Tsurfaces. Typically, in

experiment these quantities are obtained at constant pressure and as a function of temperature. We

can however obtain these quantities straightforwardly from the above surfaces. For that purpose,

we define the thermal volume expansion VP′(T) at a constant pressure P′in an implicit way by

using the pressure surface as:

P′=P(VP′(T), T).(2.230)

For the special case of P′= 0 Pa, we define the zero pressure equilibrium (eq) volume expansion

Veq(T) by

Veq(T) = VP′=0Pa(T) (2.231)

and if additionally T= 0 K, we use:

Veq

0=Veq(T= 0 K).(2.232)

From Eqs. (2.80) and (2.231), we obtain the thermal expansion of the P=0 Pa equilibrium lattice

constant alat,eq:

alat,eq(T) = [4Veq(T)]1/3and alat,eq

0= (4Veq

0)1/3.(2.233)

We will be in particular interested in the relative expansion εeq(T) given by

ε(T) = εeq(T) = 1

alat,eq

0halat,eq(T)−alat,eq

0i,(2.234)

and the expansion coefficient αeq(T) given by:

α(T) = αeq(T) = 1

alat,eq(T)

∂alat,eq(T)

∂T .(2.235)

In Eqs. (2.234) and (2.235), we have deliberately defined in a first step expressions with the ”eq”

superscript, in order to emphasize the correspondence to the zero pressure equilibrium lattice

constant/volume. In principle, one can define the relative expansion and the expansion coefficient

as well as all following quantities for any VP′. As will be discussed in Sec. 4.1.2, we can however

62 2.4. From the free energy to materials properties

safely neglect the pressure dependence for our purposes and we have thus introduced in a second

step a simplified notation. Now, using Veq(T) and the surfaces Eq. (2.229), we can define the usual

representation for the isochoric and isobaric heat capacities by

CV(T) = Ceq

V(T) = CV(Veq(T), T) and CP(T) = Ceq

P(T) = CP(Veq(T), T),

(2.236)

and equivalently for the adiabatic and isothermal bulk modulus by:

BS(T) = Beq

S(T) = BS(Veq(T), T) and BT(T) = Beq

T(T) = BT(Veq(T), T).

(2.237)

There is an exact thermodynamic relation coupling the heat capacities and the bulk moduli [68]:

CP/CV=BS/BT.(2.238)

We will use Eq. (2.238) to obtain BSfrom BT,CP, and CV, which is computationally more

convenient than calculating the derivative of Pwith respect to Vat a constant entropy Sin

Eq. (2.229). Further, we will use

BT,0=BT(T= 0 K),(2.239)

and the pressure derivative of the isothermal bulk modulus:

B′

T,0=B′

T(T= 0 K),with B′

T(T) = ∂BT(V, T)

∂P V

.(2.240)

It is important to distinguish the isochoric (i.e., constant volume) heat capacity from the fixed

volume heat capacity CV0defined by:

CV0(T) = CV(Veq

0, T)6=CV(Veq(T), T) = CV(T).(2.241)

Finally, the free energy FPat zero pressure is given by:

FP(T) = Feq

P(T) = F(Veq(T), T).(2.242)

The determination of vacancy concentrations from F(V, T ) will be discussed in Sec. 3.1.

Chapter 3

Methodological developments

In this chapter, we discuss the main methodological developments that were obtained during the

present work:



In Sec. 3.1, we lift the restriction that the atoms are arranged on a perfect lattice and introduce

an intuitive approach to treat point defects (cf. the discussion at the end of Sec. 2.3.1).



We discuss in Sec. 3.2 the computational requirements of the TILD method (Sec. 2.3.6) and

introduce an extension, the hierarchical coarse graining UP-TILD method, which reduces

significantly the CPU time of the TILD method.



In Sec. 3.3, we show how to compute the various parts of the free energy surface in practice,

with a particular focus on the parametrization along Vand T. We also reconsider in Sec. 3.3.4

the quasiharmonic approximation using the temperature dependent dynamical matrix.



Finally, in Sec. 3.4, we address the four convergence parameters identified in Chap. 2: ksam-

pling and plane wave cutoff Ecut (Sec. 2.1.7), the augmentation grid (Sec. 2.1.8), and the

supercell size (Sec. 2.3.2). To this end, we present a method which enables efficient conver-

gence checks.

3.1 An intuitive description/treatment of point defect formation

3.1.1 Motivation

With increasing temperature, the probability that atoms leave the vicinity of their T= 0 K

equilibrium positions, {R0

I}, and that they start to penetrate through the lattice will likewise

increase. For entropic reasons, this holds true even for otherwise ideal single crystalline materials as

considered here. Thus, the translational symmetry of the crystal will be perturbed by the presence

of thermally excited point defects. Due to the dominant role of the vacancies and to enable a

convenient discussion, we will explain our approach for the treatment of point defects especially

for vacancies. The key ideas and statements apply however in a similar manner to other types of

point defects such as for instance self interstitials. The corresponding generalized expressions will

be given in Sec. 3.1.4.

Two competing free energy contributions are commonly associated with the occurrence of va-

cancies:

1) Creation/formation free energy: In order to define this contribution, let us consider the free

energy Fpof a perfect crystal, i.e., without vacancies, and the free energy Fvcorresponding

64 3.1. An intuitive description/treatment of point defect formation

to a crystal with one vacancy (single vacancy crystal). The formation free energy Ffof the

vacancy is then generally defined as

Ff=Fv−Fp,(3.1)

with Fvand Fpobtained either at the same volume or the same pressure. In this general

definition, it is implicitly assumed that the number of atoms in the perfect crystal and the

crystal containing the vacancy is equal. For this to be true, the extra atom coming from the

vacancy needs to be incorporated into the crystal at the energetically most favorable place,

for instance at the surface. Defined in this way, Ffwill – if we assume stable phases – always

be positive, since the perfect crystal corresponds already to the most favorable configuration

and therefore Fv> Fp. Put differently, considering Ffas the only contribution to vacancy

formation, a vacancy would actually never occur.

2) Configurational free energy: Let us consider a crystal consisting of Natoms. If nis the

number of vacancies then the number Wof distinct ways (b= microstates) to arrange the

vacancies in the crystal is given by [87]:

W=N(N−1)(N−2) ...(N−n+ 1)

n!=N!

(N−n)! n!.(3.2)

Equation (3.2) corresponds to the classical or Boltzmann statistics. Classical statistics is

applicable to point defect formation, since the atoms in a crystal are considered to be dis-

tinguishable as discussed in Sec. 2.3.1. The configurational free energy Fcis then given by

[87]:

Fc=−kBTln W. (3.3)

Since the number of possible configurations Wis always positive, Fcis always negative and

will therefore favor vacancy formation.

Eventually, it will be the detailed balance between Ffand Fcwhich determines how many vacan-

cies are created. Let us thus discuss the available approaches to calculate Ffand Fcwithin the

framework of the so far developed formalisms.

There are basically two approaches to calculate the formation free energy within a periodic

ansatz: the rescaled volume and the constant pressure approach. Within the rescaled volume (rv)

approach the formation free energy is defined as [30]

Ff,rv(Ωp, T) = Fv(χΩp, T;Nv, nsc)−χ Fp(Ωp, T;Nv+ 1, nsc),(3.4)

with the scaling factor χ=Nv/(Nv+ 1), with the number of atoms in the single vacancy crystal

Nv, and with the volume of the perfect crystal Ωp. Two calculations are needed in Eq. (3.4), one for

a perfect crystal and one for a single vacancy crystal, both however in the same type of supercell.

This is indicated by the dependence on the supercell size nsc which fixes the supercell according

to Eq. (2.114). The single vacancy supercell has one atom less than the perfect crystal supercell.

To account for this, Fpneeds to be rescaled by χwhich is possible due to the extensivity of the

perfect crystal free energy. It is however not apparent why the volume of the single vacancy cell

needs to be rescaled as compared to the perfect supercell volume Ωp. The new approach (derived

in Sec. 3.1.2) will allow to obtain Eq. (3.4) as an approximation of a more general set of equations,

with the volume scaling arising naturally [cf. Eq. (3.32)]. Within the second approach, the constant

3.1. An intuitive description/treatment of point defect formation 65

pressure (cp) approach, the following definition of the formation free energy is commonly used:

Ff,cp(P, T ) = Fv(Ωv

P, T;Nv, nsc)−χFpΩp

P, T;Nv+ 1, nsc

| {z }

Incomplete for P > 0

.(3.5)

Here, Ωv

P(Ωp

P) denotes the volume at which the single vacancy (perfect) supercell is at pressure P,

Eq. 2.228. Defined in this way, Ff,cp corresponds uniquely to the pressure P. However, we would

like to express Ff,cp as a function of volume to be consistent with the so far developed methodology

with F(Ω, T) [= NnF(V, T ); cf. Eq. (2.185)] being the central quantity. There exist at least two

distinct possibilities to assign a volume dependence

Ff,cp(Ωv

P, T) = Ff,cp(P, T) and Ff,cp(Ωp

P, T) = Ff,cp(P, T ),(3.6)

since in general Ωv

P6= Ωp

P. There is in fact another serious deficiency contained in Eq. (3.5). As

indicated, it is incomplete for finite pressures, i.e., P6= 0, since it misses a term proportional to P.

Both issues, the assignment of the correct volume dependence and the missing pressure term will

arise naturally as an approximation to our approach [cf. Eq. (3.26)]. This will, for instance, allow

to straightforwardly explain the necessity for including the pressure term.

We stress the significance of the difficulties with Eqs. (3.4) and (3.5). For instance, Varotsos

and Alexoupolous [88] discuss extensively the importance of distinguishing constant volume and

constant pressure vacancy calculations, point out possible ambiguities, and develop a comprehensive

set of equations. They also discuss the missing term in Eq. (3.5) and stress that it has been omitted

often in literature. Their account is however not adapted to the periodic ansatz. The motivation

in the present study was therefore to provide an accurate and straightforward approach that is

compatible with the periodic boundary conditions as commonly employed in DFT calculations.

Further, it should be more intuitive and general than the rescaled volume and constant pressure

approaches which start directly by defining a formation free energy. Before introducing the new

approach in the following section, we should briefly comment on two further aspects related to

vacancy calculations.

The first comment refers to the configurational free energy Fc. It is generally obtained from

Eq. (3.2) using Stirling’s approximation and it reads [87]:

Fc≈ −kBT[n−nln(n/N)].(3.7)

This approximation is well justified since nand Ncorrespond to a macroscopic crystal and higher

orders in Stirling’s expansion will be negligible. We will therefore also apply Eq. (3.7) in the next

section. Secondly, the treatment of vacancies presented here, starting with Eqs. (3.1) and (3.2),

is only applicable to non-interacting vacancies. In principle, vacancies could agglomerate to form

clusters such as for instance the so called divacancies. In aluminum however, which will be the

focus of the vacancy calculations in Sec. 4.3.6, it has been shown that divacancies do not form [89].

Self interstitials are in general repulsive, i.e., do not form clusters, due to the associated tensile

strain.

3.1.2 Volume optimized approach

The central quantity, which we are interested in, is the free energy F(Ω, T, n) of a macroscopic

crystal at fixed volume Ω, temperature T, and containing nvacancies. In order to adapt our

approach to the periodic ansatz, we introduce a large fictitious supercell with these properties and

66 3.1. An intuitive description/treatment of point defect formation

F( ,T;N,n, , )W Wv v

ntimes

F n ,T;N-nN

p v v

( - )W W

...

F ( ,T;N )

v v v

Figure 3.1: Schematic illustration of the concept of calculating the free energy F(Ω, T ;N, n, Ωv, Nv) of

a periodic crystal with vacancies. The larger box represents a supercell of volume Ω at temperature T

containing Natoms and nvacancies. A light-gray shaded box with a white circle represents a cell of volume

Ωv, containing Nvatoms, exactly one vacancy, and having the free energy Fv(Ωv, T ;Nv). The dark-gray

shaded region represents the perfect crystal without vacancies, with volume Ω −nΩv,N−nNvatoms, and

free energy Fp(Ω −nΩv, T ;N−nNv). The dashed lines indicate periodic boundary conditions.

with Natoms to represent the macroscopic crystal (Fig. 3.1). Fictitious refers to the fact that an

actual calculation of this supercell is not feasible. Beyond the supercell we have periodic boundary

conditions as introduced in Sec. 2.3.2. As indicated in Fig. 3.1, we construct a cell/box around

each vacancy which is large enough to cover its strain field and which we call the vacancy cell.

The vacancy cell contains Nvatoms, has a volume of Ωv, and a free energy Fv=Fv(Ωv, T ;Nv).

According to this construction, the crystal outside the vacancy cells can be described by a perfect

crystal with a volume Ωpand Npatoms,

Ωp= Ω −nΩv,(3.8)

Np=N−nNv,(3.9)

and with a free energy Fp=Fp(Ωp, T ;Np). We can then write the free energy of the fictitious

supercell as

F(Ω, T;N, n, Ωv, Nv) = Fp(Ωp, T ;Np) + nFv(Ωv, T;Nv)

| {z }

=: Fssc

+Fc(N, n),(3.10)

where we have defined Fssc the free energy of a single specific configuration and where Fcis given

by Eq. (3.7). Equation (3.10) and the definitions so far contain already the key conceptual ideas of

our approach. We first state the most important one which is sufficient to understand the concept

and, then, we comment on further aspects in more detail. The key idea is Eq. (3.8) which couples

the volume of the perfect crystal and of the vacancy cells. Since in equilibrium the total free energy

F(Ω, T, n) of the fictitious supercell needs to be minimal [Eq. (3.11) below], the volume of the

perfect crystal and of the vacancy cells will adjust self-consistently, i.e., until the optimum volume

for both is achieved when minimizing F(Ω, T, n). We thus refer to our approach a the volume

optimized approach. The actual minimization procedure will depend on the free energy volume

curves of the perfect crystal and of the vacancy cell. The resulting free energy F(Ω, T, n) is then

the quantity that can be used to derive the thermodynamics of a crystal with vacancies.

To allow for a detailed understanding of the volume optimized method, we need to consider the

3.1. An intuitive description/treatment of point defect formation 67

following aspects:

1) In writing Eq. (3.10), we need to take special care of vacancy configurations with overlapping

vacancy cells. Such configurations need to be accounted for, since the term Fssc must be

consistent with Fcand the latter contains all possible configurations. For nNv< N, we

can however always map overlapping onto non-overlapping configurations using the non-

interacting vacancies assumption. This guarantees that Eq. (3.10) is applicable. The cases

nNv> N and nNv=Nare discussed in points 4) and 6).

2) When speaking about the strain field of a vacancy, we refer to perturbations which cannot

be described by a change of the perfect crystal volume. The perfect crystal beyond the

vacancy cell is allowed to adapt its volume to the presence of the vacancy as expressed by

Eq. (3.8). Therefore, the effective influence range of the vacancy is not bound to the vacancy

cell. In particular, the total free energy of the fictitious supercell is affected by the vacancy

cell through the minimum free energy requirement with respect to Ωv:

∂F/∂Ωv≡0.(3.11)

3) By varying the temperature any number of vacancies nis possible. Consequently, the perfect

crystal can have any number of atoms (given by N−nNv). This is not possible in pure

geometrical terms due to the restriction by the supercell. We are however not explicitly

interested in the specific geometrical arrangement. We only need the corresponding free

energy of the perfect crystal Fpand this can be obtained for any number of atoms due to its

extensivity property

Fp(Ωp, T;Np)/Np=Fp(Vp, T; 1) =: Fp(Vp, T ),(3.12)

with

Vp(V, c) = Ωp/Np= (V−cΩv)/(1 −cNv),(3.13)

where we have defined the volume per atom V= Ω/N and the concentration of vacancies

c=n/N. We can use Eq. (3.12) to rewrite Eq. (3.10). The resulting expression

F(Ω, T;N, n, Ωv, Nv)/N = (1 −cNv)Fp(Vp(V), T) + cFv(Ωv, T ;Nv) + Fc(c)

=: F(V, T ;c, Ωv, Nv) (3.14)

with

Fc(c) = −ckBT(1 −ln c),(3.15)

is independent of Nand defines the free energy per atom F(V, T;c, Ωv, Nv) of the full supercell

consisting of the perfect crystal and nvacancy cells.

4) Let us consider the case nNv> N. The vacancy cells overlap necessarily in each configuration.

Using the assumption of non-interacting vacancies, we can map each configuration on one

which does not contain a perfect crystal part. We can then write the free energy of the

fictitious cell as:

F(Ω, T;N, n, Ωv, Nv) = nFv(Ωv, T;Nv)−Fp(nΩv−Ω, T;nNv−N) + Fc(N, n).(3.16)

The term Fp(nΩv−Ω, T;nNv−N) = Fp(−Ωp, T;−Np) needs to be subtracted, since in the

overlap region each vacancy cell contributes a free energy containing the perfect crystal free

68 3.1. An intuitive description/treatment of point defect formation

energy. Equation (3.16) can be rewritten using Eqs. (3.12) and (3.13) as

F(Ω, T;N, n, Ωv, Nv)/N =cFv(Ωv, T ;Nv)−(cNv−1) Fp(Vp, T) + Fc(c),(3.17)

which is equivalent to Eq. (3.14) and thus equivalent to the nNv< N case.

5) We can use the condition Eq. (3.11) and the general equilibrium condition with respect to

the concentration of the vacancies,

∂F/∂c ≡0,(3.18)

to explicitly determine the equilibrium vacancy cell volume Ωv,eq and the equilibrium vacancy

concentration ceq so that

F(V, T ) := F(V, T;ceq,Ωv,eq, Nv) = (1−ceqNv)Fp(Vp,eq, T)+ceqFv(Ωv,eq, T ;Nv)+Fc(ceq),

(3.19)

with Vp,eq = (V−ceqΩv,eq)/(1 −ceqNv), which is the final free energy expression of our

volume optimized approach (for nNv< N and nNv> N). The remaining unknown quantity

Nvis determined by the specific supercell used for the vacancy calculation and has to be

checked for convergence.

6) At last, we need to consider the case nNv=N. For this particular value, the vacancy cells

exactly fill the fictitious supercell. They further have a volume of Ωv= Ω/n =NvVand we

can use, instead of Eq. (3.10), the following expression for the free energy:

F(V, T ) = cFv(NvV, T, Nv) + Fc(c).(3.20)

3.1.3 Derivation of the standard approaches from the new method

Let us first consider the constant pressure approach and show that the pressure inside the vacancy

cells and inside the perfect crystal part equals the external pressure. Using Eq. (3.14) in Eq. (2.228),

we have

P=−∂F

∂V =−(1 −cNv)∂Fp

∂V p

∂V =−∂Fp

∂V p=: Pp,(3.21)

where the last equality defines the pressure Ppof the perfect bulk subsystem. Equation (3.21) shows

that Ppis equal to the externally applied pressure P. We next apply the equilibrium condition

Eq. (3.11) to Eq. (3.14):

∂F

∂Ωv= (1 −cNv)∂Fp

∂V p

Ωv+c∂Fv

∂Ωv=c∂Fv

∂Ωv−∂Fp

∂V p≡0.(3.22)

Defining Pv=−∂Fv/∂Ωv, the pressure inside the vacancy cell, we therefore have for c > 0:

Pv=Pp.(3.23)

We next Taylor expand Fp(Vp, T ) in Eq. (3.14) around c= 0

F(V, T ) = (1 −cNv)Fp(V, T) + ∂Fp(V)

∂V (NvV−Ωv) + O(c2) + cFv(Ωv, T;Nv) + Fc(c)

=Fp(V, T ) + chFv(Ωv, T;Nv)−NvFp(V, T) + Pvfi+O(c2) + Fc(c),(3.24)

3.1. An intuitive description/treatment of point defect formation 69

where we have defined the volume of vacancy formation:

vf= Ωv−NvV. (3.25)

Retaining only the terms linear in cand using the extensivity of Fp, we obtain the expression for

the free energy within the constant pressure approach [88]

F(VP, T)≈Fp(VP, T) + chFv(Ωv

P, T;Nv)−χFp(Ωp

P, T;Nv+1) + Pvf

| {z }

Correct version of Ff,cp(Ωp

P(VP), T ) from Eq. (3.5)

+Fc(c),(3.26)

where Ωp

P(VP) = (Nv+1)VP,vf

P= Ωv

P−NvVP, and where the subscript on Vand Ωvindicates that

both volumes correspond to the external pressure P. The term in the square brackets defines the

correct version of the formation free energy at constant pressure. (An example for the temperature

and volume dependence of the formation free energy of aluminum is given in Fig. 2.1.) In contrast

to Eq. (3.5) it contains a term linear in P. Based on our derivation, we can straightforwardly

analyze the necessity of this Pvfterm. It naturally arises from the Taylor expansion in Eq. (3.24)

and needs to be taken into account since it is a part of the first order term. Equation (3.26) can

be simplified by using the vacancy equilibrium condition Eq. (3.18)

∂F

∂c =Ff,cp(VP, T) + ∂Fc

∂c =Ff,cp(VP, T) + kBT−kBT(1 −ln c)≡0,(3.27)

which gives the equilibrium vacancy concentration

ceq,cp(VP, T) = exp[−βFf,cp(VP, T )],(3.28)

and which allows to separate a vacancy contribution to the free energy Fvac

F(VP, T)≈Fp(VP, T) + Fvac(VP, T ),(3.29)

with:

Fvac(VP, T ) = −kBT ceq,cp(VP, T ).(3.30)

The rescaled volume approach can be obtained from Eq. (3.26) by realizing that:

Fv(NvVP, T;Nv) = Fv(Ωv

P, T;Nv) + ∂Fv

∂Ωv

(NvVP−Ωv

P) + O([NvVP−Ωv

P]2)

=Fv(Ωv

P, T;Nv) + Pvf

P+O([vf

P]2).(3.31)

Therefore, to first order in vf

P, we have

F(V, T )≈Fp(V, T) + c[Fv(χΩp, T;Nv)−χFp(Ωp, T;Nv+1)]

| {z }

=Ff,rv(Ωp(V), T) from Eq. (3.4)

+Fc(c),(3.32)

where Ωp= (Nv+1)Vand where we have removed the pressure subscript for clarity. The quantity

in square brackets is the formation free energy of the rescaled volume approach from Eq. (3.4).

Similarly as for the constant pressure approach, we can derive

ceq,rv(V, T) = exp[−βFf,rv(V, T)],(3.33)

70 3.1. An intuitive description/treatment of point defect formation

Table 3.1: Definitions of the various free energy (FE) contributions for the perfect and single vacancy

supercell.

perfect vacancy

electronic FE for T=0 K equilibrium nuclei positions Fp,el

0Fv,el

quantum mechanical quasiharmonic FE Fp,qh Fv,qh

classical explicitly anharmonic FE Fp,clas,ah Fv,clas,ah

and thus finally:

F(V, T )≈Fp(V, T)−kBT ceq,rv(V, T).(3.34)

We have thus obtained the two standard approaches as natural approximations of our general

volume optimized method. In particular, we are now in a position to identify a hierarchy in the

approximations: First, the constant pressure approach arises by terminating the Taylor series in

the vacancy concentration in Eq. (3.24) after the first order term. The rescaled volume approach

needs a further approximation by terminating the Taylor series in the volume of vacancy formation

in Eq. (3.31) likewise after the first order term.

3.1.4 The final free energy expression and extension to other point defects

Let us rewrite for convenience the final free energy expression of the volume optimized approach,

Eq. (3.19), in a more compact form:

F(V, T ) = (1 −cNv)FpV−cΩv

1−cNv, T+cFv(Ωv, T ;Nv) + Fc(c).(3.35)

We have omitted here for clarity the ”eq” superscript as compared to Eq. (3.19), cand Ωvare

however implicitly assumed to correspond to their equilibrium values. For both supercells, the

perfect and the single vacancy supercell, we need to perform separate calculations as given by

Eq. (2.195):

Fp(V, T )≈Fp,el

0(V, T ) + Fp,qh(V, T) + Fp,clas,ah(V, T),(3.36)

Fv(Ωv, T;Nv)≈Fv,el

0(Ωv, T;Nv) + Fv,qh(Ωv, T;Nv) + Fv,clas,ah(Ωv, T;Nv).(3.37)

The corresponding free energy contributions are defined in Tab. 3.1. The calculation procedure is

the same, i.e., as discussed in Chap. 2, for both supercells except for a necessary modification of

the quasiharmonic free energy computation for the vacancy supercell. The reason is the break of

translational symmetry which obviates the use of Fourier transforms. We will discuss the necessary

changes in Sec. 3.3.3. Equation (3.35) constitutes the most complete free energy expression used

in the present work (in Sec. 4.3): The free energy of an elementary, non-magnetic crystal including

vacancies.

The volume optimized approach can be easily generalized to other point defects, such as e.g.

3.2. Accelerating DFT calculations of the anharmonic free energy: The UP-TILD method 71

self interstitials. In case of pdifferent point defect types, Eq. (3.35) changes to:

F(V, T ) = 1−

ciNi!FpV−Pp

iciΩi

1−Pp

iciNi

, T+

ciFi(Ωi, T;Ni) + Fc

i(ci).(3.38)

Here, ci, Ωi,Ni,Fi, and Fc

iare the concentration, the point defect supercell volume, the number of

atoms inside Ωi, the free energy of the point defect supercell, and the configurational free energy of

the ith point defect. Note that depending on the number of possible point defect places available in

the crystal, Fc

ican be different for different defects. For each point defect supercell the free energy

Fiis calculated similarly as in Eq. (3.37) for the vacancy supercell. Equation (3.38) corresponds

to the free energy of an elementary, non-magnetic crystal including any type of point defects.

3.2 Accelerating DFT calculations of the anharmonic free energy:

The UP-TILD method

3.2.1 Motivation

In order to motivate the necessity for a new method to calculate Fclas,ah, let us briefly sketch the

computational requirements of the two state-of-the-art approaches. Besides the TILD method,

introduced in Sec. 2.3.6, we discuss an alternative approach which is solely based on molecular

dynamics (MD) calculations. A comparison of these approaches will allow us to set up a certain

hierarchy in the performance of the methods.

Let us thus consider calculating Fclas,ah employing conventional MD simulations. With this we

mean that 1) we do not include the random Langevin term in Eq. (2.197) and that 2) we do not

make use of the quasiharmonic reference potential, i.e., we use only the full electronic free energy

Fel({RI}) in Eq. (2.191). We need to comment on both issues in more detail since either one

prevents a direct computation of the free energy. As for 1), using the original Newton’s equation of

motion allows to calculate only system properties at constant energy (microcanonical ensemble). To

enable a constant temperature calculation (canonical ensemble), as desired here, we need to couple

it to a thermostat, such as for instance to the Nos´e-Hoover thermostat [90, 91]. Such a thermostat,

however, does not contain a random element as the Langevin term and the deterministic nature

of the original Newton’s equation of motion prevails. (Note that, in order to achieve a constant

temperature, a fictitious degree-of-freedom is added to the equations of motion within the Nos´e-

Hoover scheme. The motion corresponding to this degree-of-freedom is however deterministic [78].)

As for 2), a direct calculation of the free energy without a reference potential is not possible since

the free energy is an entropic quantity [79]. This means that it depends on the phase space volume,

i.e., the partition function [cf. e.g. Eq. (2.192)], and it therefore cannot be described by a time

average like Eq. (2.198). (Note that within the thermodynamic integration such a description

becomes possible by coupling the system to a reference.) We can however compute directly the

inner energy from which the free energy can be obtained by integration with respect to T(see e.g.

Ref. [92]).

Taking the above considerations into account, we performed anharmonic free energy calculations

for a 2 ×2×2 aluminum supercell (32 atoms) using for a first test the EAM method. To reach an

accuracy of σerr <1 meV/atom [cf. Eq. (2.202)], we needed ≈107MD steps. This is consistent

with the number given in Ref. [93], where 106MD steps were found to be sufficient for a 500 atoms

supercell. (With an increasing number of atoms in the supercell, the spatial averaging improves

reducing the number of MD steps. Specifically, the number of atoms in the supercell times the

72 3.2. Accelerating DFT calculations of the anharmonic free energy: The UP-TILD method

number of MD steps is a constant, except for finite size effects.) In a second step, we have performed

the same anharmonic free energy calculations employing the TILD method. It turns out that to

reach σerr <1 meV/atom, we need ≈104steps [including the λintegration in Eq. (2.195)]. Let

us estimate the CPU requirements, if we were to perform these calculations using DFT. For that

purpose, we anticipate the results from Sec. 3.4.4.3 that for a DFT converged1calculation in a

2×2×2 aluminum supercell, we need a 4 ×4×4ksampling (2 048 kp·atom) and a plane wave

cutoff Ecut = 14 Ry. Such a calculation takes ≈1 CPU hour 2. We thus have

107CPU hours

a= 1 000 CPU years

| {z }

original MD

→104CPU hours

a= 1 CPU year

| {z }

TILD method

(3.39)

which makes clear that the TILD method shifts DFT based Fclas,ah calculations into a feasibility

range. The given numbers are however for a single point on the Fclas,ah(V, T ) surface. If we consider

calculating the full surface (≈30 points) and take into account additional DFT convergence checks

especially for the supercell size, such a study becomes computationally rather expensive.

As a final step in this section, let us explicitly write down the scaling behavior of the TILD

method. Comparing Eq. (3.40) below with the scaling of the UP-TILD method will later provide

an illustrative explanation of the essential features of the latter. The TILD method scaling stild

reads

stild =Nld(V, T )NVNTNλ·sks(asc, Ecut,high, Nhigh

k, Ne, Nit,red)

| {z }

Kohn-Sham eq. for supercell

,(3.40)

with Nλthe number of λvalues needed for the integration in Eq. (2.195). The notation for the

plane wave cutoff and the number of kpoints including the superscript ”high” will be useful in

the discussion of the UP-TILD method. We note that sqh — the scaling of the calculation of

the reference dynamical matrix D[cf. Eq. (2.191)] — is not explicitly included in sld since it is

negligible as compared to the remaining terms which are due to solving the Langevin equations of

motion, Eq. (2.197), Nld times to generate the trajectory {RI}t. Note that the latter calculation

must be performed in the computationally expensive supercell [asc =nscalat; cf. Eq. (2.187)] and

that we cannot employ Nirr

kbut we rather have to use Nk[cf. Eqs. (2.92) and (2.187)]. The

reason is that the atoms are not fixed to their symmetric positions when performing MD runs

and consequently also the symmetries for the ksampling are lost. Further, we have to compute a

trajectory for NVdifferent volumes and NTdifferent temperatures, with NVNT≈30 as mentioned

above. As pointed out in Sec. 2.3.5, the factor NTis due to the fact that molecular/Langevin

dynamics is a statistical approach and no analytical Tdependence is produced as available using

the perturbation approaches discussed in Sec. 2.3.7. The factor Nλis not significant (for the present

investigations on aluminum), since it can be reduced even to 1. The critical factor [marked red

in Eq. (3.40)] is Nld which corresponds to the above discussed ≈104Langevin dynamics steps

needed to reach statistical convergence. In fact, Nld is strongly dependent on the considered

volume and temperature, which is due to the increased phase space available at larger volumes and

temperatures. In contrast, it is rather independent of λas a consequence of using the Langevin

dynamics.

1Henceforth, it will be necessary to explicitly distinguish between DFT convergence (ksampling, plane wave cutoff,

augmentation grid, and supercell size) and statistical convergence. Note also that the numbers given here refer to a

pseudopotential calculation. Consequently, there is no augmentation grid present (cf. Sec. 2.1.8).

2As a reference, we take an AMD Opteron with a clock speed of 2.4 GHz.

3.2. Accelerating DFT calculations of the anharmonic free energy: The UP-TILD method 73

0 2000 4000 6000

TILD steps N LD

14.4

14.7

15.0

15.3

〈∂Fel

λ/∂λ〉low

t,λ (meV/atom)

0.2 meV

0 0.2 0.4 0.6 0.8 1

Energy (meV/atom)

〈∂Fel

λ/∂λ〉low

t,λ

〈∆Fel〉UP

〈∂Fel

λ/∂λ〉high

t,λ

0 10 20 30 40

UP steps N UP

12.6

12.9

13.2

13.5

〈∆Fel〉UP

λ (meV/atom)

0.2 meV

a) b) c)

Figure 3.2: Illustration of the UP-TILD method for a 23Al supercell at V= 15.7˚

A3and T= 900 K.

a) Statistical convergence of the time average ∂Fel

λ/∂λlow

t,λas a function of the number of steps Nld of

the corresponding TILD run based on low converged DFT parameters. b) Statistical convergence of the

upsampling average ∆Felup

λ[Eq.(3.42)] as a function of the number of uncorrelated TILD structures Nup

taken from the run in a). c) The λdependence of the statistically converged quantities from a) and b) and

their difference, which equals the time average ∂Fel

λ/∂λhigh

t,λfor well converged DFT parameters according

to Eq. (3.44).

3.2.2 Extending the TILD to the UP-TILD method

In order to overcome the difficulties with the computational requirements discussed in the previous

section, we introduce the following extension of the TILD method which we refer to as the UP-

TILD (UPsampled Thermodynamic Integration based Langevin Dynamics) method and which

provides an efficient scheme to coarse grain the configuration space. The UP-TILD procedure can

be structured into two main steps:

1) As a first step, we perform a ”usual” TILD run as described in Sec. 2.3.6. The crucial point

is now that instead of using well converged DFT parameters (cf. Sec. 3.2.1) we rather use

a set of low converged parameters (specifically: 23kmesh, 256 kp·atom, and 8 Ry plane

wave cutoff). This reduces the time for a MD-step down to ≈120 s, i.e., a gain by a factor

of ≈30 compared to the 1 hour calculation using highly converged DFT parameters. With

this time reduction, the first run can be easily extended to several thousand steps in order

to obtain the desired statistical accuracy. We denote the corresponding trajectory including

Nld structures and the time average as:

TILD run with

low parameters −→ {RI}low

Eq. (2.198)

−−−−−−−→ ∂Fel

∂λlow

t,λ

.(3.41)

An example for the statistical convergence of ∂Fel

λ/∂λlow

t,λis shown in Fig. 3.2a.

2) In a second step, we need to correct ∂F el

λ/∂λlow

t,λsince we have a significant loss of accuracy

due to the reduction of the DFT convergence parameters. We refer to this correction as the

upsampling (UP). The upsampling itself can be divided into three steps:

(a) We extract from {RI}low

ta set of Nup uncorrelated structures {RI}low

tuusing Eq. (2.201).

74 3.2. Accelerating DFT calculations of the anharmonic free energy: The UP-TILD method

(b) We then calculate the upsampling average ∆Felup

λgiven by

D∆FelEup

λ=Nup−1NUP

uh∆Fel,low

tu−∆Fel,high

tui,(3.42)

with:

∆Fel,low

tu=Fel,low({RI}low

tu)−Fel,low({R0

I}),

∆Fel,high

tu=Fel,high({RI}low

tu)−Fel,high({R0

I}).(3.43)

Here, Fel,low (Fel,high) refers to the electronic free energy calculated using the low

(high/well) converged set of DFT parameters. Note that the λdependence of ∆Felup

is hidden in the trajectory {RI}low

t, which is additionally dependent on the volume

and temperature. An example for the statistical convergence of ∆Felup

λis shown in

Fig. 3.2b and demonstrates the actual merit of the UP-TILD approach: The statistical

convergence rate is improved by about two orders of magnitude compared to the original

one for ∂F el

λ/∂λt,λkeeping the number of the computationally expensive calculations

Fel,high({RI}low

tu) small. We typically obtain statistical convergence for Nup <100.

λ/∂λhigh

t,λ, is

obtained by: D∂Fel

λ/∂λEhigh

t,λ=D∂Fel

λ/∂λElow

t,λ−D∆FelEup

λ.(3.44)

An example for the upsampling at different λvalues is shown in Fig. 3.2c. It demonstrates

a further advantage of the UP-TILD method: The λdependence of ∆Felup

λbecomes neg-

ligible. It is therefore a good approximation to neglect it completely and work with a single

∆Felup

λat some fixed λ. (We typically use λ= 0.5.)

In order to verify that the UP-TILD approach is applicable to aluminum, our target material system

(cf. Sec. 4.3), we calculated ∂Fel

λ/∂λhigh

t,λfor some volumes directly and by means of Eq. (3.44).

A representative set of our results is shown in Fig. 3.3. The figure illustrates the good performance

of the method. The directly calculated values agree within the statistical uncertainty with the

upsampled values and reproduce the volume dependence of Fclas,ah accurately. (The wiggles in the

solid orange curve are due to statistical noise.) Moreover, Fig. 3.3 shows that it is indeed crucial

to use the set of well converged DFT parameters, since the set of low converged DFT parameters

yields both substantial deviations in the absolute values and an incorrect volume dependence.

A prerequisite for the UP-TILD approach to be applicable is that the phase spaces spanned by

the set of low and high converged DFT parameters are sufficiently similar and that the difference can

be described by a nearly constant energy shift. If the phase spaces differ significantly, the method

will not be applicable. It should be also noted that the method cannot be applied to different

supercell sizes since the corresponding phase spaces have different dimensions and a mapping onto

each other is excluded by definition. In addition to computationally demanding explicit tests

(Fig. 3.3), the performance of the UP-TILD method can be checked with two implicit criteria:

A) If the two phase spaces are similar, ∆Felup

λis nearly independent of λ(Fig. 3.2c). Deviations

from this independence can be thus used as a measure of the similarity of the phase spaces.

B) An alternative measure is the number of structures needed to obtain a statistically converged

∆Felup

λfor a single λvalue.

3.2. Accelerating DFT calculations of the anharmonic free energy: The UP-TILD method 75

15.5 16.0 16.5 17.0 17.5 18.0

V (Å3/atom)

Fclas,ah (meV/atom)

low; direct

high; UP-TILD

high; direct

Figure 3.3: Classical anharmonic free energy Fclas,ah of aluminum at 900 K for the LDA functional and the

23supercell as a function of the atomic volume V. Results for low and high converged DFT parameters

are shown: low: 8 Ry, 256 kp·atom and high: 14 Ry, 2 048 kp·atom. Free energies obtained directly from a

’usual’ TILD calculation are marked as ’direct’, while ’UP-TILD’ indicates that the UP-TILD method was

employed.

3.2.3 Scaling performance of the UP-TILD method

The scaling sup of the UP-TILD method reads:

sup =NVNT{

Step 1) TILD run with low parameters

z }| {

NλNld(V, T )·sks(asc, Ecut,low , Nlow

k, Ne, Nit,red)

+Nup(V, T)·sks(asc, Ecut,high, Nhigh

k, Ne, Nit )

| {z }

Step 2) Upsampling, Nup ≪Nld }.

(3.45)

The first term in the curly brackets corresponds to step 1) and the second term to step 2) of the UP-

TILD method. Let us compare Eq. (3.45) to Eq. (3.40). In Eq. (3.40), we find that the problematic

factor Nld (≈104) is coupled to the computationally expensive calculation based on the high

converged DFT parameters sks(...high ...). In contrast, in Eq. (3.45) Nld and sks(...high ...)

appear only decoupled, which we have indicated by the different shading of the terms inside the

curly brackets. Now, Nld is coupled to the computationally inexpensive calculation sks(...low ...),

while sks(...high ...) to the small factor Nup. We need however to consider Nit iteration steps for

the self consistency cycle, Eq. (2.66), since we cannot use the wave function extrapolation [point

e) at the end of Sec. 2.3.6] for the uncorrelated structures of the upsampling. This increases the

calculation time for sks(...high ...) by a factor of 2 as compared to Eq. (3.40). We thus have

Nld ≈104, sks(...low ...)≈120 sec, Nup ≈100,and sks(...high ...)≈2 h,(3.46)

and can estimate the CPU time requirements of the UP-TILD method to ≈20 days (2 ×2×2

aluminum supercell; cf. Sec. 3.2.1). Combining these estimations with Eq. (3.39) we thus have the

76 3.2. Accelerating DFT calculations of the anharmonic free energy: The UP-TILD method

following hierarchy in the scaling performance of the three introduced methods:

107CPU hours

a= 1 000 CPU years

| {z }

original MD

→104CPU hours

a= 1 CPU year

| {z }

TILD method

→500 CPU hours

a= 20 CPU days

| {z }

UP-TILD method

(3.47)

3.2.4 Extensions of the UP-TILD method

We propose two extensions of the UP-TILD approach, the first one concerning the upgrade to a

hierarchical scheme and the second one being an easy-to-use parallelization. Let us thus start with

the hierarchical scheme which allows to further improve the convergence of the DFT calculations

and/or to improve the statistical sampling. To illustrate this, consider a DFT convergence series

with a set of values {x1, x2, x3}with successively increasing quality and computational expense:

x1< x2< x3. In a first step, we perform an efficient TILD simulation based on x1and we denote

the corresponding low DFT converged time average as ∂Fel

λ/∂λ1

t,λ. In a second step, we calculate

∆Felup,1→2

λ, the upsampling average of x1with respect to x2, by:

D∆FelEup,1→2

λ=Nup−1NUP

(Fel,x1

tu−Fel,x2

tu).(3.48)

Next, we calculate ∆Felup,2→3

λ, the upsampling average of x2with respect to x3, by:

D∆FelEup,2→3

λ=Nup′−1NUP′

(Fel,x2

tu−Fel,x3

tu).(3.49)

This quantity converges statistically even faster than ∆Felup,1→2

λ, i.e., Nup′≪Nup, and there-

fore further reduces the expensive calculations corresponding to x3. Note that in total only one

MD run (for x1) is necessary. To obtain finally the fully DFT converged time average ∂Fel

λ/∂λ3

t,λ,

we replace Eq. (3.44) by:

D∂Fel

λ/∂λE3

t,λ=D∂Fel

λ/∂λE1

t,λ−D∆FelEup,1→2

λ−D∆FelEup,2→3

λ.(3.50)

This scheme can be easily extended to higher hierarchies.

A straightforward application of the hierarchical UP-TILD approach is to a set of varying k

samplings, plane wave cutoffs, and augmentation grids. We point out however a further very useful

application. As stressed in Sec. 2.1.8, we have different possibilities to treat the high oscillations

of the Kohn-Sham wave function close to the nuclei core with varying quality and efficiency. For

instance for aluminum, we can assign x1= pp (pseudopotential) and x2= PAW (PAW method).

Let us explicitly write down for future reference the corresponding upsampling average:

D∆FelEup,pp→paw

λ=Nup−1NUP

(Fel,pp

tu−Fel,paw

tu).(3.51)

Note that due to Eq. (3.43) only energy differences enter the upsampling average but not the total

3.3. Integrated approach to thermodynamic quantities 77

energies of the pseudo- or PAW potentials. We could also extend the convergence further by includ-

ing x3= LAPW (LAPW method) and a corresponding upsampling average. Thus, the hierarchical

UP-TILD method provides an extremely efficient way to check potential parametrizations not only

for symmetric atomic structures but also for thermodynamically relevant structures.

Let us now turn to the issue of parallelization. We first point out that the second step of the UP-

TILD method, the upsampling, is trivially parallelizable since each of the uncorrelated structures

can be calculated independently of the others. In contrast, the first step, the low DFT converged

TILD run, is naturally a sequential calculation, i.e., the various structures of the trajectory need

to be calculated strictly one after the other. Even though the reduction of the DFT convergence

parameters yields a strong CPU time reduction, the remaining times are in the order of CPU weeks

and since a serial calculation is needed3this equals real time weeks. In many cases, it might be

desirable to shorten the real time at the expense of using more CPUs and we therefore propose

the following easy-to-use parallelization: Instead of performing a single TILD run, we perform in

parallel Npar TILD runs. Denoting the trajectory of the jth run as {RI}j,t, we then replace the

time average Eq. (2.198) by:

∂Fel

∂λT/t,λ

= (NparNld)−1

Npar

NLD

∂

∂λFel

λ({RI}j,ti).(3.52)

Note that the sum over the different runs amounts to averaging over an ensemble of systems, i.e.,

to performing a thermodynamic average. We have thus changed the notation from h.itto h.iT/t.

This scheme may be viewed as moving partially back to the original expression used to calculate

the anharmonic free energy Fclas,ah, Eq. (2.195). The remaining steps of the UP-TILD approach

are performed as described so far. We stress however that such a parallelization crucially relies

on an efficient equilibration scheme, as is the case in the present study [cf. point c) at the end of

Sec. 2.3.6], since only configurations from trajectories after equilibration can be used in Eq. (3.52).

3.3 Integrated approach to thermodynamic quantities

Let us finally combine the various theoretical tools presented so far into a practicable integrated

approach to calculate the free energy surface F(V, T ). To this end, we will be mainly concerned

with finding appropriate parametrizations along Vand Tfor the various contributions entering

F(V, T ). For that purpose, we need to fix the temperature window of interest, which we take to be

T= 0 K . . . Tm, where Tmis the melting temperature of the studied elements (cf. Fig. 4.1). The

volume range is then determined by the equilibrium volume at T= 0 K, Veq

0(cf. Tab. 4.1), and by

the equilibrium volume at Tm, which is typically ≈1.1Veq

0(cf. Fig. 4.4). The given parametrizations

apply equally well to the free energy contributions of the perfect and single vacancy supercell [cf.

Eq. (3.35)]. As pointed out in Sec. 3.1.4, we will however have to reconsider the calculation of the

quasiharmonic free energy for the vacancy supercell (Sec. 3.3.3). As a measure of the quality of a

parametrization we will frequently refer to the expansion coefficient α, Eq. 2.235, and to the specific

heat CP, Eq. 2.236. Being second derivatives of F(V, T) both are extremely sensible to changes in

the parametrizations. The specific numbers provided in the following are for aluminum4.

3Serial refers here to the fact that the various MD structures need to calculated one after another. The calculation

of a single structure can be parallelized in the usual manner for DFT calculations (for instance kpoint parallelization).

4As a reference for the relative values for αand CP, we take the final quantities containing all contributions at

the melting temperature of aluminum (934 K) and at zero pressure. LDA: α= 3.68 ·10−5K−1and CP= 3.73 kB;

GGA: α= 3.83 ·10−5K−1and CP= 3.83 kB. See also Tab. 4.12.

78 3.3. Integrated approach to thermodynamic quantities

3.3.1 Electronic free energy surface

Let us first focus on Fel

0(V, T ), the electronic free energy for the T= 0 K equilibrium nuclei positions

{R0

I}, with the temperature dependence entering via the Fermi-Dirac function, Eq. (2.65). The

representation of Fel

0(V, T ) becomes particularly convenient if it is divided into the temperature

independent ground state energy Eel

g,0(V) := Eel

g({R0

I}, V, T ) and a remaining part e

Fel

0(V, T ):

Fel

0(V, T ) = Eel

g,0(V) + e

Fel

0(V, T ).(3.53)

The reason for this separation is that e

Fel

0(V, T ) can be accurately described with low order poly-

nomials, while Eel

g,0(V) can be parametrized using standard equations-of-state (EOS). We studied

three different EOSs: the Murnaghan [94], the third-order Birch-Murnaghan [95], and the Vinet [96]

EOS. The latter has been found to describe theoretical and experimental data of various materials

most accurately [97]. For the CPof Al, we find negligible5differences between the three EOSs.

For α, the Birch-Murnaghan and Vinet EOS yield a similar value which is, however, 1.9% larger

than the one obtained with the Murnaghan EOS. We ascribe this to the improved nature of the

Birch-Murnaghan and Vinet EOS. For the further analysis, we use the Vinet EOS which reads [96]

Eel

g,0(V) = Eel

g,0(Veq

0) + 4BT,0Veq

(B′

T,0−1)2−Veq

0BT,0

(B′

T,0−1)2(5 + 3B′

T,0"V

Veq

01/3

−1#−3V

Veq

01/3)

×exp (−3

2B′

T,0−1"V

Veq

01/3

−1#),(3.54)

with Veq

0,BT,0, and B′

T,0defined in Eqs. (2.232), (2.239), and (2.240). We note that the choice of

the EOS parametrization plays only a role if one is interested in highest accuracy results as needed

e.g. in Sec. 4.3, while for ”usual” applications this point is less crucial. On the other hand, an

important point is that for a sensible parametrization of Eel

g,0(V) the volume range needs to be

extended below Veq

0to ≈0.9Veq

0, in contrast to the remaining free energy contributions. To obtain

the surface e

Fel

0(V, T ), we calculate e

Fel

0values on a mesh of ≈5 volume and ≈10 temperature

points. The surface is then parametrized using a two dimensional fourth order polynomial as

Fel

0(V, T ) = X

i,j

ai,jViTjwith i≥0, j ≥1,and i+j≤4,(3.55)

and with fitting coefficients ai,j. The discrete set of e

Fel

0values needed to perform the fit can be

easily obtained via DFT: Only a small unit cell is required allowing for easy convergence and the

resulting free energy surface is smooth. Inclusion of terms beyond the fourth order were found to

provide no improvement to the fit.

3.3.2 Quasiharmonic free energy surface

The temperature dependence of the quasiharmonic free energy Fqh presents no additional challenge,

due to the analytical dependence given in Eq. (2.161). In contrast, a sensible parametrization of

Fqh along Vturns out to be important. We critically test three approximations which are rather

convenient, since for each it is practically sufficient to calculate Fqh only at two different volumes.

The first two approximations begin in fact with a parametrization of the phonon frequencies:

5We call an error or effect negligible if it is below 1% for αand below 0.5% for CP(cf. footnote on previous page).

3.3. Integrated approach to thermodynamic quantities 79

2.5

3.0

3.5

4.0

4.5

CP (kB)

200 400 600 800

T (K)

α (10-5 K-1)

16 17 18

V (Å3)

-2

-1

F qh (meV)

a) linear Gr

b) linear ω

c) linear F

T m

a) c)

full

Figure 3.4: Quasiharmonic expansion coefficient

αand heat capacity CPof Al. The solid lines

show fully converged quantities. The dot-dashed

[dotted/dashed] lines represent quantities based

on a) the linear Gr¨uneisen (Gr) [b) the linear fre-

quency ω/ c) the linear free energy F] approx-

imation [see Eqs. (3.56) to (3.58) respectively

for details]. The inset shows the correspond-

ing quasiharmonic free energies Fqh (per atom)

at 900 K referenced to the fully converged Fqh.

The curves correspond to GGA results. LDA re-

sults show the same behavior. The vertical thin

dashed line marks the melting temperature Tm.

a) The volume dependence is parametrized using the linear Gr¨uneisen approximation [68]

−V

ωq,s(V)

∂ωq,s(V)

∂V =γq,s = const,(3.56)

with the Gr¨uneisen constant γq,s.Solving for ωq,s(V) yields ωq,s(V) = aq,sV−γq,s , with an

integration constant aq,s.

b) The frequencies are assumed to depend linearly on the volume:

∂ωq,s/∂V =cq,s = const.(3.57)

c) The free energy is assumed to depend linearly on the volume:

∂Fqh/∂V =c= const.(3.58)

Assumption c) is not equivalent to b), since the Fqh dependence on ωq,s is strongly non-linear.

Based on highly converged DFT free energies (cf. Sec. 3.4) for several (seven) volumes, we are

in the position to check the validity/performance of each of these approximations. The volume

dependence of these free energies is fitted to a polynomial of third order:

Fqh(V, T) =

i=0

ai(T)Vi.(3.59)

Extending the polynomial to higher orders yields only negligible changes in the thermodynamic

properties. Figure 3.4 shows the comparison with respect to the target quantities αand CP. As

can be seen, applying approximation b) yields dramatically wrong αand CPvalues at temperatures

close to the melting point Tm. At Tm, they are overestimated by 74% and 21%, respectively. In

contrast, approximation a) yields a strong underestimation, −27% for αand −7% for CP. The

best approximation is c) yielding an underestimation of −14% for αand −4% for CP. However,

the remaining error is still significant when considering the small contributions beyond the quasi-

harmonic approximation (compare to Tab. 4.12) and could, in particular, have a strong influence

on the determination of anharmonic contributions.6A correct treatment of the volume dependence

6Note that this situation would occur, if we used a harmonic, i.e., fixed volume, dynamical matrix as a reference

for our anharmonic calculations. In order to obtain the volume dependence of the anharmonic contribution, we would

80 3.3. Integrated approach to thermodynamic quantities

of the quasiharmonic free energy surface is therefore crucial.

We note a further important point: In Fig. 3.4, we included the volume dependence of the free

energy (at 900 K; inset) which results from each of the approximations. The differences in the free

energies between the various approximations are in the range of a few meV/atom, which might be

considered as relatively small. Nonetheless, these small differences lead to the strong differences

in the expansion coefficient and heat capacity. This is a very important observation and we will

return to it in Sec. 3.4.2 where we consider the error propagation in DFT thermodynamics.

3.3.3 Quasiharmonic free energy for point defects

The break of translational symmetry due to the creation of a point defect has two consequences:

1) The Fourier transforms Eqs. (2.138) and (2.139) and the following transformations leading

to the eigenvalue equation, Eq. (2.143), and to the fundamental phonon frequencies determining

the quasiharmonic free energy, Eq. (2.161), cannot be applied. 2) The symmetry considerations

regarding the real space dynamical matrix Das discussed in Sec. 2.3.4 need to be modified.

The first issue can be solved rather straightforwardly. In fact, we do not need to change into

Fourier space to obtain the frequencies. We have used this procedure in Sec. 2.3.3 deliberately for

two reasons: First, it allows an interpretation of the eigenvalues of the dynamical matrix in terms

of phonon frequencies. Second, it allows easily to extend the formalism to arbitrary wave vectors

qas performed in Sec. 2.4.1. Let us now however use a more direct approach which can be applied

also to defect supercells and which consists of solving the eigenvalue equation of the real space

dynamical matrix D:

D(V)wi=ω2

i(V)wi.(3.60)

The dynamical matrix Dis in our case a 3Nn×3Nnmatrix and we thus have 3Nneigenvectors wi

each of size 3Nnand 3Nneigenvalues ω2

i. For a perfect crystal, the ωimatch exactly the frequencies

ωGsc

,s. (Note that there are NnGsc vectors and for each sruns over 3 branches so that we have

also in reciprocal space 3Nnfrequencies ωGsc

,s as needed to conserve the degrees of freedom.) We

can therefore use directly the ωito obtain the quasiharmonic free energy, Eq. (2.161), now as

Fqh(V) = 1

3Nn

3Nn−3

i1

2~ωi(V) + kBTln h1−e−β~ωi(V)i,(3.61)

again omitting the 3 zero frequencies. We need to take special care in using Eq. (3.61) in combina-

tion with a quasiharmonic free energy obtained from a dense Fourier interpolated mesh, Eq. (2.226).

To explain this consider calculating the free energy of a crystal with vacancies using Eq. (3.35). To

calculate Fv,qh, we are forced to use Eq. (3.61) due to the above discussed symmetry break. For

Fp,qh, we might want to use Fp,qh,m, i.e., the Fourier interpolated version Eq. (2.226), in order to

improve the description at lower temperatures. This leads however to a shift in the perfect crystal

free energy with respect to Fv,qh and produces an incorrect coupling between the perfect crystal

and vacancy supercell in Eq. (3.35). To nonetheless profit from the good performance of Eq. (2.226)

at low temperatures, we introduce the following correction scheme:

F(V, T ) = e

F(V, T )−Fp,qh(V, T ) + Fp,qh,m(V, T ).(3.62)

Here, e

Fis the free energy obtained from Eq. (3.35) using Fp,qh and Fv,qh, which are all obtained

then need to subtract Fqh. If Fqh(V) was incorrectly parametrized, we could easily obtain large errors in Fclas,ah(V),

in fact even leading to a wrong sign.

3.3. Integrated approach to thermodynamic quantities 81

from Eq. (3.61). [For Fp,qh, also the equivalent expression Eq. (2.161) can be used.] Subsequently,

Fp,qh is subtracted and Fp,qh,mcalculated on a well converged mesh is added.

To perform vacancy calculations in practice, it is often indispensable to apply a generalized

version of the described correction scheme:

F(V, T ) = e

F(V, T )−Fp(V, T) + Fp,best(V, T).(3.63)

Here, e

Frefers to a free energy calculated using consistent parameters (e.g., ksampling, plane wave

cutoff, augmentation grid) for the perfect bulk, Fp, and the vacancy supercell, Fv, in Eq. (3.35).

Further, Fp,best is the perfect crystal free energy calculated using the best convergence parameters.

The correction is important for the final thermodynamic quantities (e.g., Fig. 4.17) but does not

affect vacancy related properties such as the concentration (e.g., Fig. 4.15), since the latter are

obtained directly from e

Let us now discuss the second of the above issues concerning the symmetries of the real space

dynamical matrix. As derived in Sec. 2.3.4, translational and point symmetries allow to reduce

the number of Kohn-Sham calculations needed to fully fill the perfect crystal dynamical matrix to

one. For point defects, this number increases significantly. The matrices Eq. (2.172) can however

still be used to map some of the displacements and corresponding force fields onto each other by

the rules given in Sec. 2.3.4. This applies in particular to displacements in the same atomic shell

around the point defect.

3.3.4 Extension to a temperature dependent dynamical matrix

Let us now abandon the approximation performed in Eq. (2.179) and consider the temperature

dependent dynamical matrix. The quasiharmonic formalism, Eqs. (2.181) to (2.185), changes to:

HF forces from Kohn-Sham

calculations for different Ω and Tel

| {z }

Kohn-Sham calculations

for different Ω and Tel

| {z }

DIα,Jβ(Ω, Tel)=−(MI∆R)−1Fhf

I,α(R0

1,1,...,R0

J,β + ∆R, . . . , R0

Nn,3,Ω, Tel) (3.64)

DGsc

, αβ(Ω, T el)=N−1

I,J

DIα,Jβ(Ω, Tel)eiGsc·(R0

I−R0

J)(3.65)

DGsc (Ω, Tel)wGsc

, s =(ωGsc

, s)2(Ω, Tel)wGsc

, s (3.66)

Fel

0(Ω, Tel) +

prBZ′

Gsc

s~

2ωGsc

, s(Ω, T el)+kBTnuc ln h1−exp n−β~ωGsc

,s(Ω, Tel)oi

| {z }

=Fqh(Ω, Tnuc, T el)(3.67)

≈F(Ω, Tel, Tnuc)→Tel =Tnuc =T→F(Ω, T )/Nn=F(V, T ).(3.68)

Compared to Eqs. (2.181) to (2.185), we have performed the following changes. The Hellmann-

Feynman forces Fhf

Iare now calculated from the electronic free energy Fel({RI}). The latter is

82 3.3. Integrated approach to thermodynamic quantities

temperature dependent through the Fermi-Dirac function, Eq. (2.65), and this renders also the

forces temperature dependent. We have denoted the corresponding temperature by Tel and we will

refer to it as the electronic temperature. This enables a distinction with the nuclei temperature

Tnuc, which explicitly enters the quasiharmonic free energy, and it will prove convenient for the

following discussion and the subsequent parametrization. The dependence of Fhf

Ion Tel carries

over to the real space dynamical matrix D, the reciprocal dynamical matrix DGsc , and the phonon

frequencies ωGsc

,s, on which we have thus dropped the ”0K” superscript. The quasiharmonic free

energy Fqh likewise depends implicitly on Tel, besides its explicit dependence on Tnuc. As indicated,

at the end of a calculation the electronic and nuclei temperature need to be set equal to the actual

external temperature T. The difference in the free energy resulting from Eqs. (3.64) to (3.68)

as compared to the free energy resulting from Eqs. (2.181) to (2.185) is fully contained in the

quasiharmonic free energy and we therefore write:

∆Fqh(Ω, T )←∆Fqh(Ω, Tnuc, Tel) = Fqh(Ω, T nuc, T el)−Fqh,0k(Ω, Tnuc).(3.69)

It will proof useful (Sec. 4.2) to investigate also derived thermodynamic properties by separating

their temperature dependence as (T)→(Tel, T nuc). In particular, we will be interested in the

temperature dependencies of the isobaric heat capacity CPand the free energy FP, for which we

consider:

Fqh(Ω, T nuc;Tel)→(CP(Tnuc;Tel),

FP(Tnuc;Tel).(3.70)

As indicated by the semicolon, CPand FPare calculated from a quasiharmonic free energy surface

at a fixed Tel. Equation (3.70) should be contrasted with the fully consistent approach, i.e., the

one required to obtain the final thermodynamic properties:

Fqh(Ω, T) = Fqh(Ω, Tnuc =T, Tel =T)→(CP(T),

FP(T).(3.71)

An important point is that, while we have

FP(T) = FP(Tnuc =T;Tel =T),(3.72)

the heat capacity does not obey this relation in general:

CP(T)6=CP(Tnuc =T;Tel =T).(3.73)

The reason for this inequality is the fact that CPis a derivative quantity thus depending on the

curvature of the quasiharmonic free energy surface, which is different for Fqh(Ω, T nuc;Tel) and

Fqh(Ω, T). This issue will become more clear in Sec. 4.2.

A calculation required to obtain ∆Fqh(Ω, T) scales as:

sqh =NVNT·sks(alat, Ecut, Nirr,pr

k, Npr

e, Nit)

| {z }

Kohn-Sham eq. for primitive cell

+NVNT·sks(asc, Ecut, Nirr

k, Ne, Nit)

| {z }

Kohn-Sham eq. for supercell

.(3.74)

Comparing Eq. (3.74) to Eq. (2.187), we see that the number of Kohn-Sham calculations for the

supercell increases as compared to the T= 0 K dynamical matrix formalism, since we need to

account for the temperature dependence of the Hellmann-Feynman forces.

3.3. Integrated approach to thermodynamic quantities 83

In practice, we found the following scheme to be particularly convenient to calculate ∆Fqh(Ω, T ).

For a fixed volume we determine Don a mesh of 5 different Tel. Using the standard procedure

we calculate the corresponding frequencies so that we have a set of frequencies for each of the 5

different Tel. We then fit

ωGsc

, s(Tel) = aGsc

, s +cGsc

, s(Tel)2+dGsc

, s(Tel)3,(3.75)

to the set of frequencies separately for each Gsc vector and branch s. The Tel and Tnuc dependencies

are calculated using the standard expression:

Fqh(Tnuc, T el) =

prBZ′

Gsc

s~

2ωGsc

, s(Tel) + kBTnuc ln h1−exp n−β~ωGsc

,s(Tel)oi.(3.76)

The volume dependence is treated as described in Sec. 3.3.2 and ∆Fqh(Ω, T ) is eventually deter-

mined from Eq. (3.69).

3.3.5 Anharmonic free energy surface

Before discussing the critical volume and temperature parametrization of the anharmonic free

energy surface Fclas,ah(V, T ), let us reconsider the equations of the TILD approach in view of the

results of the previous section. The electronic free energy of the coupled system Fel

λ, Eq. (2.191),

depends in fact on the electronic temperature Tel

Fel

λ({RI}, Tel) = (1 −λ)hFel

0(Tel) + UD(Tel)U/(2M)i+λFel({RI}, Tel),(3.77)

(neglecting now the volume dependence) and this carries over to the partition function, Eq. (2.192),

which is additionally explicitly depending on the nuclei temperature Tnuc:

Zλ(Tel, Tnuc) = ZdRIe−Fel

λ({RI},Tel)/(kBTnuc)/(Λ3NnNn!).(3.78)

The final equation for the anharmonic free energy, Eq. (2.195), reads then

Fclas,ah =Z1

dλZ−1

λ(Tel, Tnuc) (Λ3NnNn!)−1

×ZdRIe−Fel

λ({RI},Tel)/(kBTnuc)[Fel({RI}, Tel)−Fel

0(Tel)−UD(Tel)U/(2M)],

(3.79)

where we have used the original definition of the thermal average, Eq. (2.193), and where we have

explicitly stated each temperature dependence. The electronic temperature Tel enters through the

Fermi-Dirac function, Eq. (2.65), while the nuclei temperature Tnuc enters through the Langevin

dynamics in particular through Eq. (2.200). We note that the reference for the thermodynamic

integration can be chosen arbitrarily and we therefore could remove the temperature dependence

of the dynamical matrix by replacing D(Tel)→D0k. The other temperature dependencies need

however to be respected and, since T=Tel =Tnuc must be fulfilled at the end of a calculation,

we have in principle to perform the molecular dynamics simulations at a Tnuc which corresponds

to the Tel used to determine Fel({RI}). Previous studies ignored this subtle dependence. Using

84 3.4. Towards highly accurate DFT free energies

the UP-TILD method, we are for the first time able to investigate the actual influence of Tel on

Fclas,ah (Sec. 4.3.5).

Let us now discuss the Vand Tparametrization of Fclas,ah which turns out to be a critical

point in calculating derivative quantities such as αor CP. The problem has three aspects:

a) A technical one: Even though tools are available to sample the phase space effectively (e.g.,

the UP-TILD method), reducing the statistical error σerr, Eq. (2.202), to much less than 1

meV/atom quickly becomes computationally prohibitive, especially for high temperatures.

Thus, in practice, the anharmonic free energy surface will contain non-negligible statistical

noise.

b) A physical one: Despite the statistical noise, we can clearly identify that the temperature

dependence of the surface contains higher order contributions at least up to the cubic term

(see Figs. 4.13b and 4.16). Moreover, our results also suggest that V-Tcoupling terms have

non-negligible contributions.

c) A fundamental one: In contrast to the temperature dependence of the quasiharmonic free

energy [Eq. (2.184)], no analytical equation for the full anharmonic free energy, i.e., containing

all orders in the expansion Eq. (2.130), is known.

Due to the lack of an analytical expression, it looks tempting to use a polynomial series to expand

Fclas,ah(V, T ). Based on the discussion above and extensive convergence checks, it became clear

that this is critically hindered by an interplay of a) and b): In order to take higher orders into

account, we need a sufficiently large polynomial basis. However, increasing the polynomial basis,

the parametrization becomes unphysical due to the statistical noise in the data used for the fit. We

will therefore derive in Sec. 4.3.4 a conceptually different and physically motivated approach which

solves this problem.

3.4 Towards highly accurate DFT free energies

In Chap. 2, we have discussed the origin of the key convergence parameters in DFT based thermo-

dynamic calculations (ksampling, plane wave cutoff, augmentation grid, and supercell size). In this

section, we will address the actual convergence behavior of thermodynamic properties (specifically

αand FP) for the studied elements (cf. Tab. 4.1). We focus hereby on some issues (Secs. 3.4.1 to

3.4.3), which we found to be particularly important and which we therefore discuss in detail. In

Sec. 3.4.4, we eventually give an overview and a brief discussion of the full list of DFT parameters,

on which the results presented in Chap. 4 are based, including also the statistical issues for the

anharmonicity calculations.

3.4.1 Efficient convergence: Decoupling procedure and a supercell size based

technique

A quantity, we identified to be crucial in achieving a high accuracy in thermodynamic calculations,

is the size of the grid used to calculate the augmentation charges (henceforth labeled augGrid)

within the PAW method. (The augmentation charges correspond to ρaug from Sec. 2.1.8.) To

allow a proper investigation of the augGrid, it is important to separate the dependence of the

augGrid from the grid used to store the wave function and charge density coefficients (henceforth

labeled basicGrid), which is determined by the plane wave cutoff. [This dependence is typically

introduced for simplicity, since then only a single parameter (the cutoff) needs to be controlled.]

3.4. Towards highly accurate DFT free energies 85

0 100 200 300 400

gp/atom (103)

100

110

α/αconv (%)

±2%

L2,Ag

L1L2,Cu Figure 3.5: Convergence of the expansion coefficient αwith

respect to the augGrid size (see text) in terms of grid

points per atom (gp/atom) for Cu and Ag. The values are

scaled with respect to αconv which corresponds to 432 ·103

gp/atom. The lines L1(similar for Cu and Ag on this

scale) and L2,Cu/Ag correspond to grid sizes being 8 and 64

times larger than the basic grid at a converged cutoff (see

Tab. 3.2). The values were obtained using a 33supercell

and a kmesh of 7 ·103kp·atom.

To account for this, we performed the augGrid convergence tests by keeping the well converged

cutoffs from Tab. 3.2 fixed.

The influence of the augGrid size on the expansion coefficient αis demonstrated in Fig. 3.5 for

two elements with hard, i.e. strongly localized, augmentation charges, Cu and Ag. The example

of Cu shows that working with an augGrid 8 times larger than the basicGrid7(line L1in Fig. 3.5)

results in a −24.3% error in α. [Note that we determine the error at a specific augGrid value by the

maximum deviation detected before convergence, i.e., in this case the error is given by the second

augGrid value (44 ·103) and not by the first (28 ·103; line L1).] Even by making the augGrid 64

times larger than the basicGrid8(line L2in Fig. 3.5), the error is still 11.3%. In the case of Ag,

the larger equilibrium lattice constant yields a larger basicGrid. For this case, an augGrid 64 times

larger than the basicGrid reduces the error in γto −4.4%.

In order to fully reveal the significance of these results for the augGrid convergence, let us return

to the above discussed (artificial) correlation with the plane wave cutoff. Consider that we were

using the standard coupled procedure for checking convergence, i.e., having the plane wave cutoff

(basicGrid) as the only parameter and the augGrid determined automatically. As a consequence, we

would find that extremely large cutoffs are necessary to reduce the errors and to obtain reasonable

thermodynamic quantities. For instance, to obtain an error <2%, we would need a cutoff >500 eV

(in contrast to the converged cutoff of 290 eV) even when using an extra high precision calculation

(cf. footnote 8). Since however the computational requirements increase much more rapidly with

the size of the basicGrid than with the size of the augGrid, such a convergence test would be in

fact not even feasible. It is therefore crucial to realize that actually only the augGrid causes the

the slow convergence rather than the basicGrid and that a decoupled treatment makes calculations

much more feasible.

The decoupled treatment is however only the first step, since, even if we apply this procedure,

a direct calculation of the augGrid convergence rate employing a well converged supercell size for

all investigated elements is still computationally very expensive. Using, for instance, a 33supercell

would require more than 10,000 CPU-hours (AMD Opteron with a clock speed of 2.4 GHz). In

order to tackle this challenge, we introduce in a second step a scaling procedure, which allows an

efficient and accurate determination of the convergence rate. The key idea of this approach is as

follows: Let us assume a general physical target quantity, say g, and two convergence parameters

c1and c2. We are interested in the convergence rate of gwith respect to c1at a converged value

7In vasp [46] (the software package used for the PAW calculations presented in this work), such an augGrid size

is achieved for a high precision calculation (flag: PREC=HIGH).

8This corresponds in vasp to a high precision calculation with an additional support grid (flag: AD-

DGRID=TRUE).

86 3.4. Towards highly accurate DFT free energies

210 240 270 300

Ecut (eV)

100

110

α/αconv (%)

0 100 200 300 400

gp/atom (103)

100

110

α/αconv (%)

cconv

2= 33sc

csmall

2= 13sc

a) b)

rescaled

Figure 3.6: Illustration of the supercell size based convergence technique. a) AugGrid convergence of the

expansion coefficient αfor Ag for the 13and 33supercell (sc). The blue squares show the convergence of

the 13sc after rescaling it using Eq. (3.80) with s= 2. The dotted (dashed) line indicates ca

1= 28 ·103

(cb

1= 54 ·103) gp/atom. b) Similar to a), but with the plane wave energy cutoff Ecut as the convergence

parameter and with a scaling parameter of 1.

of c2=cconv

2, i.e. g(c1, cconv

2). For most cases, computing g(c1, cconv

2) directly is highly expensive.

However, we have found that in many cases the following approximation can be utilized to obtain

an accurate description of g(c1, cconv

2):

g(c1, cconv

2)≈g(ca

1, cconv

2) + s∆g, (3.80)

∆g=g(c1, csmall

2)−g(ca

1, csmall

2).

Here, sis a linear scaling parameter defined as

s=g(cb

1, cconv

2)−g(ca

1, cconv

g(cb

1, csmall

2)−g(ca

1, csmall

2)(3.81)

and ca

1,cb

1(ca

16=cb

1), and csmall

2are fixed (but not fully converged) parameters. The idea is then

to study in a first step the convergence of g(c1, csmall

2) in detail. Therefore, csmall

2is chosen such as

to provide a computationally inexpensive calculation. In a second step, the scaling parameter sis

calculated by performing only two (expensive) calculations for cconv

2:g(ca

1, cconv

2) and g(cb

1, cconv

2).

Based on these data, g(c1, cconv

2) is approximated by Eq. (3.80).

To be more specific, let us consider gto be the expansion coefficient α,c1to be the augGrid

size, and c2to be the supercell size. Further, csmall

2corresponds to a 13and cconv

2to a 33supercell.

We set ca

1= 28 ·103gp/atom and cb

1= 54 ·103gp/atom. The resulting scaling parameter is

s≈2, which means that the changes in the convergence dependence of αare two times larger for

cconv

2than for csmall

2. Using swe can thus extrapolate the changes for csmall

2to the ones for cconv

As a representative example, Fig. 3.6a shows the results we obtained for Ag. As can be seen, the

computed values closely follow the scaling relation according to Eq. (3.80). Thus, to obtain a densely

sampled convergence dependence for large supercells only two expensive calculations are needed

and a gain in computational efficiency of at least one order of magnitude can be easily achieved.

We also studied, how the scaling parameter depends on the chemical species and the supercell size.

Our results show a small chemical dependence (<10%). The supercell size dependence shows a

significant scaling parameter but only when increasing a 13to a 23supercell. We further verified

that the same approach can be applied to determine the plane wave cutoff convergence of large

supercells (Fig. 3.6b). For this case, a scaling parameter generally close to 1 is found. For the k

3.4. Towards highly accurate DFT free energies 87

∆α

∆FP

-0.3

0.0

0.3

-0.3

0.0

0.3

∆α(T m) (10-5K-1)

100 1000

-0.3

0.0

0.3

100 1000

gp/atom (103)

100 1000

-2

∆FP (T m) (meV/atom)

-2

0 300 600 900 1200

T (K)

maxDev

L3, L4

α(T) (10-5K-1)

Al Pb Cu

-0.5

L1L4

-5%

a) b)

Figure 3.7: Illustration of the influence of the augGrid on thermodynamic properties (GGA). a) AugGrid

convergence (logarithmic scale) of the expansion coefficient αand the isobaric free energy FPfor all inves-

tigated elements at the melting temperature Tm. The presented values, ∆αand ∆FP, are referenced with

respect to the corresponding quantities with the highest employed augGrid. Line L1[L2] (falls together with

the left y-axis [dotted line]) corresponds to an augGrid at a converged cutoff for a standard [extra] high

precision calculation (cf. footnote 7 [8]). Line L3(dot-dashed line) corresponds to the augGrid size used in

the present study (Sec. 4.1). For the noble metals, Cu, Ag, and Au, line L4indicates the highest considered

augGrid. The orange solid horizontal lines indicate the 5% error for α. The values were obtained using a

kmesh of 7 ·103kp·atom and a 13supercell. The scaling procedure [Eq. (3.80)] was used to rescale the

values to correspond to large supercells. The gray arrows emphasize the chemical trend for the size of the

error among the transition metals. Al and Pb are separated to distinguish them from the transition metals.

b) The temperature dependence of αfor the noble metals. The solid lines represent the converged quantities

[L3and L4in a)] and the dashed lines correspond to the maximum deviation (maxDev) from a). Note that

for the purpose of a convenient representation, the curves corresponding to the mixed approach (Fig. 4.5b)

are shown.

sampling, starting from a 13supercell [parameter csmall

2in Eq. (3.80)] is not sufficient to guarantee

a reasonable approximation, we rather need to apply a 23supercell. The reason is the fact that

using a 23supercell guarantees kmeshes which are commensurable to the converged 43supercell.

The scaling parameter is ≈1.

The convergence rate of αand FPwith respect to the augGrid size is summarized in Fig. 3.7a

for all studied metals. (For the critical elements, we had in fact to extend the range of the augGrid

size to even larger values than shown in Fig. 3.5.) An inspection of Fig. 3.7a shows that the

convergence rate correlates inversely with the hardness (localization) of the augmentation charge

of the specific element: Among the transition metals, αand FPconverge slower when filling up the

dshell and when reducing the atomic radius (indicated by the gray arrows). Thus, Cu, having the

hardest augmentation charge, is the most sensitive element in the present study. In contrast, Al

and Pb, where only sand pelectrons form the chemical bonds, exhibit a relatively low dependence

on the augGrid. In Fig. 3.7b, the explicit influence of an unconverged augGrid mesh is shown for

the expansion coefficient of the critical elements. We stress here again that such a situation (i.e.,

88 3.4. Towards highly accurate DFT free energies

the uncertainty given by the dashed lines) would occur for a standard high precision calculation

(see footnote 7). To avoid the error, we used for the final calculations (Sec. 4.1) the augGrid size

of 432 ·103gp/atom (L3), which yields a coefficient similar to that originating from the highest

augGrid size used in our convergence tests (L4) (solid lines in Fig. 3.7b).

3.4.2 Error propagation in thermodynamic calculations

A further convergence parameter, which we found to be crucial in predicting thermodynamic prop-

erties of metals with very high accuracy, is the ksampling of the electronic dispersion. The

difficulties with the ksampling become particularly apparent, if we are interested in calculating

Fqh,0kas intended in Sec. 4.1. To obtain Fqh,0k, we need to compute the dynamical matrix from

an electronic free energy surface corresponding to T= 0 K, i.e., the electronic ground state. The

associated Fermi-Dirac function shows at this temperature a sharp change at the Fermi energy [µin

Eq. (2.65)], which causes the sums over the kvectors [e.g., Eq. (2.87)] to converge extremely slowly.

A first necessary step to tackle this difficulty is the replacement of the Fermi-Dirac function by an

artificial electron occupation function as given for instance by the Methfessel-Paxton scheme [98].

In this scheme, an artificial temperature Tart controlling the electron occupation numbers is in-

troduced. Increasing Tart improves the ksampling convergence, but the forces on the atoms still

correspond to the (desired) T= 0 K electronic ground state. However, to guarantee the latter,

one needs to ensure that the artificial electronic entropy term associated with Tart does not exceed

1 meV/atom [99]. We thus used the Methfessel-Paxton scheme based on a sufficiently low artificial

temperature of kBTart = 0.1 eV for the convergence checks presented in this section and for the

final calculations (Sec. 4.1).

Despite the usage of Tart, a careful convergence study of the ksampling is important to assess

the introduced error in the final quantities. To account for this aspect, we investigated in detail

the convergence behavior for all elements, similarly as for the augmentation grid. Concerning the

presentation of the results, we follow however a different route now, in order to reveal an important

property of errors in thermodynamic calculations. Let us thus start with discussing the errors in

αand FPat a fixed absolute temperature (400 K; see Fig. 3.8), in contrast to the augGrid where

we looked directly at the melting temperature Tm, i.e., at a varying absolute temperature (cf.

Tab. 4.1). Figure 3.8 reveals the important fact that we have a different group of critical elements

than found for the augGrid: Al and Pb exhibit a strikingly worse convergence behavior for α(note

the logarithmic scale) as compared to the transition metals. For instance for Al, the convergence of

αshows even at the largest considered kmesh (190·103kp·atom) a significant gradient. For Pb, the

smallest considered ksampling 2·103kp·atom yields an error in αas large as −25% (0.7·10−5K−1).

For the free energy FPof Al and Pb, we find instead a comparable (or slightly worse) convergence

behavior as for the transition metals.

Let us now, in a second step, extend the discussion from T=400 K to T=Tmfocusing first on

FP. For that purpose, we use in Fig. 3.9a a different representation of the errors than in Fig. 3.8:

For each element, we extract only the (absolute) error in FP, denoted by Ferr

P, corresponding to

the ksampling marked by the vertical dot-dashed line in Fig. 3.8 (the ksamplings actually used

in Sec. 4.1). We then plot this error versus the melting temperature Tmof the respective element.

The result is given by the dashed line in Fig. 3.9a. We see that all elements have a comparable Ferr

of ≈1 meV/atom in accordance with the conclusion drawn from Fig. 3.8. The similar procedure

is applied to obtain Ferr

Pat Tmand the result is given by the solid line in Fig. 3.9a. For this

temperature, we find a significantly different behavior of Ferr

P: It increases (in average) strongly

with the melting temperature, in fact showing even a linear dependence. As a consequence, the

elements with a high melting temperature have likewise a high error in FPat Tm.

3.4. Towards highly accurate DFT free energies 89

-0.04

0.00

0.04

-0.04

0.00

0.04

∆α(T=400 K) (10-5K-1)

∆α

∆FP

-0.04

0.00

0.04

10 100

kp⋅atom (103)10 10010 100

-3

∆FP (T=400 K) (meV/atom)

-3

3Al Pb Cu

-0.7

-1%

-18

-5

0.1

Figure 3.8: ksampling con-

vergence of the expansion co-

efficient αand the free en-

ergy FPfor all studied ele-

ments at T= 400 K. The

x-axis starts at the small-

est investigated kmesh of

2·103kp·atom and the verti-

cal dot-dashed line indicates

kmeshes used for the calcu-

lations presented in Sec. 4.1.

Additionally, the 1% error of

αis shown by the orange

solid horizontal lines. The

values were obtained using

the plane wave cutoffs from

Tab. 3.2, an augGrid size of

432 ·103kp·atom, and a 23

supercell.

600 1200 1800 2400

Tm (K)

Ferr

P(meV/atom)

T = Tm

T = 400 K

600 1200 1800 2400

Tm (K)

x(Tm) /x(400 K)

c) x = Ferr

x = αerr

600 1200 1800 2400

Tm (K)

αerr (10-7K-1)

T = Tm

T = 400 K

Pb Al Ag

Pd Rh

Ir b)

Figure 3.9: Illustration of error propagation in DFT thermodynamic calculations. a) The magnitude of the

error in the free energy, Ferr

P(=largest deviation before convergence), at the ksamplings marked with the

vertical dot-dashed lines in Fig. 3.8 plotted versus the melting temperature Tmof the respective element.

The error is obtained at T= 400 K (dashed line; corresponding to the values in Fig. 3.8) and at T=Tm.

b) As a) but for the error in the expansion coefficient αerr. c) Ratio of the curves in a) (solid line) and b)

(dotted line).

0.0 0.1 0.2 0.3

Amplitude of eigenvector (Å)

-30

Fel (meV/atom)

-0.3

0.0

0.3

0.6

0.9

∆Fel (meV/atom)

kBT = 934 K=Tm

Error

kBT = 400 K

Figure 3.10: Origin of the Tdependence of Ferr

Pfrom

Fig. 3.9a. The thick blue solid line shows the electronic

free energy Fel along the transversal eigenvector at the L

point for aluminum (23supercell). The thin blue horizon-

tal lines indicate some of the corresponding phonon energy

levels. The thick dashed horizontal lines show the energies

corresponding to 400 K and the melting temperature of Al.

The orange solid lines show how the difference ∆Fel be-

tween two Fel curves (7·103and 16·103kp·atom) increases

with the displacement (referenced such that, at each T, the

mean value of both energies is set to zero). All energies are

based on the quasiharmonic approximation.

90 3.4. Towards highly accurate DFT free energies

In order to understand the increase of Ferr

Pwith temperature, it is in fact sufficient to consider

only a single element and to perform an analysis of its electronic free energy Fel. This issue is

illustrated in Fig. 3.10 for aluminum. The thick blue solid line shows the electronic free energy

along an eigenvector of the dynamical matrix and the horizontal solid lines indicate some of the

corresponding phonon energy levels. Further, the thick dashed horizontal lines represent the two

temperatures of interest. From Fig. 3.10, it becomes apparent that, with increasing temperature,

the spatial extension of the phonon displacements also increases. On the other hand, with increasing

spatial displacement, the error in Fel increases in a strong non-linear fashion (orange solid lines).

Therefore, higher energy/temperature phonons are stronger affected by an error in Fel than lower

energy ones. This will in turn affect the quasiharmonic free energy – the main contribution to FP

– much stronger at high temperatures than at lower ones.

Let us now return to the discussion of αand analyze how the error in this quantity propagates

with temperature. We recall that, based on Fig. 3.8, we have identified Al and Pb as the metals

with the slowest convergence rate for αat T=400K. The corresponding error αerr is now visualized

in Fig. 3.9b (dashed line) using again the alternative representation. Based on the results obtained

for Ferr

Pand the knowledge from Sec. 3.3.2 that small changes in FP(a few meV/atom) can have

a significant effect on α, we might expect that the trend for αerr changes significantly at Tm. In

particular, we expect that αerr of the high melting metals (Rh, Ir) will increase stronger than for Al

and Pb. Figure 3.9b and more clearly Fig. 3.9c show however that this is not the case: The ratio

αerr(Tm)/α(400 K) is nearly constant for the various elements, in contrast to the linearly increasing

ratio for Ferr

P. This means that the originally (at 400 K) largest errors for Al and Pb are still the

largest at Tm.

In order to understand this behavior, we need to carefully analyze the type of the errors assessed

in this section and in Sec. 3.3.2: 1) The error Ferr

P(this section) refers to the difference between two

FPcurves (for instance for two ksamplings) at some temperature (here, either 400 K or Tm) when

both curves are set equal at 0 K (absolute free energies are physically not relevant). The important

point is that from the knowledge of Ferr

Palone, we can only conclude that, at some fixed temperature,

the two corresponding free energy surfaces F(V, T) are shifted by a constant. We have however no

information about the local Vand Tdependence of the two F(V, T) surfaces. 2) In contrast, the

error assessed in Sec. 3.3.2 (in particular, inset of Fig. 3.4) refers directly to differences in the local

Vand Tdependence. To distinguish the two types of errors, let us call 1) a global error and 2) a

local error. Next, we need to recall that the expansion coefficient αdepends on the local structure of

F(V, T ) (Sec. 2.4.2). Thus, α(and other thermodynamic properties) will be much stronger affected

by a local error in the free energy than by a global one and this explains that the results of this

section are in fact not in contradiction to Sec. 3.3.2. Let us summarize this finding schematically by

Global error in F(V, T ) Local error in F(V, T )

Derived thermodynamic properties

Small effect Large effect

and by the key message: To estimate the error in derived thermodynamic quantities, it is not

sufficient to know the magnitude of the error in F(V, T), we also need to know the nature of the

latter.

3.4. Towards highly accurate DFT free energies 91

Γ X K ΓL

Γ X K ΓLΓXK ΓL

Eω (meV)

3·103 kp·atom 7·103 kp·atom 16·103 kp·atom

Figure 3.11: ksampling convergence

of the phonon dispersion ωq,s [Eω=

hν =hω/(2π) with hthe Planck

constant] along some high symme-

try directions for Ir, Al, and Pb for

the 43supercell and the GGA func-

tional. The insets enlarge the un-

physical phonon instabilities (imag-

inary frequencies; see text).

3.4.3 The origin of unphysical imaginary frequencies and their treatment

In addition to the study of the convergence of the averaged quantities, αand FP, discussed in

Secs. 3.4.1 and 3.4.2, it is necessary to analyze the convergence of the phonon dispersion ωq,s. This

analysis is important for two reasons: 1) The first concerns our interest to evaluate the theoretical

phonon dispersion directly and thus the comparison with experiment (Sec. 4.1.3). If we, knew that

αand FPare well converged, we could not directly conclude that this applies similarly to ωq,s,

since errors in the latter might cancel in average [Eq. (2.226)] and thus not be visible in αand

FP. 2) The second reason is the convergence rate for αand FP. Suppose that we found it to be

very slow and that, as a consequence, we needed to employ computationally expensive convergence

parameters to obtain a good accuracy. However, this slow convergence rate might be caused by only

a small fraction of the phonon frequencies (for instance imaginary frequencies introduced below)

and a careful analysis and treatment might allow to use computationally less expensive parameters.

With these aspects in mind, we calculated the convergence behavior of ωq,s for all parameters

and studied elements. Let us first focus on the ksampling and further on three representative cases

shown in Fig. 3.11 for a detailed discussion. Among the transition metals, the dispersion of Ir shows

the slowest convergence. Nevertheless, the convergence is still rapid and no qualitative changes are

observed. In contrast to all transition metals, the convergence of Al and Pb is more complex: For

some kmeshes an anomalous behavior appears in the vicinity of the Γ point (indicated by the

squares in Fig. 3.11). In these regions, the phonon frequencies are imaginary. (For convenience, the

imaginary frequencies are shown as negative frequencies in Fig. 3.11.) In principle, the presence

of imaginary frequencies indicates that the structure is unstable against deformations along the

associated phonon wave vector. Put differently, this means that the structure corresponds to a

saddle point on the electronic free energy surface rather than to a minimum. The imaginary

frequencies found for some ksamplings are, however, not related to a genuine structural instability,

but are unphysical and caused by not fully converged parameters. In the case of Al, the instability

occurring at 7·103kp·atom can be removed by enlarging the ksampling to 16·103kp·atom. For Pb,

the instability is more persistent and occurs even for large samplings (16 ·103kp·atom). To ensure

that the instability is not caused by a too small ksampling, we further increased the ksampling

(55 ·103kp·atom; not shown in Fig. 3.11), but found no changes in the phonon dispersion.

In order to show that the instability of Pb is indeed fictitious, the remaining critical convergence

92 3.4. Towards highly accurate DFT free energies

Γ X K ΓL

ΓXK ΓL

53 sc, 500 atoms

Eω (meV)

23 sc, 32 atoms 33 sc, 108 atoms 43 sc, 256 atoms Pb

Figure 3.12: Supercell (sc) size convergence of the phonon dispersion for Pb and Pd. The kmesh size was

32 ·103kp·atom for the 53supercell of Pb and 7 ·103kp·atom for the other supercells (Pb and Pd). Exact

wave vectors Gsc are marked by dark blue ticks.

parameter, the supercell size, needs to be considered. To this end, we calculated the dependence of

the phonon dispersion on the supercell size for the various elements. The dispersion of Pb is by far

the most sensitive in this respect. In fact, while a 43supercell was sufficient to obtain convergence

for the other elements, we had to extend the investigations to a 53supercell (500 atoms) for Pb.

Figure 3.12 illustrates the dependence of the Pb dispersion on the supercell size. There are no

phonon instabilities for a 23and 33supercell. Increasing the size to 43, an instability occurs in the

vicinity of the Γ point in the direction Γ to K. This is the supercell size that was employed for the

above discussed ksampling convergence. One can now see that increasing the supercell size further

to 53almost completely removes the imaginary frequencies. We expect a complete disappearance

of the imaginary frequencies when going to even larger supercells, since previous calculations based

on the linear response method [100] and experiment [101] report no instability.

Let us discuss a second illustrative example for the influence of the supercell size: the phonon

dispersion of Pd shown in Fig. 3.12. We find that, except for the vicinity of the Γ point, the

dispersion of all supercells shows a very smooth sinusoidal and thus simple dependence along the

wave vector as found for elements in which the nearest neighbor interactions dominate (cf. the

discussion for Cu, Ag, and Au in Sec. 4.1.3). In fact, due to the overall ”well-behaved” phonon

dispersion without any instabilities found for the 23supercell, one might be easily mislead and

consider this supercell size as the converged one. (Note that this dispersion is different from

the one for Pb for the same cell size, since the latter shows already departures from the simple

dependence and the influence of longer ranged interactions.) However, increasing the supercell size

to 33, a small instability occurs in the vicinity of the Γ point along the L direction and a larger one

along the K direction. The instability disappears again for the 43supercell.

A closer inspection of this feature reveals that there is an intricate interplay between the sam-

pling of the phonon Brillouin zone and the occurrence of anomalies in the phonon dispersion as

a consequence of long range interactions. To explain this we need to recall the procedure (and

its limits) used to obtain a dense phonon dispersion along the high symmetry directions. Orig-

inally, phonon frequencies are only available at the Nn(=number of atoms) exact wave vectors

which are fully captured by the used supercell (denoted Gsc in Sec. 2.3). Since using only the Gsc

vectors would yield a very coarse sampling for reasonable supercell sizes (cf. the dark blue ticks

in Fig. 3.12), we introduced in Sec. 2.4.1 the Fourier interpolation scheme which allows to obtain

an arbitrarily dense sampling. The important point now is the fact that the Fourier interpolation

3.4. Towards highly accurate DFT free energies 93

13233343

supercell size

0.0

0.1

0.2

0.3

∆α(T m) (10-5K-1)

13233343

supercell size

-20

-10

∆FP (T m) (meV/atom)

a) b)

0.9

Figure 3.13: Supercell size convergence of the expansion coefficient αand the isobaric free energy FPfor Cu,

Pb, and Pd at the melting temperature Tm. The shown values, ∆αand ∆FP, are referenced with respect to

the 43supercell. The used k-point and augGrid mesh were consistently for all supercell sizes 7 ·103kp·atom

and 432 ·103gp/atom, respectively.

corresponds to a sinus-like basis and that the number of basis functions actually included in the

interpolation is determined by the number of available Gsc vectors, i.e., with increasing super-

cell size also the number of basis functions increases. If only small supercell sizes are accessible

the Fourier interpolation is therefore best suited for describing sinusoidally shaped dispersions as

found, for instance, for the 23supercell of Pd. The ”true” phonon dispersion of Pd shows however

strong deviations from this simple dependence in the vicinity of the Γ point along the K direction.

The reason is an experimentally verified [101] anomaly, which will be discussed in more detail in

Sec. 4.1.3 (cf. Fig. 4.3). Based on this discussion, we can explain the observed dependence of the

Pd phonon dispersion on the supercell size (Fig. 3.12) as follows: For the 23supercell, the number

of included Fourier basis functions is so small that none of them matches the highly oscillatory

dependence (along the wave vector) which is needed to describe the sharp changes of the anomaly.

Put differently, all basis functions available in the 23supercell are orthogonal to each of the basis

functions which would be necessary to describe the shape of the anomaly. The situation changes

for the 33supercell, where one (fast oscillating) basis function, which is captured by the supercell,

is also needed to describe the anomaly. Since this is however only a single basis function it can-

not reproduce the anomaly properly. In fact, it strongly overshoots the dependence causing the

observed ”dip” with imaginary frequencies. For the 43supercell, further basis functions with a

yet faster oscillatory dependence are included and they are able to improve the description of the

anomaly significantly even removing the imaginary frequencies.

The occurrence of imaginary frequencies due to insufficient convergence requires special care

when calculating thermodynamic quantities. A possible approach would be to calculate the ther-

modynamic quantities exclusively for highly converged parameters, for which the unphysical insta-

bilities are fully absent. However, even in the presence of these instabilities sufficiently converged

thermodynamic quantities can be obtained by neglecting all imaginary frequencies when calculat-

ing the quasiharmonic free energy according to Eq. (2.226). We found this choice/approximation

to have little effect on the free energy and derived quantities in most cases, since the phase space

around the Γ point is negligible compared to the full Brillouin zone. An example of how well this

approach works is given in Fig. 3.13, where the convergence of αand FPfor Cu, Pb, and Pd

is shown. Pb and Pd, for which imaginary frequencies are present (for the 43and 33supercell,

respectively), show a similarly good convergence behavior as Cu without such frequencies, thus

indicating that the neglect of unphysical imaginary frequencies is a reasonable approximation.

94 3.4. Towards highly accurate DFT free energies

Table 3.2: Details for the PAW calculations of Fqh,0k;Ecut is the plane wave cutoff, ”augGrid” the aug-

mentation grid in grid points (gp) per atom, and Dthe dynamical matrix; see also text.

Pb Al Cu Ag Pd Rh Au Pt Ir

supercell 53|43|

atoms 500 |256 |

Ecut (eV) 150 250 290 |270 | | 250 |

augGrid (103gp/atom) |432 |

kdensity (103kp·atom) 32 16 |7|

kmesh for supercell 4343|33|

electron occupation |Methfessel-Paxton, 0.1 eV |

displacement for D|0.01 a0|

qmesh for primitive BZ |163|

3.4.4 Computational details

3.4.4.1 Details for Sec. 4.1, Systematic study

For the LAPW+lo reference calculations (Sec. 4.1.2), we used the wien2kcode [102]. The muffin

tin (mt) radius9Rmt [= 3

p3ΩI/(4π)] was held constant when varying the volume of the crystal V.

For each element, Rmt was adjusted such that the spheres were nearly touching at the smallest V.

The resulting sphere radii are ranging from 2.3a0to 2.5a0. A 243kmesh (= 6 ·103kp·atom) was

used for Al, a 203kmesh (= 8·103kp·atom) for Pb, and an 183kmesh (= 14·103kp·atom) for the

other elements. The product RmtKmax was set to 10, where Kmax (= p2meEcut/~2) corresponds

to the magnitude of the largest reciprocal electronic wave vector. [In LAPW+lo calculations, it is

more convenient to specify RmtKmax rather than Kmax (or Ecut) as a consequence of the matching

condition between plane waves and atomic orbitals at the sphere surface [49].] The maximum

ℓmax value for the waves inside the atomic spheres [similar to the number of expansion coefficients

in Eq. (2.96) within the PAW method] was set to ℓmax = 12. The wien2kcode employs an

additional plane wave cutoff Gmax for the Fourier expansion of the charge density, which we set

to Gmax = 14 a−1

0. Relativistic effects were included fully within the atomic spheres and using the

scalar relativistic approximation in the valence region.

The PAW calculations were performed using the vasp [46] code and the potentials provided

with this package [53]. To calculate Fel

0(T), we used the finite temperature formulation of DFT [26]

(cf. Sec. 2.1.4). The electronic dispersion was sampled using the Monkhorst-Pack scheme [103] and

akmesh of 323(= 33 ·103kp·atom). The details for the calculations of Fqh,0kare compiled in

Tab. 3.2. The convergence issues of the DFT related parameters have been discussed in Secs. 3.4.1

to 3.4.3. For the calculation of the dynamical matrix, we used the direct force constant method

(cf. Sec. 2.3.4). We studied displacements (for the finite difference of the forces) in the range of

0.0005 a0to 0.1a0for all elements. We find, in general, that the thermodynamic properties are

rather insensitive (examples given in Fig. 3.14) with respect to a variation in the displacement. In

9We use the notation customary in the corresponding literature (e.g., Ref. [49]) and give in brackets reference to

our notation from Secs. 2.1.7 and 2.1.8.

3.4. Towards highly accurate DFT free energies 95

0.001 0.01 0.1

Displacement (a0)

0.00

0.05

0.10

∆α(T m) (10-5K-1)

0.001 0.01 0.1

Displacement (a0)

-0.6

-0.3

0.3

0.6

∆FP (T m) (meV/atom)

a) b)

0.05%

Figure 3.14: Dependence of a) the expansion coefficient αand b) the isobaric free energy FPon the displace-

ment used to calculate the dynamical matrix (a0=Bohr radius) for Al, Pb, and Pd at the respective melting

temperature Tm. The shown values, ∆αand ∆FP, are referenced with respect to αand FPat 0.01 a0

(marked by the vertical dot-dashed line) and the horizontal lines mark the relative values with respect to

these quantities. A ksampling of 7 ·103kp·atom and a 13supercell size were used for the calculation.

particular, we observe a plateau in the region close to 0.01 a0which indicates the harmonic region,

while at smaller and larger values numerical noise and anharmonic effects, respectively, become

visible (departure from the constant dependence). Therefore, a value of 0.01 a0was employed

for the calculations in Sec. 4.1. For the sampling of the phonon dispersion, we used likewise the

Monkhorst-Pack scheme [103] and a well converged qmesh of 163.

3.4.4.2 Details for Secs. 4.2 and 4.3.5, Temperature dependent dynamical matrix

In order to simulate the Tdependence of the dynamical matrix, the artificial electron occupation

function (Methfessel-Paxton scheme) was replaced by the physical Fermi-Dirac distribution. The

associated difficulty is that, predominantly at lower temperatures, the ksampling convergence

becomes significantly worse. We therefore investigated even larger ksamplings (up to 187 ·103

kp·atom for Al and up to 131 ·103kp·atom for Rh) than in Sec. 3.4.2. The results for the target

quantity, CP, are summarized in Fig. 3.15. For Rh, we obtain a reasonable convergence, whereas

for Al (light blue solid lines) we have a remaining scatter even though utilizing denser samplings. In

principle, the resulting accuracy for the CPcontribution in Al would be sufficient for most purposes,

since the uncertainty of ±0.1kB(±2.6% of the full heat capacity) is rather small. In Sec. 4.3, we

will however reveal that the contributions due to other excitation mechanisms are similarly small

and it would be therefore desirable to narrow the uncertainty in this contribution further.

For that purpose, we studied the influence of the Tdependent dynamical matrix using also the

linear response method (a complementary method to the direct forces constant method) as imple-

mented in the abinit code [104–107]. To utilize the abinit code we replaced the PAW description

of the core electrons by the pseudopotential description employing exactly the same pseudopoten-

tial as in Sec. 4.3. See therefore the discussion in the following subsection. We calculated phonons

(and eigenvectors) at wave vectors corresponding to a 23supercell and Fourier transformed this

mesh into its real space representation. Using Eq. (2.223), the real space representation was back

transformed onto a dense (163) mesh in reciprocal space. We find that a faster convergence rate

is obtained with the linear response method (dark blue solid lines in Fig. 3.15; note the smaller k

meshes), which allowed us to estimate slightly lower bounds for the thermodynamic properties of

Al (see Tab. 4.3). All details regarding the final calculations of the Tdependent dynamical matrix

(presented in Secs. 4.2 and 4.3.5) are compiled in Tab. 3.3.

96 3.4. Towards highly accurate DFT free energies

0500 1000 1500 2000

Tel (K)

-0.4

-0.3

-0.2

-0.1

0.1

CP (kB)

Al, LR, 47...64

Al, DFC, 55...187

Rh,DFC, 32...131aa

Figure 3.15: ksampling convergence of the CPcon-

tribution due to a Tdependent dynamical matrix for

Rh and Al at the respective melting temperature.

The representation is equivalent to Fig. 4.10a, but

referenced here with respect to the consistently cal-

culated heat capacity CP(T=Tm) for a convenient

visualization of the results for both elements. ”DFC”

denotes direct force constant method (PAW calcula-

tions) and ”LR” linear response (the latter only for

Al and utilizing pseudopotentials). The numbers in

the legend give the range of investigated ksamplings

in 103kp·atom. The thickness of the lines increases

with increasing ksampling. The vertical lines denote

the melting temperatures of the elements.

Table 3.3: Parameters for the calculations of the influence of the Tdependent dynamical matrix presented

in Sec. 4.2 (Rh) and in Sec. 4.3.5 (Al). ”pp”, ”DFC”, and ”LR” denote the pseudopotential, the direct force

constant, and the linear response method, respectively. See text and Tab. 3.2 for further details (e.g., units

of augGrid and ksampling).

code core Dmethod supercell Ecut (eV) augGrid kdensity qmesh

Rh vasp PAW DFC (0.01 a0) 23270 432 ·103131 ·103163

Al abinit pp LR b=23191 (14 Ry) — 64 ·103163

3.4.4.3 Details for Sec. 4.3, Anharmonicity and vacancies

The calculations of the full excitation spectrum of aluminum, presented in Sec. 4.3, were performed

using the norm-conserving pseudopotential formalism. We were able to construct10 accurate pseu-

dopotentials reproducing well the PAW results at a lower converged cutoff: 191 eV (14 Ry) vs.

250 eV (18 Ry). The actual quality of our pseudopotentials will be assessed in Sec. 4.3.2 using the

PAW based thermodynamic results from Sec. 4.1 as a reference. Besides speeding up the calculations

due to the lower cutoff, the pseudopotential formalism allowed us to utilize the s/phi/nx code [80],

which has various features implemented supporting anharmonicity calculations (see Secs. 2.3.5 and

2.3.6). Details regarding the pseudopotential calculations of the various free energy contributions

are listed in Tab. 3.4. In the following, we discuss briefly the most important technical aspects.

For the quasiharmonic contribution of the perfect crystal Fp,qh, we employed the linear response

method (using the abinit code and identical pseudopotentials as for s/phi/nx) as described in the

previous subsection, in order to profit from the faster ksampling convergence. This method allowed

us also to calculate frequencies on a qmesh corresponding to a 43supercell, while still employing

high ksamplings (for the electronic dispersion). Further, we used a rather high cutoff for Fp,qh,

to provide a well defined reference for the anharmonic contribution Fp,clas,ah. This turned out

to be important due to the small magnitude of the latter. For the calculation of Fp,clas,ah itself,

we employed the UP-TILD method as discussed in Sec. 3.2. The necessary Langevin dynamics

10We generated the Al pseudopotentials using the fhi98pp code [56] and the Hamann construction scheme [108].

The 3sand 3pelectrons were treated as valence electrons. Cutoff radii of 1.25 and 1.40 Bohr radius were used for

the sand dcomponents of the GGA pseudopotential. The remaining pcomponent and all the components of the

LDA pseudopotential were constructed using the default cutoff radii of fhi98pp. The dcomponent was used as a

local orbital.

3.4. Towards highly accurate DFT free energies 97

Table 3.4: The used plane wave cutoffs Ecut,kdensities, and some further information for the pseudopotential

(pp) calculations of the free energy contributions entering Eqs. (4.9) to (4.11). Note that Fp,el

0and Fv,el

have been separated according to Eq. (3.53). Example kmeshes corresponding to a one atom unit cell (k

mesh 1 atom) are also given. Further: ζ=friction parameter, dt=time step; see also text for details.

identical pp identical pp

vacancy perfect

code sc Ecut kmesh kdensity comments

(Ry) (kp·atom) 1 atom

Ep,el

g,0abinit — 14 186,624 573

Fp,el

0abinit — 14 3,014,284 1443

Fp,qh abinit 4334 46,656 363linear response

Fp,clas,ah s/phi/nx 2314 2,048 133λ=0.5, ζ=0.01, dt=10 fs

Ev,el

g,0abinit 3314 186,624 573

Fv,el

0abinit 2314 186,624 573

Fv,qh s/phi/nx 2314 6,912 193direct force const., ±0.01 a0

Fv,clas,ah s/phi/nx 2314 2,048 133λ=0.5, ζ=0.01, dt=10 fs

simulations were performed using a corresponding implementation in s/phi/nx [81]. We used a

friction parameter ζ= 0.01 and a time step dt = 10 fs for the MD simulations. The number of

MD steps was adjusted to the specific volume, temperature, and supercell size, such as to achieve a

statistical error of less than 1 meV/atom. Our largest simulations consisted of ≈10 000 MD steps.

For investigating the convergence behavior of Fp,clas,ah with respect to the plane wave cutoff and k

mesh, we applied the hierarchical UP-TILD method as discussed in Sec. 3.2.4. We performed the

convergence checks for a set of selected volumes and, in particular, high temperatures (i.e., close

to the melting point), which is important to achieve a high accuracy. The corresponding atomic

configurations represent the most distorted structures, which in turn are the most sensitive with

respect to the convergence parameters. As for the volume, at least two points have to be investigated

to obtain the influence on the thermodynamic properties. Based on these considerations, we find

that a cutoff of 14 Ry and a ksampling of 2,048 kp·atom are sufficient. It is interesting to

compare these values to the converged parameters for the quasiharmonic contribution (34 Ry and

46,656 kp·atom; see Tab. 3.4): The anharmonic contribution converges considerably faster than the

quasiharmonic one. The reason is that the anharmonic contribution is based directly on the energy,

whereas the quasiharmonic one relates to the second derivative in the potential energy surface.

We investigated the λdependence [Eq. (2.195)] in detail for the perfect bulk and vacancy

calculations by studying λvalues at extreme conditions, such as temperatures close to the melting

point, sufficiently large volumes, and λvalues close to 0 and 1. Specifically, we computed a dense set

of λvalues at 900 K, for a small and a large volume for both the perfect bulk and the vacancy cell.

The results, shown in Fig. 3.16a, reveal that the perfect bulk and the vacancy cell show nearly the

same λdependencies. Moreover, we see, even at this high temperature, only negligible deviations

from linearity for the smaller volume. For the larger volume, deviations are found for λvalues close

to 0 and 1, which partially compensate each other. The consequence of the increase in the number

of λvalues in the integration in Eq. (2.195) is shown in Fig. 3.16b. For the larger volume, the

free energy obtained from an integration based on the dense λsampling (λset2 in Fig. 3.16b) is

0.9 meV/atom higher than one based on λ= 0.5 only. Since the free energy at the smaller volume

does not change upon inclusion of more λvalues, we observe in total a small positive shift in the

98 3.4. Towards highly accurate DFT free energies

0.0 0.2 0.4 0.6 0.8 1.0

-10

〈∂Fel

λ/∂λ〉high

t,λ (meV/atom)

vac; V = 15.7 Å3

vac; V = 17.7 Å3

perf; V = 15.7 Å3

perf; V = 17.7 Å3

16.0 16.5 17.0 17.5

V (Å3/atom)

Fp,clas,ah (meV/atom)

23 sc, λ=0.5

23 sc, λ set1

23 sc, λ set2

33 sc, λ=0.5

Veq

0Veq

a) b)

Figure 3.16: Results for the convergence tests of the anharmonic contribution in aluminum. a) λdependence

of the ensemble average ∂Fel

λ/∂λhigh

t,λfor the perfect (perf) and vacancy (vac) supercell for a small and a

large volume each. The results were obtained for a 23supercell, the LDA-functional, and a temperature of

900 K. The statistical error σerr ≈0.5 meV/atom, Eq. (2.202), is of a similar size as the symbols indicating

the calculated values. The solid and dashed lines are guides to the eye. b) Explicitly anharmonic free energy

Fp,clas,ah at 900 K for the LDA functional. Results for the 23and 33supercell (sc) and for three different λ

samplings, λ= 0.5, λset1= {0.1,0.3,0.5,0.7,0.9}, and λset2=set1+{0.02,0.98}, used for the integration in

Eq. (2.195), are shown. The symbols represent the calculated values, the corresponding vertical lines show

the statistical error, and the dashed and dotted lines in-between are a guide for the eye. The vertical black

dashed lines indicate the 0 K (including zero-point vibrations) and the melting temperature equilibrium

volumes of LDA, Veq

0and Veq

Tm, respectively.

slope of the Fp,clas,ah(V) curve. This shift decreases αby −1.6% and CPby −0.6% (cf. footnote

4 on page 77). These changes are opposite to the changes (2.1% for αand 0.8% for CP) caused

by the negative shift of the slope of Fp,clas,ah(V), which occurs when the supercell size is increased

from 23to 33(see Fig. 3.16b). For the calculations in Sec. 4.3, we used the 23supercell at λ= 0.5.

The influence of the vacancies was taken into account using the volume optimized approach

(Sec. 3.1). To calculate e

Fv,el

0, we employed a 23supercell using the same volumes, temperatures,

and parametrization as for e

Fp,el

0. (Note that the details regarding the parametrization of the various

free energy contributions have been discussed in Sec. 3.3.) As for the ksampling, we investigated

three meshes: 143(87,808 kp·atom), 163(131,072 kp·atom), and 183(186,624 kp·atom). We find

that the electronic term yields negligible (cf. footnote 5 on page 78) contribution to αand CP

for all kmeshes. To calculate Fv,qh, we used the direct force constant method using a positive

and negative displacement of ±0.01 Bohr radius to obtain the Hellmann-Feynman forces. We

investigated a 23supercell with two different kmeshes, 53(= 4,000 kp·atom) and 63(= 6,912

kp·atom), and a 33supercell with a 33kmesh (= 2,916 kp·atom). We find negligible differences

between all three calculations for αand CP. Note that, as discussed in Sec. 3.3.3, the calculation of

the quasiharmonic free energy contributions, Fp,qh and Fv,qh, requires special attention to ensure a

consistent treatment of the vacancy supercell with the corresponding bulk part. To account for this,

we employed the correction scheme Eq. (3.62) for the supercell and Eq. (3.63) for the plane wave

cutoff. The anharmonic free energy of the vacancy supercell Fv,clas,ah was treated in exactly the

same manner as the perfect crystal counterpart. In particular, we ensured the applicability of the

UP-TILD procedure to the vacancy cell by calculating two free energy points at 900 K for different

volumes directly. Further, we tested the supercell size convergence by increasing the supercell from

23to 33and found also here only negligible effects.

3.4. Towards highly accurate DFT free energies 99

3.4.4.4 Details for Sec. 4.4, Quantum mechanical anharmonicity

The quantum mechanical phonon shift of aluminum (third and fourth order) was investigated for

the longitudinal phonon branch at the L point in a 23supercell. We further fixed the volume

to alat = 4.054 ˚

A (low temperature equilibrium lattice constant). The tensors were calculated as

derivatives from the T= 0 K electronic potential energy surface. For the third order shift, we

calculated the real space tensor using the finite difference method (displacement 0.01 Bohr radius)

for the Hellmann-Feynman forces (cf. Sec. 2.3.7). We implemented a search routine to determine

all symmetry equivalent pairs of displacements and reduced the actual Kohn-Sham calculations

to the irreducible part of the tensor (reduction ≈105→103). We then Fourier transformed the

real space tensor using the phonon basis as determined from the (quasi)harmonic calculations for

aluminum presented in Sec. 4.1. The phonon shift was eventually calculated from the expressions

derived in Ref. [68] using the operator renormalization method [cf. also Eq. (2.214)]. In our

calculations, we did not experience difficulties with divergences so that we could set the generic

infinitesimal parameter for determining the principle value [γin Eq. (2.219)] to 0. The fourth order

term was treated directly in reciprocal space by performing the finite difference in the phonon basis

(using energies rather than forces). The used DFT related parameters were: LDA, PAW method,

Ecut = 250 eV, 131 ·103kp·atom, Methfessel-Paxton [98] occupation with 0.1 eV. For the critical

parameter, the ksampling, we investigated three different meshes 123(= 55 ·103kp·atom), 143

(= 88 ·103kp·atom), and 163(= 131 ·103kp·atom) for the fourth order tensor, which is the more

sensitive one being a higher order derivative. We find a variation in the fourth order shift of only

0.2 meV at the melting temperature among these kmeshes (cf. Fig. 4.18).

Chapter 4

Results: Selected topics

In this chapter, the key results of the thesis are discussed:



Section 4.1 presents a systematic study of the two dominating free energy contributions,

Fel

0and Fqh,0k, for the full range of fcc metals and for an extensive set of thermodynamic

quantities. Based on the highly converged DFT results (Sec. 3.4), we are able to address the

quality of two popular xc functionals (cf. Sec. 2.1.6) in predicting thermodynamic properties.



For two representative elements, Rh and Al, we extend in Sec. 4.2 the quasiharmonic descrip-

tion to the Tdependent dynamical matrix formalism. The corresponding phonon shift, its

influence on Fqh,0k→Fqh, and on derived thermodynamic properties are investigated.



For Al, the set of physical excitation mechanisms is further extended (Sec. 4.3) by including

the classical anharmonic and vacancy contribution, which becomes feasible by employing

our new point defect formalism (Sec. 3.1) and the UP-TILD method (Sec. 3.2). With these

calculations, we are able to tackle a long standing debate about the dominating contributions

beyond the quasiharmonic approximation to the high temperature heat capacity of aluminum.



In Sec. 4.4, the anharmonic contribution in aluminum is investigated using a complementary

method: quantum mechanical perturbation theory. This approach allows to validate the

classical results of Sec. 4.3 at low temperatures.



Finally (Sec. 4.5), the highly accurate DFT results for aluminum are used as a reference to

evaluate the quality of three state-of-the-art EAM parametrizations in predicting thermody-

namic properties. We investigate the question whether the hierarchy found in the complexity

of the construction procedures of the EAM potentials correlates with their predictive power.

4.1 Assessing DFT accuracy in predicting thermodynamic prop-

erties of metals

4.1.1 Motivation

Throughout Sec. 4.1, we focus on the dominant free energy contributions which are given by

Eqs. (2.184) and (3.53):

F(V, T ) = Fel

0(V, T ) + Fqh,0k(V, T ) = Eel

g,0(V) + e

Fel

0(V, T ) + Ezp,0k(V) + T Sqh,0k(V).(4.1)

This free energy expression includes the quasiharmonic approximation, Eq. (2.133), and the D0k

approximation, Eq. (2.179). In order to classify the corresponding calculations correctly, we need

to comment on two important issues with respect to previous studies:

100

4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals 101

207

Lead

[Hg]6p

601

82Pb

201

Mercury

[Xe]4f 5d 6s

14 10 2

80Hg

204

Thalium

[Hg]6p

81Tl

112

Cadmium

[Kr]4d 5s

10 2

48Cd

115

Indium

[Cd]5p

49In

119

Tin

[Cd]5p2

50Sn

Nickel

[Ar]3d 4s

8 2

28Ni

Cobalt

[Ar]3d 4s

7 2

27Co

Silicon

[Ne]3s 3p

2 2

14Si

Zinc

[Ar]3d 4s

10 2

30Zn

Gallium

[Zn]4p

31Ga

Germanium

[Zn]4p2

32Ge

Aluminum

[Ne]3s 3p

934

13Al

197

Gold

[Xe]4f 5d 6s

1338

14 10

79Au

195

Platinum

[Xe]4f 5d 6s

2045

14 9

78Pt

192

Irdium

[Xe]4f 5d 6s

2720

14 7 2

77Ir

108

Silver

[Kr]4d 5s

1235

47Ag

106

Palladium

[Kr]4d

1827

46Pd

103

Rhodium

2236

[Kr]4d 5s

45Rh

Copper

[Ar]3d 4s

1358

29Cu

= non magnetic fcc metals

Figure 4.1: Part of the periodic table

showing in gray the investigated ele-

ments. In each field the upper left num-

ber gives the atomic mass in atomic

units. Below the electron configuration,

the atomic number and symbol, and the

element name are shown. The values

are from Ref. [113]. For the investigated

elements, the melting point in Kelvin is

additionally given (bold number in up-

per right corner; Ref. [14]).

1) The D0kapproximation is not an approximation which is only performed in the present

work. It has been implicitly assumed in previous studies, without however being explicitly

mentioned. With respect to our main goal of providing a complete ab initio description of

elementary non-magnetic metals, we have discussed the full necessary formalism (including

for instance the FEBO derivation; Sec. 2.1.3) and are, therefore, now in the position to check

the validity of approximations made in previous studies.

2) DFT calculations of Fqh,0k, in particular for elementary materials, are nowadays standard

and several studies are available (see e.g., Refs. [109–111] and further references provided in

the following sections). The key points that distinguish and motivate this study in comparison

to previous ones are:



Asystematic study of an extensive set of materials properties up to the melting point

for a large number of materials with the same crystal structure. In contrast, previous

ab initio studies mostly concentrated on one or two elements and a few properties. We

have in particular chosen all metals which are non-magnetic and crystallize in the fcc

structure (cf. Fig. 4.1). Such a systematic approach allows to investigate for instance

chemical trends. A comparable systematic study is the pioneering work of Moruzzi et

al. [112], who investigated fcc and bcc metals combining LDA and a Debye treatment of

the lattice dynamics. The essential difference, however, is that in our study we replace

the empirical Debye treatment by a fully consistent ab initio approach.



Highly converged results with respect to ksampling, plane wave cutoff, augmentation

grid, and supercell size and the assessment of the remaining error provided in Sec. 3.4.

As discussed further in Sec. 3.4, these DFT parameters become critical when considering

the quasiharmonic approximation particularly at high temperatures.



The study of all elements and properties using the two complementary xc functionals:

LDA and GGA-PBE. In combination with the well controlled numerical convergence,

this approach allows to analyze the performance of these functionals.

The calculations presented in Sec. 4.1 were performed using the PAW method with the quality of

the PAW potentials assessed in Secs. 4.1.2 and 4.1.4. The used DFT convergence parameters are

compiled in Sec. 3.4.4.1 (in particular Tab. 3.2).

102 4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals

4.1.2 Electronic free energy and zero-point vibrations at T= 0 K

At T=0 K, two free energy contributions are present: Eel

g,0(V) and Ezp,0k(V). The dominant term

is the electronic ground state energy Eel

g,0(V), which is thus discussed first. We calculated Eel

g,0(V)

for all elements and fitted the Vinet equation of state, Eq. (3.54), to the results. Parameters

for the equilibrium lattice constant, bulk modulus, and its derivative are listed in Tab. 4.1. The

additional superscript ”e” emphasizes the exclusion of zero-point vibrations, Ezp,0k(V), at this

stage. Table 4.1 contains also the results of our LAPW+lo reference calculations (cf. Sec. 3.4.4)

for Eel

g,0(V). In general, a remarkably good agreement between the PAW and LAPW+lo results is

found. For the transition elements, the PAW lattice constant (bulk modulus) is consistently 0.1%

(−2.2%) larger (smaller) than the corresponding LAPW+lo prediction. This is true for LDA and

GGA. For Al and Pb, we find for both functionals a slightly smaller (larger) PAW lattice constant

(bulk modulus). These results indicate that the approximations intrinsically connected to the used

PAW potentials (cf. Sec. 2.1.8), i.e.,



finite number of basis and projector functions,



frozen core approximation,



scalar relativistic effects only implicitly included,

are justified and that the PAW potentials can be employed for the further calculations of ther-

modynamic properties. It would be however desirable to find out the explicit influence of these

differences between PAW and LAPW+lo on thermodynamic quantities. Corresponding tests are

presented in Sec. 4.1.4.

For simplicity, experimental data being extrapolated to T= 0 K are frequently directly com-

pared with ab initio Eel

g,0(V) values without taking zero-point vibrations into account. However,

the experimental data correspond to the free energy F(V, T =0 K) and thus contain a contribution

due to zero-point vibrations. To estimate the error related to their neglect, we fitted in a second

step F(V, T =0 K) = Eel

g,0(V) + Ezp,0k(V) to the Vinet equation of state. The results indexed with

a superscript ”i” are also given in Tab. 4.1. They show that the inclusion of zero-point vibrations

has only a minor effect on the lattice constant. The average increase is ≈0.1%. In contrast, for

the bulk modulus the effect is significantly stronger (reduction of ≈2%). In particular, the Al

GGA-bulk modulus softens considerably by about 4%. The change in the equilibrium properties

can be directly understood by an analysis of Ezp,0k(V). Note first that the value of Ezp,0kat alat,

which we find to be in average 25 meV/atom, affects only the absolute free energy. To account

for the change in the lattice constant, the linear term in the volume dependence of Ezp,0kmust

be included. For the considered elements, the magnitude of the phonon frequencies decreases with

volume (cf. Fig. 4.9) which in turn yields a decreasing zero point energy. The consequence is a

modified energy-volume dependence with an equilibrium volume shifted to larger lattice constants.

The bulk modulus is indirectly affected by the linear term in Ezp,0ksince it softens with increasing

volume. Inclusion of the second order term has also an explicit effect on BT,0, since the latter is a

second order derivative. For all considered elements, both effects, the implicit and explicit, lead to

a softening of BT,0.

An additional effect included in the experimental data are ambient pressure conditions (100 kPa),

whereas theoretically zero pressure conditions are more straightforward to assess. To test the im-

portance of pressure we performed corresponding calculations. The results show that the effect of

pressure on the lattice constant and bulk modulus is systematically more than an order of magni-

tude smaller than the changes due to zero-point vibrations. They show further that the ambient

4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals 103

pressure conditions are also irrelevant for the study of materials properties at higher temperatures.

We therefore concentrate here and in the remainder of Chap. 4 only on values obtained at 0 Pa.

The comparison between experiment and theory shows that LDA/GGA under-/overestimates

the experimental value of the lattice constant alat,eq

0and over(under)estimates the value of the bulk

modulus BT,0. This behavior is a well established trend observed in numerous previous studies. For

the set of metals investigated in this study, we find that LDA underestimates alat,eq

0on average by

−0.7% and GGA overestimates it on average by 1.8%. The error in BT,0, being a second derivative

of F(V, T ), is much larger in magnitude (LDA-average: 11.6%, GGA-average: −13.7%) and is

inversely correlated to the error in alat,eq

0(Fig. 4.2). The inverse relation can be explained by the

volume dependence of the total energy [Eq. (3.54)] causing a monotonous decrease of BT,0with

increasing volume. Particularly for GGA, an increase of the error with the number of d-electrons

among the 4dand 5dtransition metals is apparent.

4.1.3 Phonon dispersion

In order to allow an accurate and consistent comparison between experimental and theoretical

phonon dispersions, the latter have to be computed at the experimental temperature. The temper-

ature enters the calculation of ωq,s at three distinct points depending on the considered approxi-

mation:

1) Using the quasiharmonic approximation, Eq. (2.133), and the D0kapproximation, Eq. (2.179),

the phonon frequencies ωq,s are volume dependent, but not explicitly temperature dependent.

For the zero pressure condition as considered here, they will be a function of the temperature

dependent equilibrium volume Veq(T), Eq. (2.231), and thus have an implicit temperature

dependence:

ωq,s =ωq,s(Veq(T)).(4.2)

2) As discussed in Sec. 3.3.4, going beyond the D0kapproximation, the phonon frequencies

become dependent on the electronic temperature Tel due to the dependence of the dynamical

matrix Don Tel. We therefore have an explicit temperature dependence:

ωq,s =ωq,s(Tel).(4.3)

3) Finally, as discussed in Sec. 2.3.5, going beyond the quasiharmonic approximation leads to

interaction among phonons (their creation and annihilation), which also causes a temper-

ature dependent frequency shift. In particular, performing molecular/Langevin dynamics

(Secs. 2.3.6 and 3.3.5) the frequency shift will explicitly depend on the electronic temperature

Tel and the nuclei temperature Tnuc which both determine the nuclei motion [cf. Eqs. (3.77)

to (3.79)]:

ωq,s =ωq,s(Tel, T nuc).(4.4)

[Note that at the end of a calculation Tel =Tnuc =Tmust be set, cf. Eq. (3.68) and the

following discussion.]

The most dominant contribution to the temperature dependence of ωq,s is given by Eq. (4.2). This

temperature dependence is fully included in the phonon dispersion results presented in this section.

The dependencies given by Eqs. (4.3) and (4.4) will be considered in Secs. 4.2 and 4.4, respectively,

for a smaller set of elements. Our results based on Eq. (4.2) for both xc functionals, LDA and

GGA, are shown in Fig. 4.3a.

104 4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals

-2 -1 0 1 2 3

∆ai

0 (%)

-30

-20

-10

∆Bi

T,0 (%)

Rh Ir

Pt Pd

LDA

GGA

Figure 4.2: Correlation between the deviation from

experiment for the lattice constants ∆ai

0and the

bulk moduli ∆Bi

T,0(Tab. 4.1). All quantities contain

the influence of zero-point vibrations. The dashed

lines are a guide for the eye and emphasize the lin-

ear trend for each functional.

Table 4.1: Results for the equilibrium lattice constant alat,eq

0, bulk modulus BT,0, and its derivative B′

T,0

(cf. Sec. 2.4.2) for the projector-augmented wave (PAW) and the LAPW+lo method. For aluminum, the

pseudopotential (pp) results (cf. Sec. 4.3) and the EAM results (cf. Sec. 4.5) are also included. The EAM

parametrizations are Mei-Davenport (MD) [62], Zope-Mishin (ZM) [63], and Ercolessi-Adams (EA) [64].

The superscript ”e” (”i”) indicates whether zero-point vibrations are excluded (included). The deviation

from experiment is labeled ∆ai

0and ∆Bi

T,0. LAPW+lo values marked with an asterisk are estimated val-

ues obtained using the (relative) zero-point contributions from the PAW results. All experimental lattice

constants are taken from Ref. [14] (T= 5 K). The experimental bulk moduli are taken from Ref. [68] (low

temperature values) for Al, Pb, Cu, Ag, Au and from Ref. [114] (Pd, T→0 K), Ref. [115] (Rh, T→0 K),

and Ref. [116] (Pt, Ir, T→0 K). The available experimental B′

T,0are taken from Ref. [117] (collection of

experimental T→0 K data).

LDA GGA EAM

LAPW PAW pp LAPW PAW pp ZM MD EA

alat,eq,e

0(˚

A) 3.986 3.985 3.968 4.043 4.041 4.050 4.050 4.049 4.032

alat,eq,i

0(˚

A) 3.999∗3.998 3.981 4.032 4.056∗4.054 4.063 4.059 4.063 4.042

∆ai

0(%) -0.8∗-0.8 -1.3 0.6∗0.6 0.8 0.7 0.8 0.2

T,0(GPa) 83 84 82 77 78 75 79 76 82

T,0(GPa) 79∗80 78 79 74∗75 72 77 73 80

∆Bi

T,0(%) 0.0∗1.3 -1.2 -6.3∗-5.0 -8.9 -2.5 -7.6 1.3

B′

T,04.5 4.6 4.6 4.7 4.7 4.7 4.6 4.3 4.1 4.8

Exp.

Pb Cu

LDA GGA LDA GGA

LAPW PAW LAPW PAW LAPW PAW LAPW PAW

alat,eq,e

0(˚

A) 4.881 4.875 5.047 5.030 3.522 3.524 3.638 3.637

alat,eq,i

0(˚

A) 4.885∗4.879 4.905 5.052∗5.034 3.529∗3.530 3.602 3.645∗3.644

∆ai

0(%) -0.4∗-0.5 3.0∗2.6 -2.0∗-2.0 1.2∗1.2

T,0(GPa) 52 53 39 40 188 183 141 136

T,0(GPa) 51∗52 49 39∗40 183∗178 142 137∗132

∆Bi

T,0(%) 13.2∗15.6 -13.5∗-11.1 28.5∗25.4 -3.4∗-7.0

B′

T,04.7 5.0 5.5 4.2 5.6 5.1 5.1 5.3 5.1 5.1

Exp. Exp.

Continued.

4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals 105

Table 4.1 (continued).

Ag Au

LDA GGA LDA GGA

LAPW PAW LAPW PAW LAPW PAW LAPW PAW

alat,eq,e

0(˚

A) 4.008 4.016 4.152 4.165 4.054 4.062 4.163 4.174

alat,eq,i

0(˚

A) 4.014∗4.022 4.061 4.159∗4.172 4.057∗4.066 4.065 4.167∗4.179

∆ai

0(%) -1.2∗-1.0 2.4∗2.7 -0.2∗0.0 2.5∗2.8

T,0(GPa) 140 136 91 88 193 189 139 134

T,0(GPa) 137∗133 109 89∗86 190∗186 180 136∗131

∆Bi

T,0(%) 26.8∗23.1 -17.6∗-20.4 5.1∗2.8 -25.0∗-27.6

B′

T,05.8 5.7 5.9 5.8 5.9 5.9 5.8 5.9 6.0 6.0

Exp. Exp.

Rh Pd

LDA GGA LDA GGA

LAPW PAW LAPW PAW LAPW PAW LAPW PAW

alat,eq,e

0(˚

A) 3.762 3.767 3.839 3.842 3.847 3.854 3.949 3.954

alat,eq,i

0(˚

A) 3.766∗3.770 3.798 3.844∗3.847 3.851∗3.858 3.879 3.954∗3.959

∆ai

0(%) -0.8∗-0.7 1.2∗1.3 -0.7∗-0.5 1.9∗2.1

T,0(GPa) 321 313 260 252 230 223 169 165

T,0(GPa) 316∗308 269 255∗248 226∗219 195 166∗162

∆Bi

T,0(%) 12.8∗10.0 -8.8∗-11.4 20.2∗16.5 -11.5∗-13.8

B′

T,05.3 5.2 5.4 5.3 5.7 5.5 5.7 5.7

Exp. Exp.

Ir Pt

LDA GGA LDA GGA

LAPW PAW LAPW PAW LAPW PAW LAPW PAW

alat,eq,e

0(˚

A) 3.819 3.819 3.879 3.877 3.900 3.906 3.976 3.977

alat,eq,i

0(˚

A) 3.822∗3.821 3.835 3.882∗3.880 3.903∗3.909 3.909 3.978∗3.980

∆ai

0(%) -0.3∗-0.3 1.2∗1.2 -0.2∗0.0 1.8∗1.8

T,0(GPa) 406 401 349 342 308 301 246 245

T,0(GPa) 402∗397 366 345∗338 304∗297 288 243∗242

∆Bi

T,0(%) 8.7∗7.3 -6.8∗-8.6 4.7∗2.4 -16.3∗-16.6

B′

T,05.2 5.1 5.2 5.2 5.5 5.5 5.4 5.6

Exp. Exp.

106 4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals

123

123 1234

1234

4.000 Å

4.055 Å

4.884 Å

5.043 Å 3.541 Å

3.658 Å

Eω (meV)

3.777 Å

3.854 Å

3.867 Å

3.971 Å 4.036 Å

4.195 Å

ΓX K ΓL

3.827 Å

3.886 Å

ΓX K ΓL

3.910 Å

3.982 Å

ΓX K ΓL

4.081 Å

4.201 Å

ν (THz)

LDA

GGA

Exp.

Cu@TRT

Au@TRT

Ag@TRT

Pd@TRT

Rh@TRT

Pt@90 K

Ir@TRT

Al@80 K Pb@100 K

4.033 Å

4.914 Å 3.614 Å

Eω (meV)

3.803 Å

3.888 Å 4.078 Å

ΓX K ΓL

3.840 Å

ΓX K ΓL

3.910 Å

ΓX K ΓL

4.078 Å

ν (THz)

LDA

GGA

Exp.

Cu@TRT

Au@TRT

Ag@TRT

Pd@TRT

Rh@TRT

Pt@90 K

Ir@TRT

Al@80 K Pb@100 K

Figure 4.3: Phonon dispersions ωq,s [Eω=hν =hω/(2π) with hthe Planck constant] along high symmetry

directions. The insets in the Pd and Pt phonon dispersions magnify the Kohn anomalies (see text; the

dashed line is a guide for the eye). A 53supercell was used for Pb and a 43supercell for all other elements

(see Sec. 3.4 for further details). The experimental values are taken from Ref. [118] for Rh, from Ref. [119]

for Ir, and from Ref. [101] for all other elements. Al and Pb are slightly separated to emphasize the

periodic table correspondence of the other elements and the dispersion of Pb is rescaled to visualize it

properly. a) Calculated at the DFT obtained equilibrium lattice constant (given in the legends) for the given

temperature (TRT =room temperature). b) Calculated at the experimental lattice constant (see Sec. 4.1.8).

The gray shading is included to allow a convenient distinction to a).

4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals 107

Generally, both functionals show a good agreement with experimental data. LDA, yielding

larger phonon frequencies than GGA, overestimates the experimental data in most cases, while

GGA underestimates it. A systematic trend we observe is that one can estimate from the compar-

ison of different xc functionals approximate error bars for the theoretical calculations with respect

to experimental data. Small deviations between LDA and GGA, as e.g. for Al, thus indicate small

error bars and hence a high predictive power, while larger deviations as for Ag or Au indicate lower

prediction accuracy.

The phonon dispersion of Cu, Ag, and Au is comparatively ”simple” and can be described

accurately already with nearest neighbor interactions. The dispersion of these elements has been

the subject of previous theoretical studies (Cu [120–123], Ag [124], and Au [125, 126]). For Ag and

Au, however, only the LDA formalism has been used in these investigations. The available data

from the literature is in good agreement with our results. The remaining elements exhibit more

complex phonon dispersions. Anomalies, i.e., deviations from the ”simple” dependence, are caused

by interactions of the phonons with the electronic Fermi surface. To resolve these anomalies in

the phonon dispersion, force constants including long range inter-atomic interactions need to be

captured, i.e., large supercells as used in the present study are crucial. As pointed out in Ref. [118],

the dispersion relations of these elements ”constitute a severe test for any theoretical treatment”.

We first focus on Pd and Pt. The anomalies in the vicinity of the Γ point (enlarged in the

insets of Fig. 4.3a) have been attributed to virtual Kohn transitions [101]. Figure 4.3a shows that

these Kohn anomalies are strong for the GGA calculated dispersions, whereas the LDA dispersions

show almost no deviation from linearity. In order to identify why LDA behaves differently, we

repeated the calculations of the phonon dispersion at the same (experimental) lattice constant for

LDA and GGA (cf. Fig. 4.3b). For Pt, both functionals yield a comparable anomaly under these

conditions. We therefore conclude that, for Pt, the xc functional dependence is mainly related to an

effect of the atomic structure on the Fermi surface rather than due to changes in the Fermi surface

caused by LDA/GGA. For Pd however, this argument does not hold, since calculating the phonon

dispersion at the same lattice constant yields a stronger pronounced anomaly for GGA than for

LDA. Previous theoretical studies on Pd [120, 127] based on the linear response method have not

reported the anomaly. However, in a recent theoretical study [128] on the electron-phonon coupling

in Pd, the anomaly has been resolved and correlated to a distinct peak in the phonon line widths.

The phonon dispersion of Rh and Ir shows anomalies in almost all branches. These pronounced

anomalies originate from sharp peak-like features (caused by the d-states) in the electronic Fermi

surface. Both elements have been the subject of only a few experimental and theoretical (LDA)

investigations [118, 119, 129]. Our LDA phonon dispersions agree well with the LDA results from

Refs. [129] and [119], where a sophisticated supercell approach has been employed. In particular,

the approach was based on calculations for various supercells (also non-cubic), which were then

combined using an interpolation scheme to generate a phonon dispersion including the information

from all supercells. This procedure allowed to include also long-ranged force constants.

The phonon dispersion of Al has been in the focus of numerous studies [109, 120, 121, 126, 130].

It is, at first glance, similar to the phonon dispersion of Cu, Ag, and Au. However, a detailed

inspection reveals anomalies for most branches. This is consistent with theoretical findings in

Ref. [68]. Our results for the LDA and GGA phonon dispersion of Al in Fig. 4.3 accurately

reproduce these anomalies. It is noteworthy that a 43supercell is the minimum cell size to resolve

these anomalies.

Pb exhibits the most complex phonon dispersion among the investigated elements. This ”com-

plexity” became already apparent in Sec. 3.4.3 when inspecting the convergence with respect to the

supercell size. The theoretical dispersions computed for a large 53supercell (500 atoms) reproduce

108 4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals

0 300 600 900

DFT

mix

ε (%)

0 300 600 900 1200 0

0500 T (K)

1000 1500 20000 600 1200 1800 2400

0 200 400 6000

GGA

LDA

Exp.

Al Pb

Figure 4.4: Linear thermal expansion ε, Eq. (2.234), as a function of temperature T. The solid lines

indicate results obtained fully from DFT (LDA/GGA). The dashed lines correspond to the mixed approach

as discussed in Sec. 4.1.8. The gray shaded arrows show the direction of increasing error among the transition

elements (see text). The melting temperature is indicated for each element (cf. Fig. 4.1) by the vertical

dashed line. The experimental data are taken from Ref. [14].

reasonably the experiment. The LDA results for Pb agree well with a previous LDA linear response

study [100], which naturally includes long range force constants. A study performed subsequent to

our investigations [131] could even further improve the description of the Pb phonon dispersion by

including spin-orbit coupling effects. It would be interesting to investigate if spin-orbit coupling

also affects the thermodynamic properties of Pb. (In Ref. [131] only the phonon dispersion has been

calculated due to the enormous computational requirements associated with spin-orbit coupling.)

In fact, we will return to this issue in Sec. 4.1.8 as a possible explanation for the observed behavior

of the thermodynamic properties of Pb.

4.1.4 Thermal expansion

The results for the thermal expansion εand the expansion coefficient αare shown in Figs. 4.4 (solid

lines) and 4.5a. (Note that the lower parts of Figs. 4.5 to 4.8 will only be discussed in Sec. 4.1.8.)

Let us first return to the evaluation of the PAW potentials as discussed in Sec. 4.1.2. We focus on

the expansion coefficient, since it is the most sensitive thermodynamic quantity. To estimate the

PAW parametrization error, the following procedure is applied: In Eq. (4.1), the LAPW+lo results

(Tab. 4.1) are used for the description of the T= 0 K energy surface Eel

g,0(V), while the remaining

contributions are obtained from PAW. The resulting expansion coefficient for GGA is shown in

Fig. 4.5a (dot-dashed lines) and should be compared with the orange solid line. We find for all

elements only a small effect strongly supporting the argument that the employed PAW potentials

4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals 109

0 300 600 900

pp, GGA (Al only)

α (10-5K-1)

0 200 400 6000

GGA

LDA

Exp.

0 300 600 900 1200 0

0500 T (K)

1000 1500 2000

α (10-5K-1)

LAPW-Eel

g,0, GGA

αlin, LDA

0 600 1200 1800 2400

Al Pb Cu

0 300 600 900

α (10-5K-1)

0 300 600 900 1200 0

0500 T (K)

1000 1500 20000 600 1200 1800 2400

0 200 400 6000

GGA

LDA

Exp.

Al Pb

Figure 4.5: Thermal expansion coefficient α, Eq. (2.235). The melting temperature is indicated for each

element (cf. Fig. 4.1) by the vertical dashed line. The experimental data are taken from Ref. [14]. a) The

LDA and GGA α(solid lines) are calculated fully from DFT. Additionally, αlin the expansion coefficient

based on the linear approximation Eq. (3.58), the expansion coefficient calculated from a free energy surface

with Eel

g,0(V) from LAPW+lo (see text), and the pseudopotential (pp) result for the Al α(not discussed

until Sec. 4.3.2) are shown (dashed, dot-dashed, and dotted lines). b) The LDA and GGA αare calculated

using the mixed approach as discussed in Sec. 4.1.8. The gray shading is included to allow a convenient

distinction to a). This applies likewise to Figs. 4.6b to 4.8b in the following.

110 4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals

are indeed reliable for thermodynamic calculations.

We now turn to the comparison of our DFT results for εand αwith experiment. In performing

this comparison, one should keep in mind that both are differential quantities, which might mask

errors in the absolute values of the lattice constants (listed in Tab. 4.1). Figures 4.4 and 4.5a reveal

on average a better agreement for the LDA than for the GGA results with GGA predicting a too

large expansion. The deviation between the GGA results and experiment shows a chemical trend

among the transition metals which is indicated in Fig. 4.4 by the gray arrows. Filling up the d-shell

and increasing the atomic radius enlarges the error. The two most prominent examples are Ag and

Au, where in particular the GGA αshows a significant deviation from experiment.

In general, we find for both, εand α, that most experimental data lie in between the LDA and

GGA results. The probability for this to occur is however not as high as found for the phonon

dispersion which is most apparent for α(Fig. 4.5). For instance, while the experimental phonon

dispersion of Pt is enclosed by the LDA and GGA results, the expansion coefficient of Pt fulfills

this only up to a certain temperature. Above this temperature both the LDA αand the GGA α

increase much faster than the experimental data. We attribute this to the fact that the expansion

coefficient is highly sensitive with respect to the free energy surface and that small changes in

F(V, T ) significantly affect αin particular at high temperatures. To support this we have included

in Fig. 4.5, the LDA expansion coefficient αlin which was calculated from a free energy surface based

on the linear approximation Eq. (3.58) (dashed lines). The curves for αlin are to be compared with

the fully converged LDA α(blue solid lines) based on the interpolation Eq. (3.59), which includes

additionally higher order terms as compared to Eq. (3.58). The corresponding change in F(V, T ) is

below 1 meV/atom. To see this, compare the dashed line with the zero line in the inset in Fig. 3.4.

This free energy is for Al, the situation however does not change significantly for the other elements.

In contrast, the change when going from αlin to the converged αis very strong in particular at

high temperatures. In fact, the difference between αlin and αshows the same chemical trend as the

deviation from experiment and is thus again most pronounced for Au. The discrepancy between

the fully converged DFT results and experimental data will be investigated further in Sec. 4.1.8,

where we present an approach allowing to detect which free energy contribution actually causes

these large deviations.

Previous ab initio studies on the thermal expansion of metals are rare. For the elements

investigated here, we have found LDA studies for Al [109, 110], Cu [122], and Ag [124]. In Ref. [122],

a combined LDA and GGA study of thermodynamic properties for Cu has been performed. These

data are in good agreement with our corresponding LDA/GGA data. However, the influence of

the second order term on αhas been neglected. For instance in Ref. [109], the presented expansion

coefficient shows only a linear behavior at temperatures above 300 K and not the steeper increase

as we observe upon inclusion of the second order term.

4.1.5 Heat capacity

The results for the heat capacity at constant pressure CPand constant volume CVare shown in

Fig. 4.6a in comparison with experimental data. We focus first on CPwhich is the experimentally

measured quantity. Similar conclusions as in the previous section for the thermal expansion can be

drawn: LDA yields an (astonishingly) good agreement with experiment and the GGA error shows

the same chemical trend as in the case of the expansion coefficient. Hence, the largest deviations

are again found for the noble metals Ag and Au. We note that our heat capacity results compare

well with available previous calculations (Cu [122], Ag [124]).

Our approach allows to directly determine the electronic contributions, Cel

P, to the heat capacity

(dot-dashed lines in Fig. 4.6a). For Al, Pb, Cu, Ag, and Au the electrons make only a small

4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals 111

0 300 600 900

CV0,GGA

CP (kB)

600 1200 0

0 100 200 300

T (K)

0 100 200 300

1000 20001000 2000 0 100 200 300

050 100 150

CV, LDA

Cel

P, LDA

300 600

GGA

LDA

Exp.

600 800

3.6

4.0

Al Pb

(Al only)

0 300 600 900

CP (kB)

600 1200 0

1000 20001000 2000

0 100 200 300

T (K)

CP (kB)

0 100 200 300

300 600

GGA

LDA

Exp.

050 100 150

Al Pb Cu

Figure 4.6: Isobaric heat capacity CP, Eq. (2.236), at P= 0 Pa as a function of temperature Tin units of

the Boltzmann constant kB. The temperature axis is split by the vertical dotted line into two parts to allow

a convenient representation. The melting temperature is indicated for each element (cf. Fig. 4.1) by the

vertical dashed line. The experimental data are taken from Ref. [132]. a) The LDA and GGA CP(solid

lines) are calculated fully from DFT. Additionally, the constant volume heat capacity CV, Eq. (2.236), for

LDA (dashed lines), the electronic contribution, Cel

P, to the LDA heat capacity (dot-dashed lines), and the

GGA fixed volume heat capacity CV0[Eq. (2.241)] for Al discussed in Sec. 4.3.1 (thin solid line) are shown.

The inset for Al is a zoom into the high temperature region visualizing the experimental scatter which will

be also discussed in Sec. 4.3.1. The additional experimental references are given in Fig. 4.17. b) The LDA

and GGA CPare calculated using the mixed approach as discussed in Sec. 4.1.8.

112 4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals

contribution to the heat capacity over the full temperature range up to the melting point. For

instance at 900 K, the contribution for Al, Cu, Ag, and Au is ∼0.1kBand ∼0.2kBfor Pb. For

Pb, however, already at 600 K the melting point is reached. The contribution at 600 K is ∼0.1kB.

This order of magnitude agrees for instance with findings in Ref. [122] (for Cu) and Ref. [124] (for

Ag). In contrast to these elements, Cel

Pbecomes significant for Pd, Pt, Rh (∼0.4kBat 900K), and

Ir (∼0.3kBat 900 K). For Pd, Pt, Rh, and Ir, the origin of the significant electronic contribution

are large electronic entropy effects due to a high electronic density of states near the Fermi level

caused by partially occupied dstates (dstates peak). Results for the electronic density of states of

Pd, Pt, Rh, and Ir can be found for instance in Ref. [133]. Particularly interesting in this respect

is a comparison of the temperature dependence of Cel

P(dot-dashed lines in Fig. 4.6a). For Pd, Cel

increases up to ≈0.5kBat 1000 K and decreases above this temperature until it reaches ≈0.4kBat

the melting temperature (1827 K). This indicates that the dstates peak has been filled at least up

to its half and that excitations of further electrons become energetically unfavorable. For Pt, we

find a linearly increasing Cel

Pup to its melting point. For Ir and in particular for Rh, the electronic

contribution increases even stronger than linearly up to the melting point. This indicates that half

filling of the dstates peak is not reached before melting for Rh and Ir.

Another noteworthy feature is observed for Al, for which the DFT calculated CPfor both xc

functionals agrees excellently with experiment up to ≈70% of the melting temperature. For higher

temperatures, the experimental heat capacity increases stronger than the theoretical prediction.

The possible origins of these deviations will be investigated in detail in Sec. 4.3. In focusing on

this temperature region, we will also have to discuss in more detail the up to now problematic

experimental/theoretical situation.

Let us turn to the heat capacity at constant volume CV. It falls together with CPin the low

temperature region (up to ≈300 K), whereas it significantly deviates from the latter at higher

temperatures. The reason for the faster increase of CPis directly correlated to the strong increase

of the expansion coefficient at high temperatures (Fig. 4.5a): CPis obtained from the derivative of

the entropy with respect to temperature, Eq. (2.229). The important point is that this derivative

needs to be calculated at constant pressure, i.e., in the direction of the thermal expansion on the

entropy surface S(V, T ). Thus, the behavior of the thermal expansion/expansion coefficient will

directly affect the behavior of CPand easily enable values of above 4kBfor this quantity. In

contrast, the derivative needed to calculate CVis obtained from S(V, T) in the direction of the

temperature axis and the maximum contribution that can be obtained from the quasiharmonic

entropy surface is only 3kB(Dulong-Petit rule). The reason why also CVcan go beyond the

classical limit of 3kB, particularly visible for Rh and Ir, is the contribution of the electrons. Their

influence modifies the curvature of the entropy surface and thus also CV. Note however that we

have in general 3kB+Cel

P6=CV. This inequality can be again explained by considering the direction

of differentiation: Cel

Pcorresponds to the electronic contribution obtained from a derivative in the

direction of the thermal expansion. The electronic contribution contained in CVcorresponds to a

derivative in the direction of the temperature axis.

4.1.6 Bulk modulus

In experiment, the adiabatic bulk modulus BSis typically measured, while in theory the isothermal

bulk modulus BT[cf. Eqs. (2.229) and (2.237)] is more straightforward to assess. The two bulk

moduli coincide however only at T= 0 K, whereas at higher temperatures BSand BTdeviate

strongly as exemplified for the GGA bulk modulus of Al in Fig. 4.7a. To allow for an unbiased

comparison, we therefore computed the full temperature dependence of BSfrom Eq. (2.238) for all

elements. In Fig. 4.7a, we display however only the temperature interval up to 300 K (except for

4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals 113

Pt), due to the missing experimental data for comparison at higher temperatures.

In comparing with the available experimental data, we find a similar behavior as observed

already for the thermal expansion and the heat capacity. Compare for instance BSand αalong

the column Cu, Ag, Au: For Cu, GGA yields a better BSand a (slightly) better α. For Ag, LDA

yields a good description of both, BSand α, while GGA starts deviating. For Au finally, GGA

deviates strongly, whereas LDA clearly shows a better agreement with experiment. It is important

to point out that in order to achieve such a consistent comparison between theory and experiment

for ε,α,CP, and BS, the inclusion of the second order term in the parametrization of Fqh,0k(cf.

the corresponding discussion for αlin in Sec. 4.1.4) when calculating BSis crucial. To show this,

Fig. 4.7 contains also Blin

S(dashed lines), i.e., the LDA adiabatic bulk modulus resulting from a

calculation based on the approximation Eq. (3.58). The curves for Blin

Sdeviate strongly from the

fully converged LDA BSresulting from Eq. (3.59) (blue solid lines) and they would not allow to

derive a reasonable description of the bulk modulus.

4.1.7 Free energy

We finally come to the free energy at constant (zero) pressure FP. The results are shown in Fig. 4.8a

in comparison with values obtained from the calphad approach [7]. For the elementary materials

considered here, calphad interpolates calorimetrically measured free energies as a function of

temperature. In order to compare our DFT free energies Fdft

P(T) to the calphad free energies

Fcal

P(T), we shift both using:

Fdft

P(T) = Fdft

P(T)−Fdft

P(Tref),(4.5)

Fcal

P(T) = Fcal

P(T)−Fcal

P(Tref).(4.6)

We follow the calphad approach and chose a finite temperature as the reference (Tref = 200 K).

The reason to use a temperature different from 0 K is the lack of accurate experimental data at

low temperatures, which causes the calphad free energies to diverge in this region. To allow for

a convenient comparison, we included also the difference free energy ∆FP,

∆FP(T) = e

Fcal

P(T)−e

Fdft

P(T),(4.7)

for both xc functionals in Fig. 4.8a.

The results for the free energy follow closely the trends found for the thermal expansion, the

heat capacity, and the bulk modulus. In general, the LDA results are in good agreement with

the calphad data, whereas the GGA error exhibits the same chemical trend among the transition

metals: Filling up the dshell and increasing the atomic radius, the GGA deviation from experiment

increases and cumulates again in significant errors for Pt, Ag, and Au. In fact, for Au we find that

the contribution from the second order term in Eq. (3.59) (cf. the discussion for αlin and Blin

Sin

the preceding sections) is so strong that the (GGA) free energy starts to diverge at ≈1000 K. This

is the reason why we have cut the corresponding curve at this temperature. The divergence occurs

consistently also for the thermodynamic quantities discussed so far, it however did not become

apparent yet due to the chosen scale in Figs. 4.4 to 4.7.

At this stage, we want to highlight the consistent agreement between the DFT results and

experiment (i.e., the same qualitative and quantitative trends) observed for all thermodynamic

quantities: ε,α,CP,BS, and FP. This is remarkable, since in experiment – in contrast to the

theoretical description where all quantities are rigorously connected due to their derivation from

the free energy surface – largely different setups and techniques are employed. To be more specific,

114 4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals

0 100 200 300

0.92

0.94

0.96

0.98

1.00

GGA

LDA

Exp.

0.96

0.98

1.00

0.96

0.98

1.00

Blin

S , LDA 0.96

0.98

1.00

BS/BS(T=0K)

0 100 200 300

0.92

0.94

0.96

0.98

1.00

0 100 200 300

0.96

0.98

1.00

200 300

T (K)

0.96

0.98

1.00

BS/BS(T=0K)

0 100 200 300

0.96

0.98

1.00

Blin

S , GGA

BT, GGA

600 1200

0.85

0.90

0.95

Al Pb Cu

0 100 200 300

0.96

0.98

1.00

0.96

0.98

1.00

0.96

0.98

1.00

0.96

0.98

1.00

BS/BS(T=0K)

0 100 200 300

0.92

0.94

0.96

0.98

1.00

0 100 200 300

0.96

0.98

1.00

200 300

T (K)

0.96

0.98

1.00

BS/BS(T=0K)

0 100 200 300

0.92

0.94

0.96

0.98

1.00

GGA

LDA

Exp.

600 1200

0.85

0.90

0.95

Al Pb Cu

Figure 4.7: Relative adiabatic bulk modulus BS, Eq. (2.237), at P= 0 Pa as a function of temperature T

up to 300 K. The bulk modulus is scaled with respect to its T= 0 K value as given in Tab. 4.1. The inset

shows the high temperature region for Pt. The experimental data are taken from Ref. [15] (Al), Ref. [134]

(Pb), Ref. [135] (Cu), Ref. [136] (Ag, Au), Ref. [137] (Pd), and Ref. [138] (Pt). a) The LDA and GGA

BS(solid lines) are calculated fully from DFT. Additionally, Blin

Sthe adiabatic bulk modulus based on the

linear approximation Eq. (3.58) for LDA (all elements; blue dashed lines) and GGA (Al only; orange dashed

line) and the isothermal bulk modulus BTfor GGA Al (orange dot-dashed line) are shown. b) The LDA

and GGA BSare calculated using the mixed approach as discussed in Sec. 4.1.8.

4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals 115

300 600 900 -80

-40

-80

-40

∆FP (meV/atom)

200 300 400 500 600

-80

-40

300 600 900 1200

-80

-40

500 1000 1500 2000

T (K)

600 1200 1800 2400

-0.3

-0.2

-0.1

200 300 400 500 600 -1.6

-1.2

-0.8

-0.4

GGA

LDA

CALPHAD

300 600 900 1200

-1.6

-1.2

-0.8

-0.4

0.0

FP (eV/atom)

600 1200 1800 2400

-1.6

-1.2

-0.8

-0.4

0.0

500 1500 2000

Al Pb Cu

Ecut

2, 2 meV

8, 3meV 5, 2 meV

4, 3 meV 1, 0 meV5, 2 meV

1, 2 meV

1, 1 meV 2, 0 meV kp

300 600 900 -80

-40

-80

-40

∆FP (meV/atom)

200 300 400 500 600

-80

-40

300 600 900 1200

-80

-40

500 1000 1500 2000

T (K) 1500 2000

600 1200 1800 2400

-0.3

-0.2

-0.1

200 300 400 500 600 -1.6

-1.2

-0.8

-0.4

GGA

LDA

CALPHAD

300 600 900 1200

-1.6

-1.2

-0.8

-0.4

0.0

FP (eV/atom)

600 1200 1800 2400

-1.6

-1.2

-0.8

-0.4

0.0

500 1500 2000

Al Pb Cu

Figure 4.8: Free energy FPat P=0 Pa as a function of temperature Tin comparison with values obtained

with the calphad method [7] (using thermocalc version Q [139] and the ”pure4-sgte Pure Elements

Database”). On the left axis, the free energies e

FPshifted according to Eq. (4.5) and on the right axis, the

difference free energy ∆FPaccording to Eq. (4.7) are shown. The curves for ∆FPare shaded as a guide for

the eyes. The melting temperature is indicated for each element (cf. Fig. 4.1) by the vertical dashed line.

a) The LDA and GGA free energies are calculated fully from DFT. Additionally, for each element the error

(in meV/atom ≈0.023 kcal/mol) in FPat the corresponding melting temperature based on the convergence

tests in Sec. 3.4 is given. The first number refers to the error from the ksampling (kp) and the second to

the error due to the plane wave cutoff Ecut. The error due to the augmentation grid is consistently zero and

thus not shown. The incomplete curve for the GGA ∆FPis discussed in the text. b) The LDA and GGA

FPare calculated using the mixed approach as discussed in Sec. 4.1.8.

116 4.1. Assessing DFT accuracy in predicting thermodynamic properties of metals

εand αare typically measured simultaneously and FPis mostly derived from CPmeasurements.

Nonetheless, we still have three distinct classes of quantities measured in separate experimental

setups all leading to a consistent picture when compared to the DFT results:

1) Thermal expansion εand its coefficient α.

2) Isobaric heat capacity CPand free energy FP.

3) Adiabatic bulk modulus BS.











Consistent agreement between

experiment and DFT results for

a large set of metals.

The question which arises now is: Can one employ these systematic results in order to determine

the source of the – for some elements substantial – error in the theoretical data, in particular for

the GGA results? We address this question in the following section.

4.1.8 Comparison between theory and experiment: Mixed approach

It remains an aim of our study to use the consistent and systematic trends observed so far, in

order to derive strategies for an improved materials description. In this context, it is interesting

to take also the T= 0 K DFT parameters and their deviation from experiment into account. It

is a remarkable fact that the largest error observed in FP, which is caused by the divergence in

the GGA description of Au, coincides with the largest deviation from experiment at T= 0 K (see

Fig. 4.2/Tab. 4.1). Vice versa, the small errors in the GGA T= 0 K quantities for Al (but also

for Cu, Ir, and Rh) are consistent with relatively small errors in the respective free energies. Note

further that GGA gives systematically a too low free energy, whereas the free energy from LDA

is almost always too high. This behavior, which in a similar fashion is also seen in the derived

thermodynamic quantities, is likewise correlated to the observed trends for the T= 0 K data

(Fig. 4.2).

In order to quantitatively analyze the influence of the errors in the T= 0 K parameters,

different approaches have to be employed depending on the considered quantity. For the phonon

dispersion, the DFT calculated equilibrium lattice constant (Sec. 4.1.3) can be simply replaced by

the experimental lattice constant at the given temperature [121, 140, 141]. Figure 4.3b shows the

effect for all elements. The difference between the LDA and GGA results is systematically reduced

and thus the mean deviation from experiment decreases slightly. This effect is most prominent for

GGA. A further interesting consequence is that replacing the fully theoretical lattice constant by the

experimental one changes the qualitative behavior of LDA and GGA: While in Fig. 4.3a (theoretical

lattice constant) the LDA/GGA curves give rise to an upper/lower bound of the experimental data,

in Fig. 4.3b (experimental lattice constant) the opposite behavior is found. The same trend has

earlier been reported for a small subset of the here investigated systems, Al [121] and Cu [121, 122].

In order to estimate how the error in the T= 0 K quantities affects thermodynamic properties,

the following approach is introduced: In a first step, we use the experimental values for alat,eq

0,BT,0,

and B′

T0as given in Tab. 4.1 in order to construct the exact (experimental) free energy volume curve

Fexp(V, T =0K) from the Vinet equation of state, Eq. (3.54). For elements, where no experimental

data are available for B′

T,0, we use the average of the LDA and GGA value given in Tab. 4.1. In a

second step, we use Fexp to derive a mixed experimental-theoretical free energy surface Fmix by:

Fmix(V, T) = Fexp(V, T =0 K) + e

Fel

0(V, T ) + TSqh,0k(V).(4.8)

Note that Fexp contains already the influence of zero-point vibrations and is therefore not addition-

ally included in Eq. (4.8). Finally, in a third step, Fmix(V, T) is used to derive all thermodynamic

4.2. Beyond the conventional scheme: Temperature dependent dynamical matrix 117

properties. In the following, we call this approach of combining the experimental free energy Fexp

with the fully ab initio calculated thermal energy contribution e

Fel

0+TSqh,0kthe mixed approach.

We applied the mixed approach to ε,α,CP,BS, and FP[Figs. 4.4 (dashed lines), 4.5b, 4.6b,

4.7b, and 4.8b, respectively]. In general, i.e., for all elements and all quantities, the same effect as

observed for the phonon dispersion is found: The mixed approach flips the results for LDA and

GGA, i.e., if GGA predicted a lower magnitude of a quantity than LDA using the fully DFT based

approach, it predicts a higher magnitude using the mixed approach. As for the agreement with

experiment, the following changes due to the mixed approach are observed:



For the transition elements, one needs to distinguish between LDA and GGA:

–For LDA, the agreement with experiment mostly worsens upon the application of the

mixed approach. (One exception is the FPof Rh for which it improves.)

–In contrast, the GGA agreement with experiment generally and substantially improves.

This becomes most evident for Au, where the fully DFT based GGA free energy FP

diverges (cf. Sec. 4.1.7). The mixed approach does not only allow to obtain FPup to

the melting temperature, but it also provides an outstanding agreement with experiment.



Applying the mixed approach to Al only negligibly affects the agreement with experiment,

since the original T= 0 K quantities are already in good agreement with experiment.



In the case of Pb, the mixed approach strongly decreases the agreement with experiment for

both LDA and GGA for almost all quantities.

We have thus identified the main source of the error in the GGA description of thermodynamic

properties of transition metals: It originates from the errors already inherent in the description

of the T= 0 K energy surface. This is an important statement and it implies that the entropic

contributions can be predicted with a high accuracy. The situation is different for LDA. Here, we

have already a good description of the thermodynamics of the transition metals using the fully DFT

based approach, while the mixed approach worsens the description. This leads to the conclusion

that, for LDA, we have a cancellation of the errors in the T= 0 K and entropic quantities.

Improving only one yields a decrease in the accuracy of the combined quantity. In summary, fully

DFT based calculations yield a better description for LDA, whereas the mixed approach is more

suitable for GGA.

This conclusion holds also for Al which can be viewed as a special case with negligible error.

However, as pointed out above, the results for Pb show a considerably different behavior. As

a possible explanation, we refer to the discussion of the phonon dispersion of Pb at the end of

Sec. 4.1.3. In Ref. [131], it was found that spin-orbit coupling (not included in our study) improves

the description of the phonon dispersion of Pb. We can therefore only speculate that the thermo-

dynamic properties are also affected by spin-orbit coupling and that including them would yield a

description consistent with the results for the transition elements.

4.2 Beyond the conventional scheme: Temperature dependent dy-

namical matrix

Let us now extend the calculations presented in Sec. 4.1 beyond the D0kapproximation. For that

purpose, the explicit dependence of the dynamical matrix Don the electronic temperature Tel is

included as discussed in Sec. 3.3.4. This extension leads to an explicit temperature dependence of

118 4.2. Beyond the conventional scheme: Temperature dependent dynamical matrix

the phonon frequencies [Eq. (4.3)], modifies therefore the quasiharmonic free energy (see Sec. 3.3.4),

and consequently also the resulting thermodynamic properties. Out of the nine elements studied

in Sec. 4.1, we focus now on two representative elements: aluminum and rhodium, which stand for

the elements with a small and with a large electronic contribution, respectively (cf. Fig. 4.6a).

We first discuss rhodium and in particular its LDA phonon frequencies shown in Fig. 4.9. To

allow a convenient representation, the Tel dependence of the frequencies at four representative q

vectors/branches (marked with the arrows in Fig. 4.9a) is displayed in Fig. 4.9b. We find that

most frequencies decrease with Tel and only a few (≈10%) show the opposite behavior as for the

2/3 KTpoint. The predominant decrease of the frequencies can be traced back to a decrease of

the force constants with increasing Tel. It can be thus concluded that the influence of electrons

softens the inter-atomic force constants at higher temperatures. The magnitude of the change in the

frequencies is in the range of 1 meV at the melting temperature (2236K), which might be considered

as relatively small (3% . . . 5%) when compared to the absolute values (Fig. 4.9a). One needs

however to use a different reference for a sensible comparison, which is the implicit temperature

dependence of the phonon frequencies caused by the thermal expansion Veq(T). Figure 4.9c shows

this comparison for two qvectors. Here, indeed a different situation is found: The magnitude

of the shift caused by the Tel dependence is ≈25% of the shift due to the implicit temperature

dependence, i.e., due to quasiharmonicity. Since however the latter is an important contribution to

thermodynamic properties, a decisive questions is: To what extent do the electronic phonon shifts

affect thermodynamic properties?

In order to answer this question it is useful to consider first thermodynamic quantities obtained

from a quasiharmonic free energy surface at fixed Tel as discussed in Sec. 3.3.4. An example is given

in Fig. 4.10a, which shows the CP(Tnuc;Tel) dependence on the (respectively fixed) value for Tel

(thick solid lines). The nuclei temperature Tnuc is set to Tmsince this yields the largest contribution

and visualizes the effect best. We find that LDA and GGA yield a slightly different qualitative

temperature dependence, but the overall change in CP(Tnuc;Tel) when going from Tel = 0 K to

Tel =Tmis for both similarly small (<0.05 kB). Let us now, in a second step, contrast this result

with the influence of Tel on the consistently calculated CP(T), Eq. (3.71). For that case, the effect

is considerably stronger [dots marked with CP(T=Tm)−CP(Tnuc =Tm;Tel = 0 K) in Fig. 4.10a]:

The contribution is 0.29 kB, i.e., 6 times larger.

To explain this rather unintuitive feature, the temperature dependence of the isobaric free

energy FPas shown in Fig. 4.10b needs to be considered. The dashed lines visualize the Tnuc

dependence of FP(Tnuc;Tel) for four different Tel values. The heat capacity CP(Tnuc;Tel) (thick

solid lines in Fig. 4.10a) corresponds roughly to the second derivative along FPwith respect to

temperature. (This is not exactly true since the first derivative should be done at constant volume

and not pressure, but the essential conclusions are still the same.) The fully consistent CP(T)

corresponds instead to the second derivative of the consistently calculated FP(T) (thick solid line

in Fig. 4.10b). The latter is however constructed by choosing at a certain Tnuc the FP(Tnuc;Tel)

surface for which Tnuc =Tel as illustrated by the filled circles in Fig. 4.10b. This leads to a

considerably different curvature and consequently to a different second derivative. We stress the

importance of this result: In order to fully include the influence of the Tel dependence of the

dynamical matrix on derivative thermodynamic quantities, it is not sufficient to calculate the

dynamical matrix and hence the quasiharmonic free energy at a single, possibly sufficiently high

Tel. It is rather necessary to calculate several D(Tel) and to obtain the consistent quasiharmonic

free energy Fqh(Ω, T nuc =T;Tel =T) as discussed in Sec. 3.3.4.

Taking the fully consistently calculated CP(T) into account, let us investigate its influence on

the final heat capacity and compare it to the other contributions, in particular to the quasiharmonic

4.2. Beyond the conventional scheme: Temperature dependent dynamical matrix 119

ΓX K ΓL

Eω (meV)

LDA

Exp.

0500 1000 1500 2000

Tel (K)

-1

Eω (meV)

0500 1000 1500 2000

T (K)

-4

-2

Eω (meV)

XTa

2/3KT

Al, LL

a) b) c)

XT2/3KT

Veq(T)

Tel

Figure 4.9: Illustration of the influence of temperature on the phonon frequencies of rhodium. The plotted

results are for LDA. The GGA dependencies show a similar behavior. a) Quasiharmonic phonon dispersion

of rhodium (as in Fig. 4.3). The qvectors and branches, to which the phonon shifts shown in b) and c)

correspond to, are marked with the arrows. The subscripts stand for transversal (T) and longitudinal (L)

modes. b) Phonon shifts for rhodium due to the influence of the electronic temperature Tel [Eq. (4.3)]

at the qvectors marked in a). Additionally, the shift for aluminum at LLis shown (black solid line). The

dashed (dotted) vertical line corresponds to the melting temperature of rhodium (aluminum). c) Comparison

between the temperature dependence shown in b) (thin lines, marked with Tel) and the implicit temperature

dependence of the phonon frequencies due to the thermal expansion [Eq. (4.2); thick lines, marked with

Veq(T)] for two qvectors.

0 500 1000 1500 2000

Tel (K)

0.1

0.2

0.3

CP(Tnuc;Tel) (kB)

LDA

GGA

0500 1000 1500 2000

Tnuc (K)

-3

-2

-1

P(Tnuc;Tel) (meV/at.)

P(T)

Tel=1/4Tm...4/4Tm

500 1000 1500 2000

T (K)

CP(T) (kB)

el + qhela

el + qha

elaa

qh − ha

qhel− qh

a) b) c)

CP(T=Tm) −

CP(Tnuc=Tm;Tel=0K)

Figure 4.10: Illustration of the influence of the Tel dependent phonon shift on thermodynamic properties of

rhodium. The dashed vertical line corresponds to the melting temperature of rhodium. a) The dependence

of the isobaric heat capacity CP(Tnuc;Tel) from Eq. (3.70) on Tel with Tnuc =Tm. The curves are referenced

with respect to CP(Tnuc =Tm;Tel =0K). Additionally, the consistently calculated CP(T) [Eq. (3.71)] at Tm,

similarly referenced, is shown. b) The dependence of the isobaric free energy FP(Tnuc;Tel) from Eq. (3.70)

on Tnuc for four different electronic temperatures (dashed lines): Tel = 1/4Tm

, 2/4Tm

, 3/4Tm

, and 4/4Tm.

The thickness of the lines increases with temperature. Additionally, the consistently calculated free energy

FP(T) (solid line) from Eq. (3.71) is included and the dots indicate the crossing points between FP(T) and

the other curves, which exactly fall together with the fixed Tel temperatures of FP(Tnuc;Tel). All curves

are referenced with respect to FP(Tnuc;Tel = 0 K) and correspond to the LDA functional c) Consistently

calculated isobaric heat capacity CPfor the LDA functional in comparison with experiment (black dots;

Ref. [132]). The thick solid line corresponds to CPincluding electronic (el) and quasiharmonic contributions

based on a temperature dependent dynamical matrix (qhel). The dotted line corresponds to a CPbased on

the electronic and quasiharmonic (qh) contribution but within the D0kapproximation [Eq. (2.179)]. The

pure electronic contribution is indicated by the dot-dashed line, the quasiharmonic minus the harmonic (h)

contribution by the thin solid line, and qhel−qh by the dashed line.

120 4.3. Beyond the quasiharmonic approximation: Anharmonicity and vacancies in Al

Table 4.2: The contribution of the various mechanisms to the heat capacity of rhodium at Tm. ”Full”

denotes CPincluding all mechanisms and the other notation is as in Fig. 4.10c.

full h el qh−h qhel−qh

CP(kB) 5.36 2.99 1.41 0.67 0.29

CP(%) 100 56 26 13 5

one. For that purpose, Fig. 4.10c shows the high temperature heat capacity of rhodium. Indeed,

the contribution to CPcaused by the Tel dependent phonon shift (dashed line) is ≈1/2 of the

contribution originating from the thermal expansion, which corresponds to the quasiharmonic minus

the harmonic contribution (thin solid line). Using the total heat capacity (thick solid line) as

a reference, the contribution due to the Tel dependent phonon shift is in the range of 5%. A

quantitative comparison of the influence of the various investigated mechanisms is given in Tab. 4.2.

We now turn to aluminum. For its phonon frequencies, we find that most of them show a

shift with a magnitude <0.05 meV at Tm(934 K). This is <5% of the shift observed for Rh

at its melting temperature of 2236 K. An interesting finding is the fact that if considering both

elements at the same absolute temperature, say 934K, the shift for Al is considerably closer in size

to that for Rh. This is illustrated in Fig. 4.9b for the LLpoint. It should however not be expected

that, by (artificially) extending the melting temperature of Al, we would achieve the same order

of magnitude as for Rh also at higher temperatures. The phonon shift sensibly depends on the

electronic density of states and this is significantly different for Al and Rh. In fact, we find for

some Al phonon shifts a curvature indicating that the magnitude of the shift will remain constant

or maybe even decrease with temperature, in contrast to the linearly increasing magnitude for Rh.

The Tel dependent phonon shift of Al will be contrasted with the thermal expansion caused shift

in Sec. 4.4, where we will be able to compare both also to the shift caused by anharmonicity. The

influence of the Tel dependent dynamical matrix on the thermodynamics of Al will be discussed in

Sec. 4.3.5 together with the results for the anharmonic and vacancy contribution.

4.3 Beyond the quasiharmonic approximation: Anharmonicity and

vacancies in Al

4.3.1 Motivation

The fact that, at elevated temperatures, the heat capacity CPof metals strongly deviates from the

harmonic prediction — i.e., the heat capacity at fixed volume CV0(cf. thin solid line in Fig. 4.6a

for Al) — is well known. Indeed, this was pointed out almost 90 years ago in a seminal work by

Born and Brody [142]. In this and in many subsequent studies, various mechanisms such as the

thermal expansion, the occurrence of anharmonicity, the electronic degrees of freedom [142], or

the formation of vacancies [143] have been proposed to explain the non-linear increase. Ab initio

methods as those used to obtain our results (Sec. 4.1) allow to accurately determine the contribution

of two of the proposed mechanisms (Fig. 4.6): the thermal expansion which is captured by the

quasiharmonic approximation and the electronic degrees of freedom captured by the temperature

dependent DFT approach. However, the subtle balance between further contributions, such as

explicit anharmonicity and vacancies, has not yet been resolved in the literature even for the

simple elementary metals as considered in the present study [144].

4.3. Beyond the quasiharmonic approximation: Anharmonicity and vacancies in Al 121

Let us focus on aluminum which is a prototypical example. It has been studied intensively

in the past decades due to its industrial importance (light weight, corrosion resistance) and the

availability of single crystals. Measuring the high temperature heat capacity, however, turns out

to be a challenge and despite numerous measurements using various techniques the obtained data

scatter largely as shown in the inset of Fig. 4.6a and in Fig. 4.17a below. A quantitative assessment

of the subtle influences to the free energy/heat capacity has therefore been considerably hampered.

Selected examples of the corresponding literature shall illustrate this fact: Initially, the contribu-

tion to the high temperature heat capacity due to thermally activated vacancies was believed to

exceed the anharmonic contribution and to cause a non-linear temperature dependence [143]. In

1968, Brooks and Bingham [145] measured the heat capacity of aluminum using dynamic adiabatic

calorimetry. Reducing the experimental heat capacity from constant pressure to fixed volume and

employing a Debye model, the authors interpreted the non-linear increase in their experimental

data as arising mainly from explicit anharmonicity. In 1985, Ditmars et al. [13] measured the

enthalpy of aluminum by means of isothermal phase-change calorimetry and Shukla et al. [146]

calculated from these results the isobaric heat capacity. Shukla et al. then employed the same

reduction scheme as Brooks and Bingham [145] to obtain the fixed volume heat capacity. In a

subsequent theoretical analysis, they went beyond the Brooks and Bingham approach [145] by

employing empirical potentials rather than a simple Debye model to calculate the fixed volume

heat capacity. The comparison of the experimental and theoretical data suggested that, in con-

trast to the original work by Brooks and Bingham [145], the vacancy contribution dominates the

anharmonicity. More recently, in 2004, Forsblom et al. [93] calculated the explicit anharmonicity

contribution to the fixed volume heat capacity employing the embedded atom method (EAM; see

Sec. 2.2). Their results show, in contrast to the study in Ref. [146], that the contribution due

to explicit anharmonicity can well be of a magnitude similar to the one obtained for the vacancy

contribution from Ref. [146]. The precise value could not be assessed, since it varied significantly

between the used potential parametrizations.1

In order to resolve the ongoing controversy and to identify the key mechanisms, two critical

issues have to be addressed: a) First, all previous theoretical studies have been restricted to fixed

volume calculations, i.e., only CV0was assessed. This restriction constitutes a serious limitation,

since it requires the use of the above mentioned reduction scheme [93, 145, 146] in order to compare

to experimental data at constant pressure. This reduction involves model assumptions and experi-

mental input parameters such as the temperature dependent bulk modulus, thermal expansion, and

the thermal expansion coefficient. Such an approach is therefore a potential source for uncontrolled

errors/error cancellation. b) Second, all previous studies were based on a simple Debye model or on

empirical potentials, with Forsblom et al. [93] employing the most sophisticated empirical method,

the EAM approach. In Sec. 4.5, we will be able to judge explicitly whether the EAM method is

suitable for determining the subtle influence of high temperature mechanisms by comparing to the

ab initio results presented in the following. At this stage, we can only judge implicitly from the

EAM results of Forsblom et al. [93], which indicate that the presently available empirical potentials

might indeed fail in this respect due to the scatter between predictions from different potentials.1

The aim of the investigations presented in Sec. 4.3 is to overcome both limitations. For that pur-

pose, we consider the complete volume and temperature dependent free energy surface of aluminum

including vacancies and anharmonicity. As discussed in Sec. 2.4, the explicitly calculated volume

1Forsblom et al. [93] employed three potential parametrizations and obtained at the melting temperature an

anharmonic contribution to the fixed volume heat capacity of 0.06 kB, 0.03 kB, and −0.05 kB(cf. Fig. 2 in Ref. [93];

kB: Boltzmann constant). Shukla et al. [146] estimated a vacancy contribution to the fixed volume heat capacity of

0.08 kBat the melting point [cf. Eq. (15) and Fig. 4 in Ref. [146] and our discussion in Sec. 4.3.6]

122 4.3. Beyond the quasiharmonic approximation: Anharmonicity and vacancies in Al

dependence provides access to properties at constant pressure which corresponds to the typical

experimental condition. The DFT approach allows on the other hand for an unbiased parameter-

free derivation of the desired quantities, in contrast to the difficulties occurring for the empirical

approaches. In fact, our DFT results will eventually provide a test-bed to evaluate the EAM

potentials a posteriori (Sec. 4.5). Specifically, we computed F(V, T) using the volume optimized

approach and the UP-TILD method according to Eqs. (3.35) to (3.37),

(4.9)

F(V, T ) = (1−cNv)FpV−cΩv

1−cNv, T+cFv(Ωv, T ;Nv) + Fc(c),(4.10)

(4.11)

Fp(V, T ) = Fp,el

0(V, T ) + Fp,qh(V, T) + Fp,clas,ah(V, T )

Fv(Ωv, T;Nv) = Fv,el

0(Ωv, T;Nv) + Fv,qh(Ωv, T;Nv) + Fv,clas,ah(Ωv, T ;Nv)

the various free energy contributions being defined in Tab. 3.1 and the remaining quantities in

Sec. 3.1.2. Further, following the philosophy developed in Sec. 4.1 all free energy contributions

were calculated using both xc functionals: LDA and GGA-PBE. To allow a convenient discussion,

we focus in the following on: CP,α, and FP. The heat capacity is the main quantity of interest. The

expansion coefficient is equally sensitive as CP, experimentally however more accurately accessible,

which will allow a cross check of our results (cf. Sec. 4.3.7). The free energy is included to show

the fact that small absolute FPvalues can have significant effects on derived properties (cf. the

curves for the vacancies in Fig. 4.12 below).

4.3.2 Pseudopotential evaluation

The technical aspects of the calculations in the present section, Sec. 4.3, were discussed in detail in

Sec. 3.4. It was also pointed out why it is advantageous to employ the pseudopotential rather than

the PAW method in the case of aluminum DFT calculations as considered here. However, due to

missing reference data, the actual evaluation of the used pseudopotential remained an open issue.

Now, that the validated PAW results from Sec. 4.1 are available, we are in a position to fill this

gap. The evaluation is split into four steps:

1) Let us first consider the T= 0 K results given in Tab. 4.1. We could in principle compare

directly to the LAPW+lo results here. For consistency with the following steps, the differences

to the PAW results are given instead. For both pseudopotentials, LDA and GGA, a rather

good agreement is found: The lattice constant is under(over)estimated only by −0.4% (0.2%)

for LDA (GGA). The bulk modulus deviates by −2.4% (−3.8%) for LDA (GGA).

2) A more severe test concerns thermodynamic quantities. To perform this test, we calculate the

electronic and quasiharmonic free energy contributions entering Eq. (4.1) using the pseudopo-

tential method, derive the thermodynamic quantities, and compare with the corresponding

PAW results (Figs. 4.4 to 4.8). Note that this test differs from the evaluation of the PAW

method against the LAPW+lo method (Sec. 4.1.4) where only a single term from Eq. (4.1) was

considered. Here, the pseudopotential result falls together with the PAW result on the given

scales, except for α, the most sensitive quantity (Fig. 4.5). We therefore mentioned/displayed

the pseudopotential (pp) result only in Fig. 4.5. For the quantities of interest here and focus-

ing on the GGA results at Tm, we find that the α(CP) computed using the pseudopotential

4.3. Beyond the quasiharmonic approximation: Anharmonicity and vacancies in Al 123

method overestimates the PAW result by 4.8% (1.4%) or 0.18 ·10−5K−1(0.07kB). For FP,

an underestimation of 0.9% or 3 meV/atom is found.

3) It is necessary to determine the source of the errors given in 2). For that purpose, we perform

a pseudopotential calculation exchanging Eel

g,0(V) [Eq. (4.1)] by the corresponding PAW data,

i.e., the T= 0 K errors given in 1) are removed. We then find that for instance the error in

αfor GGA at Tm[from 2)] reduces from 4.8% to 1.2%. This means that 3/4 of the error in

the thermodynamic properties given in 2) are caused by the T=0 K errors given in 1). In a

similar manner, the errors arising from Fqh,0kand e

Fel

0were separated out and the error in

Fel

0was found to be negligible as compared to Fqh,0k. Thus, the remaining 1/4 of the errors

in the thermodynamic properties is coming from the quasiharmonic free energy Fqh,0k.

4) It would be desirable to extend the tests performed in 2) beyond a comparison to quasihar-

monic properties for the following reason: The quasiharmonic free energy is obtained from the

dynamical matrix (Sec. 2.3.3) which depends on the local electronic free energy surface in the

vicinity of the T= 0 K equilibrium nuclei positions. The atomic structures determining the

anharmonic free energy, the quantity of major interest in the following sections, will however

typically correspond to structures far from equilibrium. A direct evaluation of these struc-

tures by performing an extra PAW molecular dynamics run would be computationally rather

expensive. The extension of the UP-TILD method discussed in Sec. 3.2.4 provides here an ef-

ficient alternative. We thus calculated ∆Felup,pp→paw

λ, Eq. (3.51), and find also for this test

that the pseudopotential and PAW results agree very well: ∆Felup,pp→paw

λ≈0.5meV/atom

(at 900 K and λ= 0.5).

The technical results of the tests given in points 1) to 4) have a very important consequence for the

physical results discussed in Secs. 4.3.3 to 4.3.7: The T=0 K energy (3/4) and quasiharmonic free

energy (1/4) have been found to be the dominating sources for the errors of the pseudopotential

properties while the remaining free energy contributions have been identified to yield far smaller

errors. This is important since the absolute values of these ”remaining” contributions will turn

out to be very small compared to the quasiharmonic contribution (see Fig. 4.11 below). Based on

the tests presented here, we have thus ensured that the small absolute numbers of the ”remaining”

contributions are reliable. In contrast, the final curves based on all free energy contributions

(Fig. 4.17 below) do contain the errors given in 2). The corresponding error bars are therefore

included in these figures.

4.3.3 Explicit anharmonicity: Direct ab initio calculations

A key result of the calculations presented in Sec. 4.3 is that the dominating contribution to thermo-

dynamic properties of aluminum is contained in the quasiharmonic description of the perfect crystal.

All remaining contributions can be considered as small perturbations on top of the quasiharmonic

result. This is demonstrated schematically in Fig. 4.11, while Fig. 4.12 contains a more detailed de-

scription and Tab. 4.3 gives actual numbers at the melting point. In this and the following section,

we concentrate on discussing the results for explicit anharmonicity, for the influence of Tel on the

nuclei motion (quasi- and anharmonic; Sec. 4.3.5), and vacancies (Sec. 4.3.6), since the electronic

and quasiharmonic contribution were intensively treated in Sec. 4.1. In Sec. 4.3.7, all contributions

are combined and the resulting thermodynamic properties are compared to experiment.

Let us thus focus on the classical anharmonic free energy of the perfect crystal. Previous ab

initio studies of the anharmonic free energy are rather scarce [130, 147]. One exception is the

work by Voˇcadlo and Alf`e [130], who determined the melting curve of aluminum under extreme

124 4.3. Beyond the quasiharmonic approximation: Anharmonicity and vacancies in Al

0 200 400 600 800

T (K)

CP (kB)

0 200 400 600 800

T (K)

0.0

0.1

-2%

vac

sum

sum q

Figure 4.11: Illustration of the dominant role played by the quasiharmonic (qh=q+h) contribution for the

example of the heat capacity. The right diagram magnifies the ”smaller” contributions. See Fig. 4.12/Tab. 4.3

below for further details.

-0.1

0.0

0.1

CP (kB)

-3%

0 200 400 600 800

T (K)

-0.2

0.0

0.2

α (10-5 K-1)

-5%

-4.0

-2.0

0.0

2.0

FP (meV/atom)

vac

sum

700 800 900

-0.08

-0.04

0.00

0.02%

0.01%

T m

LDA-FPGGA-FP

(meV) (%) (meV) (%)

full −300.5 100.0−312.5 100.0

h−284.4 94.6−294.8 94.3

q−14.0 4.7−15.1 4.8

el −4.4 1.5−4.6 1.5

ah 2.3−0.8 2.0−0.6

vac 0.0 0.0−0.1 0.0

qhel−qh <|0.1|0.0<|0.1|0.0

ahel−ah <|0.1|0.0<|0.1|0.0

LDA-CPGGA-CP

(kB) (%) (kB) (%)

full 3.73 100.0 3.83 100.0

h2.97 79.6 2.97 77.5

q0.76 20.4 0.86 22.5

el 0.13 3.5 0.13 3.5

ah −0.16 −4.2−0.12 −3.3

vac 0.05 1.4 0.07 1.8

qhel−qh <|0.02|<|0.5|<|0.02|<|0.5|

LDA-αGGA-α

105K) (%) (1

105K) (%)

full 3.68 100.0 3.83 100.0

qh 3.75 101.8 3.89 101.6

el 0.06 1.5 0.06 1.5

ah −0.25 −6.7−0.31 −8.1

vac 0.12 3.4 0.20 5.1

qhel−qh <|0.05|<|1.3|<|0.05|<|1.3|

Figure 4.12/Table 4.3: Influence of the various investigated excitation mechanisms on the free energy FP,

the heat capacity CP, and the expansion coefficient αof aluminum.

Figure 4.12: The electronic (el), anharmonic (ah), vacancy (vac) contribution and their sum. Thick lines

show GGA results. LDA results are either shown as thin lines or coincide with the GGA result. The dotted

lines show the el contribution from a PAW calculation (all other results are for pseudopotentials). The

right axes are scaled with respect to the ’full’ GGA values at the melting temperature Tm(indicated by the

vertical dashed line) given in Tab. 4.3. The inset shows an enlargement of the vacancy contribution to FP

at high temperatures.

Table 4.3: The quasiharmonic contribution (qh) is split for FPand CPinto its harmonic (h) and ”quasi” (q)

part, i.e., the thermal expansion. The contribution due to inclusion of the Tel dependence in the dynamical

matrix is indicated by ”qhel−qh”. The effect of including the Tel dependence in the anharmonic contributions

is indicated by ”ahel−ah” (FPonly).

4.3. Beyond the quasiharmonic approximation: Anharmonicity and vacancies in Al 125

15.5 16.0 16.5 17.0 17.5 18.0

V (Å3/atom)

Fp,clas,ah (meV/atom)

LDA

GGA

Veq,GGA

0Veq,GGA

Veq,LDA

0Veq,LDA

0 200 400 600 800

T (K)

Fp,clas,ah (meV/atom)

V = 17.7Å3, alat=4.133Å

V = 16.7Å3, alat=4.054Å

V = 15.7Å3, alat=3.974Å

Figure 4.13: Explicitly anharmonic free energy for the perfect crystal Fp,clas,ah of aluminum. The calculated

values are represented by the dots/diamonds/triangles. At each point the statistical error σerr, Eq. (2.202),

is represented by the vertical solid lines. The solid lines are fits using the analytical model (see Sec. 4.3.4).

a) Volume dependence of Fp,clas,ah at 900 K for both investigated functionals. The vertical dashed (dotted)

lines indicate the 0K (including zero-point vibrations) and the melting temperature Tmequilibrium volumes

of LDA (GGA), Veq,LDA

0(Veq,GGA

0) and Veq,LDA

Tm(Veq,GGA

Tm), respectively. b) Temperature dependence of

Fp,clas,ah at three different volumes for the LDA functional. The dashed lines in-between are a guide for the

eye. GGA results show the same qualitative dependence. The melting temperature Tmis indicated by the

vertical dashed line.

temperatures and pressures. The temperature and pressure region investigated in their work is

very different from ours so that the corresponding anharmonic free energies cannot be used here.

A representative set of our results for Fp,clas,ah is shown in Fig. 4.13. Despite considerable

statistical errors, the figures provide some clear trends: First we note that Fp,clas,ah is positive in

the whole region except for a small low temperature regime, which is likely due to statistical noise.

The absolute values close to the melting temperature are ≈2 meV, which is about two orders of

magnitude smaller than Fp,qh. The volume dependence is positive with a small positive curvature.

The temperature dependence has a curvature which goes beyond second order. The results for

LDA and GGA are qualitatively similar. Comparing relative Fp,clas,ah values, i.e., referenced to

the corresponding equilibrium volume, LDA and GGA results differ by an almost constant shift.

At 900 K, this shift is ≈0.5 meV (see Fig. 4.13a).

The anharmonic contribution to the free energy has a significant influence on αand CPas

compared to the electronic contributions. Figure 4.12 shows that αis shifted downwards by ≈ −7%

and CPby ≈ −3.5%. In fact, the negative anharmonic contribution to CPcancels the positive

contribution from electronic excitations over a large temperature range yielding a close to zero net

contribution. For α, the anharmonic contribution is much stronger (roughly a factor of 5) compared

to the electronic one yielding a net decrease of the expansion coefficient. An important conclusion

from these results is that explicit anharmonicity cannot be considered as the origin of the non-linear

increase, which applies to both the expansion coefficient and the heat capacity. In fact, even the

sign is opposite to the one previously assumed [145]: For both quantities, anharmonicity results in

a reduction rather than in an increase at high temperatures.

A major source of this unexpected behavior lies in the increase of Fp,clas,ah with increasing

volume. To verify this statement, we artificially forced Fp,clas,ah(V, T ) in a separate calculation to

126 4.3. Beyond the quasiharmonic approximation: Anharmonicity and vacancies in Al

be volume independent: Ffix(V, T ) = Fp,clas,ah(V(T=0 K), T). Note that the explicit temperature

dependence is fully included. Replacing the full anharmonic free energy by Ffix makes the anhar-

monic contribution to αpractically disappear, while the anharmonic contribution to CPis reduced

by ≈60%.

4.3.4 Explicit anharmonicity: Introduction of an analytical model

In order to resolve the physical origin of the positive volume dependence, we developed an ap-

proximate analytical model. The key idea is to replace the full phonon spectrum by a single

effective frequency and describe anharmonicity by its renormalization. Within this model (mod),

the anharmonic free energy Fah,mod is expressed as:

Fah,mod(V, T) = e

Fqh(T;ωqh(V) + ωah)−e

Fqh(T;ωqh(V)).(4.12)

Here, ωqh is an effective/averaged quasiharmonic frequency, ωah the renormalization shift due to

anharmonicity, and e

Fqh(T;ω) is obtained from an adapted version of Eq. (2.161):

Fqh(T;ω) = kBTln 1−exp −~ω

kBT.(4.13)

To obtain ωqh, we use our computed quasiharmonic phonon frequencies ωGsc

, s. In particular, we

calculate the average [cf. Eqs. (2.125) and (2.161)]:

ωqh(V) = (3Nn)−1

Gsc

ωGsc

, s(V).(4.14)

Figure 4.14b (black solid line) shows the volume dependence of ωqh. The anharmonic shift ωah

in Eq. (4.12) accounts for the anharmonic free energy. Tests based on our ab initio data showed

that, in order to capture the essential qualitative features, it is sufficient to assume ωah to be

volume independent. The magnitude of this volume independent shift is used as a free parameter

to fit to the ab initio data and to explore the physics embodied in the model. The resulting

anharmonic free energy for three different ωah is shown in Fig. 4.14c. If we assume a negative

shift, Fah,mod is negative and shows a negative volume dependence. Thus, to reproduce the positive

anharmonic free energy observed in the ab initio results, the renormalization shift must be positive,

i.e., the renormalized phonons must be blue-shifted in average. By introducing a linear volume

and temperature dependence2in ωah also a good quantitative description of the full volume and

temperature dependence of the anharmonic free energy surface is achieved. The quality of this

approach is shown in Fig. 4.13 (solid lines). It was also successfully employed for the parametrization

of the corresponding vacancy contribution (Fig. 4.16).

The key result we obtain from this analysis is that the vibrational free energy and hence the

major part of the thermodynamics of aluminum at high temperatures, up to the melting point, can

be well described by renormalized phonon frequencies. Compared to the quasiharmonic frequencies,

the renormalized frequencies are in average energetically harder (blue-shifted), but exhibit a similar

volume dependence.

2We use ~ωah =a+bT +cV and obtain the following fitting parameters: LDA: a=−2.6 meV, b= 9.9·10−4

meV/K, c= 0.15 meV/˚

A3and GGA: a=−4.5 meV, b= 3.1·10−4meV/K, c= 0.27 meV/˚

A3.

4.3. Beyond the quasiharmonic approximation: Anharmonicity and vacancies in Al 127

15.5 16.0 16.5 17.0 17.5

V (Å3)

-1

Fah,mod (meV)

-0.6

0.0

0.6

hωah (meV)

h(ωqh+ ωah) (meV)

hωah= 0.6 meV

hωah= 0.3 meV

hωah= 0.0 meV

hωah= -0.3 meV

Figure 4.14: Illustration of the analytical model (see

text) describing the origin of the anharmonic free en-

ergy: a) Volume independent anharmonic frequency

ωah. b) Effective frequency resulting from the sum of

the quasiharmonic frequency ωqh and ωah. c) Model

anharmonic free energy Fah,mod at the melting tem-

perature as obtained from Eq. (4.12) and the fre-

quencies from b).

4.3.5 Influence of the electronic temperature on the nuclei motion

The influence of the Tel dependent dynamical matrix on the thermodynamics of Al was much more

challenging to obtain than the one for Rh which has been discussed in Sec. 4.2. The reason for this

difficulty is the small absolute size of this influence and the extremely sensitive dependence of the

thermodynamic properties of Al on the electronic ksampling. Based on the convergence checks

performed in Sec. 3.4, we were however able to estimate rather low limiting bounds, which are listed

in Tab. 4.12. For instance for CP, we find that the Tel dependent phonon shift yields a change with

a magnitude below 0.02 kBat Tm. This result corresponds already to the consistently calculated

CP(T) (cf. the discussion in Sec. 4.2 for Rh). Compared to the corresponding contribution for

Rh (0.3kB), this is rather low. Using however the contribution of the vacancies to the CPof Al

(0.05 kB) as a reference, we are in the same order of magnitude and the influence of the electronic

phonon shift is thus not negligible. A further reduction of the uncertainty in this contribution

would however exceed sensible computational times at present. For this study, we therefore use the

estimated limits as error bounds in the final thermodynamic properties (Fig. 4.17).

In Sec. 3.3.5, we have discussed that, similarly as the dynamical matrix, also the higher order

contributions, i.e., explicit anharmonicity, depend on the electronic temperature. To determine the

corresponding contribution, we used a further advantage of the hierarchical UP-TILD method: We

can apply the method not only to a set of varying convergence parameters as discussed in Sec. 3.2.4,

but also to a set of varying physical parameters as for instance various electronic temperatures. We

thus calculated the change in the anharmonic free energy upon varying the electronic temperature

and find it to be ≈50% of the change caused by the Tel dependence of the dynamical matrix.

The exact contribution cannot however be singled out due to the above discussed scatter in the

Tel dependence of the dynamical matrix. The reason is that the Tel dependent dynamical matrix

constitutes the correct reference potential which should be used to obtain the Tel dependence of

the anharmonic shift [see Eq. (3.79)].

128 4.3. Beyond the quasiharmonic approximation: Anharmonicity and vacancies in Al

4.3.6 Vacancies: Electronic, quasiharmonic, and anharmonic excitations

In this section, we discuss the second mechanism proposed as a source for the non-linear increase

in the experimental heat capacity of aluminum: The thermal formation of vacancies. In advance,

we comment on self interstitials and show that this other possible type of thermally activated point

defects can be excluded as a potential candidate for explaining the non-linear increase.

In aluminum (fcc crystal), two self interstitial sites are available: the tetrahedral and octahedral.

The octahedral site is known [148] to be energetically preferred and its formation energy has been

calculated from ab initio giving Ef= 3.4 eV [148]. This is considerably higher than for vacancies

(cf. Tab. 4.4 below) and would indeed yield only negligible self interstitial concentrations. However,

inspecting Ref. [148], we found that a rather small supercell (16 atoms) and ksampling (1024k·atom)

were used. Both values are critical since a self interstitial requires larger supercells than the vacancy

due to its stronger strain field (cf. with the vacancy supercells in Sec. 3.4.4.3) and since our results

show that Al is one of the most sensitive elements with respect to the ksampling (Fig. 3.8). We

therefore investigated in detail the influence of these parameters in a separate study [149] and found

indeed a different converged formation energy of Ef= 2.8eV. Even more interestingly, an additional

phonon analysis revealed that the octahedral site is in fact meta stable, i.e., a saddle point on the

energy surface. Following the relaxation path resulted in a new self interstitial configuration with a

different symmetry: two Al atoms sharing an interstitial site (dumbbell geometry). Our finding is

consistent with empirical knowledge about self interstitials in fcc metals [150]. As a consequence,

for thermodynamic properties, the dumbbell configuration needs to be considered instead of the

original octahedral site due to its energetical preference. The actual decrease to Ef= 2.5 eV does

however not change the conclusion that self interstitials in Al do not yield a noticeable contribution

to thermodynamic properties as we explicitly verified by using Eq. (3.38).

As for the vacancies, monovacancies are believed to be dominating, while the influence of

divacancies has been found to be negligible due to a negative (repulsive) binding energy [89].

Starting with the seminal work by Gillan [30], monovacancies in aluminum have been studied

extensively in the past by ab initio techniques [89, 148, 151–153]. However, the focus has been

mainly on T= 0 K properties. Finite temperature effects, except for the configurational entropy,

have not been included. One exception is a combined study [89] where the T=0 K energetics and

the harmonic contribution have been determined by DFT, while the remaining finite temperature

effects have been obtained by employing empirical potentials (referred to as the ai/ep-approach in

the following).

Our results for the full [i.e., including all free energy terms in Eqs. (4.9) and (4.11)] contribution

of the vacancies to FP,α, and CPare shown in Fig. 4.12. The contribution to FPis negative and

rather small, ≈ −0.1 meV/atom at Tm, i.e., well below 0.1% of the quasiharmonic reference. Sur-

prisingly, despite the small free energy contribution due to vacancies, they substantially contribute

to αand CPin the temperature window from ≈700 K up to Tm. This effect is related to the

strong non-linear decrease of this contribution with temperature. For both quantities, αand CP,

the anharmonic contribution is positive and amounts to ≈5% and ≈2% of the quasiharmonic

reference at Tm, respectively.

With the fully ab initio calculated results at hand, we have a reference against which the

performance/accuracy of previously suggested empirical approaches for estimating the effect of

vacancies can be tested. One such empirical estimation [146] starts from the following expression

for the vacancy contribution to the vacancy heat capacity Cvac

Cvac

P=Ef

kBT2

exp −Ef−TSf

kBT.(4.15)

4.3. Beyond the quasiharmonic approximation: Anharmonicity and vacancies in Al 129

Table 4.4: The formation energy Efand entropy of formation Sffor various approaches and combinations

of the free energy contributions used for the calculation of vacancy properties of aluminum. ai/ep indicates

values for the coupled ab initio-empirical potentials approach from Ref. [89]. The further values (also the

experimental) are obtained by fitting the vacancy concentrations over the temperature range given in Fig. 4.15

to the function exp[−(Ef−T Sf)/kBT]. The notation is as in Fig. 4.15.

Ef(eV) Sf(kB)

LDA GGA LDA GGA

volOpt; qh 0.65 0.58 0.2 0.1

volOpt; qh + el 0.65 0.58 0.2 0.1

volOpt; qh + ah + el 0.78 0.68 2.2 1.5

constP; qh + ah + el 0.78 0.68 2.2 1.5

constV; qh + ah + el 0.85 0.75 2.5 1.9

ai/ep [89] 0.78 0.61 1.6 1.3

experiment 0.75 2.4

1 1.1 1.2 1.3 1.4 1.5 1.6

T m/T

10-6

10-5

10-4

10-3

Vacancy concentration

GGA

LDA

Exp.

volOpt; qh (+ el)

volOpt; qh + ah + el

constP; qh + ah + el

constV; qh + ah + el

Figure 4.15: Equilibrium vacancy concentration of aluminum as a function of the inverse temperature multi-

plied by the melting temperature Tm. Results for the two investigated exchange-correlation functionals, LDA

and GGA, are shown. The solid lines correspond to a volume optimized (volOpt) calculation, Eq. (3.35),

with all free energy terms included in Eqs. (3.36) and (3.37). They also correspond to a calculation based on

the constant pressure (constP) approach, Eq. (3.29), including all free energy contributions, since the results

for the volOpt and constP approach fall together on the given scale. The dashed lines correspond to a volume

optimized calculation excluding the anharmonic free energy Fclas,ah, in Eqs. (3.36) and (3.37). The dashed

lines also correspond to a volume optimized calculation excluding the anharmonic, Fclas,ah, and electronic,

Fel

0, (indicated by the parentheses enclosing the electronic contribution in the legend), terms in Eqs. (3.36)

and (3.37) [see also Eq. (3.53)]. The dotted lines correspond to a calculation based on the constant volume

(constV) approach, Eq. (3.34), and including all free energy terms. The squares indicate experimental values

from Ref. [154] (differential dilatometry). The diamonds/circles indicate experimental values from Ref. [155]

(differential dilatometry/positron annihilation).

130 4.3. Beyond the quasiharmonic approximation: Anharmonicity and vacancies in Al

0 200 400 600 800

T (K)

Fv,clas,ah (meV/atom)

V=17.7Å3, alat =4.133Å

V=16.7Å3, alat =4.054Å

V=15.7Å3, alat =3.974Å

Figure 4.16: Temperature dependence of the explic-

itly anharmonic free energy Fv,clas,ah for the vacancy

cell of aluminum at three different volumes for the

LDA functional. The dots/diamonds/triangles rep-

resent the calculated values. At each point the sta-

tistical error σerr, Eq. (2.202), is represented by the

vertical solid lines. The dashed lines in-between are a

guide for the eye and the solid lines are fits using the

analytical model (see Sec 4.3.4). GGA results show

the same qualitative dependence. The melting tem-

perature Tmis given by the vertical dashed line.

Here, Efis the formation energy, which was set in Ref. [146] to the experimental value 0.66 eV from

Ref. [156]. Further, the entropy of formation Sfwas estimated to be 1.4kBbased on experimental

heat capacity data and empirical potential calculations of the heat capacity excluding vacancy

contributions. Using these values, the authors of Ref. [146] estimated an increase in CPof 0.08 kB

at Tmdue to vacancies. This approximate value based on experimental input is in surprisingly

good agreement with our explicitly calculated values of 0.05 kBfor LDA and 0.08 kBfor GGA.

We finally turn to the comparison with the ai/ep-approach, discuss the validity of the constant

volume/pressure approach, and analyze the influence of the various free energy contributions to

the vacancy concentration. Figure 4.15 summarizes the corresponding results. A comparison

with experiment (squares, diamonds, and dots) shows overall a good agreement: Using the full

approach, i.e., the volume optimized formalism including all free energy contributions, LDA and

GGA are very close to experiment and form a lower and upper bound, in consistence with our results

for other thermodynamic properties in Sec. 4.1. Further, it becomes apparent that the constant

pressure approach is a very good approximation to the volume optimized formalism, whereas the

constant volume approach yields significantly lower concentrations. To enable a comparison with

the ai/ep approach and to allow a quantitative analysis of the influence of the various free energy

contributions, we deduced from the temperature dependence of the vacancy concentrations the

formation energy Efand entropy Sf. The data compiled in Tab. 4.4 show a good agreement of

our full approach with the ai/ep approach [93]. With respect to the influence of the free energy

contributions Tab. 4.4 reveals: The electronic contribution is negligible. In contrast, the anharmonic

contribution to the vacancies has a large influence on the entropy of formation increasing it from

0.2kB(0.1kB) to 2.2kB(1.5kB) for LDA (GGA). This strong influence can be traced back to the

temperature dependence of the explicitly anharmonic free energy surface for the vacancy cell. From

Fig. 4.16, it becomes apparent that the anharmonic free energy is concave at higher temperatures

in contrast to the convex temperature dependence of the perfect crystal anharmonic free energy

(Fig. 4.13b). The corresponding difference favors the creation of vacancies.

4.3.7 Comparison between theory and experiment

So far, the terms entering Eqs. (4.9) to (4.11) have been discussed separately. Now, we construct

the full free energy surface F(V, T ) including all excitations, calculate the final thermodynamic

quantities, and compare with experiment focusing on αand CP. In contrast to the comparisons

performed in Secs. 4.1.4 and 4.1.5, we are now interested in small effects in the high temperature

region which is thus shown magnified (Fig. 4.17). The low temperature region is also magnified to

illustrate the different experimental situations in the two extreme temperature regimes. Further,

4.4. Quantum mechanical treatment of the anharmonic contribution 131

to allow an unbiased comparison, an extensive set of experimental data is included. Let us start

with CP.

Figure 4.17a shows a very good agreement between theory and experiment for the low tem-

perature heat capacity, where experimental scatter is negligible. In the temperature region up

to ≈600 K, the majority of the experimental data agree well with our ab initio results. Above

600 K, there is large experimental scatter making a fair comparison with the ab initio values dif-

ficult. A general trend is that almost all experiments performed later than 1950 (solid squares in

Fig. 4.17a) show a steeper increase towards the melting temperature making the ab initio data

a lower bound. Based on the presently available experimental measurements, a final conclusion

whether the remaining small deviation (≈0.2kBat Tm) is due to the limited accuracy of presently

available exchange-correlation functionals or due to errors in experiment is not possible. The fact

that the ab initio results form a lower bound to the scattered experimental data suggests that there

are additional sources for the heat capacity in the experimental measurements, however of varying

influence. Possible sources are imperfections within the sample such as dislocations, grain bound-

aries, impurities, etc. Systematic deviations can also be caused by the experimental setup, such as

for instance the sample holder. In particular, the cubic dependence of the radiant heat-exchange

coefficient on the temperature is an experimental challenge [168].

In contrast to the heat capacity, which is experimentally difficult to measure and shows large

scatter in the data, the temperature dependence of the thermal expansion coefficient requires the

determination of inter-atomic distances which can be obtained by X-ray measurements with high

precision. The experimental together with our ab initio data are shown in Fig. 4.17b. The ab initio

values agree with the experimental data over the full temperature range up to the melting temper-

ature Tm. Even details such as the curvature of αat high temperatures are well reproduced. It is

important to note that only the combined interplay of the electronic, quasiharmonic, anharmonic,

and vacancy contribution (black solid lines in Fig. 4.12) provides this agreement. Particularly im-

portant for the steep increase close to the melting temperature is the strongly non-linear increase

due to vacancy formation.

4.4 Quantum mechanical treatment of the anharmonic contribu-

tion

In Sec. 4.1.3, we have pointed out that there are two distinct explicit temperature dependencies

of the phonon frequencies, given by Eqs. (4.3) and (4.4), and in Sec. 4.2 we have presented our

results for the latter dependency. The first one, which is caused by phonon-phonon interactions,

is implicitly contained in the anharmonicity calculations presented in Sec. 4.3.3. Its influence on

the thermodynamic properties is therefore also implicitly included and this actually allowed us

to perform the final comparison with experimental data already in Sec. 4.3.7. As discussed in

detail in Sec. 2.3.5, it is nonetheless desirable to study the phonon shift due to phonon-phonon

interaction explicitly with a complementary method, i.e., with perturbation theory. The main

reason is the fact that the anharmonic calculations based on molecular dynamics are purely classical,

while the perturbation theory as presented in Sec. 2.3.7 fully contains quantum mechanical effects.

Corresponding calculations are however computationally extremely expensive and in order to render

the calculations presented in this section feasible, we focused on a single qvector and branch of

the phonon dispersion (LLpoint; see Fig. 4.18) at constant volume. The choice of the LLwas

motivated by the fact that a previous model potential study [176] (discussed in detail below) found

the largest shift for this qvector.

The results for the third and fourth order phonon shift are presented in Fig. 4.18. It becomes

132 4.4. Quantum mechanical treatment of the anharmonic contribution

300 400 500 600 700 800 900

T (K)

2.8

3.0

3.2

3.4

3.6

3.8

4.0

CP (kB)

GGA

LDA

Exp. >1950

Exp. <1950

0 10 20

0.00

0.03

0.06

200 300 400 500 600 700 800 900

T (K)

2.0

2.4

2.8

3.2

3.6

4.0

α (10-5 K-1)

GGA

LDA

Exp. >1950

Exp. <1950

0 10 20

0.00

0.04

Figure 4.17: a) Heat capacity CPand b) thermal expansion coefficient αincluding all investigated excitation

mechanisms compared to experiment. The melting temperature Tmis given by the vertical dashed line. At

Tm, the crosses indicate the sum of the estimated pseudopotential error (Sec. 4.3.2) and the uncertainty

due to the (qhel-qh) contribution (Sec. 4.2) for the GGA. The LDA error is ≈1/2 of the GGA error.

Experimental data older than 1950 are indicated by the open circles (Refs. [157–159] for CPand [160–163]

for α). The remaining experimental results are indicated by the filled squares (Refs. [13, 143, 145, 164–169]

and [14, 170–174]). The inset shows the low temperature region with experimental data from Refs. [13] and

[175].

4.4. Quantum mechanical treatment of the anharmonic contribution 133

ΓX K ΓL

Eω (meV)

LDA

Exp.

0 200 400 600 800

T (K)

-4

-2

Eω (meV)

sum

3. ord

4. ord

0 200 400 600 800

T (K)

-4

-2

Eω (meV)

Tel

Veq (T)

a) b) c)

Figure 4.18: a) Quasiharmonic phonon dispersion of aluminum as in Fig. 4.3. The qvector and branch

at which the anharmonic shift was calculated [b) and c)] is marked with the arrow. b) The third and

fourth order contribution to the quantum mechanical anharmonic shift. c) Comparison between the various

temperature dependencies of the phonon frequencies: the anharmonic (ah) shift corresponding to the sum

from b), the Tel dependent shift discussed in Sec. 4.2, and the shift due to the thermal expansion Veq(T).

apparent that if we consider each order separately (Fig. 4.18b), we have a strong phonon shift

which has a magnitude similar to the shift caused by the thermal expansion Veq(T) (Fig. 4.18c).

The third order leads to a negative shift and the fourth order to a positive one so that in total

both contributions tend to cancel each other. This effect of cancellation is known [86] and, as

mentioned above, it has been also calculated for aluminum [176]. In order to facilitate this rather

early (1990) study of the phonon shift, the author in Ref. [176] employed a force constant harmonic

model potential with pairwise and first neighbor three body interactions. (Note that the pairwise

interaction is harmonic but taking many body effects into account can still lead to anharmonicity.)

The provided values of −0.80 and 0.95 meV for the third and fourth order shift at room temperature

agree astonishingly well with our values of −1.30 and 1.44 meV. In order to explain this good

agreement, we need to consider how the model potential was constructed: The force constants were

fitted to experimental frequencies and to higher order elastic constants [176]. The such obtained

model potential thus implicitly contains a large pool of experimental information. In particular,

the fit to the highest order elastic constants yields information closely related to the anharmonic

phonon shifts. Based on these considerations, we can in fact consider the comparison with the

results from the model potential calculation as an indirect comparison with experimental data.

The influence of quantum mechanics is visible in the low temperature region (to ≈200 K)

of both orders. Classically, each of the shifts would start at 0 meV and increase linearly with

temperature. Quantum mechanically, we have at T= 0 K an energy shift of ≈0.5 meV and a

parabolic temperature dependence. This energy shift is a direct consequence of the zero-point

vibrations: Due to these vibrations the atoms explore already at T= 0 K the electronic free

energy surface in the vicinity of their classical equilibrium positions. The (small) contribution

of the anharmonic terms in this region manifests itself as the T= 0 K shift of the anharmonic

frequencies. Above ≈200K, the temperature dependence of the anharmonic phonon shifts changes

into a linear, classical dependence. Considering the physically relevant sum of both orders the

quantum mechanical influence cancels largely and a classical picture can be applied even at lower

temperatures. It can be thus concluded that for the high temperature region which was of interest

in Sec. 4.3.1, quantum mechanical effects are not important.

We can draw another important conclusion from the results of this section, which nicely sup-

ports the physical idea behind our model used to parametrize the anharmonic free energy surface

(Sec. 4.3.4). We found that the model is able to describe the directly calculated results for Fclas,ah

134 4.5. Achievable accuracy with empirical approaches: EAM vs. DFT

accurately, if we assume a positive shift in the (artificial) renormalization frequency ωah. This fre-

quency however corresponds to the sum of the contribution of the third and fourth order obtained

here explicitly and it indeed shows a positive shift with temperature. In fact, even quantitatively

a good agreement can be observed: To obtain the correct magnitude of ≈1 meV/atom in the

anharmonic free energy at 900 K (Fig. 4.13), a renormalization frequency of ωah ≈0.5 meV is

needed (cf. Fig. 4.14). This agrees with the results for the directly calculated anharmonic phonon

shift (cf. Fig. 4.18c at 900 K).

4.5 Achievable accuracy with empirical approaches: EAM vs.

DFT

In this final section of Chap. 4, our aim is to employ the highly accurate DFT results of Secs. 4.1

and 4.3, in order to evaluate the performance of the embedded atom method (EAM) in predicting

thermodynamic properties of metals, in particular of aluminum. The EAM approach was motivated

in Sec. 2.2 by the fact that EAM potentials allow to tackle system sizes and simulation times orders

of magnitude larger than addressable using DFT. Based on the discussions that followed Sec. 2.2,

particularly concerning the DFT calculation of anharmonic contributions, we can now remotivate

the EAM approach very specifically: Despite advanced techniques, like for instance the UP-TILD

method, the DFT calculation of anharmonic contributions in aluminum took many months of CPU

time. In contrast, using EAM potentials the CPU time consumption was in the order of minutes.

We choose the following three commonly employed EAM potentials: the parametrizations of

Mei-Davenport [62], Zope-Mishin [63], and Ercolessi-Adams [64]. A particular motivation for this

choice is that the construction procedures for these potentials follow a certain hierarchy in their

complexity. This issue is illustrated in Tab. 4.5. The Mei-Davenport potential employs 13 fitting

parameters and only a rather small set of T= 0 K reference data. The Zope-Mishin potential has

the same number of parameters, but in addition to the reference data used for the Mei-Davenport

potential, further T= 0 K ab initio results are employed. Finally, the Ercolessi-Adams potential

is the most elaborated, since it uses not only a large set of T= 0 K data, but also information

about structures relevant at higher temperatures. Additionally, it is based on significantly more

fitting parameters which should increase its flexibility. Based on this choice of potentials, we will

be able to address the question whether the quality of the thermodynamic results correlates with

the complexity of the construction procedure.

The T= 0K properties for the three potentials are shown in Tab. 4.1 (page 104). The deviation

from experimental values is small and in the range of the deviation for the ab initio results. This is

however not surprising since these quantities enter directly the potential optimization (information

about the bulk modulus is contained in the elastic constants). Let us thus turn to more advanced

tests and discuss first the EAM phonon dispersions, which are shown in Fig. 4.19 in comparison

with DFT and experiment. Figure 4.19 reveals the significantly different performance of the EAM

potentials. The Mei-Davenport potential considerably underestimates the longitudinal branches by

≈5 meV. The Zope-Mishin and Ercolessi-Adams potentials agree better with DFT/experiment,

with the first one slightly overestimating and the latter underestimating the DFT/experimental

dispersion. An important point concerns the dependence of the phonon dispersion on the qvector.

As discussed in Sec. 4.1.3, the phonon dispersion of aluminum is rich of Kohn-anomalies which

distinguish it from a ”simple” dispersion as found for instance for Cu. While these anomalies are

accurately described by DFT, none of the EAM potentials can reproduce them (cf. e.g. the low

energy branches along Γ to K and Γ to L). We note that all EAM dispersions are calculated in the

same, sufficiently large supercell (43) as used for the DFT calculations. The reason for the lack of

4.5. Achievable accuracy with empirical approaches: EAM vs. DFT 135

Table 4.5: Illustration of the increasing level of complexity in the construction procedure of the Mei-

Davenport (MD) [62], Zope-Mishin (ZM) [63], and Ercolessi-Adams (EA) [64] EAM parametrizations. Shown

are the pair potential vpair, the atomic charge density ρat, and the embedding energy femb. The explicitly

given functions contain only significant fitting parameters (p1, p2, p3) occurring in the exponents. The im-

plicit determination of femb (for ZM) is based on matching to a universal equation-of-state proposed in

Ref. [117]. Further, nfit gives the number of fitting parameters and the last column the quantities used as

a reference for the fitting procedure: alat=lattice constant, Ecoh=cohesive energy, Ef=vacancy formation

energy, c/a=ratio of the lattice vector lengths in hcp, Estack=stacking fault energy, Esurf =surface energy,

and E(V)=energy volume dependence. The T= 0 K reference data are obtained from experiment or from

ab initio [for E(V) for instance]. For the fitting of the Ercolessi-Adams potential also structures from a

molecular dynamics (MD) run have been used.

vpair(r)ρat(r)femb(ρ)nfit Reference for fitting

MD r e−rP5

i=0 r−icomplex expansion 13 T=0 K: alat,Ecoh,Ef

in (ln ρ) elastic constants

ZM r−p1−r−p2rp3e−r[1+e−r] implicit using 13 T=0 K: alat,Ecoh,Ef,c/a,

universal EOS elastic constants,

E(V): bcc, fcc, sc, hcp

EA |cubic splines through |40 T=0 K: alat,Ecoh,Ef,Estack,

|a set of fitting points |elastic constants, Esurf

T >0 K: 85 MD structures

Kohn anomalies is the fact that they are due to a complex interplay between the phonons and the

electronic Fermi surface and this interplay is not captured by the EAM parametrizations.

The performance of the EAM potentials in predicting quasiharmonic thermodynamic properties

is displayed in Fig. 4.20. To allow for an unbiased comparison, we use here the purely quasihar-

monic DFT results as a reference, which means that they are obtained from a free energy surface

with the e

Fel

0term excluded from Eq. (4.1). Averaged over the various properties, we find that the

Ercolessi-Adams potential yields the best performance. For instance, the heat capacity (Fig. 4.20c)

or the free energy (Fig. 4.20f) of this potential agree reasonably well with the DFT data. However,

the thermal expansion and consequently also the expansion coefficient (Figs. 4.20a and b) strongly

underestimate the DFT results. The situation is even more critical for the other two EAM poten-

tials which dramatically under (Zope-Mishin) and overestimate (Mei-Davenport) the DFT thermal

expansions. This shortcoming of the EAM potentials can be traced back to a wrong description of

the volume dependence of the quasiharmonic free energy. Further, we observe an interesting feature

for the low temperature heat capacity enlarged in the inset of Fig. 4.20c. Here, the Ercolessi-Adams

and Zope-Mishin potentials agree well with the quasiharmonic DFT result [solid black line], which

is a consequence of the fact that the long wavelength limit is well described by these potentials

(Fig. 4.19). There is however a small discrepancy with the experimental data. The reason is the

missing electronic contribution as shown by the dotted curve, which is the DFT result containing

this additional contribution (i.e., the e

Fel

0term being included in the calculation). The inability to

describe electronic contributions to the free energy is a general drawback of the EAM approach.

We now turn to the discussion of the subtle influences of explicit anharmonicity and vacancies

summarized and compared to the DFT results from Sec. 4.3 in Fig. 4.21 and Tab. 4.6. Figure 4.21a

136 4.5. Achievable accuracy with empirical approaches: EAM vs. DFT

shows the volume dependence of the anharmonic free energy of the perfect crystal Fp,clas,ah which

was identified in Sec. 4.3.3 to be important for deriving thermodynamic quantities. It reveals that

the Zope-Mishin potential yields not only a qualitatively wrong volume dependence (decreasing

with volume instead of increasing), but also a negative sign for the Fp,clas,ah values instead of the

correct positive one. The negative sign can be explained by the qualitatively incorrect description of

the temperature dependence of Fp,clas,ah by the Zope-Mishin potential as illustrated in Fig. 4.21b.

The wrong description of both the volume and temperature dependence of Fp,clas,ah cumulates in

a qualitatively wrong prediction of the anharmonic contribution to the heat capacity (Fig. 4.21c).

The situation is significantly better for the other two EAM potentials which describe Fp,clas,ah(V, T )

qualitatively well. The resulting anharmonic contribution to CPis in reasonable agreement with

DFT for the Ercolessi-Adams potential, while it is too large in magnitude for the Mei-Davenport

potential. For the vacancy concentrations shown in Fig. 4.21d, we find that all EAM parametriza-

tions yield a similar result which is also in reasonable agreement with DFT/experiment. A more

quantitative analysis (Tab. 4.6) reveals however that the entropy of vacancy formation (equal to

the slope of the curves in Fig. 4.21d) is correctly predicted only by the Mei-Davenport potential.

The final thermodynamic quantities based on all possible excitation mechanisms, i.e., quasihar-

monic, anharmonic, and vacancies for the EAM potentials and electronic excitations additionally for

DFT, are represented by the heat capacity in Fig. 4.21e and the expansion coefficient in Fig. 4.21f.

For the heat capacity of the Zope-Mishin and Mei-Davenport potential, we observe an interesting

situation: Whereas the purely quasiharmonic heat capacity (Fig. 4.20c) significantly deviates from

the quasiharmonic DFT heat capacity, the situation for the heat capacity including all excita-

tion mechanisms (Fig. 4.21e) is considerably improved. The latter is mainly a consequence of the

strong (but incorrect) contribution due to anharmonicity (Fig. 4.21c). Thus, the errors in the two

separately incorrectly predicted contributions (quasiharmonic and anharmonic) cancel each other

to yield a reasonable result in their sum. For the Ercolessi-Adams potential, we find a different

situation: We have a reasonable description of the separate heat capacity contributions, whereas

the resulting full heat capacity shows the worst agreement with DFT/experiment out of the three

EAM potentials. This might be surprising at first sight, but the following two reasons explain

this behavior: First, both the quasiharmonic and anharmonic heat capacity are slightly lower than

the corresponding DFT result, i.e., show a deviation with the same sign. This accumulates in a

larger deviation in the resulting heat capacity. Second, the DFT result contains additionally the

electronic contribution which pushes the heat capacity slightly upwards (cf. Fig. 4.12).

The anharmonic contribution to the expansion coefficient (not shown in Fig. 4.21) is qualita-

tively similar to the anharmonic heat capacity (Fig. 4.21c). However, the anharmonic contributions

for the Zope-Mishin and Mei-Davenport potential are not strong enough to fully cancel the wrong

quasiharmonic prediction of the expansion coefficient (Fig. 4.20b) as observed for the heat capacity.

For the Ercolessi-Adams potential anharmonicity pushes the expansion coefficient down, increasing

the disagreement with DFT/experiment. In fact, while we have identified this potential as giving

the best quasiharmonic properties, it is now (i.e., for the full excitation spectrum) the worst EAM

parametrization.

Let us comment on a rather technical issue, which however illustrates well a further important

difficulty of EAM potentials in combination with volume dependent thermodynamic properties. In

our calculations, we experienced frequently that the originally provided EAM potentials showed

an unphysical, singular behavior at higher but still relevant volumes. An example is given in

Fig. 4.21a for the case of the volume dependence of Fp,clas,ah for the Ercolessi-Adams potential.

The filled squares indicate the actually calculated values which show a kink at a certain volume. We

found also a similar behavior for the Mei-Davenport potential. For the latter, we could remove the

4.5. Achievable accuracy with empirical approaches: EAM vs. DFT 137

ΓX K ΓL

Eω (meV)

Mei-Davenp.

Zope-Mishin

Ercolessi-Adams

DFTaa

Exp.

Figure 4.19: Phonon dispersion ωq,s [Eω=

hω/(2π)] of aluminum for the three investigated

EAM parametrizations Mei-Davenport [62], Zope-

Mishin [63], and Ercolessi-Adams [64] in compari-

son with DFT (GGA, PAW) data and experiment.

A 43supercell (256 atoms) was consistently used

for all theoretical dispersions. Experimental values

are from Ref. [101].

0 200 400 600 800

0.0

0.5

1.0

1.5

2.0

ε (%)

0 200 400 600 800

α(10-5K-1)

050 100 150 200 250

0.95

0.96

0.97

0.98

0.99

BS/BS(T=0K)

DFT

Exp./CALPHAD

0 200 400

CP (kB)

200 400 600 800

T (K)

-0.3

-0.2

-0.1

0.0

FP (eV/atom)

Mei-Davenport

Zope-Mishinaa

Ercolessi-Adams

200 400 600 800

T (K)

-20

-10

∆FP (meV/atom)

0 10

0.005

0.010 DFT,

qh+el

a) b)

d)c)

e) f)

T m

Figure 4.20: Quasiharmonic thermodynamic properties of aluminum for the three investigated EAM

parametrizations (see Fig. 4.19 for references) in comparison with DFT (GGA, PAW) and experi-

ment/calphad. The DFT values do not contain the electronic contribution [except for the dotted curve in

the inset in c)]. Experimental references are the same as in Figs. 4.4 to 4.8. a) Linear thermal expansion

ε, Eq. (4.8). b) Expansion coefficient α, Eq. (2.235). c) Isobaric heat capacity CP, Eq. (2.236). The inset

shows the low temperature region. d) Adiabatic bulk modulus BS, Eq. (2.237). e) Isobaric free energy e

FP,

Eq. (2.242), shifted according to Eq. (4.5) for DFT and calphad and similarly for the EAM potentials.

f) The difference ∆FPaccording to Eq. (4.7) for DFT and similarly for the EAM potentials.

138 4.5. Achievable accuracy with empirical approaches: EAM vs. DFT

Table 4.6: The formation energy Efand en-

tropy of formation Sfof a vacancy in alu-

minum for the investigated EAM potentials

(cf. Fig. 4.19) in comparison with DFT (GGA,

pseudopotential) and experiment. The values

(also the experimental) are obtained by fitting

the vacancy concentrations over the tempera-

ture range given in Fig. 4.21d to the function

exp[−(Ef−T Sf)/kBT].

Ef(eV) Sf(kB)

qh qh+ah qh qh+ah

Mei-Davenport 0.63 0.77 −0.1 1.5

Zope-Mishin 0.68 0.71 0.0 0.1

Ercolessi-Adams 0.69 0.81 0.0 0.2

DFT 0.58 0.68 0.1 1.5

Experiment 0.75 2.4

17.0 17.5 18.0

V (Å3)

-10

-5

Fp,clas,ah (meV/atom)

0 200 400 600 800

T (K)

-10

-5

Fp,cla,ah (meV/atom)

1 1.2 1.4 1.6

T m/T

10-6

10-5

10-4

10-3

Vacancy concentration

0 200 400 600 800

T (K)

-0.6

-0.3

0.0

0.3

CP (kB)

0 200 400 600 800

T (K)

CP (kB)

Mei-Davenport

Zope-Mishin

Ercolessi-Adams

0 200 400 600 800

T (K)

α(10-5K-1)

DFT

Exp.

a) b)

d)c)

e) f)

T m

Figure 4.21: Explicitly anharmonic, vacancy, and the resulting full, i.e., with all excitation mechanisms,

properties of aluminum for the investigated EAM potentials (cf. Fig. 4.19) in comparison with DFT (GGA,

pseudopotential; cf. Sec. 4.3) and experiment. Experimental data are from Ref. [155] for d), from Refs. [132]

and [13] for e), and from Ref. [14] for f). a) Volume dependence of the anharmonic free energy of the perfect

crystal Fp,clas,ah. The curves are fits through the calculated values based on the model from Sec. 4.3.4. The

actually calculated values are shown only for the Ercolessi-Adams potential (filled squares with the dotted

line as a guide for the eye) to visualize the kink in the volume dependence discussed at the end of Sec. 4.5.

The other, not shown calculated values fall together with the fits on the given scale. b) As a), but showing

the temperature dependence. c) The anharmonic contribution to the isobaric heat capacity. d) Vacancy

concentration. e) Isobaric heat capacity CP. f) Expansion coefficient α.

4.5. Achievable accuracy with empirical approaches: EAM vs. DFT 139

unphysical volume dependence: Since this potential is provided in an analytical form, we enlarged

the originally introduced interaction cutoffs [62] which resulted in a potential with a smooth volume

dependence. In contrast, the situation is more problematic for the Ercolessi-Adams potential which

is provided numerically [64] and therefore not tunable. We solved here the problem by including

only the uncorrupted free energy points in our calculations. (For instance, the green solid line in

Fig. 4.21a indicates a fit based on the values before the kink.)

From the previous discussions, we draw one major conclusion: Among the investigated po-

tentials, an EAM potential yielding overall good thermodynamic properties cannot be identified.

Moreover, a clear correlation between the complexity of the construction procedure (see discussion

at the beginning of this section) and the accuracy in the thermodynamic properties is not visi-

ble. One might argue that the most complex potential, the Ercolessi-Adams one, yields in average

the best thermodynamic quantities. This is however not as clear as we would have expected, in

particular, due to the fact that it contains T > 0 K information.

The most interesting and important question is whether the failure of the EAM potentials

is 1) an inherent deficiency of the EAM concept or if 2) it applies only to the here employed

parametrizations. We cannot give a decisive answer based only on our results, but we can provide

some valuable hints: An argument supporting 1) is the missing correlation discussed above. If the

EAM approach was, in principle, able to yield accurate thermodynamic properties, we would expect

that increasing the complexity of the potentials leads to a systematic improvement. This is not

the case. Inspecting Ref. [64], we found however an issue which still leaves room for hypothesis 2).

The reference calculations for the Ercolessi-Adams potential, in particular the 85 MD structures,

were not obtained from a ”pure” ab initio scheme, but rather from a simplified tight binding

approximation (developed in Ref. [177]), which is only based on ab initio input. This approach

was necessary to allow the ab initio MD simulations at that time (1994). The tight binding

method is however not well suited to describe metallic systems, particularly free-electron systems

as aluminum, since it is based on a localized electron picture, i.e., the electrons are assumed to sit

close to their host atoms. In contrast, the reference calculations used for the Zope-Mishin potential

were obtained using ”pure” ab initio calculations (LAPW method). It is therefore reasonable to

assume that the originally (at the beginning of this section) claimed increasing complexity in the

construction procedure of these EAM potentials (including the quality of the reference calculations)

does not fully hold. We thus propose to modify the Ercolessi-Adams potential such that it is based

on ”pure” ab initio results (as for instance provided by the UP-TILD method in this study).

The evaluation of the corresponding thermodynamics could provide further valuable insight in the

general nature and quality of the EAM concept.

Chapter 5

Conclusions

The present work has addressed a major challenge of ab initio assisted materials design: The

efficient determination of highly accurate materials properties at finite temperatures. For that

purpose, various methods have been developed which allow to compute the contribution of all rele-

vant excitation mechanisms to the thermodynamic properties of non-magnetic elementary metals.

The developments have been extensively benchmarked by focusing on a set of experimentally well

investigated metals. Key findings/messages of this thesis are summarized as follows:

1) Today’s EAM parametrizations are not suited for an accurate prediction of thermodynamic

properties of metals.

Based on the extensive set of thermodynamic DFT data computed for aluminum, the perfor-

mance of three state-of-the-art EAM parametrizations in yielding thermodynamic properties

was evaluated (Sec. 4.5). None of the EAM potentials is able to yield a satisfactory agreement

with DFT for all considered free energy contributions/thermodynamic quantities (Figs. 4.20

and 4.21). For instance, the Ercolessi-Adams potential [64] yields the best quasiharmonic

(Fig. 4.20c) and anharmonic (Fig. 4.21c) heat capacity compared to DFT. However, the

resulting total heat capacity of this potential shows the worst agreement. This rather unin-

tuitive feature can be explained by an error cancellation in the heat capacity contributions

predicted by the other potentials. Moreover, a clear correlation between the complexity of

the EAM potentials and their predictive power could not be identified. Nonetheless, we were

able to suggest a way for improving today’s EAM potentials (see end of Sec. 4.5).

2) Extension of the ”usual” phase space by the convergence parameter space boosts your DFT

calculations.

With the hierarchical UP-TILD method (Sec. 3.2), a conceptually new approach has been

opened to combine accuracy and performance in computing thermodynamic averages. The

basic idea is to extend the phase space spanned by the set of atomic coordinates (”usual”

phase space) with the space of DFT convergence parameters (e.g. kpoint mesh, plane

wave cutoff). The sampling, needed for instance for calculating the anharmonic free energy,

is then performed in a hierarchical manner in this extended space. This approach reduces

considerably the number of computationally expensive calculations. In our study, we achieved

a CPU time reduction of ≈30, but we expect that an even larger reduction is possible

after optimizing and fully automating the procedure. We therefore believe that a possible

future route for improving existing DFT codes could be an automated calculation of physical

quantities in the extended space.

140

141

3) The volume optimized or the constant pressure approach should be used for accurate vacancy

results.

Our volume optimized approach (Sec. 3.1) provides an intuitive understanding of the dilute

limit point defect concept and allows to derive the typically applied standard approaches

as approximations. In the new approach, the usually applied separation of the vacancy

contribution by means of a formation free energy was abandoned. The cell containing the

point defect and the perfect bulk have to be treated as a coupled system. For vacancies in

aluminum, the commonly employed fixed volume approximation results in large error bars in

the predicted concentrations (up to 50% too small). The other approximate approach, the

constant pressure approach, yields basically the same vacancy concentrations as the general

volume optimized method. Hence, for accurate vacancy calculations, one of the latter should

be applied.

4) LDA and GGA provide an ab initio confidence interval for experiment.

The only approximation in a numerically fully converged DFT calculation is the exchange-

correlation (xc) functional. We studied the accuracy of two popular xc functionals, LDA and

GGA-PBE, in predicting a wide range of thermodynamic material properties for a large and

comparable set of elementary non-magnetic metals (Sec. 4.1). The most important result

is that, for all thermodynamic quantities studied here, the two functionals (LDA and GGA)

approximate an confidence interval for experiment and may be thus used to estimate ab initio

error bars. While further tests are needed to check the validity for other materials classes,

the existence of such a relation would allow to estimate error bars which are solely based on

DFT calculations. The availability of such error bars is a key prerequisite to make truly ab

initio based materials simulations.1

5) The anharmonic heat capacity is negative and the vacancy contribution is positive in alu-

minum.

The developed techniques allowed us to settle a long standing debate about which exci-

tation mechanism dominates thermodynamic quantities in aluminum close to the melting

point (Sec. 4.3). This issue has been raised first by Born and Brody [142] almost a century

ago and has been heavily debated since then in both experimental and theoretical studies

[143, 145, 146]. We found that – in contrast to common belief – explicit anharmonicity gives

rise to a negative contribution to the isobaric heat capacity. On the other hand, the heat

capacity contribution due to vacancies increases in a strongly non-linear fashion at high tem-

peratures. We therefore conclude that, among these two contributions, only the vacancies

can be considered as the origin of the steep positive increase in the Al heat capacity observed

in experiment.

6) Todays ab initio methods, smart statistical approaches, and computer power enable highly ac-

curate predictions up to the melting point.

Possibly the most important finding of the present work is the recognition that today’s com-

puter codes and today’s hardware enable a new level of accuracy in simulating realistic phys-

ical processes at finite temperatures. Traditionally, ab initio approaches such as DFT have

1Note that the property to form bounds to experiment originates from the T= 0 K under(over)estimation of the

equilibrium lattice constant by LDA (GGA) (cf. the discussion in Sec. 4.1.8). This phenomenon is well known and

investigated in the ab initio community. The crucial point made here is the recognition that this T= 0 K property

translates into an equivalent finite temperature property. The latter fact is however of utmost importance for the

metal research community.

142

been applied to calculate T= 0 K properties and the extension to T > 0 K has been ham-

pered by significantly more expensive calculation times. Including all relevant excitation

mechanisms into the free energy, we showed for the example of fcc aluminum that T > 0 K

quantities are accessible even up to the melting point. Indeed, it turned out that for some

thermodynamic quantities the achievable accuracy is in the range of experimental scatter.

Thus, for further progress and for a decisive comparison between ab initio and experiment,

improved experimental techniques are called for.

The results and insights gained in this study are expected to provide a firm basis for the further

development of ab initio assisted metal design. The importance of the presented investigations has

been emphasized, e.g. in Ref. [178]. They can however be only a very first step towards the final

goal of metals design on the computer. Further critical points that have to be addressed are:



We investigated only the temperature independent xc energy functional [e.g., Eq. (2.72)].

Results for a temperature dependent LDA xc functional are available [179, 180] so that its

influence should be investigated in a forthcoming study.



The chemical trends observed in the GGA thermodynamic properties resulted in large errors

in particular for the noble metals Ag and Au. The origin of these large errors could be traced

back to the inability of present day xc functionals to describe the T= 0 K potential energy

surface. It is now necessary to determine the physical mechanism causing the errors in the

potential energy surface. A possible source might be related to the closed shell structure of

the noble metals. For such elements, so called van der Waals, i.e., dipol-dipol, interactions

become important, which are not captured by standard DFT functionals (LDA/GGA). Recent

developments incorporating these interactions into the DFT approach [181] should therefore

be considered.



A very interesting but also computationally expensive extension would be to include the full

excitation spectrum, which was considered here only for Al, also for the other fcc metals. In

this respect, one open question is whether the explicitly anharmonic contribution to the heat

capacity will turn out to be systematically negative.



The quantum mechanical perturbation theory calculations were carried out only for a single

phonon frequency. These calculations should be extended to other phonon frequencies. Even-

tually, if a sufficient number of frequencies is available, a quantum mechanical anharmonic

free energy can be computed (corresponding equations are given in Ref. [68]) and compared

to the classical anharmonic free energy obtained in Sec. 4.3.3.



Finally, the insight achieved here should be combined with the following related and very

active research areas:

–Inclusion of magnetic excitations [182, 183].

–Extension to alloys which requires the description of configurational entropy [184].

–Coupling with homogenization methods allowing to treat polycrystalline materials [185].

–Atomistic description of (martensitic) phase transitions.

The advancements of these issues will likely result in a new generation of simulation tools that

are fully capable of providing finite temperature properties of real materials with hitherto not

achievable accuracy and predictive power. The availability of such techniques will open new and

exciting routes in computational materials design.

Appendix A

Supplement

A.1 Technical details

A.1.1 Scaling function

We use frequently a symbolic scaling function s(and also sel,sdft,sks,sqh) to characterize the

dimensions of a problem. In case of matrices, for instance,

s=N1·(M1×M1) + N2·(M2×M2) (A.1)

means that the corresponding problem consists of solving N1times (indicated by ”·”) the eigenvalue

equation of a matrix of size M1×M1and (indicated by ”+”) of solving N2times the eigenvalue

equation of a matrix of size M2×M2. In contrast, in cases where no matrix diagonalization is

directly involved to solve the problem, as for instance for the optimized Kohn-Sham equation,

Eq. (2.92), the scaling function indicates the dependence of the CPU time and memory, which are

needed to solve the problem, on various parameters. Due to these two different characters of the

scaling functions, equations involving both types, as for instance Eq. (2.112), should be understood

only symbolically as a means of describing the dimensions of a problem. In the same spirit, we do

not include constants or negligible terms, as for instance in Eq. (2.187).

A.1.2 Atomic and reduced units

We use in general SI units in the present work. In order to be consistent with customary notation

used in the field of DFT, we give the expression for the plane wave cutoff Ecut, Eqs. (2.88) and

(2.89), in reduced units [186]

|Ga0|=a0|G|,ka0|=a0|k|, Ecut

h=Ecut/Eh, Ecut

r=Ecut/Ry,(A.2)

with the atomic units: the Bohr radius a0, the Hartree energy Eh=~2/(mea2

0) [not to be confused

with the Hartree energy functional Eh, Eq. (2.60)], and the Rydberg energy Ry = Eh/2 [186].

A.1.3 Various definitions

Within the Dirac notation, an abstract, i.e., without a reference to a coordinate system, vector v

in Hilbert space is denoted as |vi. Its dual counterpart (which is simply the complex conjugate

v∗in the real space representation) is denoted as hv|. A scalar product for vectors |wiand |vi

is denoted hw|vi. In case of discrete vectors, hw|vi=Pi,j w∗

ivjwith wiand vicomponents in a

143

144 A.1. Technical details

specific representation. In case of continuous functions, for instance in the real space representation,

hw|vi=Rw∗(r)v(r′)drdr′.

The commutator [ ˆ

A, ˆ

B] of two operators ˆ

Aand ˆ

Bis given by:

[ˆ

A, ˆ

B] = ˆ

Aˆ

B−ˆ

Bˆ

A. (A.3)

The exponential function of an operator ˆ

Ais defined by its polynomial series

eˆ

A|ψi=ˆ

1 + ˆ

A|ψi+1

2ˆ

A(ˆ

A|ψi) + 1

6ˆ

A(ˆ

A|ψi)) + ... (A.4)

with a general wave function ψ. If |aiis an eigenfunction of ˆ

A, it follows from Eq. (A.4):

ha|eˆ

A|ai=eha|ˆ

A|ai.(A.5)

The trace of an operator ˆ

Adefined by X

ihbi|ˆ

A|bii,(A.6)

where {bi}is a basis and the sum runs over all basis elements, has the property

ihbi|ˆ

A|bii=X

jhgj|ˆ

A|gji,(A.7)

which is true for any other basis {gj}[71].

An integral over nuclei coordinates {RI}is written for short as:

ZdRI:= Z...ZdR1dR2. . . dRNn.(A.8)

A.1.4 Linear response approach to the dynamical matrix

The basic idea (see Ref. [187] for details) behind the method of calculating phonon frequencies from

linear response is that the harmonic force constants are entirely determined by their static linear

electronic response. In fact, within the Born-Oppenheimer approximation, the lattice distortion

associated with a phonon can be seen as a static perturbation acting on the electrons. In Ref. [187],

it is shown that the reciprocal dynamical matrix Dqfor a certain qvector can be calculated from

the linear variation of the external potential ∂v(r)/∂Uq,α, the second derivative of the external

potential ∂2v(r)/(∂Uq,α∂Uq,α) at q= 0, and the linear electronic response ∂ρ(r)/∂Uq,α as:

Del

q, αβ(q) = 1

M1/2

IZ∂ρ(r)

∂Uq,α ∗∂v(r)

∂Uq,β

dr+δij Zρ0(r)∂2v(r)

∂Uq=0,α ∂Uq=0,β

dr.(A.9)

The superscript ”el” indicates that Eq. (A.9) describes only the electronic contribution to the

dynamical matrix. The additional contribution due to the nucleus-nucleus interaction can be

obtained using the Ewald sum. The variation in the external potential v(r) and the electron

density ρ(r) is obtained upon a lattice distortion of the form

Uα(R) = Uq,α exp(iq·R),(A.10)

A.1. Technical details 145

where Ris a primitive lattice vector and where Uα(R) is the displacement of the nucleus at R

into the direction of α(note that, as always in the present thesis, the equations are for elemen-

tary materials). In Eq. (A.9), MIis the mass of the considered element, ρ0(r) is the unper-

turbed electron density, and [ .]∗denotes the conjugate complex. The variations ∂v(r)/∂Uq,α and

∂2v(r)/(∂Uq,α∂Uq,α) follow directly from the specific form of the external potential and the distor-

tion. The calculation of the linear electronic response ∂ρ(r)/∂Uq,α is more complex and requires

the application of density-functional perturbation-theory (DFPT).

In DFPT, the lattice distortion is superimposed on the unperturbed external potential which

produces a change ∆v:

v→v+ ∆v . (A.11)

This perturbation causes in turn a change ∆veff in the effective Kohn-Sham potential,

veff →veff + ∆veff ,(A.12)

which can be used to calculate the change/response in the electron density by means of first order

perturbation theory. Since the veff depends on the the electron density ρthrough the Hartree and

exchange-correlation potential, the change in ρwill in turn affect veff , which has to be recalculated.

The procedure needs to be continued until self consistency is reached and a converged electronic

response ∂ρ(r)/∂Uq,α has been obtained.

A disadvantage of the linear response method is the fact that for each qvector a separate

Kohn-Sham calculation is needed. However, no supercell has to be introduced as is the case for

the direct force constant method. It suffices to employ the primitive unit cell to obtain the linear

electronic response at a certain qpoint. The actual number of calculations is determined by the

(converged) density of the qvectors inside the first Brillouin zone and the given crystal symmetries.

For crystals with small unit cells, the linear response method is more convenient than the direct

force constant method. On the other hand, for sufficiently large primitive unit cells the direct force

constant method is better suited. More specifically, ”sufficient” means that the primitive unit cell

has reached a size for which finite size effects are negligible. In such a case, the supercell can be

set equal to the primitive unit cell which favors the direct force constant approach.

An important technical point is the complexity inherent in a DFPT implementation, which

requires profound methodological and theoretical skill and is in general very time consuming and

error-prone. In contrast, the implementation of the direct force constant method, which can be

performed ”on top” of an existing ab initio code, is relatively straightforward.

146 A.2. Frequently used notation

A.2 Frequently used notation

In Tabs. A.1 and A.2 (on the following pages), we summarize frequently used abbreviations and

symbols, respectively.

Table A.1: Frequently used abbreviations.

Abbreviation Description

ah (explicitly) anharmonic (contribution)

augGrid augmentation grid; for augmentation charges in the PAW method, Sec. 3.4.1

basicGrid basic grid; used for representing plane waves, Sec. 3.4.1

BZ first Brillouin zone

DFT density functional theory

DFC direct force constant (method); for the calculation of the dynamical matrix

EAM embedded atom method

EOS equation-of-state

fcc face-centered-cubic (lattice)

FEBO free energy Born-Oppenheimer (approximation); Sec. 2.1.3

FES (electronic) free energy surface

GGA generalized gradient approximation

gp/atom grid points per atom; unit for the augmentation grid, Sec. 3.4.1

kp·atom kpoints times number of atoms in the supercell; Sec. 2.1.7

LAPW+lo linearized augmented plane waves plus local orbitals

LDA local density approximation

LR linear response (method);for the calculation of the dynamical matrix

MD molecular dynamics

PAW projector augmented wave (method)

PES (electronic) potential energy surface

PBE Perdew-Burke-Ernzerhof (parametrization) [37]; for the GGA functional

pp pseudopotential

prBZ first Brillouin zone of the primitive unit cell; Sec. 2.3.2

qh quasiharmonic (contribution)

sc supercell

scBZ first Brillouin zone of the supercell; Sec. 2.3.2

TILD thermodynamic integration based Langevin dynamics; Sec. 2.3.6

UP-TILD upsampled TILD (method); Sec. 3.2.2

vac vacancy

xc exchange-correlation (functional)

A.2. Frequently used notation 147

Table A.2: Important and frequently used symbols listed in the order: small Latin, capitalized Latin, small

Greek, and capitalized Greek characters.

Symbol Description

a0Bohr radius

a1,a2,a3primitive lattice vectors, Eq. (2.76)

a′

1,a′

2,a′

3conventional fcc unit cell lattice vectors, Eq. (2.115)

alat,alat,eq (equilibrium) lattice constant

b1,b2,b3vectors spanning the first BZ of the primitive lattice, Eq. (2.79)

bsc

1,bsc

2,bsc

3vectors spanning the first BZ of the supercell lattice, Eq. (2.122)

cvacancy concentration

cν,k+Gexpansion coefficients of the Kohn-Sham wave function in reciprocal space

eelementary charge

fi,fν,kKohn-Sham occupation numbers, Eq. (2.65) and Sec. 2.1.7

femb embedding function of an EAM parametrization

~reduced Planck constant

kelectronic wave vector inside first BZ

kBBoltzmann constant

meelectron mass

nnumber of vacancies

nsc size of supercell in one dimension, Eq. (2.117)

pνstatistical weights of electronic wave function

qwave vector of nuclei displacements

ricoordinates of electrons

s,sel,sdft,

aa.sks,sqh

scaling functions used to indicate the computational requirements of a problem;

see Sec. A.1.1

s1,s2,s3supercell lattice vectors, Eq. (2.114)

vexternal potential, Eq. (2.49)

veff,veff

Geffective Kohn-Sham potential, in real and reciprocal space, Eq. (2.58) and

Sec. 2.1.7

vfvolume of vacancy formation, Eq. (3.25)

vpair pair potential of an EAM parametrization

wGsc

, s,wGsc

, s eigenvectors of the reciprocal dynamical matrix DGsc and its components

BS,BTadiabatic, i.e., at constant entropy S, and isothermal bulk modulus, Eq. (2.229)

BT,0,B′

T,0BTand the pressure derivative of BTat T= 0 K

Continued.

148 A.2. Frequently used notation

Table A.2 (continued).

Symbol Description

CPisobaric heat capacity, i.e., at constant pressure P, Eq. (2.229)

CV,CV0heat capacity at constant and fixed volume; see Sec. 2.4.2

D,DIα,Jβ real space dynamical matrix and its elements, Eq. (2.132)

D0K,D0K

Iα,Jβ T= 0 K approximation to the real space dynamical matrix, Eq. (2.179)

DGsc ,DGsc

, αβ reciprocal dynamical matrix at wave vector Gsc and its elements, Eq. (2.141)

D0K

Gsc ,D0K

Gsc

, αβ T= 0 K approximation to the reciprocal dynamical matrix, Eq. (2.141)

Eξeigenvalues of the main Hamiltonian ˆ

Eωenergy corresponding to the phonon frequency ω

Ecut plane wave cutoff energy

Ecut,high,Ecut,low highly and low converged cutoff, used for the development of the UP-TILD

method (Sec. 3.2)

Efenergy of vacancy formation

Eel

νeigenvalues of the electronic Hamiltonian ˆ

Hel

Eel

g({RI}), Eel

g,0T= 0 K ground state of ˆ

Hel and Eel

g,0=Eel

g({R0K

I})

Eh[ρ], Exc[ρ]Hartree [Eq. (2.60)] and exchange-correlation [Sec. 2.1.6] energy functional

Enuc

ν,µ eigenvalues of the nuclei Hamiltonian ˆ

Hnuc

Ezp,Ezp,0kenergy of zero-point vibrations [Eq. (2.161)] and its D0kapproximation

F(V, T )full free energy surface including all excitations, Eq. (2.2)

FPfree energy at constant pressure P

FPFPreferenced with respect to FP(T= 200K) to allow a convenient compar-

ison with calphad data, Eqs. (4.5) and (4.6)

∆FPFPreferenced at each Twith respect to the calphad free energy, Eq. (4.7)

Fcconfigurational free energy of point defects, Eq. (3.7)

Fclas,ah classical explicitly anharmonic free energy, Eq. (2.195)

Fp,Fvfree energy of the perfect crystal and vacancy supercell

Fp,clas,ah

,Fv,clas,ah Fclas,ah for the perfect crystal and the vacancy supercell

Fel({RI}), Fel

0electronic free energy and Fel

0=Fel({R0K

I})

Fel

0temperature dependent part of Fel

0, Eq. (3.53)

Fel[{pν},{ψν}]free energy functional, Eq. (2.41)

Fqh,Fqh,0kquasiharmonic free energy [Eq. (2.161)] and its D0kapproximation

Fvac free energy contribution due to vacancies, Sec. 3.1.3

Continued.

A.2. Frequently used notation 149

Table A.2 (continued).

Symbol Description

Fhf

I,Fhf

I,α Hellmann-Feynman forces and their components, Eq. (2.164)

Fhf,0k

I,Fhf,0k

I,α Fhf

Iand Fhf

I,α within the D0kapproximation, Eq. (2.179)

F[ρ], Fks[ρ]main DFT functional, Eq. (2.48), and the Kohn-Sham functional, Eq. (2.55)

G,Gsc reciprocal lattice vector of the primitive unit cell and the supercell, Eqs. (2.78)

and (2.121)

Hmain Hamiltonian including all interactions, Eq. (2.6); Ψξ,Eξ

Hel electronic Hamiltonian, Eq. (2.15); ψν,Eel

Hks Kohn-Sham Hamiltonian, Eq. (2.63); ϕi,ǫi

Hnuc

νnuclei Hamiltonian, Eq. (2.17); Λν,µ,Enuc

ν,µ

Hqh

,ˆ

H3ord

,ˆ

H4ord quasiharmonic, 3. and 4. order Hamiltonian, Eqs. (2.135), (2.209), and (2.210)

Hrks reduced Kohn-Sham Hamiltonian, Eq. (2.101)

MImass of nucleus I

Ne,Nnnumber of electrons and nuclei

NV,NTnumber of Vand Tpoints used to parametrize free energy surfaces

Nk,Nirr

knumber of (irreducible) kvectors used for the ksampling

Nhigh,Nlow number of kvectors for a highly and low converged ksampling, used for the

development of the UP-TILD method (Sec. 3.2)

Nmesh number of mesh points (1-dim); used for estimations of the CPU requirement

Nit,Nit,red number of electronic iterations in a Kohn-Sham calculation and the reduced

number (wave function extrapolation; see Sec. 2.3.6)

Npnumber of atoms in the perfect crystal [volume optimized approach; Eq. (3.9)]

Nvnumber of atoms in the vacancy supercell

Nld number of steps in a Langevin dynamics simulation

Npw number of plane waves

O(U3)terms of third and higher order in the Taylor expansion of Fel

Ppressure

PGsc ,PGsc

, s momentum of plane wave with vector Gsc and its components (branches)

PGsc ,ˆ

PGsc

, s quantum mechanical version of PGsc

Rprimitive lattice vector, Eq. (2.75)

RI,R0K

Icoordinates of nuclei and their T= 0 K equilibrium positions

∆Rdisplacement used for finite differences

Sentropy

Continued.

150 A.2. Frequently used notation

Table A.2 (continued).

Symbol Description

Sqh,Sqh,0kquasiharmonic entropy [Eq. (2.161)] and its D0kapproximation

Sfentropy of vacancy formation

Ttemperature

Tel,Tnuc electronic and nuclei temperature, Sec. 3.3.4

Tel,ˆ

Tnuc electronic and nuclei kinetic energy operator, Eqs. (2.7) and (2.8)

Tmmelting temperature

UI,UI,α classical displacement of nucleus Iout of R0K

Iand its components; U(R0K

I) is

also used

UGsc ,UGsc

, s classical reciprocal displacement of nuclei and its components (branches)

UGsc ,ˆ

UGsc

, s quantum mechanical version of UGsc

Vvolume per atom; equivalent to volume of primitive unit cell

Veq

0equilibrium volume at T= 0 K

Veq(T)thermal expansion of the equilibrium volume

Vel electron-electron repulsion operator, Eq. (2.9)

Ve-n electron-nucleus attraction operator, Eq. (2.11)

Vnuc Ewald contribution, scalar, Eq. (2.50)

Vnuc nucleus-nucleus repulsion operator, Eq. (2.10)

Zfull partition function including all excitations, Eq. (2.3)

ZIproton number of nucleus I

αthermal expansion coefficient, Eq. (2.235)

βthermodynamic beta, β= 1/(kBT)

εrelative thermal expansion, Eq. (2.234)

ǫ0electric constant

ǫi,ǫν,keigenvalues of the Kohn-Sham Hamiltonian ˆ

Hks in real and reciprocal space

ζfriction parameter in a Langevin dynamics simulation

λcoupling parameter in the thermodynamic integration method

ρelectronic charge density, Eq. (2.45)

ρat atomic charge density for an EAM parametrization

ρaug augmentation electronic charge density (PAW method)

σerr statistical error in molecular/Langevin dynamics calculations, Eq. (2.202)

ϕi,ϕν,keigenfunctions of the Kohn-Sham Hamiltonian ˆ

Hks in real and reciprocal space

Continued.

A.2. Frequently used notation 151

Table A.2 (continued).

Symbol Description

ψνeigenfunction of electronic Hamiltonian ˆ

Hel

ωq,s phonon dispersion

ωGsc

, s,ωiphonon frequency at wave vector Gsc and branch s

ω0k

Gsc

, s,ω0k

ifrequencies within the D0kapproximation

∆ωqm,ah

Gsc

, s quantum mechanical anharmonic phonon shift, Eq. (2.213)

∆ω3ord

Gsc

, s, ∆ω4ord

Gsc

, s third and fourth order phonon shift, Eqs. (2.214) and (2.215)

Λν,µ eigenfunctions of nuclei Hamiltonian ˆ

Hnuc

Φ3ord, Φ4ord third and fourth order real space tensor, Eqs. (2.206) and (2.207), with

components Φ3ord

Iα,Jβ,Kγ and Φ4ord

Iα,Jβ,Kγ,Lδ and with the Fourier transformed

elements Φ3ord

Gscs, Gsc′s′,Gsc′′s′′ and Φ4ord

Gscs, Gsc′s′,Gsc′′s′′,Gsc′′′s′′′ , Eqs. (2.211) and

(2.212)

Ψξeigenfunctions of the main Hamiltonian ˆ

Ωvolume of a supercell

ΩIvolume of a sphere around atom I; PAW and LAPW+lo method

Ωpvolume of the perfect crystal in the volume optimized approach, Eq. (3.8)

Ωvvolume of the vacancy supercell

Bibliography

[1] Z. Zhao, F. Roters, W. Mao, and D. Raabe, Adv. Eng. Mater. 3, 984 (2001).

[2] D. Raabe and F. Roters, Int. J. Plasticity 20, 339 (2004).

[3] Prof. W. Bleck, Head of the Department of Ferrous Metallurgy and Speaker of the Collabo-

rative Research Center ”Stahl – ab initio” (SFB761), University RWTH Aachen, Germany;

private communication (04/19/2009).

[4] Prof. G. Gottstein, President of BV MatWerk, the German Umbrella Association of Materials

Science, and Head of the Institute of Physical Metallurgy and Metal Physics, University

RWTH Aachen, Germany; private communication (04/21/2009).

[5] Dr. H.-P. Schmitz, Center of Materials Excellence, ThyssenKrupp Steel AG; private commu-

nication (04/27/2009).

[6] Prof. J. Hirsch, Research and Development Center, Hydro Aluminium Deutschland GmbH;

private communication (07/13/2009).

[7] L. Kaufman and H. Bernstein, Computer Calculation of Phase Diagrams (Academic Press,

New York, 1970).

[8] T. B. Massalski, Metall. Trans. A 20, 1295 (1989).

[9] P. E. A. Turchi, I. A. Abrikosov, B. Burton, S. G. Fries, G. Grimvall, et al., CALPHAD 31,

4 (2007).

[10] J. L. Basdevant and J. Dalibard, The Quantum Mechanics Solver: How to Apply Quantum

Theory to Modern Physics (Springer, Berlin / Heidelberg, 2006).

[11] K. Ohno, K. Esfarjani, and Y. Kawazoe, Computational Materials Science: From Ab Initio

to Monte Carlo Methods (Springer, Berlin, 1999).

[12] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

[13] D. A. Ditmars, C. A. Plint, and R. C. Shukla, Int. J. Thermophys. 6, 499 (1985).

[14] Y. S. Touloukian, R. K. Kirby, R. E. Taylor, and P. D. Desai, Thermal Expansion: Metallic

Elements and Alloys, vol. 12 of Thermophysical Properties of Matter (IFI/Plenum, New York,

1975).

[15] G. N. Kamm and G. A. Alers, J. Appl. Phys. 35, 327 (1964).

152

[16] F. A. Berezin and M. A. Shubin, The Schr¨odinger Equation (Kluwer Ac. Publ., Dordrecht /

Boston / London, 1991).

[17] I. W. Sudiarta and D. J. W. Geldart, J. Phys. A: Math. Theor. 40, 1885 (2007).

[18] G. Farin and D. Hansford, Practical Linear Algebra: A Geometry Toolbox (A K Peters,

Wellesley, 2005).

[19] D. V. de Jager and J. T. Bradley, PageRank: Splitting Homogeneous Singular Linear Systems

of Index One, ICTIR (2009), pp. 17-28.

[20] I. Ipsen and R. M. Wills, Analysis and Computation of Google’s PageRank, 7th IMACS Inter-

national Symposium on Iterative Methods in Scientific Computing, Fields Institute, Toronto,

Canada, 5.-8. May 2005, URL http://www4.ncsu.edu/∼ipsen/ps/slides imacs.pdf.

[21] M. Born and J. R. Oppenheimer, Ann. Phys. 389, 457 (1927).

[22] B. Grabowski, diploma thesis, Universit¨at Paderborn (2005).

[23] D. Marx and J. Hutter, Mod. Methods Algorithms Quantum Chem. 1, 301 (2000).

[24] M. Suzuki, Commun. Math. Phys. 57, 193 (1977).

[25] J. Cao and B. J. Berne, J. Chem. Phys. 99, 2902 (1993).

[26] N. D. Mermin, Phys. Rev. 137, A1441 (1965).

[27] M. Levy, Proc. Nat. Acad. Sci. USA 76, 6062 (1979).

[28] E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983).

[29] R. G. Parr and W. Young, Density-Functional Theory of Atoms and Molecules (Oxford

Science Publications, New York, 1989).

[30] M. J. Gillan, J. Phys. Condens. Matter 1, 689 (1989).

[31] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).

[32] M. Levy, Phys. Rev. A 26, 1200 (1982).

[33] C. A. Ullrich and W. Kohn, Phys. Rev. Lett. 87, 093001 (2001).

[34] H. Englisch and R. Englisch, Physica A 121, 253 (1983).

[35] M. Levy and J. P. Perdew, in Density Functional Methods in Physics, edited by R. M. Dreizler

and J. Providencia (Plenum Press, New York, 1985), pp. 11–30.

[36] K. Capelle, Braz. J. Phys. 36, 1318 (2006).

[37] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).

[38] S. Curtarolo, D. Morgan, and G. Ceder, CALPHAD 29, 163 (2005).

[39] J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).

[40] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).

153

[41] U. von Barth, Phys. Scr. T109, 9 (2004).

[42] J. P. Perdew and S. Kurth, in A Primer in Density Functional Theory, edited by C. Fiolhais,

F. Noqueira, and M. A. L. Marques (Springer, Berlin, 2003), pp. 1–55.

[43] N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New

York, 1976).

[44] F. Bloch, Z. Phys. A 52, 555 (1928).

[45] C. Kittel, Introduction to Solid State Physics (Wiley, 2004).

[46] G. Kresse and J. Furthm¨uller, Phys. Rev. B 54, 11169 (1996).

[47] M. Bockstedte, A. Kley, J. Neugebauer, and M. Scheffler, Comput. Phys. Commun. 107, 187

(1997).

[48] E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, New York, 1974).

[49] S. Cottenier, Density Functional Theory and the Family of (L)APW-Methods: A Step-by-

Step Introduction (Instituut voor Kern- en Stralingsfysica, K.U.Leuven, Belgium, 2002), URL

http://www.wien2k.at/reg user/textbooks/.

[50] S. Bl¨ugel and G. Bihlmayer, in Computational Nanoscience: Do It Yourself, edited by J. Gro-

tendorst, S. Bl¨ugel, and D. Marx (John von Neumann Institite for Computing, J¨ulich, 2006),

pp. 85–129.

[51] P. E. Bl¨ochl, Phys. Rev. B 50, 17953 (1994).

[52] N. A. W. Holzwarth, A. R. Tackett, and G. E. Matthews, Comput. Phys. Commun. 135, 329

(2001).

[53] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).

[54] M. Marsman and G. Kresse, J. Chem. Phys. 125, 104101 (2006).

[55] M. Torrent, F. Jollet, F. Bottin, G. Z´erah, and X. Gonze, Comput. Mater. Sci. 42, 337 (2008).

[56] M. Fuchs and M. Scheffler, Comput. Phys. Commun. 119, 67 (1999).

[57] L. Kleinman and D. M. Bylander, Phys. Rev. Lett. 48, 1425 (1982).

[58] P. E. Bl¨ochl, C. J. F¨orst, and J. Schimpl, Bull. Mater. Sci. 26, 33 (2003).

[59] P. M. A. Sloot, D. Abramson, A. V. Bogdanov, J. J. Dongarra, A. Y. Zomaya, et al., Com-

putational Science - ICCS 2003 (Springer, Berlin / Heidelberg / New York, 2003).

[60] M. Upmanyu, Z. T. Trautt, and B. B. Kappes, Mater. Sci. Forum 467-470, 715 (2004).

[61] M. S. Daw, S. M. Foiles, and M. I. Baskes, Mater. Sci. Rep. 9, 251 (1993).

[62] J. Mei and J. W. Davenport, Phys. Rev. B 46, 21 (1992).

[63] R. R. Zope and Y. Mishin, Phys. Rev. B 68, 024102 (2003).

[64] F. Ercolessi and J. B. Adams, Europhys. Lett. 26, 583 (1994).

154

[65] B. Lee and K. Cho, Surf. Sci. 600, 1982 (2006).

[66] K. Kadau, T. C. Germann, P. S. Lomdahl, R. C. Albers, J. S. Wark, et al., Phys. Rev. Lett.

98, 135701 (2007).

[67] J. E. Drut, R. J. Furnstahl, and L. Platter, arXiv:0906.1463v2 [nucl-th] (2009), URL

arxiv.org/abs/0906.1463/.

[68] D. C. Wallace, Thermodynamics of Crystals (Dover, New York, 1998).

[69] M. Born, Dynamik der Kristallgitter (Leipzig / Berlin, 1915).

[70] M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford, London, 1954).

[71] L. E. Ballentine, Quantum Mechanics: A Modern Development (World Scientific, London,

1998).

[72] K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997).

[73] H. Hellmann, Einf¨uhrung in die Quantenchemie (Deuticke, Leipzig, 1937).

[74] R. P. Feynman, Phys. Rev. 56, 340 (1939).

[75] J. Harris, R. O. Jones, and J. E. M¨uller, J. Chem. Phys. 75, 3904 (1981).

[76] P. Pulay, Mol. Phys. 17, 197 (1969).

[77] W. Nolting, Statistische Physik, Grundkurs Theoretische Physik (Springer, Berlin / Heidel-

berg, 2002).

[78] D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applica-

tions (Academic Press, San Diego, 2002).

[79] J. M. Haile, Molecular Dynamics Simulation: Elementary Methods (Wiley, New York, 1997).

[80] S. Boeck et. al, http://www.sphinxlib.de/.

[81] L. Ismer, Ph.D. thesis, Universit¨at Paderborn (2008).

[82] W. F. van Gunsteren and H. J. C. Berendsen, Mol. Phys. 45, 637 (1982).

[83] L. Verlet, Phys. Rev. 159, 98 (1967).

[84] A. Bergstr¨om, Normat 20, 138 (1972).

[85] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill,

New York, 1971).

[86] M. Zoli, Phys. Rev. B 41, 7497 (1989).

[87] H. V. Keer, Principles of the Solid State (Wiley, New York, 1993).

[88] P. A. Varotsos and K. D. Alexopoulos, Thermodynamics of Point Defects and Their Relation

with Bulk Properties, vol. 14 of Defects in Solids (North-Holland, Amsterdam, 1986).

155

[89] K. M. Carling, G. Wahnstr¨om, T. R. Mattsson, N. Sandberg, and G. Grimvall, Phys. Rev. B

67, 054101 (2003).

[90] S. Nos´e, J. Chem. Phys. 81, 511 (1984).

[91] W. G. Hoover, Phys. Rev. A 31, 1695 (1985).

[92] W. Nolting, Spezielle Relativit¨atstheorie, Thermodynamik, Grundkurs Theoretische Physik

(Springer, Berlin / Heidelberg / New York, 2002).

[93] M. Forsblom, N. Sandberg, and G. Grimvall, Phys. Rev. B 69, 165106 (2004).

[94] F. D. Murnaghan, Proc. Nat. Acad. Sci. USA 30, 244 (1944).

[95] F. Birch, Phys. Rev. 71, 809 (1947).

[96] P. Vinet, J. Ferrante, J. H. Rose, and J. R. Smith, J. Geophys. Res. [Solid Earth Planets]

92, 9319 (1987).

[97] R. E. Cohen, O. G¨ulseren, and R. J. Hemley, Am. Mineral. 85, 338 (2000).

[98] M. Methfessel and A. T. Paxton, Phys. Rev. B 40, 3616 (1989).

[99] G. Kresse, M. Marsman, and J. Furthm¨uller, Vienna Ab Initio Simulation Package: VASP

the Guide, URL cms.mpi.univie.ac.at/vasp/vasp.pdf.

[100] S. de Gironcoli, Phys. Rev. B 51, 6773 (1995).

[101] K.-H. Hellwege and J. L. Olsen, eds., Metals: Phonon States, Electron States and Fermi

Surfaces, Landolt-B¨ornstein (Springer, Berlin, 1981).

[102] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz, WIEN2k, An Augmented Plane

Wave + Local Orbitals Program for Calculating Crystal Properties (Technische Universit¨at

Wien, Austria, 2001).

[103] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976).

[104] X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux, M. Fuchs, et al., Comput. Mater. Sci. 25,

478 (2002).

[105] X. Gonze, G.-M. Rignanese, M. Verstraete, J.-M. Beuken, Y. Pouillon, et al., Z. Kristallogr.

220, 558 (2005).

[106] X. Gonze, Phys. Rev. B 55, 10337 (1997).

[107] X. Gonze and C. Lee, Phys. Rev. B 55, 10335 (1997).

[108] D. R. Hamann, Phys. Rev. B 40, 2980 (1989).

[109] A. Debernardi, M. Alouani, and H. Dreyss´e, Phys. Rev. B 63, 064305 (2001).

[110] A. A. Quong and A. Y. Liu, Phys. Rev. B 56, 7767 (1997).

[111] A. Siegel, K. Parlinski, and U. D. Wdowik, Phys. Rev. B 74, 104116 (2006).

[112] V. L. Moruzzi, J. F. Janak, and K. Schwarz, Phys. Rev. B 37, 790 (1988).

156

[113] P. A. Tipler, Physik (Spektrum, Heidelberg, 1994).

[114] J. A. Rayne, Phys. Rev. 118, 1545 (1960).

[115] E. Walker, J. Ashkenazi, and M. Dacorogna, Phys. Rev. B 24, 2254 (1981).

[116] R. E. Macfarlane, J. A. Rayne, and C. K. Jones, Phys. Lett. 20, 234 (1966).

[117] J. H. Rose, J. R. Smith, F. Guinea, and J. Ferrante, Phys. Rev. B 29, 2963 (1984).

[118] A. Eichler, K.-P. Bohnen, W. Reichardt, and J. Hafner, Phys. Rev. B 57, 324 (1998).

[119] R. Heid, K.-P. Bohnen, K. Felix, K. M. Ho, and W. Reichardt, J. Phys. Condens. Matter 10,

7967 (1998).

[120] S. Y. Savrasov and D. Y. Savrasov, Phys. Rev. B 54, 16487 (1996).

[121] F. Favot and A. dal Corso, Phys. Rev. B 60, 11427 (1999).

[122] S. Narasimhan and S. de Gironcoli, Phys. Rev. B 65, 064302 (2002).

[123] L. Voˇcadlo, D. Alf`e, G. D. Price, and M. J. Gillan, J. Chem. Phys. 120, 2872 (2004).

[124] J. Xie, S. de Gironcoli, S. Baroni, and M. Scheffler, Phys. Rev. B 59, 965 (1999).

[125] A. A. Quong, Phys. Rev. B 49, 3226 (1994).

[126] R. Bauer, A. Schmid, P. Pavone, and D. Strauch, Phys. Rev. B 57, 11276 (1998).

[127] T. Takezawa, H. Nagara, and N. Suzuki, Phys. Rev. B 71, 012515 (2005).

[128] I. Y. Sklyadneva, A. Leonardo, P. M. Echenique, S. V. Eremeev, and E. V. Chulkov,

J. Phys. Condens. Matter 18, 7923 (2006).

[129] R. Heid, K.-P. Bohnen, and K. M. Ho, Phys. Rev. B 57, 7407 (1998).

[130] L. Voˇcadlo and D. Alf`e, Phys. Rev. B 65, 214105 (2002).

[131] A. dal Corso, J. Phys. Condens. Matter 20, 445202 (2008).

[132] Y. S. Touloukian and E. H. Buyco, Specific Heat - Metallic Elements and Alloys, vol. 4 of

Thermophysical Properties of Matter (IFI/Plenum, New York, 1970).

[133] P. E. A. Turchi, V. Drchal, and J. Kudrnovsky, Phys. Rev. B 74, 064202 (2006).

[134] D. L. Waldorf and G. A. Alers, J. Appl. Phys. 33, 3266 (1962).

[135] W. C. Overton and J. Gaffney, Phys. Rev. 98, 969 (1955).

[136] J. R. Neighbours and G. A. Alers, Phys. Rev. 111, 707 (1958).

[137] D. K. Hsu and R. G. Leisure, Phys. Rev. B 20, 1339 (1979).

[138] S. M. Collard and R. B. McLellan, Acta Metall. Mater 40, 699 (1992).

[139] Thermo-Calc Software, Stockholm, URL http://www.thermocalc.com/.

157

[140] A. van de Walle and G. Ceder, Phys. Rev. B 59, 14992 (1999).

[141] A. dal Corso and S. de Gironcoli, Phys. Rev. B 62, 273 (2000).

[142] M. Born and E. Brody, Z. Phys. 6, 132 (1921).

[143] T. E. Pochapsky, Acta Metall. 1, 747 (1953).

[144] Y. Kraftmakher, Phys. Rep. 299, 79 (1998).

[145] C. R. Brooks and R. E. Bingham, J. Phys. Chem. Solids 29, 1553 (1968).

[146] R. C. Shukla, C. A. Plint, and D. A. Ditmars, Int. J. Thermophys. 6, 517 (1985).

[147] D. Alf`e, G. A. de Wijs, G. Kresse, and M. J. Gillan, Int. J. Quantum Chem. 77, 871 (2000).

[148] P. J. H. Denteneer and J. M. Soler, J. Phys. Condens. Matter 3, 8777 (1991).

[149] T. Gomoll, diploma thesis, Technische Fachhochschule Berlin (2008).

[150] P. H. Dederichs, C. Lehmann, H. R. Schober, A. Scholz, and R. Zeller, J. Nucl. Mater. 69/70,

176 (1978).

[151] A. De Vita and M. J. Gillan, J. Phys. Condens. Matter 3, 6225 (1991).

[152] R. Benedek, L. H. Yang, C. Woodward, and B. I. Min, Phys. Rev. B 45, 2607 (1992).

[153] K. M. Carling, G. Wahnstr¨om, T. R. Mattsson, A. E. Mattsson, N. Sandberg, et al.,

Phys. Rev. Lett. 85, 3862 (2000).

[154] R. O. Simmons and R. W. Balluffi, Phys. Rev. 117, 52 (1960).

[155] T. Hehenkamp, J. Phys. Chem. Solids 55, 907 (1994).

[156] W. Triftsh¨auser, Phys. Rev. B 12, 4634 (1975).

[157] E. D. Eastman, A. M. Williams, and T. F. Young, J. Am. Chem. Soc. 46, 1178 (1924).

[158] S. Umino, Sci. Rep. Tohoku Imp. Univ., Ser. 1 15, 597 (1926), (cited from Ref. [168]).

[159] A. Avramescu, Z. Tech. Phys. (Leipzig) 20, 213 (1939).

[160] K. Honda and Y. Okubo, Sci. Rep. Tohoku Imp. Univ., Ser. 1 13, 101 (1924), (cited from

Ref. [14]).

[161] F. L. Uffelmann, Philos. Mag. 10, 633 (1930), (cited from Ref. [14]).

[162] F. C. Nix and D. MacNair, Phys. Rev. 60, 597 (1941).

[163] A. J. C. Wilson, Proc. Phys. Soc. 54, 487 (1942).

[164] R. A. McDonald, J. Chem. Eng. Data 12, 115 (1967).

[165] A. J. Leadbetter, J. Phys. C: Solid State Phys. 1, 1481 (1968).

[166] U. Schmidt, O. Vollmer, and R. Kohlhaas, Z. Naturforsch. A A25, 1258 (1970).

158

[167] D. I. Marchidan and M. Ciopec, Rev. Roum. Chim. 15, 1005 (1970), (cited from Ref. [168]).

[168] P. D. Desai, Int. J. Thermophys. 8, 621 (1987).

[169] Y. Takahashi, T. Azumi, and Y. Sekine, Thermochim. Acta 139, 133 (1989).

[170] R. M. Nicklow and R. A. Young, Phys. Rev. 129, 1936 (1963), (cited from Ref. [14]).

[171] P. G. Strelkov and S. I. Novikova, Prib. Tekh. Eksp. (USSR) 5, 105 (1957), (cited from

Ref. [14]).

[172] P. D. Pathak and N. G. Vasavada, J. Phys. C: Solid State Phys. 3, L44 (1970), (cited from

Ref. [14]).

[173] B. von Gu´erard, H. Peisl, and R. Zitzmann, Appl. Phys. 3, 37 (1974), numerical differentation

of the given experimental expansion data; strongly scattering points were taken out.

[174] F. R. Kroeger and C. A. Swenson, J. Appl. Phys. 48, 853 (1977).

[175] K. O. McLean, Ph.D. thesis, Iowa State University (1969), (cited from Ref. [14]).

[176] M. Zoli, Philos. Mag. Lett. 62, 203 (1990).

[177] O. F. Sankey and D. J. Niklewski, Phys. Rev. B 40, 3979 (1989).

[178] G. Grimvall, Physics 2, 28 (2009).

[179] F. Perrot and M. W. C. Dharma-Wardana, Phys. Rev. A 30, 2619 (1984).

[180] D. G. Kanhere, P. V. Panat, A. K. Rajagopal, and J. Callaway, Phys. Rev. A 33, 490 (1986).

[181] D. C. Langreth, M. Dion, H. Rydberg, E. Schr¨oder, P. Hyldgaard, and B. I. Lundqvist,

Int. J. Quantum Chem. 101, 599 (2004).

[182] F. K¨ormann, A. Dick, B. Grabowski, B. Hallstedt, T. Hickel, and J. Neugebauer, Phys. Rev. B

78, 033102 (2008).

[183] F. K¨ormann, A. Dick, T. Hickel, and J. Neugebauer, Phys. Rev. B 79, 184406 (2009).

[184] W. A. Counts, M. Friak, D. Raabe, and J. Neugebauer, Acta Mater. 57, 69 (2009).

[185] M. Friak, W. A. Counts, D. Raabe, and J. Neugebauer, Phys. Status Solidi B 245, 2636

(2008).

[186] I. Mills, T. Cvitas, K. Homann, N. Kallay, and K. Kuchitsu, Quantities, Units and Symbols

in Physical Chemistry (Blackwell Science, 1993).

[187] P. Giannozi, S. Gironcoli, P. Pavone, and S. Baroni, Phys. Rev. B 43, 7231 (1991).

159

List of Publications

1. B. Grabowski, L. Ismer, T. Hickel, and J. Neugebauer, ”Ab initio up to the melting point:

Anharmonicity and vacancies in aluminum”, Phys. Rev. B 79, 134106 (2009).

See also: G. Grimvall, Physics 2, 28 (2009).

2. J. Neugebauer, B. Grabowski, F. K¨ormann, A. Dick, J. von Pezold, M. Friak, and T. Hickel,

”Ab initio based multiscale modeling of engineering materials: From a predictive thermody-

namic description to tailored mechanical properties”, Asia Steel 2009 Proceedings (2009).

3. T. Hickel, A. Dick, B. Grabowski, F. K¨ormann, and J. Neugebauer, ”Steel design from fully

parameter-free ab initio computer simulations”, Steel Res. Int. 80, 4 (2009).

4. D. Lencer, M. Salinga, B. Grabowski, T. Hickel, J. Neugebauer, and M. Wuttig, ”A map for

phase-change materials”, Nat. Mater. 7, 972 (2008).

5. F. K¨ormann, A. Dick, B. Grabowski, B. Hallstedt, T. Hickel, and J. Neugebauer, ”Free

energy of bcc iron: Integrated ab initio derivation of vibrational, electronic, and magnetic

contributions”, Phys. Rev. B 78, 033102 (2008).

6. T. Hickel, M. Uijttewaal, B. Grabowski, and J. Neugebauer, ”Determination of symmetry

reduced structures using a soft phonon analysis for magnetic shape memory alloys”, J. Phys.:

Condens. Matter 20, 064219 (2008).

7. B. Grabowski, T. Hickel, and J. Neugebauer, ”Ab initio study of the thermodynamic prop-

erties of nonmagnetic elementary fcc metals: Exchange-correlation-related error bars and

chemical trends”, Phys. Rev. B 76, 024309 (2007).