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Cite this: Phys. Chem. Chem. Phys.,
2023, 25, 14799
Computational workflows for perovskites: case
study for lanthanide manganites†
Peter Kraus, *
ab
Paolo Raiteri
b
and Julian D. Gale
b
Robust computational workflows are important for explorative computational studies, especially for
cases where detailed knowledge of the system structure or other properties is not available. In this work,
we propose a computational protocol for appropriate method selection for the study of lattice
constants of perovskites using density functional theory, based strictly on open source software. The
protocol does not require a starting crystal structure. We validate this protocol using a set of crystal
structures of lanthanide manganites, surprisingly finding N12+Uto be the best performing method for
this class of materials out of the 15 density functional approximations studied. We also highlight that +U
values derived from linear response theory are robust and their use leads to improved results. We
investigate whether the performance of methods for predicting the bond length of related gas phase
diatomics correlates with their performance for bulk structures, showing that care is required when
interpreting benchmark results. Finally, using defective LaMnO
3
as a case study, we investigate whether
the four shortlisted methods (HCTH120, OLYP, N12+U, PBE+U) can computationally reproduce the
experimentally determined fraction of Mn
IV+
at which the orthorhombic to rhombohedral phase
transition occurs. The results are mixed, with HCTH120 providing good quantitative agreement with
experiment, but failing to capture the spatial distribution of defects linked to the electronic structure of
the system.
1 Introduction
One of the prerequisites for predicting any material properties
using computational methods is to have a good starting point for
the structure of the material. As shown by the series of blind crystal
structure prediction benchmarks, predicting experimentally-
observed crystal structures reliably and without prior informa-
tion about the lattice is still a difficult and open question.
1
However, suppose one wants to determine a property of several
materials from a single material class with a reasonably well-
defined structural motif; for instance the phonon modes of
perovskites. In this case, it may be more beneficial to determine
an energy minimum that sacrifices agreement with the
experimental structure, but allows us to calculate the phonon
spectrum using a trusted method. This is especially important
in density functional theory (DFT) when applied to solid state
systems, as it has been repeatedly shown that the use of a
combination of density functional approximations (DFAs) may
be required to obtain the best possible description of the
material.
2,3
In this work, we discuss a computational workflow for model-
ling structures of perovskites using DFT. Several stipulations are
specified for the workflow:
(1) The structures obtained have to be energy minima. This
ensures that they can be subsequently used to calculate the
second derivatives at the same level of theory, which is impor-
tant for determining most thermochemical properties, a range
of mechanical properties, and the dielectric behaviour of the
system.
(2) Only the crystal system and atomic composition of the
unit cell are known a priori. This ensures the workflow can be
applied to study novel materials for which an experimental
crystal structure is not yet determined.
(3) Only open source tools may be included in the workflow.
This ensures the workflow can be applied by anyone, regardless
of their academic affiliation or if it is for commercial purposes.
The workflow is shown schematically in Fig. 1. Specifically,
we aim to find an affordable DFA capable of predicting the
a
Institute for Material Science and Technology, Technische Universita
¨t Berlin,
Hardenbergstr. 40, 10623 Berlin, Germany.
E-mail: peter.[email protected]
b
Curtin Institute for Computation, School of Molecular and Life Sciences, Curtin
University, GPO Box U1987, Perth, WA 6845, Australia
†Electronic supplementary information (ESI) available: The mash code is avail-
able at https://github.com/PeterKraus/mash. Version 1.0 used in this work is
archived under DOI: https://doi.org/10.5281/zenodo.7492808. The complete code
archive including all Quantum ESPRESSO calculation input and output files, as
well as postprocessing scripts used to generate the figures in this manuscript, is
available on Zenodo under DOI: https://doi.org/10.5281/zenodo.7874704.See
DOI: https://doi.org/10.1039/d3cp00041a
Received 4th January 2023,
Accepted 10th May 2023
DOI: 10.1039/d3cp00041a
rsc.li/pccp
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structures of orthorhombic LnMnO
3
perovskites (where Ln =
{La, Pr, Sm, Eu, Yb}). We choose to use DFT, as the ligand field
effects present in several of the selected materials make the use
of classical potentials more complex.
4
While the workflow may
be used with any DFA, for the systems studied here, the
computation of Hartree–Fock exchange would be prohibitively
expensive. Therefore, our choice of methods is effectively
restricted to the generalised gradient approximation (GGA)
family. We focus on lanthanide manganite perovskites, as they
show interesting catalytic properties
5
and the behaviour of their
dielectric properties in operando is known.
6
The structures
reported here can therefore be re-used in a further study of
the electronic conductivity of these perovskites. Additionally,
structures of their orthorhombic forms have been accurately
determined using neutron powder diffraction,
7–11
or low-
temperature X-ray diffraction,
12
allowing for a comparison with
high-quality experimental data. This results in a case study that
is of a manageable size, yet challenging for electronic structure
calculations due to the presence of lanthanides (relativistic
effects) as well as manganese (multiple spin states). Finally,
we verify that DFAs that are able to predict the crystal structures
of stoichiometric perovskites with a reasonable accuracy con-
tinue to do so for defect structures, as well as rhombohedral
analogues. To address this last point, we focus on the defect-
induced orthorhombic to rhombohedral phase transition in
LaMnO
3
, which has been studied experimentally in detail.
7,13
2 State of the art
Of the five perovskites investigated here, the LaMnO
3
is certainly
the most computationally studied material. The series of publica-
tions by Gavin and Watson is perhaps the most relevant here: the
first publication focuses on determining the best DFA for deter-
mining lattice parameters and electronic structure of the stoichio-
metric perovskite,
14
before applying their methodology (PBEsol+U)
to study surfaces and oxygen defects.
15,16
Our work differs from
theirs in two key points: (i) we do not use the experimental crystal
structure as a starting point, and (ii) we investigate the bulk effects
caused by La-vacancies, as opposed to surface effects caused by
oxygen defects. In fact, several other computational studies (using
PW91) focus on the surface of the perovskite
17,18
rather than its
bulk behaviour, which is not surprising given its catalytic activity.
5
The computational studies of PrMnO
3
include a study of the
electronic structures, surface relaxation, and oxygen defect for-
mation energies of its cubic and orthorhombic structures (using
PW91+U),
19
as well as a particularly interesting comparative study
of CaMnO
3
and PrMnO
3
using inelastic neutron scattering and
DFT (with PBE).
20
The experimental phonon spectrum of PrMnO
3
,
as well as the IR and Raman spectra, are fairly well reproduced by
these calculations.
20
Both studies use the experimental structure
as a starting point in their calculations.
For SmMnO
3
, several studies have focused on the effect of
dopants in the structure. Dopant segregation on the surfaces of
SmMnO
3
and LaMnO
3
has been investigated (using PW91+U)
and explained as a function of cation size mismatch.
21
The
metal-to-insulator transition induced by H-doping has been
investigated in several perovskites (using PBEsol+U), including
SmMnO
3
.
22
Most other computational research focuses on the
surface behaviour of, and kinetics over, the mullite phase of
SmMn
2
O
5
, which seems to be a more active catalyst than the
perovskite SmMnO
3
.
23,24
The remaining two perovskites considered here, EuMnO
3
and YbMnO
3
, are relatively poorly studied. Perhaps the most
relevant work concerning either of the two is the recent study of
charge transfer between Eu and Mn.
25
The study (carried out
using PBEsol+U) focuses on the changes in structure of
EuMnO
3
driven by different magnetic moments of the two
elements. The experimental structure is again used as a starting
point for the calculations.
A look at the methods employed in the computational
studies of the five perovskites listed above reveals that PBEsol+U
is the most widely applied one, followed by the related PW91+U.
In both cases, a DFA of the GGA family (PBEsol
26
and PW91
27
)is
Fig. 1 Proposed computational workflow for geometries of perovskites. An initial list of DFAs is ranked and sorted using a benchmark of bond lengths of
lanthanide diatomics (1). At the same time, mash (2) is used to produce initial trial structures of perovskites. These structures are used to determine+U
corrections (3) for the top ranked methods from the initial benchmark. A reduced set of DFAs, including methods with +Ucorrections, is then benchmarked
using perovskite structures (4). Having obtained a final ranking, the performance of the best-performing DFAs is evaluated on a LaMnO
3
case study (5).
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combined with a so-called Hubbard term (+U),
28
an energy
‘‘penalty’’ for electronic states, tailored to reproduce the metal-
lic as well as insulating behaviour of materials.
29
The
approaches for choosing appropriate values of the +Ucorrection
are described in detail elsewhere.
30
Given the frequent use of
GGA+Uin the literature, it is somewhat surprising that a bench-
mark study of DFAs focused on predicting octahedral tilting in
perovskites
31
omitted +Uvariants of the studied GGAs completely.
Instead, ‘‘hybrid’’ DFAs that include a fraction of Hartree–Fock
exchange (HFx) were included, and they were found to outperform
the (uncorrected) GGAs across the board.
31
GGA+Uis generally
considered to be sufficient for determining ground-state properties
(structure, band gap) of perovskites, but the determination of more
complex quantities, such as the pressure-induced insulator to
metal transition in LaMnO
3
,wouldrequiretheuseofhybrid
DFAs.
32
In such cases, a hybrid functional with a screened HFx
should be applied,
3
with the range-separation (screening) constant
tuned accordingly for each studied system.
32
However, for the
prediction of lattice constants, even the (uncorrected) GGAs were
shown to perform comparably to hybrid DFAs.
31
3 Computational methods
3.1 DFT calculations
All DFT calculations were carried out using the pw.x code from the
Quantum ESPRESSO package (versions 6.6 and 7.0).
33,34
The DFAs
were computed using the built-in interface with the exchange–
correlation functional library libxc (compiled against libxc version
4.3.4)
35
throughout this work. Basis sets and pseudopotentials
from the SSSP Efficiency pseudopotential library (version 1.1, PBE
parametrisation)
36,37
were used whenever possible. For calcula-
tions involving the exchange-hole dipole moment dispersion
correction (XDM),
38,39
the projector-augmented wave pseudopo-
tentials from the PSlibrary (version 0.3.1) were used.
40
The
computational details for the diatomic, single unit cell, and
supercell calculations of the studied systems are included in the
respective Results sections for clarity.
3.2 Starting structures
A key step in the proposed computational workflow is to be able
to determine reasonable starting structures of orthorhombic
perovskites using only their atomic composition. For this
purpose, we have developed mash.
41
Our tool is heavily
inspired by the excellent program SPuDS by Lufaso and
Woodward,
42
and for most applications we would recommend
SPuDS (or the recently developed python interface, PySPuDS)
over mash. However, while SPuDS is available for download free
of cost, it is not open source, so we chose to develop and use
mash (version 1.0) for this work.
Given the formula of the perovskite (ABX
3
), the requested
Glazer tilt system, and output file format (xyz file, cif file, or
pw.x-compatible input template), mash generates multiple trial
unit cells according to the following protocol:
1. The elements in the formula supplied (ABX
3
) are parsed,
and based on the anion (X), combinations of possible integer
charges of the cations are deduced.
2. The mendeleev library
43
is used to obtain the ionic radii of
A, B, and X (denoted r
A
,r
B
,r
X
) for all possible spin and charge
combinations, and the perovskite tolerance factors t
44
and t
45
are calculated from these radii and the number of A-site atoms
in the unit cell (n
A
):
t¼rAþrX
ffiffiffi
2
pðrBþrXÞ
t¼rX
rBnAnArA=rB
ln rA=rB
ðÞ
3. For each of these candidate structures, if to4.18, the
candidate is likely to be a perovskite, otherwise the candidate
structure is discarded. The tolerance factor tis used instead of
Goldschmidt’s factor tas thas been shown to outperform tin
predicting perovskite stability.
45
4. The Glazer tilt angle fis calculated from t, using a
polynomial fit to the data of Lufaso and Woodward.
42
This tilt
was shown to be approximately correct for orthorhombic perovs-
kites (tilt system a
b
+
a
).
42
The tilt angle is currently applied
indiscriminately for all implemented tilt systems in mash (cubic
a
+
a
+
a
+
, orthorhombic a
b
+
a
,rhombohedrala
a
a
).
5. The appropriate supercell is generated using Wyckoff
positions, as described in ref. 46. All candidate structures are
saved in separate output files.
Using mash for the set of five perovskites, we obtain a set
of 13 starting structures: 2 each for LaMnO
3
and PrMnO
3
(Ln
III
Mn
III
with high and low Mn spin states); 3 each for
SmMnO
3
, EuMnO
3
and YbMnO
3
(also including Ln
II
Mn
IV
).
3.3 Linear-response +Uvalues
As discussed previously, GGA+Uis a popular level of theory
applied to study solid state systems, including perovskites.
However, the +Uvalues for a given element are not universal,
and would ideally be fitted for each compound, each DFA, and
each property of interest.
30
As such, many parametrisations of
GGA+Umay exist for a given system, requiring benchmarking.
14
To address this issue, while obtaining reasonable +Uvalues
without further a priori information, we have decided to use +U
values obtained using linear-response theory, as described by
Cococcioni and Kulik.
47,48
In this approach, +Uis obtained
from the difference of the inverted self-consistent response
function wfrom the non-interacting response w
0
on each site,
where the response functions ware calculated as the partial
derivatives of the total energy Ewith respect to localised
potential shifts of strength a:
U¼w01w1
wIJ ¼@2E
@aI@aJ
The linear response functions ware obtained here by a linear
regression to Escalculatedusingavalues in the range of 0.08 eV.
The atomic orbitals are orthogonalised using the ortho-atomic
keyword in Quantum ESPRESSO. Previous works have shown
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that the choice of appropriate orbital projections is crucial, as
the calculated +Uvalues might differ by up to an order of
magnitude when using ortho-atomic or atomic Hubbard
projectors.
49
For a multiple-site system, the w,w
0
, and by
extension U, become matrices, where the off-diagonal elements
U
IJ
correspond to the response at site I due to perturbations on
site J. The +Us for sites in a multiple-site system are therefore
the diagonal elements of the matrix, U
II
. Note that strictly
speaking, such +Ucorrections, as implemented in Quantum
ESPRESSO, correspond to U
eff
=UJ, where Uwould be the on-
site Coulomb and Jwould be the on-site exchange interactions.
However, in the current work, the +Uvalues correspond to the
diagonal elements of the U
eff
matrix.
The +Uvalues in this work are calculated using the linear-
response approach for each of the three elements (except La) in
each of the five perovskites for four of the DFAs studied (PBE,
PBEsol, WC, and N12), using the unit cells obtained with mash.
The calculated values are shown in Table 1. For comparison,
literature GGA+Uvalues include the PBE+Uvalues of 4.5 eV for
Mn 3d and 5.5 eV for O 2p to model LaMnO
3
,
14
a PW91+Uvalue of
3.1 eV for Mn 3d to model PrMnO
3
,
19
a PW91+Uand PBEsol+U
value of 4.0 eV for Mn 3d in SmMnO
3
,
21,22
and the AFLOW
standard values of 4.0 eV for Mn 3d, 5.5 eV for Pr 4f, 6.4 eV for
Sm 4f, 5.4 eV for Eu 4f, and 6.3 eV for Yb 4f.
50
While there is some
consistency between functionals for a given element in the dataset
showninTable1(thefewexceptionsaretheN12+Uvalues for Mn
and O in LaMnO
3
and Mn in PrMnO
3
), the values obtained from
the linear-response approach are generally quite different from the
literature values, including the AFLOW standard.
50
3.4 Defective LaMnO
3
supercell lattices
The defective supercells used to investigate the phase transition
between orthorhombic and rhombohedral LaMnO
3
can be cre-
ated by removing La atoms from the systems being modelled.
Each La atom removed from the system has to be compensated
by an increase in the oxidation state of 3 Mn atoms from III+ to
IV+, in order to maintain charge neutrality. In other words, upon
removal of a single La atom from the unit cell of orthorhombic
LaMnO
3
, the fraction of Mn
IV+
is 75%. In order to obtain smaller
fractions of Mn
IV+
, a supercell has to be used. For instance, with a
232 orthorhombic supercell (La
48
Mn
48
O
144
), each La defect
would correspond to a 6.25% increase in the fraction of Mn
IV+
.
However, choosing just four La atoms to remove out of the 48 La
atoms in the 2 32supercell(i.e. in combinatorial notation
n
k
¼48
4
) results in 194580 possible combinations; with
onemoreLadefect(i.e. 48
5
) the number of combinations
increases to 1712304. To constrain the number of supercells to a
manageable number, we use defective orthorhombic and rhom-
bohedral supercells, derived from the respective 2 32and
241 stoichiometric supercells (both La
48
Mn
48
O
144
), with
1 to 4 La defects spanning the range from 6.25% to 25% Mn
IV+
.
As our calculations use periodic boundary conditions, many
of the above defective supercells are related to each other by
symmetry. Rather than explicitly considering the symmetry
relations in the supercell, we instead calculate the potential
energy of a set of equivalent ABO
3
defective supercells, using
the effective medium theory calculator from the ase.calculator-
s.emt module of the package ase (version 3.21).
51
For conve-
nience, the parameters for Al, Ni and O have been used as A, B,
and O, respectively. Defective supercells with the same
potential energies (to within 6 decimal places) are considered
symmetry-equivalent, therefore, all but one of each set is
removed from further consideration. The resulting number of
unique defective orthorhombic (Pnma) and rhombohedral (R3c)
supercell lattices, generated by ase, is shown in Table 2. These
supercell lattices provide the sets of A-site positions which are
to be removed from the LaMnO
3
supercells prior to their
evaluation by DFT.
4 Results and discussion
4.1 Equilibrium bond lengths in lanthanide diatomics
The usual way of selecting an appropriate DFA in computa-
tional chemistry in general is by benchmarking several candi-
date methods using a related, but simpler system.
52
This
approach is also often applied in the solid state.
53
Instead, we
first focus on calculating the equilibrium bond lengths (r
e
)in
diatomic lanthanide oxides and halides, in analogy with ref. 54.
The reference r
e
values for 17 lanthanide diatomics are avail-
able, including other spectroscopic constants, from calculations
Table 1 GGA+Uvalues, in eV, as calculated using linear-response theory
Perovskite Element PBE+UPBEsol+UWC+UN12+U
LaMnO
3
La 4f — — — —
Mn 3d 5.689 5.701 5.702 6.877
O 2p 8.002 7.977 8.019 9.215
PrMnO
3
Pr 4f 3.802 3.848 3.845 3.875
Mn 3d 3.008 3.810 3.504 6.172
O 2p 6.237 6.320 6.332 6.079
SmMnO
3
Sm 4f 3.826 3.891 3.973 3.999
Mn 3d 6.940 6.806 6.910 6.345
O 2p 6.599 6.909 5.994 6.334
EuMnO
3
Eu 4f 2.883 2.919 2.912 2.811
Mn 3d 6.618 6.648 6.646 6.581
O 2p 5.446 5.478 5.499 5.300
YbMnO
3
Yb 4f 4.733 4.781 4.766 4.710
Mn 3d 6.795 6.817 6.825 6.763
O 2p 5.757 5.793 5.813 5.611
Table 2 Number of unique defective lattices generated using ase from
the Pnma and R3cstoichiometric supercells. The sum of weights of each
defective lattice Pw
ðÞ
as well as the total number of combinations ( n
k
where n=48andkA{1, 2, 3, 4}) included for comparison
Supercell Pnma R3c PwðÞ n
k
%Mn
IV+
La
48
Mn
48
O
144
1 1 — — 0.00%
La
47
Mn
48
O
144
1 1 1 48 6.25%
La
46
Mn
48
O
144
21 18 47 1128 12.50%
La
45
Mn
48
O
144
213 138 1081 17296 18.75%
La
44
Mn
48
O
144
2479 1448 16215 194580 25.00%
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using a composite method exceeding the CCSD(T) level of
theory.
55
Our calculations are performed by placing the two
atoms 1 Å apart in a 10 10 10 Å box, and then relaxing the
structure with the BFGS algorithm using the default convergence
criteria (change in energy between steps dEo10 mRy, maximum
force Fo1mRya
01
) and a range of DFAs. The calculations are
spin-unrestricted, with a G-centered 3 43 Monkhorst-Pack
k-point grid, and energy cut-offs of 65 Ry and 780 Ry for the
wavefunction and density, respectively. While this computational
approach may be unusual for isolated gas-phase systems, it was
used to maintain consistency with the method used for extended
systems, below. For completeness, comparison with results per-
formed with G-point only and the Makov–Payne correction for
isolated systems
56
are shown in Fig. S1 (ESI†).
The results, obtained with the SSSP Efficiency pseudopotential
library (version 1.1, PBE parametrisation) are shown in Fig. 2. The
distribution of the absolute deviations from the reference r
e
is
shown using box plots. The set of 17 diatomics is shown in blue; a
subset of the 5 La-containing diatomics is shown in orange. We
note that the RMSD of the reference data from the best known
experimental values of r
e
is below 10 mÅ for the whole dataset, and
below 3 mÅ for the La-subset.
55
Therefore, the significantly larger
deviations in the bond lengths that we obtain across the board in
our calculations would be spectroscopically significant. The best
overall performers are the PBE, N12, B86BPBE, and OLYP DFAs.
We note that in general, the La-containing subset is more challen-
ging than the overall set, as in most cases the median of the
deviations in the larger set (navy) is well below the first quartile of
the La-containing subset. The exceptions to this behaviour are the
two DFAs developed especially for solids: PBEsol and WC, which
are the best performers for the La-containing subset. The N12 and
PBE DFAs also show good performance for the La-containing
subset, coming in 3rd and 4th place, while also yielding consistent
results for the full dataset. Therefore, the methods chosen for
determination of +Ucorrections (see Fig. 1) are WC, PBEsol, N12
and PBE. On the other hand, the worst performer is BLYP – in fact,
only 13 out of the 17 diatomics could be optimised with this DFA.
While DFAs and their implementations in computational
codes are often benchmarked extensively,
36
this is not often the
case for the basis sets and core electron representations used in
solid-state calculations (pseudopotentials, projector-augmented
waves, etc.).
37
For completeness, the effect of pseudopotentials
and their (lack of) parametrisation for the corresponding DFAs on
the above results is evaluated by comparing the results of five of the
well-performing DFAs (PBE, N12, OLYP, WC, and PBEsol), with
results from calculations using the PBEsol parametrisation, both
using the SSSP Efficiency library (version 1.1), as well as to the set of
optimised norm-conserving Vanderbilt pseudopotentials from the
SG15 ONCV library (version 1.2, archive dated Feb. 2020).
57
While
norm-conserving pseudopotentials generally require higher cutoff
values than their ultrasoft counterparts, the benchmarks performed
as part of the SSSP project
36
show the properties of Mn, which is the
hardest of the studied elements, are well-converged with the SG15
ONCV pseudopotential above 60 eV for the wavefunction cutoff.
Note that this is below the 65 eV cutoff suggested by the SSSP
Efficiency library for the ultrasoft GBRV pseudopotential for Mn.
58
The difference in results (see Table S1, ESI†) between the two
parametrisations of SSSP Efficiency (PBE vs. PBEsol) is negligible,
below 1 mÅ in the RMSD of r
e
in all five cases. We can therefore
conclude that the effect of any DFA-related ‘‘mis-parametrisation’’ of
the pseudopotentials on their performance in bond lengths is
marginal. However, comparison between the SSSP Efficiency and
the SG15 ONCV libraries shows the choice of an appropriate
pseudopotential library is crucial, as the RMSDs of r
e
for the La-
containing diatomics obtained with the SG15 ONCV library are at
least double that of the SSSP Efficiency library, if not higher.
Finally, we discuss the effect of dispersion corrections. The
results, obtained with the projector-augmented wave pseudo-
potentials from the PSlibrary (version 0.3.1) are shown in Fig. 3.
Neither XDM (a
1
= 0.3275, a
2
= 2.7673)
38,39
nor the D3(BJ)
dispersion correction (a
1
= 0.4289, s
8
= 0.7875, a
2
=
4.4407)
38,59,60
can systematically improve on the results of the
PBE DFA for this dataset.
4.2 Lattice constants of lanthanide manganites
In the second step of our computational workflow, we compare
the computed crystal structures of the five lanthanide
Fig. 2 Absolute deviations of r
e
calculated with DFAs w.r.t. CCSD(T) refer-
ence values
55
plotted as a box plot (whiskers span 0th to 100th percentile, box
spans 1st to 3rd quartile) for a set of 17 lanthanide diatomics (blue). Results for
the La-subset containing 5 diatomics are shown in orange. The medians are
denoted by vertical lines in navy and red, respectively.
Fig. 3 Effect of dispersion correction on the distribution of absolute
deviations from reference r
e
. Colours are as in Fig. 2.
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perovskites against experimental crystal structures obtained
from room temperature diffraction studies of LaMnO
3
from
ref. 7, PrMnO
3
from ref. 8, SmMnO
3
from ref. 12, EuMnO
3
from
ref. 9, and YbMnO
3
from ref. 10. Note that the use of low-
temperature data, where available,
10,12,61
does not significantly
affect the results below. The combined cell and geometry
relaxations are started from the orthorhombic unit cells as
calculated by mash, using the BFGS algorithm with 1 atm cell
pressure as the convergence threshold, while restricting the cell
to maintain an orthorhombic shape. All calculations are per-
formed using spin-unrestricted DFT with the wavefunction and
density cut-offs of 65 Ry and 780 Ry, respectively. A G-centered,
343 Monkhorst-Pack k-point grid was used. As the effects
of DFA-related parametrisation of the pseudopotentials on the
bond lengths were shown above to be negligible, the PBE
parametrisation of the SSSP Efficiency pseudopotential library
(version 1.1) has been used with each DFA in this work. If
multiple candidate structures are generated by mash for a single
perovskite, the successfully optimised supercell with the lowest
deviation in the figure of merit is used for comparison.
There are several ways of comparing crystal structures
quantitatively. Three methods are implemented in the program
critic2,
62,63
including approaches based on the radial distribu-
tion functions (RDF), and powder diffraction patterns (POW-
DER). With both of these methods, a dissimilarity score of 0
corresponds to an identical structure. The results obtained
when using the POWDER method are shown in Fig. 4. When
considering the overall dataset (blue), three DFAs (WC, PBEsol,
N12) of the best-performing set in the diatomics benchmark
perform rather poorly, barely improving on the mash-generated
starting structures. For these three DFAs, the most challenging
structure is the SmMnO
3
perovskite (POWDER score 40.72),
which is predicted by mash rather well (POWDER score B0.15).
For LaMnO
3
(orange), only the PW86PBE DFA shows an
improvement over mash – it is also the best-performing DFA
for the whole set. On the other hand, when we use the RDF
method in critic2 (see Fig. S2, ESI†), the DFAs are grouped
much closer together with no clear best performer. With the
exception of BLYP, all other DFAs significantly improve over the
starting structures generated with mash, approximately halving
the dissimilarity score. This qualitative disagreement between
the two methods for comparison of crystal structures in critic2
is puzzling, as both metrics should include the effects of lattice
constants as well as the atomic positions within the lattice.
Given the issues identified with the above approach, we
resorted to a simpler figure of merit, i.e. the relative RMSD of
the computed lattice constants from the experimental reference
values. The resulting relative signed deviations of computed
lattice constants are shown as violin plots in Fig. 5. The relative
RMSDs are shown as blue dots for the whole dataset and orange
dots for LaMnO
3
, respectively. The relative RMSDs for the five
perovskites are significant, and even in the best cases are still
above 3% of the lattice constant. From the change in the shape
and narrower width of the violin plots, we can see that the
structures generated by mash are generally improved by opti-
misation using a DFA. However, in the particular case of
LaMnO
3
, this optimisation may actually lead to a higher RMSD
than that obtained with the guess generated by mash. Critically,
three of the DFAs that performed well for La-containing dia-
tomics (WC, PBEsol, and N12, cf. Fig. 2) perform rather poorly
here, both for the whole dataset and for LaMnO
3
. This is
particularly surprising for the WC and PBEsol DFAs, which
were designed for use in solid state applications. The perfor-
mance of a DFA in predicting the gas-phase bond lengths of
lanthanide diatomics is therefore a rather poor predictor of
performance in determining the lattice constants of lanthanide
perovskites. For the overall dataset, there is no clear best
performer, with several DFAs yielding a relative RMSD of
B3%. For LaMnO
3
, the only DFAs able to outperform mash
with a relative RMSD below 1.9%, are OLYP and HCTH120.
The effect of +Ucorrection on the performance of selected
DFAs is shown in Fig. 6. The four DFAs for which the +U
correction has been determined have been chosen based on
Fig. 4 Dissimilarity scores calculated with the POWDER method in critic2
using computed and experimental crystal structures of lanthanide per-
ovskites. Values obtained for the initial structures generated by mash
included for comparison. Average score for a set of five perovskites shown
in blue, standard deviation of the set indicated using error bars, and the
score for the LaMnO
3
perovskite shown in orange.
Fig. 5 Violin plot of relative signed deviations of computed lattice constants
of lanthanide perovskites from experimental reference values. The deviations
of lattice constants obtained with mash are included for comparison. Relative
RMSDs for the set of five perovskites shown as blue dots, and the relative
RMSDs for the LaMnO
3
perovskite shown as orange dots.
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their performance for diatomics, which, as we now know, may
not have been the best criterion. Based on the results in Fig. 6,
the +Ucorrection determined using linear response theory can
systematically reduce the overprediction in lattice constants of
lanthanide perovskites, with the violin plots becoming much
more symmetric around zero. In fact, for the whole dataset,
N12+Uand WC+Uare the two best performing methods from
those studied, with relative RMSD of 2.4% and 2.7%, respec-
tively. Finally, as shown in Fig. S3 (ESI†) neither the XDM
dispersion correction (evaluated with several of the studied
DFAs) nor the D3(BJ) dispersion correction (evaluated with PBE
DFA) provides a systematic improvement in the performance of
the uncorrected functionals. In most cases, the dispersion
corrections shift the violin plots of the signed deviations to
higher values.
4.3 Defect-induced phase transition in LaMnO
3
Finally, we use the above lattice constant benchmarks to select
methods for prediction of the defect-induced phase transition
between orthorhombic (Pnma) and rhombohedral (R3c)
LaMnO
3
. This case study requires a reasonable performance
in predicting geometries. However, a good prediction of relative
energies of the Pnma and R3csupercells is required for quanti-
tative agreement with experimental data. Note that unlike in
some theoretical studies of perovskite phase transitions, where
the effects of bulk pressure are investigated,
64,65
here we focus
on predicting the phase transition caused by defects in the
structure of the perovskite. For this case study, we shortlisted
the DFAs OLYP and HCTH120, as they were the best performers
for LaMnO
3
lattice constants, as well as N12+U, which was the
best performer in the overall lattice parameter benchmarks.
Finally, we have also included PBE+U, to allow comparison
between a ‘‘non-empirical’’ DFA (PBE) and a systematically
optimised DFA (N12).
To compare our results with experimental data, we turn to
the investigation of Wold and Arnott, who prepared a series of
lanthanum manganites with Mn
IV+
fractions between 0.02 and
30%, and determined their phase transition temperatures.
13
We note that a non-zero fraction of Mn
IV+
in a pure La–Mn–O
system can be caused by both A-site defects (La
1x
Mn
III
13x
Mn
IV
3x
O
3
) as well as B-site defects in isolation (LaMn
III
14x
Mn
IV
3x
O
3
), or a
combination of the two, leading to excess O in the structure
(e.g. LaMn
III
12x
Mn
IV
2x
O
3+x
). The experimental trend (black, Fig. 7),
obtained for defective perovskites of the latter (LaMnO
3+x
) type,
shows two regimes: a non-linear section below 21% Mn
IV+
, and
a linear trend above this percentage.
13
The non-linear section is
attributed to the destruction of the long-range bond-ordering
effect in orthorombic manganites
66
caused by an unequal
occupation of the e
g
orbitals in Mn
III+
(d
z
2
occupied, d
x
2
y
2
empty). The bond-ordering effect is increasingly disrupted as a
function of increasing Mn
IV+
, causing the non-linearity and an
abrupt change in the overall behaviour. The linear section
above 21% Mn
IV+
is caused by the decreased size of Mn
IV+
compared to Mn
III+
, allowing an easier reorganisation into the
denser R3cstructure. A linear regression of the transition
temperatures in highly-defective structures
13
predicts the R3c
to be more stable than Pnma at T= 0 K when the fraction of
Mn
IV+
sites reaches 43.7(7)%.
The first analysis of the performance of the DFAs is under-
taken using defective supercells derived from the geometry of the
relaxed stoichiometric LaMnO
3
supercell. The appropriate Pnma
and R3csupercells are optimised (cell and geometry) with each DFA,
with convergence criteria of dEo1mRy, Fo10 mRy a
01
and cell
pressure o1 bar. The orthorhombic unit cells were constrained to
Pnma symmetry with the lattice parameters a,b,crelaxed. The
rhombohedral unit cells were constrained to monoclinic symmetry
(a 2 41 supercell generated from the La
6
Mn
6
O
18
rhombohedral
unit cell is not rhombohedral anymore, as b=2a) with the lattice
parameters a,b,cas well as the angle g=+ab relaxed. Then, the
defective supercells are created by removing the A-site positions
using the supercell lattices generated using ase (shown in Table 2).
Finally, single-point energy calculations of each of the defective
supercells is performed with the corresponding DFAs. All of these
calculations were spin-unrestricted, with a single k-point at the G
point, and with a 65 Ry and 780 Ry cutoff for the wavefunction and
density, respectively.
Fig. 6 Violin plot comparing the performance of DFAs with +Ucorrec-
tions in predicting lattice constants of lanthanide perovskites. Colours are
as in Fig. 5. The results for DFAs with +Ushown in cyan for clarity.
Fig. 7 Difference in the energies of the defective rhombohedral (R3c) and
orthogonal (Pnma) supercells, per mole of Mn, as a function of Mn
IV+
sites.
Results calculated using single-point energies, with OLYP (orange),
HCTH120 (green), N12+U(red), and PBE+U(blue). Experimental data,
adapted from ref. 13, shown in black.
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The results obtained from single point calculations are
shown in Fig. 7, with the difference in the single point energies
of the structures DE(R3cPnma) normalised by the number of
Mn atoms in the supercell (n= 48). As expected, all four DFAs
predict the correct stability ordering in the stoichiometric case,
where the Pnma phase is the energetically favoured one. For the
OLYP (orange) and HCTH120 (green) methods, the Pnma -R3c
transition point can be determined by linear regression of the
data followed by extrapolation. The linear regressions are
reasonable with R
2
40.99 in both cases, and the transition
points obtained (39.9% and 47.5%, respectively) are in a good
agreement with the result obtained by extrapolating the experi-
mental data (43.7(7)%). However, the qualitative behaviour of
the two DFAs is a little different, as OLYP overstabilises the
Pnma phase compared to HCTH120 and experiment.
The situation is very different with both GGA+Umethods.
First, the convergence of the self-consistent cycle with both
GGA+Umethods is very slow for some of the defective struc-
tures, in several cases requiring more than 100 iterations and
often requiring adjustment of the mixing parameters. Second,
the stability trend obtained is clearly non-linear, with the highly
defective Pnma structures calculated to be significantly more
stable than their R3ccounterparts. This can be partially
explained by looking at the mean defect distances in the
lowest-energy Pnma structures, shown in the left panel of
Fig. 8. Both OLYP and HCTH120 clearly prefer to maximise the
separation between La defects. With PBE+Uand N12+U,anin-
plane arrangement of defects is preferred, leading to a smaller,
but not necessarily minimal mean defect distance in the lowest
energy structures. The structures preferred by the +Umethods
therefore offer further qualitative support for the presence of the
long-range bond-ordering effect below 21% Mn
IV+
.
66
The differ-
ences between the +Uresults and the other two DFAs are likely a
consequence of the self-interaction error in OLYP and HCTH120,
leading to an overly delocalised electronic structure. The +U
correction reduces the self-interaction error, and localises the
defective state. For the R3cstructures (see right panel of Fig. 8),
no such trend is observed, and the mean defect distance does not
seem to be a good predictor of stability. In fact, several defective
supercell lattices are consistently predicted to be among the
minimum structures, regardless of the DFA used. In conclusion,
none of the four DFAs fully capture the phase transition beha-
viour. While reasonable agreement with the experimental
Pnma -R3ctransition point is obtained using OLYP and
HCTH120, the experimentally observed non-linearity is not repro-
duced. By contrast, the long-range bond-ordering effect can be
observed with GGA+Umethods, which, however, do not seem to
predict a phase transition at all.
One reason for the poor agreement of our calculations with
experimental data may be due to our comparison of total energies
obtained from calculations (EBU(0 K)) with free energies
obtained from the experimental data. A computational determi-
nation of Gibbs free energies (G=HTS), which ought to be
directly comparable to the experimental data, would require
the calculation of the cell pressure and volume (pV)foreach
defective lattice, obtaining the enthalpy (H=U+pV); and, more
importantly, the calculation of thermochemical corrections for
each defective lattice, which would then allow us to calculate the
canonical partition function (Q) and therefore the entropy (S)for
both the R3cand Pnma structures. Both steps would require a
significant further computational expense. However, if we
assume the pV term is approximately constant between a R3c
structure and a Pnma structure with the same number of defects,
it will cancel out, and we can use the Helmholtz free energy
(A=UTS). We cannot account for vibrational effects without
further calculations, involving cell relaxation and determination
of the phonons, which would be expensive for such large super-
cells. However, we can account for the energy lowering effect that
any other structures, with total energy E
i
=E
0
+DE
i
, lying near the
structure with the lowest total energy E
0
, have on the internal
energy Uat a finite temperature T. We can also determine the
entropy term Sarising due to the degeneracy w
i
of each of these
structures (see Table 2). For this, we use the following equations,
U¼E0þ1
QX
i
DEiebDEi
S¼UE0
TþkBln Q
Q¼X
i
wiebDEi
where bis the Boltzmann factor 1/k
B
Tand k
B
is the Boltzmann
constant. Note that E
0
is determined for each cell symmetry and
number of defects separately. We can then evaluate the difference
in the Helmholtz free energies of the two cell symmetries,
DA(R3cPnma)/n, at a finite temperature, with results at
1000 K shown in Fig. S4 (ESI†) (circles). By comparing to the
total energies (DE(R3cPnma)/n, shown as dots), we can see the
magnitude of the effect is rather small, but increases as a
Fig. 8 Comparison of the mean defect distances in the orthorhombic
(Pnma) and rhombohedral (R3c) defective supercell lattices with the
number of A-site defects, kA{2,3,4}. The lowest energy structures (up
to 5 kJ mol
1
above the minimum energy structure) obtained with PBE+U
(blue), N12+U(red), OLYP (orange), and HCTH120 (green) are indicated by
dots.
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function of the number of defects, and may become appreciable
at 430% Mn
IV+
. However, the sign of the correction is different
for PBE+U(blue) when compared to HCTH120 (green), which we
again attribute to the preference of PBE+Ufor an ordered in-
plane arrangement of defects, which leads to a smaller entropy in
the orthorhombic case, and therefore S(R3cPnma)40.
Another reason for the poor performance of the DFAs in the
above analysis may be due to the use of single point calcula-
tions, with only the stoichiometric Pnma and R3csupercell
relaxed. In light of the computational workflow introduced in
the Introduction, we investigate the effect the relaxation of the
atoms and the cell on the predicted phase transition behaviour.
For this purpose, we relax a subset of the defective supercells
with total energies within 5 kJ mol
1
of the lowest single-point
energy, for each combination of supercell geometry, DFA, and
number of defects. The number of relaxed supercells is shown
in Table 3. As above, the Pnma supercells were constrained to
an orthorhombic geometry (a,b,crelaxed), while the R3c
supercells were constrained to a monoclinic shape (a,b,c,g
relaxed). All of these relaxations were spin-unrestricted, with a
single k-point at the Gpoint, using convergence criteria of
dEo1mRy, Fo10 mRy a
01
and cell pressure o1 bar, and with
a 65 Ry and 780 Ry cutoff for the wavefunction and density,
respectively.
The results are shown in Fig. 9. In all cases, the relaxation of
the structures leads to a smaller energy difference between the
R3cand Pnma cells, shifting any predicted phase transition to a
lower fraction of Mn
IV+
sites. This is most likely due to the
additional relaxation of the R3csupercells, when compared to
the Pnma supercells, as shown in Fig. S5 (ESI†). As in the single-
point energy results, a linear regression of the HCTH120 and
OLYP results fits all four datapoints with R
2
40.99. The relative
discrepancy between the predicted transition points obtained
with the two GGA methods shrinks to about 10%. Notably, the
results of HCTH120 calculations (green) are in an excellent
quantitative agreement with the experimental data, especially
when also considering the results for the stoichiometric super-
cell (which is also relaxed): out of the four methods studied, it is
the only method which reproduces the non-linearity in the
experimental data at lower fractions of Mn
IV+
sites. However,
this agreement may be coincidental, given that HCTH120 does
not stabilise the in-plane ordering of vacancies in the orthor-
hombic structures (no long-range bond-ordering effects). The
behaviour of the GGA+Umethods is again different from the
former two DFAs. Both methods still fail to predict the Pnma -
R3cphase transition. The trends shown in Fig. S5 (ESI†) show
that while the relaxation energy increases almost linearly with
the number of Mn
IV+
defects in the R3cstructure, it plateaus at
around 0.4 Ry in the Pnma structure. While little is known
about the behaviour of the N12 DFA in solids, the uncorrected
PBE as well as PBE+UDFAs are known to over-bind (see also the
shape of the violin plots in Fig. 6). It may well be that in this
case PBE+Uover-stabilises the orthorhombic cells. One remedy
may be the use of a dispersion-corrected GGA+U, such as
PBE+U+XDM, which has been reported to improve upon PBE+U
for cell volumes and formation enthalpies of uranium
compounds.
67
To conclude, in order to obtain agreement with
the experimental data, it is crucial to allow the defective super-
cells to relax from the parent stoichiometric geometry. Quanti-
tiative agreement with experiment is in this case obtained with
HCTH120. However, it might be the case of ‘‘the right answer
for the wrong reasons’’, as the qualitative behaviour related to
the electronic structure of the system is not captured. Con-
versely, with PBE+Uand N12+U, the electronic structure trends
are partially captured, but the results are not accurate for
quantitative use.
We could identify several reasons for the apparent failure to
predict the experimentally observed phase transition beha-
viour. First, it might be necessary to keep the R3cdefective
structures constrained to a rhombohedral shape (a=b,g=
1201), in order to avoid over-relaxation; this can be achieved by
doubling the supercell size with a nominal doubling of the
computational cost. Second, method selection has been based
on performance for lattice constants of orthorhombic cells,
without considering the apparently more important rhombo-
hedral case, or their relative energies; the key challenge here is
gathering enough benchmark-quality lattice parameters of the
Table 3 Number of relaxed defective supercell lattices, where kis the
number of A-site defects
DFA
Pnma R3c
k=12341234
OLYP 1 1 6 4 1 3 1 3
HCTH120 1 3 5 9 1 2 1 4
PBE+U1 1211344
N12+U1 1211343
Fig. 9 Effect of cell and geometry optimisations (circles) on the differ-
ence in rhombohedral (R3c) and orthogonal (Pnma) supercell energies as a
function of Mn
IV+
sites. Single point energy results (dots) included for
comparison. Colours as in Fig. 7.
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relevant phases, as the increased computational cost of the
benchmark itself would be marginal. Third, thermochemical
corrections and zero-point energies are not considered in the
current study; including the calculation of numerical second
derivatives as a routine step of the workflow would, with the
current supercell size (B330 atoms), be extremely costly; the
zero-point effects might, however, be approximated using a
smaller supercell. Fourth, the transition point should ideally
not be extrapolated, but interpolated from calculated data; as
already noted above, an increase in kfrom 4 to 5 increases the
number of combinations n
k
more than 8-fold, so adding an
additional 5th or 6th defect would require strategies for
identification of the lowest-energy structure candidates by
methods other than direct comparison. Fifth, Mn
IV+
sites might
be also created by B-site defects or O-site non-stoichiometry in
the perovskite. Finally, a more robust analysis might be possi-
ble if the phase transitions of more than one system is
considered, to avoid bias from experimental data.
5 Conclusions
In this work, we proposed a computational workflow compatible
with the investigation of novel materials, for which crystal struc-
turesmaynotbeavailable.Wehaveintroducedtheprogram
mash, capable of predicting candidate structures for cubic, orthor-
hombic, and rhombohedral perovskites, which can then be used
as a starting point for further work. Based on both the POWDER
score using critic2
62,63
and direct lattice constant comparisons
with experimental data, the performance of mash is remarkable, in
some cases better than that obtained with subsequent geometry
optimisation using DFT.
We have also evaluated whether the standard computational
approach to method selection based on benchmarking using a
simple system leads to a good performance for a related, more
complex system. Good performance of a method for predicting
equilibrium geometries of diatomic lanthanide molecules does
not correspond to a good performance in predicting lattice
constants of lanthanide perovskites. It may be that the systems
chosen here are simply too different (extended vs. isolated) even
when the same property (geometries) is being evaluated.
The best performing DFAs in this work for predicting the
lattice constants of orthorhombic lanthanum-containing perovs-
kites are HCTH120 and OLYP. For lanthanide perovskites in
general, the best performance is surprisingly obtained by N12+U.
Additionally, all GGA+Uresults improve upon the performance of
the base functional, confirming that the linear response
framework
47,48
for determining +Ucorrections is robust, provided
that an appropriate projection of the Hubbard manifold is used.
49
The use of dispersion corrections with the DFAs studied does not
lead to a systematic improvement in the predicted lattice constants
ofthebulksystemsstudied;wenotethatdispersioneffectsmaybe
crucialwhencomparingtotalenergies.
We have evaluated whether the above three DFAs can be
used to predict a more complex observable, such as the phase
transition in defective LaMnO
3
as function of the fraction of
Mn
IV+
sites. The results are encouraging, especially with
HCTH120, which was predicted to perform well based on our
benchmarks. We highlight the importance of cell relaxation in
such studies. However, while numerical agreement with the
experimental data
13
could be obtained using HCTH120, some
qualitative aspects related to the electronic structure, including
the description of long-range bond-ordering effects stabilising
in-plane orientation of defects,
66
was not achieved. On the
other hand, GGA+Umethods are able to capture at least some
of these qualitative aspects.
Finally, we provide several pointers for further work in this
area. More targeted and thorough benchmarking is necessary
for method selection in extended systems, in analogy with the
recent transformative work in molecular systems (see e.g. ref. 68
and 69). The use of screening methods faster than DFT is
necessary to filter through the combinatorial space in defective
supercells, thus allow focusing computational time on a smal-
ler set of more challenging calculations.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
PK thanks the Forrest Research Foundation for funding. JDG
thanks the Australian Research Council for funding. This work
was supported by resources provided by the Pawsey Supercom-
puting Centre (project f97) and the National Computational
Infrastructure (project f97), with funding from the Australian
Government and the Government of Western Australia.
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