Fully relativistic calculations of EPR
and paramagnetic NMR parameters
for heavy atom compounds
vorgelegt von
M.Sc. Sebastian Gohr
geb. in Berlin
Von der Fakult¨at II - Mathematik und Naturwissenschaften
der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
– Dr. rer. nat. –
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Andreas Grohmann
Gutachter: Prof. Dr. Martin Kaupp
Gutachter: Prof. Dr. Markus Reiher
Tag der wissenschaftlichen Aussprache: 15.12.2017
Berlin 2018
For all my kind teachers
in science, school, and – most importantly – life
Acknowledgements/Danksagungen
“Science cannot tell us a word about why music
delights us, of why and how an old song can
move us to tears.”
— Erwin Schr¨odinger
First of all, I would like to express my sincere gratitude to Prof. Dr. Martin Kaupp who
gave me the possibility to write this thesis under his supervision and introduced me to
the “Mausefalle”. His seemingly endless patience and “open-door policy” made me feel
very welcome in his group. I am particularly grateful for his remarks on the different
manuscripts and for his great support to visit various interesting and stimulating confer-
ences, especially the 2015 Summer School on Actinide Science & Application.
Furthermore, I would like to thank Prof. Dr. Markus Reiher who kindly agreed to be the
second referee for this thesis.
Special thanks are due to my coworkers over the past years. Especially but not exclusively:
Arobendo, my conference mate since 2013 (the best one!) who consistently believed(/s)
in my scientific abilities; Shadan, my office colleague for some very enlightening scientific
discussions & explanations and for always enduring my keyboard typing style; Peter for
the stimulating discussions and all his input; Toni for never being short of (DFT) ad-
vice and for his comments on my theory chapter; Alexey for always coming up with a
helpful equation when I’ve been stuck with some theory; Ladislav for his introduction
to pNMR and his respective Matlab script; Vladimir for his help with the design of
the tungstoenzyme model structures; Caspar for the Boulder experiences; Matthias H.
for the informal discussions with beer and Zirbenschnaps; Sascha for organizing most
of our regular boardgame afternoons/evenings and of course the entire boardgame crew
(Sascha,Toni,Martin E.,Johannes,Robert,Kolja,Caspar,Matthias H.) for the countless
fun afternoons/evenings. Rulle and Freddy, thank you for your valuable hints in Novem-
ber 2017. Moreover, I am very thankful to the good souls in our group: Nadine for her
support with all the tedious university bureaucracy as well as the organization of our con-
I
stant supply with beverages, and Heidi for always having solutions for all the technical
problems (both software and hardware). In this regard, I am also thankful to our cluster
administrators Norbert Paschedag and especially Sven Grottke who always try their best
to make or keep everything smooth and running.
Of course, there were also many other coworkers outside our group, who accompanied me
on my way: Trond Saue who took time on a conference in 2014 to introduce me to the
details of relativistic quantum chemistry; Felix Kraffert who vividly explained an experi-
mentalist’s point of view on EPR to me; Bernd Schimmelpfennig for his friendly welcome
when I visited Karlsruhe; Jan V´ıcha, if you don’t know what to do outside science some
day, you can always be a tourist guide on Suomenlinna. Thank you for all the nice (and
often non-scientific) discussions; Michael Patzschke thank you for your feedback on the
theory part and particularly for your always encouraging feedback on my posters and
talks in Mariapfarr! It was much needed at that time.
Moreover, I took great benefit from the research visits of Olga Malkina & Vladimir Malkin,
and Stanislav Komorovsk´y: the many hours of discussions, explanations and patience in
answering all(!) my questions helped me a lot in understanding the scope and the chal-
lenges of relativistic magnetic resonance theory much better.
Words of gratitude are also due to the Studienstiftung des deutschen Volkes, which not
only supported my research financially but also gave me the possibilities to take part in a
lot of scientific and social activities to acquire new skills and most importantly: meeting
so many awesome people. In particular I would like to mention here those, who gave
the word “flocculent”aa very special meaning: Friederike & Andreas, Nadja, Elisabeth,
Michael (MCDF), and Manuel.
None of this would have been possible without a) Prof. Dr. Beate Paulus and Prof. Dr.
Peter Schwerdtfeger who introduced me to Theoretical Chemistry and encouraged me to
follow this path, and b) our unbeatable undergraduate study team, namely Andi, Jan,
Kai, Moritz, and Stefan. I can say with absolute certainty that I would not have been
able to achieve my study results (and hence all the subsequent success) without you!
Unfortunately, it is always a challenging task to acknowledge everybody. I am undoubt-
edly missing here many more people (e.g. Larissa, Daniel, Alexandra, and Tim to just
name a few), whom I also owe my gratitude.
Thank you
aFlocculent [Chemistry]: Containing or made up of small particles that have been aggregated together.
II
Stellvertretend f¨ur alle meine Lehrer und Dozenten der vergangenen 23 Jahre m¨ochte ich
mich an dieser Stelle bei meiner ersten Grundschullehrerin, Frau (Sokolowski-)Krewer,
bedanken, die maßgeblich meinen Hang zur Neugier und zur (kreativen) Probleml¨osung
gef¨ordert hat und mir vor 10 Jahren mit einer “E=mc2Briefmarke” zum Abitur grat-
ulierte. Ohne den Enthusiasmus, den viele meiner Lehrer in den 13 Schuljahren an den
Tag gelegt haben, w¨are ich sicherlich nicht an diesen Punkt gekommen.
Da das Leben aber nicht nur aus Lernen besteht, ist es mir eine Freude, an dieser
Stelle auch diejenigen zu erw¨ahnen, denen ich f¨ur Dinge dankbar bin, die manchmal
noch viel wichtiger sind: Dominik f¨ur die unz¨ahligen Male, die wir den großen Wagen
haben passieren lassen; Marc daf¨ur, dass Du die Definition von Loyalit¨at in einer Freund-
schaft bist; Patrick, weil Du einfach “der” Nachbar bist; Terence daf¨ur, dass Du dutzende
Seiten dieser v¨ollig fachfremden(!) Arbeit gelesen hast und nat¨urlich f¨ur die Anf¨ange des
“Wololo”; Tim &Kerstin f¨ur die Ehre, eure Hochzeit als Trauzeuge zu begleiten; Katrin,
weil Du nie m¨ude geworden bist, mir Deine ehrliche Meinung zu sagen; Carmen, weil
Du mich wieder zum Turnen gebracht hast, Cindy f¨ur all die gemeinsamen Entdeckungs-
touren durch Berlin (sowie euch beiden f¨ur euren Versuch in Tegel 2012); Matthias P.
daf¨ur, dass Du mich so herzlich in die Frankfurter Apfelweinkultur eingef¨uhrt hast.
Bei den Mitgliedern der 2. Knaben m¨ochte ich mich f¨ur zwei tolle Jahrzehnte, inklusive
einiger legend¨arer Weihnachtsfeiern, bedanken. F¨ur die kontinuierliche Motivation am
fr¨uhen Morgen geb¨uhrt auch dem “harten Kern” der TU Hochschulsportgruppe (Turnen)
mein expliziter Dank.
Zweifelsohne ist auch diese Liste nicht vollst¨andig, daher m¨ochte ich mich an dieser Stelle
auch noch einmal ganz generell bei all jenen bedanken, die mich ¨uber viele Jahre begleitet
haben und mit denen ich so ungemein viele sch¨one Momente und Erinnerungen teilen darf
(z.B. Larissa,Sonja,Michael T.,Anja P. und viele weitere).
Abschließend m¨ochte ich mich ganz besonders bei meiner gesamten Familie bedanken,
meiner kleinen Schwester, meiner Cousine, meiner Oma sowie meiner Tante und meinem
Onkel. Aber insbesondere bei meinen Eltern, die mir mein Studium ¨uberhaupt erst
erm¨oglicht haben. Die unerm¨udlichen “Motivationsversuche” in der anf¨anglichen Schulzeit
und euer Vertrauen auch in schwierigen Zeiten, gepaart mit nie ¨ubertriebenen Leistungser-
wartungen, haben mich hierher gebracht. Diese Arbeit ist vor allem auch euch gewidmet.
Danke
III
Abstract
In the thesis at hand, a 4-component relativistic density functional theory (DFT) approach
is applied within the ReSpect program package to calculations of magnetic resonance
parameters for various open-shell systems containing heavy 4d and 5d transition metals.
Such calculations represent a relatively uncharted territory. Therefore, the present work
starts with a comprehensive benchmark study on the calculation of electron paramagnetic
resonance (EPR) parameters, more precisely g-tensors and hyperfine coupling constants
(HFCCs), for 17 small 4d1and 5d1transition metal complexes (S=1
2). This study em-
phasizes the importance to use a 4-component framework for the inclusion of effects that
are attributable to special relativity. Moreover, a recommended computational protocol
is provided based on thorough examination of different functionals with varying amounts
of exact-exchange (EXX) admixture and basis-set combinations. The protocol is after-
wards validated on six medium- to large-scale iridium and platinum complexes with 43
to 133 atoms.
The benchmark progress is then extended to systems containing tungsten-sulfur bonds
to fine-tune the protocol for its subsequent application to the active-site structure of
tungstoenzymes. Tungsten is the heaviest known element that plays a well-defined role in
nature and even though several investigations have been performed to elucidate the EPR-
active W(V) intermediate state with its variety of available structure proposals, the true
structure is still up to debate. A set of 13 different structure proposals is thus investigated
in the present work. Distinct statements on the nature of the W(V) state are presented
based on comparisons of calculated and experimental EPR spectra.
Finally, the hitherto gained knowledge and insight is successfully applied to the calculation
of nuclear magnetic resonance (NMR) chemical shifts for open-shell systems (pNMR).
Three S=1
2systems are assessed before zero-field splitting (ZFS) is included for three
(di-)ruthenium complexes (S= 1 and S=3
2). This is the first time that the 4-component
matrix Dirac-Kohn-Sham (mDKS) approach is tested for such cases.
V
Zusammenfassung
In der vorliegenden Arbeit wird ein relativistischer 4-Komponenten Matrix-Dirac-Kohn-
Sham Dichtefunktionaltheorie-Ansatz (mDKS-DFT) im Rahmen des Programmpakets
ReSpect auf die Berechnung von Magnetresonanzparametern f¨ur offenschalige 4d und
5d ¨
Ubergangsmetallsysteme angewendet. Da solche Berechnungen noch durchaus als Neu-
land bezeichnet werden k¨onnen, beginnt diese Arbeit mit einer umfassenden Benchmark-
studie zur Berechnung von Parametern der Elektronenspinresonanz (ESR) f¨ur 17 kleine
4d1und 5d1¨
Ubergangsmetallkomplexe (S=1
2). Die Studie hebt die Notwendigkeit der
ad¨aquaten Ber¨ucksichtigung von Effekten, die aus der speziellen Relativit¨atstheorie fol-
gen, hervor. Basierend auf einer detaillierten Auswertung der Anwendung von verschiede-
nen Funktionalen sowie Basissatz-Kombinationen wird ein standardisiertes Protokoll f¨ur
die Berechnung von ESR Parametern empfohlen. In der Folge wird das Protokoll bei der
Anwendung auf sechs gr¨oßere Iridium- und Platinkomplexe (43 bis 133 Atome) ¨uberpr¨uft.
Die so erzielten Fortschritte werden anschließend auf Systeme erweitert, die Wolfram-
Schwefel Bindungen beinhalten, um das vorgeschlagene Protokoll speziell f¨ur die Anwen-
dung auf Wolframenzyme zu optimieren. Als schwerstes Element in nat¨urlich vorkom-
menden Enzymen nimmt Wolfram im ¨
Okosystem eine gesonderte Stellung ein. Obwohl
große Anstrengungen unternommen wurden, um die Struktur im aktiven Zentrum des
ESR-aktiven W(V) Intermediats aufzukl¨aren, konnte sie bis heute nicht eindeutig bes-
timmt werden. Daher wurden 13 der (experimentell) vorgeschlagenen Strukturen im
Rahmen dieser Arbeit genauer untersucht. Ein Vergleich der berechneten ESR-Spektren
mit den experimentell verf¨ugbaren Daten erm¨oglicht so Aussagen ¨uber spezifische Eigen-
schaften des W(V) Zustands.
Im letzten Projekt werden die bisher gewonnenen Erkenntnisse geb¨undelt und die ESR
Berechnungen auf chemische Verschiebungen in Kernspinresonanzspektren (NMR) f¨ur of-
fenschalige Systeme ausgeweitet (pNMR). Zun¨achst werden dazu drei S=1
2Verbindun-
gen analysiert und die bisherigen Methoden nochmals ¨uberpr¨uft und validiert. Ab-
schließend wird die 4-Komponenten mDKS-DFT Methode erstmals auch auf drei
(Di-)Rutheniumkomplexe angewendet (S= 1 und S=3
2), bei denen zus¨atzlich Nullfel-
daufspaltungen (ZFS) zu ber¨ucksichtigen sind.
VII
Contents
Listofpublications..................................XIII
Copyright .......................................XV
Listofabbreviations .................................XVII
1. Introduction 1
2. Theory 7
2.1. Theory of electron paramagnetic resonance (EPR) . . . . . . . . . . . . . . 7
2.2. Theoretical foundations for the quantum-chemical calculations . . . . . . . 13
2.3. Prerequisites (atomic units) . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4. 4-Componenttheory .............................. 16
2.4.1. 1-Electronpart ............................. 16
2.4.2. Externalfields.............................. 17
2.4.3. Spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.4. Relativistic energy spectrum . . . . . . . . . . . . . . . . . . . . . . 20
2.4.5. 2-Electronpart ............................. 21
2.4.6. Spin-orbit coupling: spin-other orbit (SOO) and spin-same orbit
(SSO)contributions........................... 22
2.4.7. Restricted kinetically balanced (RKB) basis . . . . . . . . . . . . . 23
2.5. 4-Component theory in DFT . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6. 2-Componenttheory .............................. 28
2.6.1. 1-Electronpart ............................. 28
2.6.2. 2-Electronpart ............................. 30
2.6.3. Picture change error . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7. 4-component EPR and NMR parameter calculations . . . . . . . . . . . . . 31
2.7.1. g-Tensor................................. 32
2.7.2. Hyperfine-coupling (HFC) tensor . . . . . . . . . . . . . . . . . . . 35
2.7.3. HFC tensor: finite-size nuclei . . . . . . . . . . . . . . . . . . . . . 38
2.7.4. NMR nuclear shielding tensor . . . . . . . . . . . . . . . . . . . . . 38
IX
Contents
2.7.5. Spin-orbit contributions to zero-field splittings for axially symmetric
systems in a 4-component framework . . . . . . . . . . . . . . . . . 39
2.8. 2-component DKH calculation for g- and HFC tensors . . . . . . . . . . . . 40
2.9. 1-Component EPR parameter calculations . . . . . . . . . . . . . . . . . . 42
2.9.1. g-Tensor................................. 43
2.9.2. HFCtensor ............................... 45
2.10.pNMRshifttheory ............................... 47
2.11. Spin polarization and spin contamination . . . . . . . . . . . . . . . . . . . 48
3. General computational details 51
3.1. 4-component EPR parameter calculations . . . . . . . . . . . . . . . . . . . 51
4. Validation of 4-component relativistic DFT calculations for EPR g- and
hyperfine coupling tensors with hybrid functionals 55
4.1. Introduction................................... 55
4.2. Additional computational details . . . . . . . . . . . . . . . . . . . . . . . 56
4.3. Benchmarkstudy................................ 60
4.4. Larger iridum(II) and platinum(III) complexes . . . . . . . . . . . . . . . . 70
4.5. Conclusions ................................... 78
Appendix A. Additional tables for the benchmark study 79
5. Tungstoenzymes 99
5.1. Introduction................................... 99
5.2. Additional computational details . . . . . . . . . . . . . . . . . . . . . . . 101
5.3. Validation on model complexes for tungstoenzymes . . . . . . . . . . . . . 102
5.3.1. Introduction...............................102
5.3.2. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3.2.1. Structures...........................104
5.3.2.2. Effect of the structure on EPR parameters . . . . . . . . . 104
5.3.2.3. Effect of the functional . . . . . . . . . . . . . . . . . . . . 108
5.3.2.4. Basis-set effects . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.3. Effect of sulfido vs. oxo substitution in [WX(bdt)2]−and [WX(edt)2]−,
X=O,S ................................112
5.3.4. Conclusions ...............................112
X
Contents
5.4. Identifying the W(V) state of tungsten oxidoreductase enzymes by compu-
tational EPR investigations . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4.1. Introduction...............................114
5.4.2. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.4.2.1. Model building for the structures . . . . . . . . . . . . . . 119
5.4.2.2. Comparison of structures . . . . . . . . . . . . . . . . . . 124
5.4.2.3. Comparison of computed and experimental EPR data . . 127
5.4.3. Conclusions ...............................133
Appendix B. Additional tables and figures for the study of tungsten model
complexes 135
Appendix C. Additional tables and figures for the study on the W(V) state of
tungsten oxidoreductase 139
6. NMR shifts of paramagnetic heavy-metal complexes 151
6.1. Additional computational details I . . . . . . . . . . . . . . . . . . . . . . . 152
6.2. Referenceshieldings...............................154
6.3. Iridium and platinum complexes . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3.1. [Ir(II)ClN(CHCHPtBu2)2]........................156
6.3.2. trans-Ir(II)[η2-OC(CF3)2PtBu]2.....................159
6.3.3. [Pt(III)I2(IPr)2]+.............................161
6.4. Ruthenium complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.4.1. Additional computational details II . . . . . . . . . . . . . . . . . . 164
6.4.2. Expected accuracy for ZFS . . . . . . . . . . . . . . . . . . . . . . . 164
6.4.3. [Ru(II)(PNPtBu)Cl] ...........................165
6.4.4. [Ru(II)
2(O2C-p-tolyl)4-(THF)2]0.....................169
6.4.5. [Ru(II/III)
2(O2C-p-tolyl)4-(THF)2]+...................174
6.5. Conclusions ...................................179
Appendix D. Additional tables and figures for the pNMR studies 181
7. Final conclusions and outlook 187
Bibliography 191
XI
List of publications
1. [1] S. Gohr, P. Hrob´arik, M. Repisk´y, S. Komorovsk´y, K. Ruud, and M. Kaupp,
”Four-Component Relativistic Density Functional Theory Calculations of EPR g-
and Hyperfine-Coupling Tensors Using Hybrid Functionals: Validation on Transition-
Metal Complexes with Large Tensor Anisotropies and Higher-Order Spin-Orbit Ef-
fects”, J. Phys. Chem. A,2015, 119(51), 12892-12905 (DOI: 10.1021/acs.jpca.5b10996).
2. [2] S. Gohr, P. Hrob´arik, and M. Kaupp, ”Four-Component Relativistic Density
Functional Calculations of EPR Parameters for Model Complexes of Tungstoen-
zymes”, J. Phys. Chem. A,2017, 121(47), 9106-9117 (DOI: 10.1021/acs.jpca.7b08768).
XIII
Copyright
•Chapter 4 together with Appendix A with all tables and graphics therein and in
parts also chapter 1 and 3 are reproduced with permission according to the “ACS
Policy on Theses and Dissertations” (21/06/2017) from S. Gohr, P. Hrob´arik, M.
Repisk´y, S. Komorovsk´y, K. Ruud, and M. Kaupp, J. Phys. Chem. A,2015,
119(51), 12892-12905 (DOI: 10.1021/acs.jpca.5b10996). Copyright 2015 American
Chemical Society. (Figures 4.1 and 4.2, and Tables 4.5, 4.6, and 4.8 were produced
by P.H.)
•The first half of Chapter 5 (sections 5.1 to 5.3) together with Appendix B with all
tables and graphics therein are reproduced with permission according to the “ACS
Policy on Theses and Dissertations” (21/06/2017) from S. Gohr, P. Hrob´arik, and M.
Kaupp, J. Phys. Chem. A,2017, 121(47), 9106-9117 (DOI: 10.1021/acs.jpca.7b08768).
Copyright 2017 American Chemical Society.
XV
List of abbreviations
1c 1-component
2c 2-component
4c 4-component
a.u. atomic units
addl. additional
ADF Amsterdam Density Functional (program package)
AMFI atomic mean-field integrals
AOR aldehyde oxidoreductase
ASP aspartic acid
avgd. averaged
B3LYP Becke, 3-parameter, Lee & Yang & Parr (global hybrid DFT functional)
B3PW91 Becke, 3-parameter, Perdew & Wang (global hybrid DFT functional)
BOA Born-Oppenheimer approximation
bdt benzene-1,2-dithiolate
BP Breit-Pauli
BP86 Becke & Perdew 1986 (DFT functional)
BPH Breit-Pauli Hamiltonian
c.c. complex conjugated
calcd. calculated
CASPT2 complete active space with second-order perturbation theory
cf. confer (= compare/see)
CGO common gauge origin
CGS centimetre gram second (unit system)
CI configuration interaction
COSMO conductor-like screening model
CPKS coupled-perturbed Kohn-Sham
CPU central processing unit
DC Dirac-Coulomb
DCB Dirac-Coulomb-Breit
XVII
List of abbreviations
DFT density functional theory
DHF Dirac-Hartree-Fock
dip dipolar
DKH Douglas-Kroll-Hess
DKS Dirac-Kohn-Sham
DMRG density matrix renormalization group
DMSOR dimethyl sulfoxide reductase
ECP effective core potential
edt ethane-1,2-dithiolate
EPR electron paramagnetic resonance
ES-4 Pyrococcus strain ES-4
ESC elemination of the small component
ESR Elektronenspinresonanz
expt. experimental
ext. external
EXX exact-exchange
eZ electron Zeeman
FC Fermi-contact
FDH formate dehydrogenase
Fig(s). Figure(s)
FMDH formylmethanofuran dehydrogenase
FN finite-size nuclei
FOR formaldehyde oxidoreductase
fpFW free-particle Foldy-Wouthuysen
FW Foldy-Wouthuysen
GAPOR glyceraldehyde-3-phosphate
GGA generalized gradient approximation
GIAO gauge-including atomic orbitals
GLU glutamic acid
GTO Gaussian-type orbitals
HC-SO spin-orbit hyperfine correction
HF Hartree-Fock (theory)
HFC hyperfine coupling
HFCC hyperfine coupling constant
HFS hyperfine splitting
HIS histidine
XVIII
HOSO higher-order spin-orbit
IGLO-II / IGLO-III basis sets
LDA local density approximation
LSDA local spin-density approximation
MAG magnetic (property module of MAG-ReSpect)
mDKS matrix Dirac-Kohn-Sham
mdt 1,2-dimethyl-1,2-dithiolene
Me2pipdt 1,4-dimethylpiperazine-2,3-dithione
MPT molybdopterin (cofactor)
NBO natural bond order
NMR nuclear magnetic resonance
NPA natural population analysis
NR nonrelativistic
nrel nonrelativistic
nZ nuclear Zeeman
opt. optimized (structure)
OZ orbtial Zeeman
PBE Perdew Burke Ernzerhof (DFT functional)
PBE0 Perdew Burke Ernzerhof (global hybrid DFT functional)
PC pseudocontact a)
PCE picture-change error
PCM polarizable continuum model
pdt 1,2-diphenyl-1,2-dithiolate
Pf Pyrococcus furiosus
pg. page
pNMR paramagnetic NMR
PN point-size nuclei
PNP nitrogen pincer ligand
ppm parts per million
ppt parts per thousand
PSO paramagnetic nuclear spin - electron orbit
QED quantum electrodynamics
RASSI restricted active space state interaction
ref(s). reference(s)
ReSpect relativistic spectroscopy (program package)
RHF restricted Hartree-Fock (theory)
XIX
List of abbreviations
RI resolution of the identity
RKB restricted kinetic balance
RMB restricted magnetic balance
RMC relativistic mass correction
RMSD root-mean-square deviation
ROHF restricted open-shell Hartree-Fock (theory)
SCF self-consistent field
SD spin-dipolar
sec(s). section(s)
SH Spin Hamiltonian
SI syst`eme international d’unit´es
SO spin-orbit
SO-GC spin-orbit gauge correction
SOC spin-orbit coupling
SOMF spin-orbit mean-field (approximation)
SOMO singly occupied molecular orbital
SOO spin-other orbit
SOS sum-over-states
SSO spin-same orbit
struct. structure
Tab(s). Table(s)
THF tetrahydrofuran
THR threonine
Tl Thermococcus litoralis
TMS tetramethylsilan
Tp* hydrotris(3,5-dimethylpyrazol-1-yl)
TZ triple zeta (basis set)
TZVP triple zeta valence plus polarization function (basis set)
UHF unrestricted Hartree-Fock (theory)
VMD Visual Molecular Dynamics (program)
VTZ valence triple zeta (basis set)
w.r.t. with respect to
WOR4 / WOR5 tungsten oxidoreductases
X2C exact 2-component (method)
XO xanthine oxidase
XX
1. Introduction
“One thing I have learned in a long life: that
all our science, measured against reality, is
primitive and childlike – and yet it is the most
precious thing we have.”
— Albert Einstein
Nuclear magnetic resonance (NMR) spectroscopy is one of the most important structure
determination techniques to date.[1] Every undergraduate chemistry student comes across
this methodology; especially in organic chemistry where the nuclear spin of hydrogen
(I=1
2)allows 1H NMR spectra to be measured for essentially all compounds. However,
undergraduate studies are usually aimed at diamagnetic molecules since the situation –
and therefore the interpretation – becomes much more complicated in paramagnetic sys-
tems where additional interactions between nuclear and electron spins arise (paramagnetic
nuclear magnetic resonance – pNMR).[2,3] The most relevant (p)NMR parameters that
can be read from experimental spectra are the chemical shift σand the nuclear spin-spin
coupling constant J.
Whilst NMR is concerned with nuclear spin flips, it is also possible to ‘measure’ the anal-
ogous electron spin flips using electron paramagnetic resonance (EPR) spectroscopy.[4,5]a
The most relevant EPR parameters are the g-tensor g, the nuclear-electron hyperfine cou-
pling tensor A, and the electron-electron zero-field splitting tensor D. The natural restric-
tion of EPR to paramagnetic compounds can be used as an advantage in systems where
surrounding diamagnetic molecules would interfere with NMR measurements. Thus, fre-
quent EPR applications involve nowadays metal-active sites in enzymes/proteins[7–11] and
shortlived states in the photosystem II.[12,13] The first successful EPR experiment dates
back to the year 1945,[14] followed by the first hyperfine structure observation in 1949.[15]
Thereafter, the development of spectra interpretation and theoretical calculations went in
close correspondence. The concept of the effective Spin Hamiltonian (SH) emerged in the
aRequired energies: ∼0.0001 cm−1for NMR and ∼0.3 cm−1for EPR (both at B0= 0.33 T).[6]
1
1. Introduction
early 1950s as a tool to provide a phenomenologically correct description of EPR spectra
in a rather uncomplicatedbmanner.[16–22] Quantum-mechanically calculated EPR Spin
Hamiltonian parameters (and also NMR parameters) provide thus a connection between
a spectrum and the system’s electronic structure. Nevertheless, quantum-chemical de-
scriptions of EPR spectra are still lacking behind their diamagnetic NMR analogues.[1,23]
On one hand, this can be explained by the smaller experimental applicability and the
thus smaller community. On the other hand, additional problems arise for a quantum-
chemical description of the complicated electronic structure due to the unpaired electron
spin(s). Hence, the interest in these calculations stagnated after the development of the
Spin Hamiltonian but gained momentum again in the 1990s when accurate g-tensor calcu-
lations were published by Lushington et al.[24,25] Shortly after, density functional theory
(DFT) calculations of isotropic hyperfine coupling constants were reported for organic
radicals and achieved comparable qualities to coupled-cluster results at the CCSD(T)
level.[26,27] The successful developments in the EPR field stimulated also efforts on the
pNMR side, where accurate computational predictions are especially helpful:[2,3,28,29] Rele-
vant pNMR interactions are a combination of EPR and diamagnetic NMR phenomena.[30]
Structural information can be extracted similarly as for usual NMR spectra but also infor-
mation on nuclear-electron couplings are available, particularly small hyperfine coupling
effects of light ligands, which are often not resolved in EPR spectra. However, the addi-
tional interactions lead also to additional problems, as for example to strongly displaced
chemical shifts or paramagnetic relaxation enhancements that create a region around the
paramagnetic center in which signals can be broadened beyond detectability.[31] Figure 1.1
provides an overview on different couplings that appear between electron spins, nuclear
spins, and an external magnetic field B0.
Heavy elements such as 4d and 5d transition metals or even lanthanides and actinides rep-
resent a special challenge for calculations in this context:[32,33] the treatment of effects of
special relativity,[34]c i.e. effects that arise due to the finite speed of light (c≈137 in a.u.).
Some prominent examples[35–37] for the importance of relativistic effects outside NMR or
EPR spectroscopies are the yellow color of gold,[38,39] the melting point of mercury,[40] or
the voltage of lead batteries.[41] More directly relevant for the topic of this work are heavy-
atom effects on chemical shifts that induce significant shift changes for neighboring nuclei
and have initially been mentioned already in 1967.[42] They have grown to a topic of wide
bNeglecting many of the additional but less relevant effects contained in the full physical Hamiltonian.
cIn this work, “relativistic effects” will always refer to effects due to special relativity. General relativity
will not be considered.
2
Figure 1.1.: Magnetic resonance energy level diagram for a hypothetic system with two
unpaired electrons (S= 1) and two NMR-active, distinguishable I=1
2nuclei.
B0is aligned along the z-axis. aiso >0, and D > 0. “eZ” = electron Zeeman
effect. “nZ” = nuclear Zeeman effect (including the chemical shift). “ZFS” =
zero-field splitting. “HFC” = hyperfine coupling. Nuclear spin-spin coupling
not considered. Figure not to scale.
3
1. Introduction
interest,[43] and are particularly pronounced for actinide complexes.[44] And even more
important, spin-orbit (SO) effects dominate the EPR g-tensor, which is thus intrinsically
a relativistic property, even for light elements. In addition, relativistic influences on the
HFC include also considerable spin-free effects besides the aforementioned SO coupling.[45]
A large variety of approaches is available today to quantum chemists who wish to inter-
pret magnetic resonance spectra. For example density functional or multireference theory
with nonrelativistic or relativistic approximations at different levels (1-component or 2-
/4-component). Different combinations of these methodologies are available in a variety of
programs, albeit some combinations are not yet feasible. Only small systems of around 15
atoms can, e.g., be treated with high-level multireference ab initio methods at the Douglas-
Kroll-Hess (DKH) relativistic level.[46–49] Also restricted active space state interaction
(RASSI)[50] and density matrix renormalization group (DMRG)[51,52] HFC calculations
are subject to such limitations. Furthermore, similar restrictions apply to coupled-cluster
and configuration-interaction calculations of g-[53,54] and HFC tensors.[55] Therefore, EPR
parameter calculations for larger systems will have to rely on the more efficient DFT
methods for some time to come. Different implementations are available for g- and HFC
tensors considering SO effects by leading-order perturbation theory,[45,56–61] and by 2-
component variational treatments either at the quasirelativistic spin-restricted zero-order
regular approximation (ZORA) or at the Douglas-Kroll-Hess (DKH) level.[62–65] Kramers-
unrestricted 2-component DKH methods have been successfully applied at the beginning
of this century.[66,67] Subsequently, a resolution-of-the-identity Dirac-Kohn-Sham (DKS-
RI) method,[68] and finally 4-component DFT calculations of EPR parameters became
available.[69,70] The latter approach includes higher-order spin-orbit effects in a variational
framework and it avoids the additional operator transformations that are required in 2-
component frameworks to avoid picture-change effects.[71–74] In addition, spin polarization
is included in the aforementioned 4-component approach. A more detailed discussion on
the different methodologies as well as the Spin Hamiltonian follows in section 2.
In view of all these challenges, the objective of the work at hand lies in the examina-
tion of the 4-component relativistic DFT code ReSpect[75] for the calculation of EPR
and pNMR parameters on open-shell systems of 4d and 5d transition metals in order
to establish computational protocols for such systems and to pave the way for possible
future studies on systems containing even heavier elements such as the actinides. A com-
prehensive benchmark and validation study of EPR parameters is presented in section 4
to build a foundation for the subsequent EPR-related investigations on tungstoenzymes
4
2. Theory
“While I am describing to you how Nature
works, you won’t understand why Nature works
that way. But you see, nobody understands
that.”
— Richard Feynman
2.1. Theory of electron paramagnetic resonance (EPR)
Electron paramagnetic resonance (EPR) is a spectroscopic technique that probes one
(or more) unpaired electrons with a total spin S=∑isi= 0 in an external magnetic
field B. It is related to nuclear magnetic resonance (NMR) where the nuclear spin Iis
probed. First, a short introduction to the experimental considerations of EPR as well as
the effective Spin Hamiltonian, used to describe the experimental spectra, will be given.
This section is – if not stated otherwise – based on refs. 4, 5, and 76.
The first EPR measurements have been performed in 1944 by Zavoisky (USSR).[14] EPR is
based on the interaction of electron spins with an external magnetic field and their (molec-
ular) environment. The electron spin was observed in 1922 by Otto Stern and Walther
Gerlach in their famous Stern-Gerlach experiment[77,78] and postulated three years later
by Goudsmit and Uhlenbeck in 1925.[79] The concept of the electron spin appears in quan-
tum mechanics as an angular momentum without any classical analogue. Therefore, care
has to be taken if one would like to imagine spin as the ”Eigendrehimpuls” of the electron
around its own axis. The very basic description of an EPR experiment is the separation
of the usually energetically degenerated spin states αand βdue to an applied external
magnetic field (Figure 2.1 on page 9). In addition, the sample is irradiated at microwave
frequencies to cause a spin-flip from αto β. This will give rise to a measurable absorption
spectrum.
First, let us briefly reconsider the interpretation of spin as found in most textbooks tar-
geted at experimentalists. The spin vector sis indicated by its three Cartesian components
7
2. Theory
sx,sy, and sz. The magnitude of sis defined as
|s|=√s(s+ 1) (2.1)
with the electron spin quantum number s=1
2in the case of an electron. The αand
βspin states are defined as eigenvalues msof ˆszwith numerical values of +1
2and −1
2
(in a.u.), respectively. It is common practice to measure s2and sz, hence sis usually a
vector with a known szcomponent but with unknown sx, and sycomponents, which can
be drawn as a cone where all possible orientations of slie on the cone. For α, such a cone
will point to a positive z-direction and for βit will point towards a negative z-direction.
As long as there is no particular preferential direction for zdue to an external perturba-
tion, the αand βspin states will be energetically degenerate. However, electrons bear a
magnetic moment µ, since a particle with an angular momentum (s), mass mand charge
qwill always generate such a magnetic moment, which is in our case more precisely the
electronic spin magnetic dipole moment (the right-hand side is given in atomic units, i.e.
~= 1)
µe=−geµB
~s=−geµBs(2.2)
where µBis the Bohr magneton (often also found as βe), a constant µB=e~
2me=eh
4πme=
9.274 J
Tand gethe free-electron gvalue (ge= 2.0023193043617...)a,b The latter is re-
quired to account for deviations between the behavior of a quantum object and a classical
charged particle. In an applied relativistic treatment at the level of Dirac’s equation (see
sec. 2.4.1) it would be calculated to be ge= 2. The small additional corrections can
be recovered with the sophisticated quantum electrodynamics (QED) theory, making it
not only experimentally one of the most well determined constants but also from a com-
putational point of view. In case of interacting electrons, as for example in a molecule,
theoreticians usually provide the gvalue as g-shift ∆g, i.e. the deviation from gein ppt:
g=ge+ ∆g(2.3)
Note that this notation assumes a parallel orientation of the magnetic moment and the
external magnetic field along one of the Cartesian axes. A general description is provided
through a matrix notation:
g=ge1+ ∆g(2.4)
aThe gfactor describes a single number, e.g. the isotropic gvalue, while the g-tensor refers to a
3×3 matrix g. Note that the gvalue is sometimes also called gyromagnetic factor or dimensionless
magnetic moment.
bSince ge= 1, it is sometimes also called the anomalous gfactor of the electron.
8
2.1. Theory of electron paramagnetic resonance (EPR)
We can rewrite eq. (2.2) by introducing the gyromagnetic ratio (sometimes also called
magnetogyric ratio) γe=e
2mege=geµB
~
a.u.
=geµB:
µe=−γes(2.5)
If an external magnetic field Bis applied to the free electron, the energy will be perturbed
by this field:
E=−µ·B=γes·B(2.6)
Traditionally, the magnetic field is aligned to the z-axis, so that the energy difference
between the two spin states is easily accessible with respect to the magnetic field B0
along the z-axis.
E=γeB0ms−→ E±=±1
2γeB0(2.7)
This splitting of the spin states is called the (electron) Zeeman effect, it is represented in
Figure 2.1. Note that the β-spin state is energetically below the αstate.
Figure 2.1.: Zeeman effect for a free electron according to eq. (2.7). Figure adapted from
ref. 76.
As already mentioned, also nuclei can possess a spin I. The magnetic moment is then
given analogous to eq. (2.2):
µN=gNµN
~IN=γN·IN(2.8)
with |I|=√I(I+ 1), µN=e~
2mpthe nuclear magneton for a proton, γNdescribes the
gyromagnetic ratio, and gNthe nuclear gfactor that is characteristic for every isotope
and can be either positive (e.g. 1H) or negative (e.g. 15N).
9
2. Theory
As for the electron spins, the nuclear spins are also influenced by an external magnetic
field. This nuclear Zeeman effect is important in NMR measurements.
E=−gNµNB0mI(2.9)
with mI=−I, (−I+ 1), ..., I. For EPR, the required energies for absorption and sponta-
neous emission processes are too high to cause any transitions between different nuclear
spin states (∆mI= 0). NMR transitions occur at much lower frequencies. However, in-
teractions between electron and nuclear spin moments give rise to the hyperfine coupling
(HFC) interaction. Due to µN≪µBand usually high enough B0fields (high-field ap-
proximation), the hyperfine interaction will be much smaller than the electron Zeeman
interaction. Therefore, these interactions split the individual lines from the electron Zee-
man interaction into (2I+ 1) lines.
In a nonrelativistic picture (the relativistic picture will be introduced later in this chapter),
two contributions can be distinguished for the hyperfine splitting:
1. aiso – (Fermi-) contact contribution: Interactions that happen inside the nuclear
volume. Quantum mechanically it is not excluded that an electron enters the nuclear
volume. This interaction does not depend on the orientation of the nuclear magnetic
moment to the electron magnetic moment and is therefore isotropic.
2. T–dipole-dipole contribution: Outside of the nucleus, the magnetic fields generated
by the electron and nuclear spins can be treated as classical magnetic dipoles inter-
acting with each other. Due to the dependence on the orientation of the magnetic
moments with respect to each other, this contribution is anisotropic.
Therefore, the total energy now amounts to (assuming just one nucleus with I > 0)
Etot =gµBB0ms+aisomsmI+s T I ,(2.10)
where Tis called the hyperfine dipolar (interaction) tensor (or hyperfine anisotropic
tensor).
T=⎛
⎜
⎝
Txx Txy Txz
Tyx Tyy Tyz
Tzx Tzy Tzz
⎞
⎟
⎠(2.11)
with two important properties: a) it is symmetric (Tij =Tji), and b) its trace is 0, in-
dependent of the choice for the reference frame. The consequence of the latter is that if
10
2.1. Theory of electron paramagnetic resonance (EPR)
the electron spin is spherically distributed around the nucleus, there will be no dipolar
contribution to the hyperfine interaction. Due to Tbeing symmetric, it is possible to
find a particular reference frame X,Y,Zsuch that all off-diagonal elements become zero.
This reference frame is referred to as the principal axes system where TXX,TY Y , and TZZ
are the principal values and X,Y,Zthe principal axes.
A third aspect, investigated in this work, is the zero-field splitting (ZFS). This effect
occurs for systems with at least two unpaired electrons and a total spin of S > 1
2. For
example, a triplet state S= 1 gives rise to three mSstates with mS= 1,0,−1 (in general:
2S+ 1). In the presence of a magnetic field B0, these three levels are going to split (but
not necessarily into three parts) and will be well separated at least in the high-field limit
(cf. e.g. ref. 4). While this separation will be orientation-dependent with regard to B0,
a splitting of the energy levels will remain also for vanishing B0. Zero-field splitting can
also give rise to very complex spectra that become difficult to analyze already for S=5
2
systems. Furthermore, in contrast to usual g-tensors (i.e. except for very heavy elements
like actinides), the ZFS parameters can vary widely for different systems.
ZFS contributions are often separated into two parts: the first can be attributed to dipolar-
dipolar (or spin-spin) interactions in analogy to the dipolar HFC interaction. This effect
dominates for light organic radicals. For systems containing heavier elements, the sec-
ond contribution becomes more important. It is attributable to spin-orbit effects and
starts to dominate for the 3d elements onward but is as important as the dipole-dipole
contributions already for O2. This means that for 4d and 5d elements, the ZFS is ac-
tually SO-induced and only slightly modified by the spin-spin coupling.[80] In addition,
the dipole-dipole term decreases with increasing distance between the unpaired electrons.
It is usually not considered in the determination of ZFS parameters from pure energy
comparisons as will be described later in sec. 2.7.5.
In analogy to the HFC tensor, an anisotropic, symmetric, and traceless ZFS tensor Dis
introduced
E(ee)=S D S .(2.12)
Dcan also be expressed in a principal axes system X,Y,Zwhere the off-diagonal
elements vanish and the electron dipolar interaction energy assumes the form
E(ee)
dip =DXXS2
X+DY Y S2
Y+DZZS2
Z(2.13)
Since Dis traceless (DXX +DY Y +DZZ = 0), the three contributions are often separated
11
2. Theory
into two independent terms Dand E:
D=DZZ −1
2(DXX +DY Y ) = 1
2(−DXX −DY Y + 2DZZ) = 3
2DZZ
E=1
2(DXX −DY Y )
(2.14)
Using these relations, eq. (2.13) can be rewritten to
E(ee)
dip =D[S2
Z−1
3S2]+E(S2
X−S2
Y)(2.15)
with
S2=S2
X+S2
Y+S2
Z=S(S+ 1) (2.16)
Eis a measure for the deviation of the electron distribution from axial symmetry. There-
fore, in cases of axially symmetric (/uniaxial) systems, Ewill vanish.
To extract all the above mentioned parameters (‘tensors’) for the different interactions
from an experimental spectrum, an effective Spin Hamiltonian (SH) is fitted to it.[81] In
this Hamiltonian, the true electron spin quantum number is replaced by an effective spin
˜
S. The latter is defined by (2 ˜
S+ 1) = ‘multiplicity’ (i.e. number of electron spin states).
For better readability, I will refrain from reproducing the ˜ in the following. The Spin
Hamiltonian contains only terms that depend either on the electron spin, the spin of the
nucleus, or both. All aforementioned terms, which are relevant for EPR transitions, are
collected in the Spin Hamiltonian below.
ˆ
HSH =ˆ
HeZ +ˆ
HHFC +ˆ
HZFS
=µBBgˆ
S+
NA
∑
A
ˆ
SA(N)ˆ
I(N)+ˆ
SDˆ
S(2.17)
where the three contributions are:
•eZ = electron Zeeman effect
•HFC = hyperfine coupling (HFS = hyperfine splitting)
•ZFS = zero-field splitting
12
2.2. Theoretical foundations for the quantum-chemical calculations
2.2. Theoretical foundations for the quantum-chemical
calculations
The theory for this subject is quite extensive and should be separated carefully, since
several theories are interwoven here. One has to distinguish between 1- and 2-particle
equations, the level of the relativistic Hamiltonian (4-, 2-, and 1-component), and finally
the theoretical approaches to EPR parameters. An overview on the different Hamiltoni-
ans is provided in Table 2.1.
Table 2.1.: Relations of the different Hamiltonians. a
Hamiltonians 4c 2c 1c nrl
1-electron Dirac FW
−−→ Pauli
FW, X=...
−
−−−−−−−→ ZORA
FW, U=U0U1...Un
−
−−−−−−−−−−−−−→ DKHn+1
FW, exact U
−−−−−−−−−−−→ “X2C”
nonrelativistic-limit
−
−−−−−−−−−−−−−−−→ LLB
(≥)2-electrons Dirac-Coulomb(-Breit) FW
−−→ Breit-Pauli
FW, X=...
−
−−−−−−−→ ZORA
FW, U=U0U1...Un
−
−−−−−−−−−−−−−→ DKHn+1
FW, exact U
−−−−−−−−−−−→ “X2C”
aFW = Foldy-Wouthuysen transformation, ZORA = zeroth order regular
approximation, nrl = nonrelativistic limit, LLB = L´evy-Leblond equation,
DKHn= Douglas-Kroll-Hess of n-th order.
One may add the relativistic Hamiltonian as a third axis to a representation of accuracy
in theoretical chemistry[82] as can be seen in Figure 2.2.
There are a number of very good reviews and text books available regarding the relativis-
tic 4-component Hamiltonian and various 2-component approximations. For this section,
I took advantage of the introductory reviews by Saue[83] and Autschbach[37,84] as well as
the book by Dyall and Fægri.[85] In addition, there is a very mathematical but undeni-
ably extensive review by Liu available[86] as well as the also extensive book by Reiher
and Wolf.[87] Finally, a broad collection of texts on this topic can also be found in a book
edited by Schwerdtfeger.[88]
13
2. Theory
Figure 2.2.: Accuracy of theoretical chemistry models, adapted from ref. 82.
2.3. Prerequisites (atomic units)
This part is adapted from Appendix A in ref. 89 and the “Appendix on Units and Di-
mensions” in ref. 90.
In general, atomic units will be used (i.e. e=me==1
4π0= 1). However, the atomic
unit of the magnetic field is not unambiguously defined: one can, e.g., choose between
an SI-based a.u. variant and a version that is based on the Gaussian-CGS system. In
SI-based a.u., the atomic unit for the magnetic field is:
1 a.u. =
ea2
0
(2.18)
and based on the Gaussian-CGS system it becomes:
1 a.u. =
ea2
0
·α=e
a2
0(4π0)c(2.19)
Regarding atomic units, the fine-structure constant αreducescto α=1
c, thus reducing
the difference between SI-based a.u. and Gaussian-CGS based a.u. to a factor 1
c, which
cα=e2
(4π0)c
14
2.3. Prerequisites (atomic units)
is for example reflected in the Bohr magneton:
SI: µB=e~
2me
=1
2a.u.
Gaussian-CGS: µB=e~
2mec=α
2a.u. = 1
2ca.u. ≈3.2·10−3a.u.
(2.20)
This is often only mentioned as an afterthought but has some profound consequences
for (electro)magnetic response theory as we will discuss here. Some books and (general)
reviews on the topic of relativistic energy calculations use the SI-based a.u., while the
papers by Komorovsk´y and Repisk´y on fully relativistic NMR and EPR calculations use
the Gaussian-CGS based version.[33,69] In this chapter the SI-based system is used and
cases with noticeable differences are highlighted.
Some important connections are (in the order Bohr magneton, magnetic moment of nu-
cleus N, vector potential from the nuclear magnetic moment, and magnetic vector poten-
tial for an electron):
µB[CGS] = 1
cµB[SI]
µN[CGS] = 1
cµN[SI]
AµN[CGS] = 4π
µ0
·AµN[SI]
AB[CGS] = AB[SI]
π[CGS] = p+1
cA[CGS] =p+A[SI] = π[SI]
(2.21)
12×2and 02×2will signify the 2-by-2 unit and zero matrix, respectively.
As in almost all literature relevant to this topic, the Einstein summation over repeated
indices is assumed, i.e.
y=
2
∑
i=1
cixi=c1x1+c2x2(2.22)
is simplified to
y=cixi.(2.23)
15
2. Theory
On a last note for the sake of completeness, we will also frequently make use of Leibniz’s
notation for derivatives (ashall signify a parameter):
df(a)
dx=df(x)
dx⏐⏐⏐⏐x=a
=df
dx⏐⏐⏐⏐x=a
(2.24)
2.4. 4-Component theory
“Since the mathematicians have invaded the
theory of relativity, I do not understand it
myself anymore.”
— Albert Einstein
2.4.1. 1-Electron part
An elegant derivation of the one-particle Dirac equation, starting from the relativistic
energy momentum relation for a free electron (E2=p2c2+m2c4) can be found in ref. 83
and shall not be reproduced here.
The conventional form of the Dirac equation reads
[βmc2+c(α·p)]ψ=i∂
∂tψ(2.25)
where the Dirac matrices αand βare introduced
α=⎛
⎜
⎝
αx
αy
αz
⎞
⎟
⎠β=(12×202×2
02×2−12×2)
αi=(02×2σi
σi02×2).
(2.26)
16
2.4. 4-Component theory
σrefers to the Pauli spin matrices and pis the usual momentum operator
σ=⎛
⎜
⎝
σx
σy
σz
⎞
⎟
⎠p=⎛
⎜
⎝
px
py
pz
⎞
⎟
⎠(2.27)
with
σx=(0 1
1 0 )σy=(0−i
i0)σz=(1 0
0−1)(2.28)
and
pi=−i~∇i,(2.29)
i.e. the αimatrices are:
αx=⎛
⎜
⎜
⎜
⎜
⎝
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
⎞
⎟
⎟
⎟
⎟
⎠
αy=⎛
⎜
⎜
⎜
⎜
⎝
0 0 0 −i
0 0 i0
0−i0 0
i0 0 0
⎞
⎟
⎟
⎟
⎟
⎠
αz=⎛
⎜
⎜
⎜
⎜
⎝
0 0 1 0
0 0 0 −1
1 0 0 0
0−1 0 0
⎞
⎟
⎟
⎟
⎟
⎠
.(2.30)
And finally, also the 4-component wave function ψhas been introduced with its large ΨL
and small ΨScomponents.
ψ=(ΨL
ΨS)=⎛
⎜
⎜
⎜
⎜
⎝
ϕL
(1)
ϕL
(2)
ϕS
(1)
ϕS
(2)
⎞
⎟
⎟
⎟
⎟
⎠
.(2.31)
The off-diagonal elements in αare responsible for a coupling of the small and large com-
ponents. βand Vdo not cause any coupling between these terms. The meaning of the
different components will become more obvious in the course of this chapter.
2.4.2. External fields
The aforementioned version of the free-particle Dirac equation naturally lacks a depen-
dence on external electromagnetic fields. Therefore, the principle of minimal electromag-
17
2. Theory
netic coupling is used to introduce those external fields:
p→p−qA=π
E→E−qφ =E−V(2.32)
where qresembles the particle charge, i.e. in case of the electron: q=−ewith ebeing
the fundamental charge. This introduces a vector potential A(not to be mixed up with
the HFC tensor A) and a scalar potential φor V. The effects due to a uniform external
magnetic field Band the nuclear magnetic moment µNof nucleus Nare contained in A:
A=AB+AµN
AB=1
2(B×rG)rG=r−r0
AµN=µ0
4π
µN×rN
r3
N
rN=r−RN
(2.33)
r0signifies an arbitrary fixed gauge origin for an external magnetic field and RNthe
position of nucleus N. The choice of a gauge origin will be discussed later in section 2.7.1.
Note that AµNis given for a point-nucleus magnetic moment. The term will become
more complicated in the case of a finite-nucleus magnetic moment as will be shown later
in section 2.7.3.
Introducing the principle of minimal coupling from eq. (2.32) into the Dirac equation
(2.25), we obtain the Dirac equation for one electron iin a classical external electromag-
netic field, here written in the time-independent version
hD(i)ψ=Eψ
hD(i) = β′mc2+c(αi·πi) + V·14×4=(V·12×2c(σi·πi)
c(σi·πi) (V−2mc2)·12×2)(2.34)
with an energy-alignment to the nonrelativistic energy scale (cf. Figure 2.4) by
β′=β−14×4=(0 0
0−2·12×2).(2.35)
18
2.4. 4-Component theory
2.4.3. Spin-orbit coupling
Spin-orbit coupling (SOC) is based on magnetic induction, which is a relativistic effect
arising from the Lorentz transformation. Saue has illustrated this very nicely in his re-
view[83] from which Figure 2.3 has been adapted. In its own rest frame, the nucleus
possesses an electrostatic potential φbut no (magnetic) vector potential A(the Born-
Oppenheimer approximation (BOA)[91] is used). However, the electron in its own rest
frame experiences a magnetic field. The Lorentz transformation from the frame of the
nucleus to the frame of the moving electron generates a non-zero vector potential. Spin-
orbit coupling is then defined as the interaction (energy) of the electron spin (dipole
moment) with this magnetic potential/field.
Figure 2.3.: Spin-orbit interaction between a clamped nucleus and an electron. The arrow
represents the Lorentz transformation from the frame of the nuclei to the
frame of the electron. Picture adapted from ref. 83.
Or in a more classical picture: SOC originates from magnetic fields that are induced by a
charge in relative motion and its interactions with the electron spin. The moving nucleus
(in the rest frame of the electron) is seen as a moving charged particle, which – according
to the laws of electromagnetism – generates a magnetic field. The electron’s spin dipole
moment will then interact with this magnetic field (an electron spin parallel to the mag-
netic field will result in a higher total energy than the antiparallel orientation since the
intrinsic magnetic moment of the electron is antiparallel to its spin).[92]
19
2. Theory
2.4.4. Relativistic energy spectrum
A note at the beginning of this section: the energy in eq. (2.34) has been aligned by
−mc2. Usually, the unaligned relativistic energy is denoted by a W, while the aligned
relativistic energy is denoted by an E.
Solving the unperturbed relativistic energy equation (W2=p2c2+m2c4), one obtains so-
lutions from −∞ to −mc2and from mc2to +∞, since p2c2≥0. This is of course also true
for the one-electron Dirac Hamiltonian as shown in Figure 2.4, where also the aligned en-
ergy E(0 < E < ∞and −∞ < E < −2mc2) is presented. Within these boundaries, the
solutions are a continuum of eigenvalues. Most important for quantum chemistry are the
discrete bound positive-energy states that appear in atoms and molecules for W < mc2
(E < 0). These bound states appear only for a particle in a potential, e.g. a Coulombic
potential attractive for electrons (VeN). A free electron would only have the positive- and
negative-energy continuum solutions.
Figure 2.4.: Eigenvalues of the one-electron Dirac Hamiltonian with indicated bound
positive-energy states. Picture adapted from refs. 37 and 93.
The bound energy states would be in the positive-energy regime without the energy align-
ment. Hence, in view of their original energy scale, they are still referred to as “bound
positive-energy states”, even though Eis negative in these cases. The positive-energy
states are ‘electrons-only’ solutions, whereas so-called positron(ic) states are negative-
20
2.4. 4-Component theory
energy solutions around 2mc2below the discrete electron(ic) states.
2.4.5. 2-Electron part
The Hamiltonian is of course “incomplete” without an electron-electron repulsion operator
ˆg(not to be confused with the EPR gvalue) and the nucleus-nucleus potential ˆ
VNN
(constant due to the Born-Oppenheimer approximation) as well as the aforementioned
nucleus-electron potential ˆ
VeN:
ˆ
HD=∑
i
ˆ
hD(i) + 1
2∑
i∑
j=i
ˆg(i, j) + ∑
i
ˆ
VeN(i) + ˆ
VNN
ˆ
VNN =1
2∑
K∑
L=K
ZKZL
RKL
14×4;ˆ
VeN(i) = −∑
K
eZK
riK
14×4
(2.36)
The easiest approach to the two-electron term is the nonrelativistic instantaneous Coulomb
term (in a.u.):
ˆgC(1,2) = e214×4
r12
=14×4
r12
(2.37)
This operator applies an instantaneous repulsion, which is not consistent with special
relativity where interactions are retarded. In Coulomb gauge (∇ · A= 0), relativistic
corrections add two more terms up to order O(c−2) to the Coulomb potential, using the
relativistic velocity operator cα:
ˆg(1,2) = 14×4
r12
−α1·α2
2r12
−(α1·r12)(α2·r12)
2r3
12
=14×4
r12
−α1·α2
r12
+[α1·α2
2r12
−(α1·r12)(α2·r12)
2r3
12 ]
= ˆgC(1,2) + ˆgGaunt(1,2) + [ˆggauge(1,2)]
ˆgBreit(1,2)
(2.38)
where the Coulomb term represents the charge-charge interactions, the Gaunt term is
responsible for current-current interactions and the gauge term contributes only to spin
free (scalar) interactions, which are usually less important. The Gaunt term adds unre-
tarded magnetic interactions to the Dirac-Coulomb Hamiltonian while the full Breit term
(i.e. including also the gauge term) describes retarded electromagnetic interactions.[94]
However, especially the retarded time would make the term for the vector potential very
21
2. Theory
complicated, since a tracking over time on the positions of the charges would be required
(mapping r2at trto r1at t). Therefore, the Breit term is used as a perturbation expan-
sion instead and only partly accounts for retardation effects.
A Hamiltonian for many-electron systems can then be obtained through various combina-
tions of the aforementioned two-electron parts with the Dirac one-electron Hamiltonian
in eq. (2.25) and the nucleus-nucleus potential in eq. (2.36), leading either to the Dirac-
Coulomb (DC), the Dirac-Coulomb-Breit (DCB), or the Dirac-Coulomb-Gaunt Hamilto-
nian. The DCB Hamiltonian is not Lorentz-invariant. However, the DCB Hamiltonian is
considered to be an excellent approximation to the (unknown) full Hamiltonian.[95]
The Dirac-Coulomb Hamiltonian is used in most applications, meaning that the Hamil-
tonian is actually not “fully relativistic” but usually still sufficient.[96]
2.4.6. Spin-orbit coupling: spin-other orbit (SOO) and spin-same
orbit (SSO) contributions
All terms of the two-electron operator give rise to additional spin-orbit interactions, even
ˆgC(1,2). As already depicted in Figure 2.3, the clamped nucleus possesses an electrostatic
potential φthat – upon Lorentz transformation to the frame of the electron – generates
a non-zero vector potential Afor the electron. An analogous interaction arises with a
second electron, which moves around the first electron (in the rest frame of the first elec-
tron). This contribution to the two-electron SOC is called the spin-other orbit (SOO)
interaction. Another two-electron SOC contribution, the spin-same orbit (SSO) interac-
tion, arises as a correction to the one-electron SOC, which originated from the orbital
angular momentum around the bare nucleus (see section 2.4.3). However, for a better
description of the electron’s total momentum we should also consider the movement of
one electron around a second electron (due to electron-electron interactions). Hence, the
SSO interaction is described by the Coulomb term while the SOO interaction is contained
in the Gaunt term.[83]
In addition, the Gaunt term also describes orbit-orbit and spin-spin interactions, whereas
the gauge term contributes only to scalar, i.e. spin-free interactions, making it usu-
ally less important for our purposes and a Dirac-Coulomb-Gaunt Hamiltonian sufficient.
SOO effects become less important in relation to the one-electron SO and SSO terms
for heavier nuclei, and studies have shown that their contributions reduce to around
22
2.4. Spin-orbit coupling: SOO & SSO
Figure 2.5.: Spin-same and -other orbit interaction between two electrons and a clamped
nucleus. The arrows represent the Lorentz transformations, first from the
frame of the right electron to the nuclei (blue arrow) and second to the frame
of the left electron (red arrows). Picture adapted from ref. 83.
2 % for 5d systems like tungsten.[56] This has been backed up by our own investigations.[97]
2.4.7. Restricted kinetically balanced (RKB) basis
Special care must also be taken to avoid a variational collapse that may arise in full 4-
component calculations with a finite basis set: the appearance of physically meaningless
states in the empty energy region between the bound electron states and the onset of the
negative energy continuum. This is possible due to inappropriately chosen large and small
components of the basis sets (thus it is sometimes also called finite basis set disease).[86]
From equation (2.34), an exact coupling between the small and large components can be
found:
ψS=X ψL
X=1
2mc [1 + E−V
2mc2]−1
(σ·π)(2.39)
The electronic solutions of the Dirac equation possess energies Eclose to 0. Therefore,
due to the 1
cfactor in this equation, ψSis called the small component with respect to ψL.
In these cases, ψSwill vanish for the nonrelativistic limit.[83]
First implementations of relativistic theory in computational codes did not enforce this
relation between the small and large components which led to serious problems in varia-
tional calculations.[98] Experience has shown that this problem can usually be prevented
by using a basis {χ}for ψLand {(σ·p)χ}for ψS(note that even though this is not the
exact coupling as shown in eq. (2.39), it has proven to be a sufficient approximation).[83]
Such 4-component basis sets are referred to as “restricted kinetically balanced” (RKB).
23
2. Theory
2.5. 4-Component theory in DFT
This section refers mostly to the respective chapters in refs. 85, 87, and 99, which sum-
marize the ideas of Rajagopal and Callaway,[100,101] and ref. 102. In addition, ref. 96, and
especially ref. 94 have been very helpful.
Relativistic DFT includes spin naturally and a classification – as for spin-restricted and
unrestricted nonrelativistic DFT – comes from approximations to the relativistic analogue
for the nonrelativistic spin density, the magnetization (density). (Relativistic) Current-
DFT describes relativistic DFT without approximations to the magnetization density,
while collinear and noncollinear DFT denote two relativistic DFT descriptions with re-
strictions to the magnetization density.
First, another variable, the 4-current Jhas to be introduced. It carries the spin informa-
tion due to the occurrence of the Pauli spin matrices σin α:
J= (j0,jx,jy,jz)
j0(r) = ρ(r) = N∫ψ†(r,r2, ..., rN)ψ(r,r2, ..., rN) dr2dr3...drN
ju(r) = N∫ψ†(r,r2, ..., rN)cαu(r)ψ(r,r2, ..., rN) dr2dr3...drN
(2.40)
Note that αacts only on the first electron, i.e. the electron at position r.cαdenotes the
relativistic velocity operator. ρ(r) has its usual meaning as electron density and juis the
current density (“flow of charge per unit time per unit area across a surface”[103]).
The Dirac-Kohn-Sham equation (including external electromagnetic fields) then reads:
ˆ
HD=
N
∑
i=1
ˆ
hD(i) +
N
∑
i=1
N
∑
j>i
ˆg(i, j) + ˆ
VeN
=β′mc2+cαi·p−eφS+ecαiAS(r)
+e2∫ρ(rj)
|ri−rj|−cαe2
c2∫j(rj)
|ri−rj|+ˆ
VXC +ˆ
VeN
(2.41)
where the sums have not been explicitly written from line 2 on for the sake of clarity.d
The two-electron part J[J] is written in a way that the Dirac-Coulomb-Gaunt part will
dβ′mc2+∑N
i=1 [cαi·p+eφS−eαiAS(r)] + ∑i∑j>i [e2∫ρ(rj)
|ri−rj|−cαe2
c2∫j(rj)
|ri−rj|]+ˆ
VXC +ˆ
VeN
24
2.5. 4-Component theory in DFT
be reproduced upon calculation of the expectation value with ⟨ψ|...|ψ⟩due to the leading
cαin front of the integral: ψScαψS=j(ri).
J[J] = ⟨ψS|ˆ
Vee|ψS⟩
=e2∫ρ(r1)ρ(r2)
|r1−r2|dr1dr2−e2
c2∫j(r1)j(r2)
|r1−r2|dr1dr2
(2.42)
with
ECoul =e2∫ρ(r1)ρ(r2)
|r1−r2|dr1dr2+E(Coul)
XC =e2∑
i∑
j>i ⟨ψS⏐⏐⏐⏐
14×4·14×4
rij ⏐⏐⏐⏐ψS⟩
EGaunt =−e2
c2∫j(r1)j(r2)
|r1−r2|dr1dr2+E(Gaunt)
XC =−e2∑
i∑
j>i ⟨ψS⏐⏐⏐⏐
αi·αj
rij ⏐⏐⏐⏐ψS⟩(2.43)
constructed from a Dirac-Coulomb-Gaunt Hamiltonian. While the Coulomb term re-
sembles very much its nonrelativistic counterpart, the current density is only present if
the Gaunt (or Breit) term is included in the Hamiltonian but, as mentioned in section
2.4.5, the Dirac-Coulomb Hamiltonian has proven to be sufficient for most applications.[96]
Therefore, only the Dirac-Coulomb Hamiltonian is applied in this work.
The nuclear potential is defined as
VeN =−∑
M
ZM
rM
14×4(2.44)
Special exchange-correlation functionals that incorporate the current density have been
developed within the Current-DFT approach. However, these have attracted only little
attention due to a lack of good approximations and the possibility to simply ignore the
current-dependence and apply the nonrelativistic functionals instead. The latter approach
also bears the advantage that the functionals are already implemented in every DFT
code. However, if the current density would be neglected entirely, we would also lose the
information on the spin so that further approximations become necessary, which a) enable
the use of the standard nonrelativistic functionals but b) still contain the relativistic spin
information.
Unfortunately, the definition of the spin is not unambiguous if relativistic effects (SOC)
play a role: the ˆ
s2and ˆsZoperators do not commute with the Dirac Hamiltonian ˆ
hD.
Therefore, eigenfunctions of ˆ
hDare no eigenfunctions of ˆ
s2and ˆsZand spin is not a
good quantum number anymore.[94] Thus, the definition of the relativistic analogue of the
nonrelativistic spin density Q=ρα−ρβis a non-trivial matter.[104]
25
2. Theory
Two widespread solutions for this dilemma are the collinear and noncollinearA Gordon
decompositionecan be used to separate the current density in a 4-spinor form of a many-
electron system described by the Dirac-Coulomb Hamiltonian into a spin-independent
part (first line) and a spin-dependent part (second line):
j(r1) = N1
2m∫[Ψ†β(p1+eAext(r1))Ψ + (p∗
1+eAext(r1))Ψ†βΨ]dr2...drN
+N~
2m∇1×∫Ψ†β(σ0
0σ)4×4
Ψ dr2...drN
=1
Nm(r1)
(2.45)
This enables us to define a SOC-induced analogue to the nonrelativistic spin density, the
magnetization density m:
m(r1) = N∫Ψ†βσ4×4Ψ dr2...drN
=N∫Ψ†β(σ0
0σ)Ψ dr2...drN
=N∫Ψ†(σ0
0−σ)Ψ dr2...drN
(2.46)
As for the nonrelativistic spin density, the magnetization will vanish for closed-shell sys-
tems. Assuming a magnetic field in z-direction, the relativistic magnetization density in
z-direction mZ(r) can be identified with the nonrelativistic spin density Q(r).[94] The non-
interacting Hamiltonian can then be defined in a way that – in addition to the electron
density – the mZcomponent agrees with the one from the interacting system. Using such
a fixed quantization axis for m(r) is known as collinear approach. However, remember
that σZdoes not commute with ˆ
hD. Therefore, this “relativistic spin density” depends
on the choice of the quantization axis (usually the zaxis), and special care has to be
taken when orienting the molecule: the collinear approach breaks rotational symmetry
and might therefore result in different energies for different orientations of the otherwise
identical molecule. One solution would be to demand that the length |m(r)|in the non-
interacting reference system agrees at each point in space with that of the true interacting
system.[102] This is known as the noncollinear approach.
In contrast to Current-DFT, these two formulations require “only” an exchange-correlation
eCf. the appendix of ref. 94 for further information.
26
2.5. 4-Component theory in DFT
spin-functional as is the case with the spin-unrestricted nonrelativistic DFT formulation.
Therefore, the functionals developed for the spin-unrestricted nonrelativistic domain are
used in calculations with the (non-)collinear relativistic spin DFT approaches. For the
noncollinear approach this means in practice that the exchange-correlation magnetiza-
tion vector is calculated at the grid points for the integration of the exchange-correlation
potential. A system of spin coordinates is considered for which the direction of the magne-
tization defines the local z-axis of the spin coordinates. Any traditional (nonrelativistic)
DFT functional is then used to obtain the exchange-correlation potential at this point.
Turning back to the original spin-coordinate system requires a respective 2 ×2 matrix
transformation of the exchange-correlation potential. Therefore, the exchange-correlation
potential will in general contribute to the diagonal as well as to the off-diagonal elements
of the Fock matrix.[66] In case of a collinear approach, no transformation is required and
the exchange-correlation potential will only contribute to the diagonal terms of the Fock
matrix.
Finally, since hybrid functionals with arbitrary amounts of exact-exchange (EXX) admix-
ture in the noncollinear formalism are employed herein, the exchange-correlation potential
is reproduced in eq. (2.47). The nonrelativistic spin potentials vXC
2×2are used and therefore
the 4-current changes to a form ρk(k= 0, x, y, z), where spin densities are used instead
of the current densities. The dependence of the spin densities on the total magnetization
vector Jis indicated with a superscript “(Jv)”. ξis a scalar coefficient that weights the
EXX admixture from pure Dirac-Hartree-Fock (ξ= 1) to pure Dirac-Kohn-Sham (ξ= 0)
according to the generalized (nonrelativistic) hybrid functional approach[105] shown in eq.
(2.49).
V(Jv)
XC =⎛
⎝
vXC
2×2[1−ξ, ρ(Jv)
k]02×2
02×2vXC
2×2[1−ξ, ρ(Jv)
k]⎞
⎠(2.47)
where the relativistic spin density is defined as:
ρu=ψ(Jv)†(σu02×2
02×2σu)ψ(Jv)u, v =x, y, z . (2.48)
Ehybrid
XC =ξ EHF
X+ (1 −ξ)EGGA
X+EGGA
C(2.49)
The electron-electron interaction term contains an additional exact-exchange interaction
27
2. Theory
term K(Jv)
4×4, which is also scaled by ξ.
V(Jv)
ee =∫ρ(Jv)
0(r′)
|r−r′|dV′14×4−ξ K(Jv)
4×4
K(Jv)
4×4ψ(Jv)
i(r) = ⎡
⎣∫ψ(Jv)
j
†(r′)ψ(Jv)
i(r′)
|r−r′|dV′⎤
⎦ψ(Jv)
j(r)
(2.50)
2.6. 2-Component theory
As discussed in section 2.4.4, the 4-component Dirac equation provides electronic as well
as positronic solutions without an a priori differentiation of those states. In chemistry,
however, we are only interested in the electronic states. By solving the full 4-component
equation, we spend computational resources on the calculation of components which we
are not interested in: the positronic solutions. Thus, it would be preferable to separate the
4-component Dirac equation into two 2-component equations, one solely for the electronic
states and the other one for the positronic states. Unfortunately, the lower and upper
components do not represent the electronic and positronic components, respectively. In
fact, these states are each described by sets of all four wave functions.[73] This problem
and the associated large potential savings have triggered the development of a variety of
approximations projecting out only the electronic solutions, summarized as 2-component
methods. There are two main schemes to obtain a 2-component equation that only
reproduces the positive-energy spectrum:
1. elimination techniques like the elimination of the small component (ESC).
2. (unitary decoupling) transformation techniques like the Foldy Wouthuysen (FW)
transformation which leads to the Pauli Hamiltonian and the Douglas-Kroll-Hess
method (DKH).
It has been shown that the ESC and the FW approaches deliver equivalent results.[106]
2.6.1. 1-Electron part
One of the first and most prominent 2-component Hamiltonians, the Pauli Hamiltonian,
is based on an approximation to the exact coupling condition in eq. (2.39), section 2.4.7.
28
2.6. 2-Component theory
A detailed derivation can, e.g., be found in ref. 85.
hPauli =V+
T
p2
2m
hnrel
−∇4
8c2
mass-velocity
+(∇2V)
8c2
Darwin
+1
4c2σ·[(∇V)×p]
spin-orbit
(2.51)
For the explanation of these different terms, mostly the descriptions outlined by Saue in
ref. 83 are used. The mass-velocity term can be understood as a first-order relativistic
correction to the kinetic energy T. The Darwin term is a correction to the electron-
nucleus interaction due to the so-called Zitterbewegung. Carrying out the time-averaging
to the resulting Zitterbewegung amplitude shows that its size can be deduced from the
electron-positron pair creation energy 2mc2. This suggests that the Zitterbewegung (and
that is where the name comes from) results in the Pauli Hamiltonian describing a relay
of electrons rather than a single electron: the field in the vicinity of an electron is strong
enough to enable the nearby creation of an electron-positron pair. The positron then
annihilates the “old” electron.
The only term that is actually 2-component in the Pauli Hamiltonian is the spin-orbit
coupling. In order to avoid a 2-component formulation it is therefore also common to
replace this part with a perturbative ansatz. Thus, the famous Breit-Pauli Hamiltonian
that is frequently used in the literature is often not a 2-component Hamiltonian but a
1-component Hamiltonian, and the SOC is subsequently described perturbatively.
Unfortunately, a problem arises from the mass-velocity term: it has no lower bound. This
almost prohibits the use of the Pauli Hamiltonian in variational calculations. In addition,
the Darwin term requires the second derivative of the (nuclear) potential, which can lead
to highly singular terms and thus complicates the use of a finite-basis approximation.
These complications are circumvented, for example, by the Douglas-Kroll-Hess (DKHn)
transformation(s) of various orders n. The DKH method utilizes a block diagonalization
of the Hamiltonian[37,73,83–85] to separate the positive- and negative-energy states:
U†(hLL hLS
hSL hSS )U=(h+0
0h−)
U†ψ=(ψ+
ψ−)
=⇒h+ψ+=E+ψ+
(2.52)
where the transformation matrix Uis constructed from stepwise unitary diagonalizations
29
2. Theory
U=...U4U3U2U1U0(2.53)
It is specific for DKH transformations that U0(the first-order Douglas-Kroll-Hess Hamilto-
nian DKH1) necessarily has to involve the free-particle Foldy-Wouthuysen (fpFW) trans-
formation,[107] resulting in even (E) and odd (O) terms:
h1= (UfpFW)†hDUfpFW =E0+E1+O1(2.54)
The subsequent transformations are then chosen in a way that the even terms remain
untouched and the odd term is reduced by one order so that a new even term is added.
h2=U†
1H1U1=E0+E1+E2+O2(2.55)
Finally, the partial sums of the even terms define the order nof the DKH Hamiltonian:
hDKHn=
n
∑
i=0
Ei(2.56)
2.6.2. 2-Electron part
The straightforward way to include two-electron relativistic corrections in the 2-component
methods will be the same transformation procedure as for the one-electron Hamiltonian
hD. In general, this can be expressed as:[83,84]
U†ˆg(1,2) U=⎛
⎜
⎜
⎜
⎜
⎝
˜g++
++ ˜g+−
++ ˜g−+
++ ˜g−−
++
˜g++
+−˜g+−
+−˜g−+
+−˜g−−
+−
˜g++
−+˜g+−
−+˜g−+
−+˜g−−
−+
˜g++
−− ˜g+−
−− ˜g−+
−− ˜g−−
−−
⎞
⎟
⎟
⎟
⎟
⎠
˜gqs
pr
·
=⟨pr|qs⟩(2.57)
where ˜g++
++ would be combined with h+to obtain the full 2-component Hamiltonian. How-
ever, the transformation procedure requires knowledge of the full 4-component integrals.
Thus, there is no gain regarding the computational costs. A frequent approximation is
to use the untransformed Coulomb interaction instead, i.e. neglecting the Breit interac-
tion(s) completely.
The two-electron interactions in all the above mentioned 2-component approximations de-
pend on the potential V, which should contain the electron-electron interactions as well
(V=Vee +...). A widespread approach is the use of model potentials for the two-electron
30
2.7. 4-component EPR and NMR parameter calculations
SO interaction in the form of atomic mean-field integrals (AMFI)[108] where atomic nu-
clear and Coulomb potentials are used to set up the model potential.
2.6.3. Picture change error
Up to now we have only transformed the Hamiltonian, i.e. the energy operator, to 2-
component forms, but property operators are often important in practice. To avoid
the introduction of (additional) errors, these will have to be transformed with the same
transformation U.[83,84] Hence, for an arbitrary operator ˆ
A([ ]++ signifies the ++ (2 ×2)
block of the resulting transformed 4 ×4 matrix):
ˆ
A2c =[U†ˆ
A4cU]++ (2.58)
This is called the picture change since we are changing from the 4-component picture to
a 2-component picture. A fast and easy approximation to this would be to simply take
the LL block of the untransformed operator:
ˆ
A2c ≈[ˆ
A4c]LL (2.59)
However, this can turn out to be a very bad approximation that may even introduce er-
rors larger than the relativistic effects themselves. These are termed picture-change error
(PCE).
2.7. 4-component EPR and NMR parameter calculations
Since g-shifts have their origin primarily in the effects of spin-orbit coupling, a good
description of SO effects becomes crucial. Especially for heavy elements with large SO
splittings or complexes where electronic states are very close together (near-degeneracy
effects), the often used leading-order perturbational approach is not sufficient and 2- or
4-component theories are necessary. It is also important to properly include spin polariza-
tion. As discussed previously, the latter is achieved by using a noncollinear approach. A
working equation for the 2-/4-component g-tensor is obtained by expanding the energy in
first order with respect to the magnetic field B, applying the Hellman-Feynman theorem,f
fThe Hellmann-Feynman theorem links the derivative of an eigenvalue with the expectation value of
the derivative of the respective operator: da
dx=⟨∂ˆ
A
∂x ⟩
31
2. Theory
multiplying withg1 = ⟨˜
Sv⟩
⟨˜
Sv⟩and comparing the equation to the respective part of the Spin
Hamiltonian (E=µBB·g·˜
S), eq. (2.17).[66]
E(Jv,B)≈E0(Jv) + ∑
u
Bu·dE(Jv)
dBu
=E0(Jv) + ∑
u
Bu⟨ψ(Jv)⏐⏐⏐⏐
∂H
∂Bu⏐⏐⏐⏐ψ(Jv)⟩
=E0(Jv) + ∑
u
Bu⟨ψ(Jv)⏐⏐⏐⏐
∂H
∂Bu⏐⏐⏐⏐ψ(Jv)⟩·µB
µB
·⟨˜
Sv⟩
⟨˜
Sv⟩
=E0(Jv) + ∑
u
µBBu·{1
µB⟨ψ(Jv)⏐⏐⏐⏐
∂H
∂Bu⏐⏐⏐⏐ψ(Jv)⟩1
⟨˜
Sv⟩}· ⟨ ˜
Sv⟩
(2.60)
Note that the 2-component total energy depends on the orientation of the total magneti-
zation J=L+Sand ˜
Svrepresents the effective spin of the molecule. In general, we thus
obtain:
guv =1
µB
1
⟨˜
Sv⟩
dE(Jv,B)
dBu⏐⏐⏐⏐B=0
(2.61)
The explicit dependence on the orientation of Jopens a possibility to obtain the principal
components of the g-tensor from three separate calculations:
E(Jx)≈E0(Jx) + Bx·µBgxx(Jx)·˜
Sx→gxx 0 0
E(Jy)≈E0(Jy) + By·µBgyy(Jy)·˜
Sy→0gyy 0
E(Jz)≈E0(Jz) + Bz·µBgzz(Jz)·˜
Sz→0 0 gzz
(2.62)
As will be discussed below, it is absolutely crucial for this approach to have a priori a
proper choice for the orientation of the molecule with respect to Jx,Jy, and Jz.
2.7.1. g-Tensor
This section (and the following one on HFC) is mainly a combination of the theory sec-
tions in refs. 66, 69 and 97.
In a 4-component (4c) formulation the energy will be orientation dependent in a non-
g⟨˜
Sv⟩is the effective spin of the system as used in the definition of the Spin Hamiltonian. Hence, it is
a constant factor provided that the multiplicity of the investigated system is known.
32
2.7. 4-component EPR and NMR parameter calculations
collinear approach and the g-tensor is a first-order property that can be obtained from
eq. (2.61). Inserting µB=1
2a.u. (see eq. (2.20)) givesh
guν =2
⟨˜
Sν⟩
dE(Jν,B)
dBu⏐⏐⏐⏐B=0
=2
⟨˜
Sν⟩⟨∂ˆ
H
∂Bu⟩B=0
.(2.63)
Next, we introduce[85] the magnetic dipole moment M, which can be expressed as a simple
expectation value by applying the Hellmann-Feynman theorem:
M=−[dE
dB]B=0
=−⟨∂ˆ
H
∂B⟩B=0
(2.64)
Since only one term in eq. (2.34) depends on B, the differentiation is straight forward
and produces
M=−⟨∂
∂B(ce
2α·(B×rG))⟩
=−⟨∂
∂B(−ce
2(α×rG)·B)⟩
=ce
2⟨ψ|α×rG|ψ⟩.
(2.65)
Comparing eqs. (2.63) and (2.64), we see that we can use eq. (2.65) to derive an expression
for the g-tensor from 4-component energy calculations:
guν =−2
⟨˜
Sν⟩·ce
2⟨ψ(Jν)⏐⏐(α×rG)u⏐⏐ψ(Jν)⟩(2.66)
Using the RKB basis set,
ψ=(ΨL
ΨS)=(CL
λχλ
CS
λ1
2c(σ·p)χλ)(2.67)
hIn Gaussian-CGS based a.u., eq. (2.63) would read guν =2c
⟨˜
Sν⟩
dE(Jν,B)
dBu⏐⏐⏐B=0.
33
2. Theory
we obtain (with αfrom eq. (2.26) and by substituting e= 1 due to the use of a.u.):
guν =−c
⟨˜
Sν⟩⟨ψ(Jν)
λ|(α×rG)u|ψ(Jν)
τ⟩
=−c
⟨˜
Sν⟩⟨(ΨL(Jν)
λ
ΨS(Jν)
λ)⏐⏐⏐⏐⏐(02×2σu
σu02×2)×(rG)u⏐⏐⏐⏐⏐(ΨL(Jν)
τ
ΨS(Jν)
τ)⟩
=−c
⟨˜
Sν⟩[⟨ΨL(Jν)
λ⏐⏐(σ×rG)u⏐⏐ΨS(Jν)
τ⟩+⟨ΨS(Jν)
λ⏐⏐(σ×rG)u⏐⏐ΨL(Jν)
τ⟩]
= + c
⟨˜
Sν⟩[⟨ΨL(Jν)
λ⏐⏐(rG×σ)u⏐⏐ΨS(Jν)
τ⟩+⟨ΨS(Jν)
λ⏐⏐(rG×σ)u⏐⏐ΨL(Jν)
τ⟩]
=c
⟨˜
Sν⟩[⟨CL(Jν)
λχλ⏐⏐⏐(rG×σ)u⏐⏐⏐CS(Jν)
τ
1
2c(σ·p)χτ⟩
+⟨CS(Jν)
λ
1
2c(σ·p)χλ⏐⏐⏐(rG×σ)u⏐⏐⏐CL(Jν)
τχτ⟩]
=c
⟨˜
Sν⟩[(1
2c)CL(Jν)∗
λCS(Jν)
τ⟨χλ⏐⏐(rG×σ)u(σ·p)⏐⏐χτ⟩
+(1
2c)†
CS(Jν)∗
λCL(Jν)
τ⟨χλ⏐⏐(σ·p)†(rG×σ)u⏐⏐χτ⟩]
=1
2⟨˜
Sν⟩[CL(Jν)∗
λCS(Jν)
τ⟨χλ⏐⏐(σ·p)(rG×σ)u⏐⏐χτ⟩
+CS(Jν)∗
λCL(Jν)
τ⟨χλ⏐⏐(σ·p)†(rG×σ)u⏐⏐χτ⟩]
(2.68)
which can be expressed in a more elegant way in a matrix representation (again using
Einstein notation and a.u.)
guν =1
⟨˜
Sν⟩(CL(Jν)∗
λCS(Jν)∗
λ)(02×2Λ†
Bu
ΛBu02×2)(CL(Jν)
τ
CS(Jν)
τ)
(ΛBu)λτ =1
2⟨χλ|(σ·p)(rG×σ)u|χτ⟩
(2.69)
or in an even more complete matrix representation with the sum over all occupied molec-
ular orbitals/states i. Then we reach an expression frequently found in respective publi-
cations:
guν =1
⟨˜
Sν⟩
occ
∑
i(CL(Jν)†
(i)CS(Jν)†
(i))(02×2Λ†
Bu
ΛBu02×2)(CL(Jν)
(i)
CS(Jν)
(i))
(ΛBu)λτ =1
2⟨χλ|(σ·p)(rG×σ)u|χτ⟩
(2.70)
34
2.7. 4-component EPR and NMR parameter calculations
In this work, the gauge origin r0in rG=r−r0is chosen according to the “common gauge
origin” (CGO) approach, i.e. it is fixed at some point in space. In heavy-element com-
pounds, 4-component g-tensor calculations have been found to be quite robust regarding
the positioning of the CGO.[109] Hence, it is common practice to place the gauge origin
at the position of the heavy nucleus.[69,110]
The g-tensor is represented by a 3 ×3 matrix with the magnetization vector changing
along the columns and the direction of the external magnetic field Bchanging along the
lines. Thus, in order to obtain the full 3 ×3 matrix, we need three different sets of basis
set prefactors Cfor the ground state(s) with x, y, z orientations of the magnetic vector
Jνand three different expectation values ΛBu. The values are easily calculated since they
only require knowledge of the Gaussian basis functions from the applied basis set and
the position(s) rof the atom(s). Knowledge of the basis set prefactors, however, is the
bottleneck here. In order to obtain them, we need to perform three independent SCF
iterations for the energy calculation, where the directions of Jνhave to coincide with
those of u(cf. eq. (2.62)). Otherwise we would not obtain the proper g-tensor element
g11 but some unspecific g11′.
Since the Pauli matrices are defined along the coordinate axes and thus fixed, the mag-
netization vectors will have to be aligned to them in order to fulfill this requirement.
Experience has shown that aligning the principal axes of a 1-component g-tensor calcu-
lation with the coordinate axes provide a good educated guess.[66] However, even though
this returns usually a sufficient agreement, in some cases – when higher-order SO effects
are important – a second calculation run, using the previously determined 4c principal
axes, may be required.[97] A suitable starting orientation is easily verified if the final prin-
cipal axes of the computed 4c g-tensor are aligned to the Cartesian axes.
Another option to fix this problem was described by Jayatilaka[111] who suggests six
linearly-independent SCF calculations with different orientations of J.
2.7.2. Hyperfine-coupling (HFC) tensor
The derivation of the 4-component hyperfine-coupling tensor equation, also a first-order
property in this framework, follows the same steps as the previous derivation for the g-
tensor. Therefore, we obtain the following definition of the hyperfine coupling tensor[70,112]
35
2. Theory
by applying the first derivative with respect to IM:
AM
uν =1
⟨˜
Sν⟩
dE(Jν,IM)
dIM
u⏐⏐⏐⏐IM=0
=1
⟨˜
Sν⟩⟨∂ˆ
H
∂IM
u⟩IM=0
(2.71)
Looking at eq. (2.34), we see that there is only one term that depends on the nuclear
spin IM(of atom M), thus we obtain:
∂
∂ IM
u(cαu·µ0
4π
µM×rM
r3
M)=µ0
4π
c γM
r3
M
∂
∂ IM
u((IM×rM)αu)
=c γM
µ0
4π
(rM×αu)
r3
M
=−c γM
µ0
4π
(αu×rM)
r3
M
=−c γM
r3
M
µ0
4π(02×2σu
σu02×2)×rM
(2.72)
Here we applied the relation of the gyromagnetic ratio γMand the nuclear magnetic
moment µMfrom eq. (2.8).
Now we can use an RKB basis set and perform an analogous derivation as previously for
the g-tensor:
AM
uν =−c γM
⟨˜
Sν⟩
µ0
4π⟨ψ(Jν)
λ⏐⏐⏐⏐(αu×rM
r3
M)u⏐⏐⏐⏐ψ(Jν)
τ⟩
=−c γM
⟨˜
Sν⟩
µ0
4π⟨(ΨL(Jν)
λ
ΨS(Jν)
λ)⏐⏐⏐⏐⏐
1
r3
M(02×2σu
σu02×2)×(rM)u⏐⏐⏐⏐⏐(ΨL(Jν)
τ
ΨS(Jν)
τ)⟩
=−c γM
⟨˜
Sν⟩r3
M
µ0
4π[⟨ΨL(Jν)
λ⏐⏐⏐(σ×rM)u⏐⏐⏐ΨS(Jν)
τ⟩+⟨ΨS(Jν)
λ⏐⏐⏐(σ×rM)u⏐⏐⏐ΨL(Jν)
τ⟩]
= + c γM
⟨˜
Sν⟩r3
M
µ0
4π[⟨ΨL(Jν)
λ⏐⏐⏐(rM×σ)u⏐⏐⏐ΨS(Jν)
τ⟩+⟨ΨS(Jν)
λ⏐⏐⏐(rM×σ)u⏐⏐⏐ΨL(Jν)
τ⟩]
=c γM
⟨˜
Sν⟩r3
M
µ0
4π[⟨CL(Jν)
λχλ⏐⏐⏐(rM×σ)u⏐⏐⏐CS(Jν)
τ
1
2c(σ·p)χτ⟩
+⟨CS(Jν)
λ
1
2c(σ·p)χλ⏐⏐⏐(rM×σ)u⏐⏐⏐CL(Jν)
τχτ⟩]
(2.73)
36
2.7. 4-component EPR and NMR parameter calculations
=c γM
⟨˜
Sν⟩r3
M
µ0
4π[(1
2c)CL(Jν)∗
λCS(Jν)
τ⟨χλ⏐⏐(rM×σ)u(σ·p)⏐⏐χτ⟩
+(1
2c)†
CS(Jν)∗
λCL(Jν)
τ⟨χλ⏐⏐(σ·p)†(rM×σ)u⏐⏐χτ⟩]
=c γM
2c⟨˜
Sν⟩r3
M
µ0
4π[CL(Jν)∗
λCS(Jν)
τ⟨χλ⏐⏐(σ·p)(rM×σ)u⏐⏐χτ⟩
+CS(Jν)∗
λCL(Jν)
τ⟨χλ⏐⏐(σ·p)†(rM×σ)u⏐⏐χτ⟩]
=1
2⟨˜
Sν⟩
µ0
4π[CL(Jν)∗
λCS(Jν)
τγM⟨χλ⏐⏐⏐⏐(σ·p)(rM×σ
r3
M)u⏐⏐⏐⏐χτ⟩
+CS(Jν)∗
λCL(Jν)
τγM⟨χλ⏐⏐⏐⏐(σ·p)†(rM×σ
r3
M)u⏐⏐⏐⏐χτ⟩]
This version can be expressed more elegantly as a matrix representation (again using
Einstein notation and a.u.)
AM
uν =1
2⟨˜
Sν⟩(CL(Jν)∗
λCS(Jν)∗
λ)(02×2Γ†
IM
u
ΓIM
u02×2)(CL(Jν)
τ
CS(Jν)
τ)
(ΓIM
u)λτ =γM
µ0
4π⟨χλ⏐⏐⏐⏐(σ·p)(rM×σ
r3
M)u⏐⏐⏐⏐χτ⟩,
(2.74)
or again in an even more complete matrix representation with the sum over all occupied
molecular orbitals/states i. Then we reach an expression frequently found in publications
using the ReSpect 4c program package:i
AM
uν =1
2⟨˜
Sν⟩
occ
∑
i(CL(Jν)†
(i)CS(Jν)†
(i))(02×2Γ†
IM
u
ΓIM
u02×2)(CL(Jν)
(i)
CS(Jν)
(i))
(ΓIM
u)λτ =γM
µ0
4π⟨χλ⏐⏐⏐⏐(σ·p)(rM×σ
r3
M)u⏐⏐⏐⏐χτ⟩(2.75)
iIn Gaussian CGS units, the µ0
4πfactor would be missing and the prefactor would be: 1
2c⟨˜
Sν⟩(due to
π=p+1
cA).
37
2. Theory
2.7.3. HFC tensor: finite-size nuclei
For the hyperfine splitting tensor the question of the definition of the nuclear magnetic
moment arises. We have a choice of using either a point electric charge and/or magnetic
moment (as used in the above derivation) or a more realistic finite-size nucleus model.
While the vector potential for a point-size nucleus was used in eq. (2.33), we introduce
a finite-size nuclear distribution described by a normalized s-type Gaussian function Gη
instead:
Gη(|R−RM|) = (η
π)3
2e−η(R−RM)2(2.76)
AµM,FN =−γM
µ0
4πIM×∇∫ ∫ ∫ Gη(|R−RM|)
|r−R|dR rM=r−RM(2.77)
The resulting difference becomes then obvious in the nuclear potential term
V(Jv,0)
2×2=V(Jv)
eN +V(Jv)
ee +V(Jv)
XC
V(Jv)
eN, PN =−∑
M
ZM
rM
12×2
V(Jv)
eN, FN =−∑
M
ZM∫ ∫ ∫ Gη(|R−RM|)
|r−R|dR12×2
(2.78)
and finally in the expression for (ΓIM
u)λτ :[70]
(ΓIM
u,FN)λτ =γM
µ0
4π⟨χλ⏐⏐⏐⏐(σ·p)(σ×∇∫ ∫ ∫ Gη(|R−RM|)
|r−R|dR)u⏐⏐⏐⏐χτ⟩(2.79)
2.7.4. NMR nuclear shielding tensor
In general, nuclear shielding is a second-order property defined as the derivative of the
energy with respect to the magnetic field and the nuclear magnetic moment.
σM
uν =d2E(B,µM)
dBudµM
ν⏐⏐⏐⏐B,µM=0
(2.80)
The solution of the corresponding linear-response equation produces rather complicated
and lengthy terms, which will not be reproduced here. The choice of a gauge origin is
38
2.7. 4-component EPR and NMR parameter calculations
found to be of crucial importance compared to the 4-component g-tensor calculations.[113]
The equations can be found in ref. 33 with a CGO approach and in ref. 113 with the
inclusion of gauge-including atomic orbitals (GIAOs) to eliminate the strong dependence
on the choice of the gauge origin for the external magnetic field. Both formalisms are
valid for closed-shell systems as well as for the orbital shielding contribution for open-shell
systems. However, in open-shell cases, the magnetization due to the free electron(s) has
to be considered, which can be done by using a noncollinear approach. This requires
three separate SCF runs in proper x,y,zdirections as already discussed for the EPR
parameters. The total pNMR shielding tensor will be discussed later in section 2.10.
2.7.5. Spin-orbit contributions to zero-field splittings for axially
symmetric systems in a 4-component framework
While we do not consider ZFS in this work in the EPR context, we will need it for pNMR
shift calculations when S > 1
2. In the case of axially symmetric systems, we have E= 0,
i.e. DXX =DY Y according to eq. (2.14), where we can set D=DZZ (DXX =DY Y = 0).
As already mentioned in sec. 2.1, in the microscopic (relativistic) context the ZFS tensor
Ddescribes both the previously discussed dipolar-dipolar (spin-spin) interaction and/or
the spin-orbit contributions. The spin-spin term becomes negligible for very heavy atoms
compared to the SO effects.
In 4-component calculations, the ground and lowest excited states are characterized by the
different projections of the total magnetization J. In the axial case, the zero-field splitting
can be obtained as energy difference between the two degenerate energy states and the
third relativistic state without an external magnetic perturbation, thereby neglecting
electronic (dipolar) spin-spin interactions, which are not accounted for. This is assumed
to be a good approximation, since 4-component calculations are usually performed only
for heavy-atom systems where the SO contributions are dominant. A general procedure
for fully relativistic ReSpect calculations may then be:[80,114]
1. Extract the three energies (two of them have to be equal) from the three SCF
calculations
2. Take their energy difference as ˜
D= ∆E=E∥−E⊥
3. Convert to cm−1(1 a.u. = 220 000 cm−1)
4. Apply the correction factor as described by van W¨ullen:[114] D=˜
D
S(S−1
2)
39
2. Theory
5. Build the tensor (where bcan be set to 0; the eigenvalues and therefore the pNMR
values will not change): ⎛
⎜
⎝
b0 0
0b0
0 0 b+D
⎞
⎟
⎠(2.81)
2.8. 2-component DKH calculation for g- and HFC
tensors
The developments of 2-component EPR parameter calculations from refs. 66, 71, and 115
will be briefly recapitulated here.
As described in ref. 71, a back-transformation is applied to the unperturbed 2-component
wave function ˜
ψ0, in order to obtain the expectation value of the 4-component property
operator ˆ
A.
⟨ψ0|ˆ
A|ψ0⟩=⟨(U2U1)−1˜
ψ0⏐⏐⏐ˆ
A⏐⏐⏐(U2U1)−1˜
ψ0⟩(2.82)
For a scalar-relativistic calculation of the hyperfine tensor, this results in expressions that
are rather complicated (compared to the 4-component expressions that were derived in
section 2.7.2). The corresponding back-transformed equations (up to c−4) read:
Auv =A(1)
uv +A(2)
uv
A(1)
uv =−2
⟨˜
Sv⟩
µ0
4πµNgNRe ⟨˜
ψ0⏐⏐⏐⏐iR (δuv
(rN·p)
r3
N
−rNv
r3
N
pu)σvQ⏐⏐⏐⏐
˜
ψ0⟩(2.83)
A(2)
uv =−2
⟨˜
Sv⟩
µ0
4πµNgN
Re ⟨˜
ψ0⏐⏐⏐⏐⏐
i{Rδuv ∑
t
rNt
r3
N
R[Q(p)pt
V
Ep+Ep′
R(p)
−R(p)V
Ep+Ep′
p′
tQ(p′)]
−RrNv
r3
N
R[Q(p)pu
V(p,p′)
Ep+Ep′
R(p′)
−R(p)V
Ep+Ep′
p′
uQ(p′)]}σv⏐⏐⏐⏐⏐
˜
ψ0⟩
(2.84)
40
2.8. 2-component DKH calculation for g- and HFC tensors
with
Q=1
√2Ep(Ep+c2)
R=√Ep+c2
2Ep
Ep=√c2(σ·p)2+c4
p=−i∇
V=∑
k
Zk
rk
(2.85)
The expression for Auv, that is already much less elegant than its 4-component counter-
part, will become even more complicated in the case of a finite-size nucleus.:[115]
Analogously, use of the back-transformation procedure results in the following equation
for the g-tensor components:
guv =4
⟨˜
Sv⟩Re ⟨ψ0(Jv)⏐⏐⏐⏐⏐∑
i
Λuv(i)⏐⏐⏐⏐⏐
ψ0(Jv)⟩
Λuv(i) = cQ(i)(Lu(i)σv(i) + ge
2I(i)δuv
+ (r(i)·∇(i)) δuv −rv(i)∇u(i))σv(i)R(i)
(2.86)
This specific approach does not recover dipole-dipole contributions from relativistic kinetic-
energy corrections to the orbital and spin Zeeman interactions.[66]
41
2. Theory
2.9. 1-Component EPR parameter calculations
While the aforementioned 2- and 4-component approaches provide reliable EPR parame-
ters even for systems with large spin-orbit effects, it is difficult to decompose these values
into specific contributing terms.[116] The latter, however, is required for more in-depth
interpretations of EPR features. Therefore, also calculations with a perturbational inclu-
sion of SO effects based on the Breit-Pauli Hamiltonian were conducted for this work, and
we will briefly discuss the principles here. Two possibilities to include spin-orbit coupling
into quantum-chemical calculations of EPR parameters have to be distinguished: a) SOC
as part of ˆ
H0where the perturbation is then restricted to the magnetic fields (2- and 4-
component methods) or b) SOC as perturbation on top of the unperturbed Hamiltonian
ˆ
H0(1-component method). The calculation of ground-state energies is more cumbersome
for option a) while option b) struggles with more complicated perturbation expressions.
The latter approach is based on the Rayleigh Schr¨odinger perturbation theory and there-
fore only suitable if the perturbation is small. This becomes particularly problematic in
cases with low-lying excited states and/or large spin-orbit effects, i.e. for (very) heavy
elements.[66]
The one-particle 2-component Pauli Hamiltonian was introduced in sec. 2.6, eq. (2.51).
The perturbative expression of the Breit-Pauli Hamiltonian (BPH) results in a multitude
of terms, which will not be reproduced here. They can, e.g., be found in Appendix F
of ref. 22. In general, the Hamiltonian can be separated into three different groups:
electron-electron, nucleus-nucleus, and nucleus-electron interactions:
ˆ
H=∑
k
ˆ
He,k +∑
l
ˆ
HN,l +∑
m
ˆ
HeN,m (2.87)
where the spin-orbit operator
ˆ
HSO =ˆ
HSO(1e) +ˆ
HSSO(2e) +ˆ
HSOO(2e) (2.88)
is a part of ˆ
He. From a methodological point of view, the method of atomic mean-field
spin-orbit integrals (AMFI)[108] is often used to calculate SO contributions in 1-component
approaches. It defines an effective one-electron SO operator that includes some – but
not all – exchange interactions between the outer valence and the core electrons.[117] The
screening by two-electron contributions is achieved by means of an average over the active
electrons (with respect to CI methods), i.e. a mean-field approximation for an effective
42
2.9. 1-Component EPR parameter calculations
one-electron spin-orbit Hamiltonian. Due to its inherent one-center approximation, the
AMFI method allows a specific activation/deactivation of SO effects for certain atoms in
the calculation. This permits an estimate of the SO contributions from specific atoms
within a molecule.
2.9.1. g-Tensor
Comparing the BPH terms with the effective Spin Hamiltonian in eq. (2.17), allows for
the identification of relevant terms for the calculation of the g-tensor. All terms in the
Spin Hamiltonian are bilinear with respect to the magnetic operators S,I, and B. To
identify BPH terms that are relevant for the g-tensor, we look for those which are linear
or bilinear with respect to Sand B.
The relation in eq. (2.89) is used with the Rayleigh Schr¨odinger perturbation theory for
first- and second-order properties to identify all relevant contributions from the Breit-
Pauli Hamiltonian.[22] This assumes already the inclusion of SOC as perturbation.
guv =1
µB
d2E
dBudSv
(2.89)
First, the spin Zeeman (SZ) interaction gives a first-order contribution to the total electron
g-tensor that resembles the free-electron gfactor ge= 2.002319. Since geis one of the
most precisely determined constants and in extraordinarily good agreement with quantum
electrodynamic (QED) calculations, the ˆ
HSZ term is usually not used in quantum-chemical
calculations, but instead the g-shift ∆gis introduced as deviation from geas already
shown in eq. (2.3): ∆g=g−ge1. This allows a reduction of errors by using the well-
determined experimental gevalue instead of some approximately calculated value. Thus,
the perturbation will be reduced to
ˆ
H′=ˆ
HZKE(S,B) + ˆ
HOZ(B) + ˆ
HSO(S) + ˆ
HSO-GC(S,B) (2.90)
where ˆ
HZKE and ˆ
HSO-GC depend bilinearly on Sand B, meaning that they will contribute
in first order to ∆g(‘diamagnetic expression’). ˆ
HOZ and ˆ
HSO on the other hand will only
contribute in terms of a cross-term as sum-over-states expression to second order (‘param-
agnetic expression’). Therefore, the full ∆gvalue is constructed from three contributions:
∆guv = ∆gZKE,uv
’1st-order’
+ ∆gSO-GC,uv
’1st-order’
+ ∆gSO/OZ,uv
’2nd-order’
.(2.91)
43
2. Theory
1. The Zeeman kinetic energy (ZKE) correction
∆gZKE,uv =1
⟨˜
S⟩⟨Ψ0⏐⏐⏐⏐⏐
∂ˆ
HZKE,uv
∂Bu⏐⏐⏐⏐⏐
Ψ0⟩(2.92)
where ⟨˜
S⟩denotes the effective spin quantum number.
2. The spin-orbit gauge correction (SO-GC)j
ˆ
HSO-GC =ˆ
HSO-GC(1e) +ˆ
HSO-GC(2e)
∆gSO-GC,uv =1
⟨˜
S⟩⟨Ψ0⏐⏐⏐⏐⏐
∂ˆ
HSO-GC,uv
∂Bu⏐⏐⏐⏐⏐
Ψ0⟩(2.93)
3. The second-order spin-orbit/orbital Zeeman (SO/OZ) term
∆gSO/OZ,uv =1
⟨˜
S⟩⎡
⎣∑
i=0 ⟨Ψ0⏐⏐⏐ˆ
HSO,v⏐⏐⏐Ψi⟩⟨Ψi⏐⏐⏐
∂ˆ
HOZ,u
∂Bu⏐⏐⏐Ψ0⟩
E0−Ei
+ c.c.⎤
⎦(2.94)
where the orbital Zeeman operator describes the interaction between the external
magnetic field and the orbital magnetic moment.
Note at this point that the 1-component MAG program(s) have only an implementation
of ∆gSO-GC(1e),uv but not for the respective two-electron term. This has been justified by
its “general smallness” and missing “computationally efficient approximations”.[118]
And a final note regarding the general computational efficiency: Since the ∆gSO/OZ,uv
term is usually the largest contributor, it cannot be excluded from g-tensor calculations.
This second-order perturbational expression can be solved as linear-response SOS term
if standard GGA functionals are applied, whereas hybrid functionals require a coupled-
perturbed Kohn-Sham (CPKS) scheme where an iterative method to obtain the first-order
wave function is used. It is thus by far the time-defining step for such calculations (the
first-order contributions are simply expectation values which are calculated within sec-
onds to minutes) and the reason why 1-component g-tensor calculations can consume
considerable amounts of computational resources.
jThis “gauge dependency” arising within the BPH has nothing to do with that of the vector potential
of the external field.
44
2.9. 1-Component EPR parameter calculations
2.9.2. HFC tensor
The derivation for the HFC contributions follows the same thoughts on a perturbative
treatment of spin-orbit effects as described above for the g-tensor. Five relevant ex-
pressions (i.e. those that depend on Sand I) can be identified from the Breit-Pauli
Hamiltonian:
ˆ
H′=ˆ
HN
FC(S,I) + ˆ
HN
SD(S,I) + ˆ
HN
HC-SO(S,I) + ˆ
HN
PSO(I) + ˆ
HSO(S) (2.95)
The hyperfine coupling contributions from second-order perturbation theory use the fol-
lowing relation:
AN
uv =d2E
dIN
udSv
(2.96)
where the full A-tensor is then separated into four different contributions:
AN
uv =AN
FC,uv
’1st-order’
+AN
SD,uv
’1st-order’
+AN
HC-SO,uv
’1st-order’
+AN
SO/PSO,uv
’2nd-order’
.(2.97)
1. The usually dominant term is the Fermi-contact (FC) hyperfine interaction, which
probes the spin density at the nucleus through a delta function:
AN
FC,uv =1
⟨˜
S⟩⟨Ψ0⏐⏐⏐⏐⏐
∂ˆ
HFC,uv
∂Iu⏐⏐⏐⏐⏐
Ψ0⟩(2.98)
2. The spin-dipolar (SD) interaction:
AN
SD,uv =1
⟨˜
S⟩⟨Ψ0⏐⏐⏐⏐⏐
∂ˆ
HSD,uv
∂Iu⏐⏐⏐⏐⏐
Ψ0⟩(2.99)
3. The contributions from the spin-orbit hyperfine correction (HC-SO) term are usually
small.
ˆ
HN
HC-SO(S,I) = ˆ
HN
HC-SO(1e)(S,I) + ˆ
HN
HC-SSO(S,I) + ˆ
HN
HC-SOO(S,I)
AN
HC-SO,uv =1
⟨˜
S⟩⟨Ψ0⏐⏐⏐⏐⏐
∂ˆ
HHC-SO,uv
∂Iu⏐⏐⏐⏐⏐
Ψ0⟩(2.100)
45
2. Theory
4. Second-order contributions arise only from the cross term(s) of spin-orbit (SO) and
paramagnetic nuclear spin - electron orbit (PSO) terms.
AN
SO/PSO,uv =1
⟨˜
S⟩⎡
⎣∑
i=0 ⟨Ψ0⏐⏐⏐ˆ
HSO,v⏐⏐⏐Ψi⟩⟨Ψi⏐⏐⏐
∂ˆ
HPSO,u
∂Iu⏐⏐⏐Ψ0⟩
E0−Ei
+ c.c.⎤
⎦(2.101)
The hyperfine coupling tensor is often decomposed into different contributing parts, as
summarized in Table 2.2 with respect to the following notation:
A=AFC1+APC1+Adip +Adip,2 (2.102)
Here, the anti-symmetric contribution Aas is neglected, since the results from quantum
chemical programs are usually symmetric. This decomposition makes it also obvious that
the asymmetric contributions originate solely from spin-orbit terms.
Table 2.2.: Overview on the different terms for a decomposition of the HFC tensor ac-
cording to eq. (2.102).
component name properties additional abbreviations
AFC Fermi-contact nonrelativistic -
isotropic
symmetric
Adip spin-dipolar nonrelativistic ASD,T,Anr
dip
traceless anisotropic
symmetric
APC pseudocontact relativistic -
isotropic
symmetric
Adip,2 SO dipolar relativistic Torb,ASO
dip
traceless anisotropic
asymmetric
46
2.10. pNMR shift theory
2.10. pNMR shift theory
NMR shifts or paramagnetic systems are usually found in a wider shift range than those
for analogous diamagnetic compounds. This signal shift is associated with the magnetic
field generated by the unpaired electron. In addition, a broadening of the peaks is ob-
served for nuclei close to the paramagnetic center. It has its origin in the effect of the
electron spin on the nuclear spin relaxation.
The general NMR chemical shift δNof nucleus Nis defined as the difference between its
(absolute) nuclear shielding σNand the shielding of the same nucleus type in a reference
molecule σref.
δN=σref −σN(2.103)
The first quantum-chemical approach used to actually compute paramagnetic NMR shifts
that utilizes g- and A-tensors has been introduced in 2004 by Moon and Patchkovskii[30]
for S=1
2systemsk
σN=σorb −µB
4kTγN
g·A†(2.104)
where all σvalues are given in ppm, µBis the Bohr magneton in J/T (= 9.27400968 J
T),
γN=gNµN
~the gyromagnetic ratio, gdenotes the 3 ×3gmatrix, and Athe respective
hyperfine coupling matrix in MHz. σorb describes the orbital shift, which is temperature-
independent and experimentally often approximated with the respective value taken from
a closely related analogous diamagnetic compound.
The respective isotropic pNMR shift is calculated using eq. (2.103) in matrix form and
eventually taking 1
3over the trace:
δ=σref −1
3Tr[σN] (2.105)
Eq. (2.104) is valid for doublet systems but can also be written in a more generalized
formlprovided that ZFS effects are small.[28,121]
σN=σorb −µBS(S+ 1)
3kTγN
g·A†(2.106)
kNote that some of the terms in this theory were already known for decades,[119] the inclusion of g-
and A-tensors appears already in the works by McConnell in the 1950s.[120]
lS(S+1)
3reduces to 1
4for S=1
2.
47
2. Theory
In 2015, Vaara et. al. published a formalism that utilizes the ZFS tensor Dto account for
arbitrary spins S > 1
2and also featured magnetic couplings within the ZFS-split manifold
of states. This formalism represents a modern quantum-chemical implementation of the
theory by Kurland & McGarvey[122] from 1970 and corrects the earlier Pennanen/Vaara[2]
expression for the magnetic couplings, consistent with a theory by Soncini and van den
Heuvel[29] from 2013.
σN
ϵ,τ =σorb
ϵ,τ −µB
kTγN∑
a,b
gϵ,a · ⟨SaSb⟩ · Ab,τ (2.107)
where ⟨SaSb⟩describes the dyadic (i.e. a tensor of rank 2) of the Boltzmann averaged
expectation values for the spin components in the limit of vanishing B0resulting from
the effective spin operator ˜
S.
Benchmark calculations in ref. 3 have shown that the inclusion of magnetic coupling
becomes negligible at high temperatures (at least above 400 K) – compared to the gen-
eral inclusion of ZFS as introduced in the 2008 theory – but can already be crucial for
medium to low temperatures (<200 K) with effects of up to 30 ppm (for the 1H shift in
a cobalt(II)pyrazolylborate at 75 K).
Finally, it should be noted that there exists another approach by J. Autschbach et.al.,[123]
which follows the ideas by Soncini et.al.[29] and does not require the use of EPR param-
eters. Instead, this theory employs the energy differences and their second derivatives.
Since this approach is not used here, the interested reader is referred to ref. 123 for further
information.
2.11. Spin polarization and spin contamination
This section is primarily based on the review in ref. 124 and the respective chapter on
spin contamination in ref. 125.
The concepts of spin polarization and contamination are best introduced in the framework
of unrestricted, nonrelativistic Hartree-Fock (UHF) calculations where molecular wave
functions are constructed from one-electron spin orbitals, which in turn are built from the
product of a one-electron spatial orbital and an α- or β-spin function. But let us first take
48
2.11. Spin polarization and spin contamination
a look at the restricted approach: in such a framework we would have either ‘paired’ or
‘unpaired’ electrons. Paired electrons are considered to share the same spatial orbital but
have different spin functions. Unpaired electrons have orthogonal spatial orbitals without
restrictions on the spin function. The overlap integralmξαβ between the spatial orbitals of
two spin orbitals is either 0 (‘unpaired’) or 1 (‘paired’) in the restricted framework. If we
take a molecule with equal numbers of α- and β-spin electrons (Nα=Nβ) and consider
only the two electrons that are highest in energy, the MS= 0 state can be achieved either
through a paired state (S= 0, MS= 0, ξαβ = 1) or an unpaired state (S= 1, MS= 0,
ξαβ = 0).
On the other hand, the overlap integral will usually be somewhere in between in an
unrestricted framework, i.e. 0 ≤ξαβ ≤1. Hence, the concept of pure ‘paired’ or ‘unpaired’
electrons breaks down for an unrestricted approach. In other words, (local) spin density
can arise in a molecule due to partial overlaps of the spatial parts of α- and β-spin orbitals
(even if the numbers of α- and β-spin electrons are equal, i.e. MS= 0). This effect is
known as spin polarization. A measure for the degree of spin polarization from formally
doubly occupied orbitals at a point rin space can be found from the α- and β-spin
densities:[105]
ρα(r)−ρβ(r)
ρα(r) + ρβ(r)
ρ(r)
(2.108)
Spin polarization is often associated with the concept of spin contamination, i.e. an
increase of ⟨ˆ
S2⟩with respect to the expected value for a pure spin state [⟨ˆ
S2⟩exact =
S(S+ 1)]. The expectation value can be written with respect to the overlap integral of
the orbitals as[125]
⟨ˆ
S2⟩UHF =⟨ˆ
S2⟩exact +Nβ−
Nα
∑
i
Nβ
∑
j
|ξαβ
ij |2.(2.109)
Large deviations of ⟨ˆ
S2⟩UHF from ⟨ˆ
S2⟩exact are an indication for strong influences of spin
microstates with different spin quantum numbers S(but the same MS).nThe use of a
single-determinant UHF approximation to describe a multi-determinental problem (two
or more spin states that mix) is then not appropriate anymore and can thus have tremen-
dous consequences on calculated values, as for example elongated bonds.[126] Nevertheless,
mThe overlap integral is usually denoted as Sαβ , however due to possible confusions with the spin
quantum number S, it is replaced here with ξαβ .
nNote that also higher than triplet states can contribute, i.e. a doublet spin multiplicity might have
contributions from (S=1
2, MS=1
2),(S=3
2, MS=1
2),(S=5
2, MS=1
2)and so on.
49
2. Theory
some properties depend on a certain degree of spin polarization: even though the unpaired
electrons of transition metal complexes of high symmetry are often considered to reside
in pure d orbitals, spin polarization usually creates significant amounts of spin density in
s orbitals, which is crucial for Fermi-contact contributions to hyperfine coupling constants
at the metal center.[56,127,128] Whilst no empirical rule exists for an acceptable difference
between ⟨ˆ
S2⟩UHF and ⟨ˆ
S2⟩exact, less than 10 % are usually considered to be unproblematic
for organic compounds.[124] Moreover, for an expected value of ⟨ˆ
S2⟩= 0.75, a deviation
of 0.05 has been shown to be usually not significant for transition metal compounds.[97,127]
Two different types of spin polarization will be distinguished in this work: a) core-shell
and b) valence-shell spin polarization. LDA and GGA functionals have been found to
underestimate the core-shell spin polarization, while Hartree-Fock theory (UHF) overes-
timates it. The HF exchange term is proportional to the number of unpaired electrons
and has a negative sign. Hence, wave functions with higher spin polarization are en-
ergetically preferred in UHF calculations.[124] Hybrid functionals are therefore suited to
find the right mixture between the two extremes by tuning the amount of Hartree-Fock
exchange admixture.[58,97,129,130] Enhancing the overall spin polarization with an increased
amount of exact exchange can easily lead to large valence-shell spin polarizations that in
turn result in spin-contamination effects.[127,131] A promising way to avoid this problem
might be the use of local hybrid functionals where valence-shell spin polarization can be
reduced while the important core-shell spin polarization is retained due to the possibility
of adjusting the HF admixture for every point in space.[131]
As we have discussed above, better relativistic descriptions become necessary in cases of
heavy-element compounds. Unfortunately, more complicated approaches are necessary
to properly describe spin polarization in 2- and 4-component DFT approaches, namely
noncollinear methods (cf. sec. 2.5). Therefore, as for other restricted 2-component
methods, the popular 2-component ZORA implementation for magnetic properties in the
Amsterdam Density Functional (ADF) program is currently only available in a
spin-restricted form and neglects spin polarization in the case of, e.g., HFCs.[110,132] An
unrestricted implementation of the 2c-DKH method that includes spin polarization has
been presented in ref. 66 (cf. sec. 2.8). Investigations on the effect of spin polarization
for g-shifts of 21 diatomic transition metal radicals (3d, 4d, and 5d) have been presented
in ref. 133 where a proper description of spin polarization effects has been found to be
crucial for (almost) all investigated systems, independent of the required treatment of
relativistic effects.
50
3. General computational details
“mpirun has exited due to process rank 0 with
PID 8829 on node node381 exiting improperly.”
— cluster.math.tu-berlin.de
The main computational work in this thesis can be separated into two individual parts:
a) structure optimizations and b) calculations of EPR parameters. Specific details will
be provided in the respective chapters, whereas some general details are discussed here.
Molecular structures were optimized either in Turbomole V6.3.1[134] or Gaussian09.[135]
The latter was solely applied for the small d1transition-metal systems in the primary
benchmarks in section 4. The BP86 GGA functional[136–138] and the PBE0 hybrid func-
tional[139,140] were applied. Quasirelativistic energy-consistent small-core pseudopoten-
tials {effective core potentials (ECPs)}[141] of def2-TZVP type or respective all-electron
def2-TZVP[142] basis sets for the ligand atoms were used. Grimme’s atom-pairwise D3
dispersion corrections[143] with Becke-Johnson damping,[144] D3(BJ), were included. The
“m4” grid size has been invoked in Turbomole. In addition, if applicable, the conductor-
like screening model (COSMO)[145] has been used in some cases to model indirect solvent
effects.
3.1. 4-component EPR parameter calculations
All 4-component calculations were done in the ReSpect[75] program package. As men-
tioned in section 2.7.1, this requires a proper orientation of the molecule along the principal
axes of the g-tensor. Therefore, a standard 4-component calculation run for open-shell
systems follows a specific scheme:
1. Perform a 1-component g-tensor calculation to obtain an educated guess for the
principal axes.
51
3. General computational details
2. Reorient the molecule to align the g-tensors principal axes with the Cartesian axes
x,y, and z.
3. Prepare three inputs for ReSpect, differing only in the orientation of the spin
quantization axis, i.e. the three orthogonal orientations of the magnetization Jv
(v=x, y, z). Use the METHOD KS-DKH2 keyword in ReSpect to obtain starting
orbitals with the Kohn Sham approach including scalar second-order Douglas-Kroll-
Hess one-electron relativistic corrections.
4. Use the previously generated orbitals in a restart with METHOD mDKS (4-component
Kohn Sham approach) to obtain the unperturbed ground-state MO coefficients.
5. Calculate the magnetic properties (g-tensor, HFC tensor, orbital shieldings) with
the MAG module in the ReSpect program using the ground-state MO coefficients.
In the process of this thesis, two different programs were applied in order to perform the
property calculations at the 1-component relativistic level:
a) The older ReSpect-MAG program[146] (not to be mistaken with the 4-component
ReSpect propgram package, which has little in common with it), for which first an
unrestricted single-point self-consistent field (SCF) calculation in, e.g., Gaussian is re-
quired. The latter was performed with tight convergence criteria (energy convergence
at 10−6a.u. and density matrix convergence at 10−8a.u.), an ultrafine integration grid
(99 radial shells, 590 angular points per shell), with a Gaussian-type finite nuclear-charge
model and the Douglas-Kroll-Hess second-order corrections (DKH2) to account for scalar
relativity. Subsequently, the orbitals were transferred to the ReSpect-MAG package by
suitable interface routines (gs2rs), where a “fine” grid with 64 radial grid points was
used. The property calculation was then performed at the second-order perturbational
level with the atomic mean-field approximation (AMFI)[147] to include spin-orbit (SO)
effects at the first-order DKH level. Picture-change effects were neglected for the orbital
Zeeman term, which is – based on previous experience – expected to be a reasonable ap-
proximation.[74] In a few cases, also scalar relativistic HFC calculations were carried out
using the DKH2-transformed HFC operators from ref. 71 with a Gaussian finite-nucleus
magnetic moment. All calculations applied a common gauge origin positioned at the
heavy nucleus. The same basis-set types as in the subsequent 4-component ReSpect
calculations were applied.
b) A second option is the ORCA package,[148] which features a built-in module for EPR
calculations. Here, the standard input applied the PBE0 functional in an unrestricted
52
3.1. 4-component EPR parameter calculations
fashion with Hirao basis sets[149] for the heavy atoms and Ahlrichs triple-ζvalence basis set
(TZV),[150] adapted for DKH calculations,[151] including additional polarization functions.
A medium-sized DFT integration grid (GRID5) and the slow convergence criteria were
used to help with the convergence, as recommended especially for transition-metal com-
plexes.[152] DKH was chosen to account for scalar relativistic effects, while the spin-orbit
mean-field (SOMF) treatment was applied to the spin-orbit coupling operator. g-Tensors
were finally calculated with a common gauge origin at the center of mass.
4-Component calculations have been performed either at the generalized gradient approx-
imation (GGA) level or with customized B3LYP-xHF and PBE0-xHF functionals with
variable amounts of exact-exchange admixture xHF. All-electron Hirao,[149] Dyall(DZ),
and Dyall(TZ)[153–156] basis sets were employed for the heavy atoms as well as IGLO-II,
IGLO-III,[157] and Dyall(VDZ)[153–156] basis sets for the light ligand atoms. No fits of
electron and/or spin densities were used. An integration grid of “adaptive” size for the
Lebedev angular points[158] was applied together with the following numbers of radial
grid points: H – Be: 50 / B – Ne: 60 / Na – Ar: 72 / K – Lu: 80 / Hf – Rn: 96 /
Fr – Cn: 128
The Gaussian model for nuclear charge distributions was applied in the SCF calculations.
In addition, a Gaussian distribution model for the magnetic moments of the nuclei was in-
voked in the calculations of hyperfine coupling constants. A common gauge origin (CGO)
at the center of charge was always applied in g-tensor calculations. This has been shown
previously to provide a sufficient approximation.[110]
Regarding 4-component calculations, the following naming convention is used to discrimi-
nate different combinations of functionals and basis sets: ⟨DFT functional⟩-xHF/⟨heavy-
atom basis set⟩/⟨ligand basis set⟩. Here, “heavy atoms” are considered to be elements
from the 3d transition metals onwards (including ligand atoms). For instance, “PBE0-
50HF/Hirao/IGLO-II” denotes a (property) calculation using the PBE0 functional with
an increased amount of EXX admixture (50 % instead of 25 % in this case) with the Hirao
basis set for the metal center(s) and IGLO-II for the light ligand atoms.
Principal components of the g-tensor are often provided as g-shifts ∆gin ppt, w.r.t. the
free-electron gvalue (ge= 2.002319) according to
∆g= (g−ge)·1000 .(3.1)
53
3. General computational details
Hyperfine coupling constants are always given in MHz and refer to the heavy atom – if
not stated otherwise.
The charts and graphs in this work were produced using Origin.[159] Molecular repre-
sentations have been prepared in VMD,[160] and illustrations of the tensors w.r.t. the
molecules were done either in Matlab[161] or Mathematica.[162] Images of superposi-
tioned structures were prepared with Maestro as included in the Schr¨
odinger 2015.3
package.[163]
Further details will be given in the respective sections, if required.
54
4. Validation of 4-component
relativistic DFT calculations for EPR
g- and hyperfine coupling tensors
with hybrid functionals
“It is nice to know that the computer
understands the problem, but I would like to
understand it too.”
— Eugene Wigner
Copyright notice: This chapter and appendix A are reproduced from S. Gohr, P.
Hrob´arik, M. Repisk´y, S. Komorovsk´y, K. Ruud, and M. Kaupp, J. Phys. Chem. A,
2015, 119(51), 12892-12905 (DOI: 10.1021/acs.jpca.5b10996) with permission according
to the “ACS Policy on Theses and Dissertations” (21/06/2017). Figures 4.1 and 4.2,
and Tables 4.5, 4.6, and 4.8 were produced by P. Hrob´arik.
4.1. Introduction
Electron paramagnetic resonance (EPR) spectroscopy[1,4,22,164] of open-shell transition-
metal complexes is an important spectroscopic tool in a variety of research fields, ranging
from a mapping of defects in solid-state materials and surfaces (e.g., in heterogeneous
catalysis)[165,166] via studies of single-molecule magnets[167–170] to those of paramagnetic
metalloenzyme sites.[171–173] Use of quantum-chemical methods to aid the evaluation and
interpretation of EPR parameters, or to elucidate the structure of new, sometimes ex-
otic species based on EPR experiments has seen tremendous developments over the past
20 years[1,174–177] as mentioned in more detail in section 1.
55
4. 4-component EPR validation studies
Initial assessments of the 4-component matrix Dirac-Kohn-Sham (mDKS) method for
smaller heavy-atom radicals and for medium-sized molybdenum(V) and tungsten(V) com-
plexes have revealed the advantages of this method.[110] The initial mDKS implementation
of EPR parameters in the 4-component ReSpect program[75] was, however, restricted
to generalized-gradient approximation (GGA)-type functionals. As the admixture of ex-
act exchange is known to be beneficial for both g-tensors[58,129,130] and in particular for
isotropic hyperfine coupling constants[28,127,128] of transition-metal complexes, the use of
global hybrid functionals is desirable also in a 4-component framework. The implemen-
tation of the 4-component mDKS method in ReSpect was thus recently extended to
hybrid functionals and is now validated in this work. Moreover, metal HFCs for 4d and
5d transition-metal complexes will be evaluated more systematically than done in the past.
Evaluating and benchmarking the optimal EXX admixture in hybrid functionals will ini-
tially be done for a larger set of previously studied small 4d1and 5d1transition-metal
complexes of the [M(E)X4]qand [M(E)X5]qtype (M = Mo, Tc, W, Re, Os; E= O, N;
X = F, Cl, Br; q= 0, -1, -2) with comparison to an extensive set of experimental data.
Similar systems have also been the focus of 2-component ZORA studies,[178,179] but with-
out a comparably systematic evaluation of the optimal EXX admixture in global hybrids.
This test set is used to derive a ”best functional” to be suggested as part of a computa-
tional protocol for applications to a wider variety of 4d and 5d systems. The approach
is then applied and tested for a selection of larger Ir(II) and Pt(III) d7complexes ex-
hibiting particularly large g-tensor anisotropies. By comparison with scalar relativistic
DKH calculations with leading-order perturbation treatment of spin-orbit (SO) effects,
the importance of higher-order SO effects will be demonstrated. In addition, it will be
shown that in some cases these effects are necessary to even reproduce qualitative features
(such as the sign of certain tensor components).
4.2. Additional computational details
Structures: The small d1complexes were optimized using Gaussian09 at the
PBE0/def2-TZVP level of theory where quasirelativistic energy-consistent small-core pseu-
dopotentials (effective-core potentials, ECPs)[141] were used for the metal centers. If not
stated otherwise, for calculations of the large iridium and platinum complexes, the exper-
imentally determined structures have been used. These have been taken from the same
references as the EPR data (cf. Table 4.7 and references therein). Due to the absence
of an experimental structure for [Pt(C6Cl5)4]−, this complex has been optimized at the
56
4.2. Additional computational details
PBE0-D3(BJ)/def2-TZVP level. To assess the influence of the input structures on the
quality of the computed EPR parameters, also structures of some larger Ir(II) and Pt(III)
complexes – for which X-ray structure data are known – were optimized at the same com-
putational level and compared to the computed spectroscopic parameters with satisfying
agreements (cf. Table 4.1)
Table 4.1.: Principal components of the ∆g- (in ppt) and A-tensors (in MHz) in selected
Ir(II) complexes computed for X-ray and fully optimized structures (4c-mDKS-
PBE0-40HF/Hirao/IGLO-II results).
∆giso ∆g11 ∆g22 ∆g33 Aiso A11 A22 A33 RMSDa[˚
A]
[Ir(Me3tpa)(η2-ethene)]2+ PBE0-D3(BJ)/gas-phase 266 -38 259 576 135 94 146 164 0.22
PBE0-D3(BJ)+COSMOb261 -39 260 561 135 94 146 164 0.24
X-ray structure 261 -36 240 579 123 79 136 155 0
expt.[180] 258 -27 263 538 - - - 138
[Ir(C6Cl5)2(cod)] PBE0-D3(BJ) 437 -120 610 820 408 370 421 434 0.12
X-ray structure 454 -126 647 842 414 374 427 440 0
expt.[181] 545 -149 788 998 not available
aDeviations of the optimized structures from the corresponding molecular solid-state
structure quantified by the RMSD values obtained using Maestro from the
Schr¨
odinger program package.[163]
bεr= 26.6
EPR parameters: g-Tensor principal axes in the small [M(E)Xn]qcomplexes were deter-
mined a priori by the C4vpoint group symmetry. The orientations of the larger complexes
were obtained by the usual procedure of a preceding 1-component calculation. It is noted
in passing, that in some Ir(II) complexes, the principal axes of the EPR tensors obtained
at the 4c-mDKS level deviated moderately from those of the input structure (cf. Fig-
ures 4.1 and 4.2). However, a reorientation of selected molecules according to these new
principal axes and subsequent 4c-mDKS calculations did not affect the computed data
by more than 10 ppt or 2 MHz for g-shifts and HFCs, respectively (less than the effect of
the metal basis sets; see below). To assess the influence of higher-order SO effects, plots
of computed g- and HFC tensor components against a ”cscaling factor”, equal to 1
ωare
provided. The factor ωscales the speed of light in the mDKS calculations as ω·cand
varies from 1 to 100, where the latter value approaches the nonrelativistic limit and ω= 1
corresponds to a fully relativistic treatment.[66,67,110]
For comparative purposes, a few 1c-DKH calculations were also performed in the ReSpect-
MAG[146] program package. In appendix A, Tables A.1 and A.2, they are, e.g., compared
with those obtained at the 4c-mDKS level.
57
4. 4-component EPR validation studies
Figure 4.1.: Visualization of the electronic ∆g-tensors (represented as polar plots of
a∑i,j rirj∆gij function; blue isosurface = positive, orange isosurface
= negative value)[162,182] and their principal axes in selected Ir(II) and
Pt(III) complexes (a green arrow indicates a positive g-shift component;
a red one indicates a negative g-shift component). Comparison of PBE0-
40HF/Hirao/IGLO-II results at the 1c-DKH and 4c-mDKS relativistic level.
58
4.2. Additional computational details
Figure 4.2.: Orientation of ∆g-tensor principal axes with respect to the molecular frame
in selected Ir(II) and Pt(III) complexes, as obtained at the 1c-DKH (arrows
in blue) and 4c-mDKS (arrows in red) level.
59
4. 4-component EPR validation studies
4.3. Benchmark study
It is known that judicious EXX admixture in hybrid functionals can improve both, the
g-tensors and in particular the isotropic metal HFCs of transition-metal complexes (see
above). For HFCs, the main issue is the description of the spin polarization of the metal
s-type core-shells (e.g., 2s and 3s orbitals for 3d centers), which is underestimated by
(semi)local functionals and enhanced by EXX admixture (unless spin contamination be-
comes an issue).[127,128] For g-tensors, the too covalent metal-ligand bonding at semilocal
DFT levels is the main factor that is corrected for by EXX admixture. In the case of
metal-centered spin density, the latter is underestimated at the LSDA or GGA levels.
EXX admixture increases the metal spin density in such cases (this is expected to hold
for all systems studied here). As the major SO contributions to the g-tensor often arise
from metal SO coupling, more EXX admixture tends to increase the g-anisotropies in such
cases[58,110] (heavy ligand atoms may modify the picture, and ligand-centered radicals be-
have in an opposite manner).[183] Compared to earlier studies on the basis of leading-order
perturbation theory for the SO contributions, the present inclusion of higher-order SO
(HOSO) contributions might diminish the optimal EXX admixture for the g-tensors, as
the HOSO effects will enhance the g-anisotropies for a given EXX value. The effects
on the isotropic HFCs are less obvious, as SO contributions may exhibit the same or
opposite sign in comparison to the Fermi-contact-type terms (with sometimes dramatic
consequences).[184]
With these considerations in mind, a test set of 17 small 4d1and 5d1complexes with
known experimental EPR data (17 g-tensors, 14 metal HFC tensors) is used, largely
adapted from previous studies,[110,178,179,185] to carefully tune the optimal EXX admix-
ture in 4-component calculations using hybrid functionals, with particular emphasis on
the very sensitive HFCs. Where possible, the experimental EPR values collected orig-
inally in the work of Patchkovskii and Ziegler[185] were replaced with those from more
recent and reliable references (cf. Table 4.2 below). The complete set of results of this
benchmark study for a large variety of functionals and basis sets, at the 1-component
second-order perturbation DKH (1c-DKH) and 4-component mDKS (4c-mDKS) levels is
collected in Tables A.3 to A.6 in appendix A. Figures 4.3 and 4.4 compare graphically
the average percentage deviations of the 4c-mDKS results from experiment with a few
selected functionals and basis sets, respectively (Tables 4.3 and 4.4 provide the average
total and percentage deviations in more detail).
60
4.3. Benchmark study
Figure 4.3.: Effect of selected DFT functionals on the average percentage deviations
of computed data from experiment for the test set of d1transition-metal
complexes (cf. Tables A.3 in appendix A and 4.4 for the numerical data;
Dyall(TZ)/IGLO-III basis set combination used). Due to their very small
g-shift values (2–15 ppt), [TcNCl4]−and [TcNF4]−are only included in the
average deviations for Aand not for ∆g.
Figure 4.3 shows clearly that pure GGA functionals, such as PBE and BP86, perform
poorly for both ∆gand HFC values (with the average percentage deviations being larger
than 20 % for both ∆g- and A-tensor components), consistent with the above analyses
and previous experience at 1- and 2-component levels.[45,58,67,110,178,179] This is particu-
larly notable for the metal HFC components (for some rhenium complexes the percentage
error exceeds 60 % at the GGA level; cf. [ReNCl4]−and [ReNBr4]−in Table A.3 in
appendix A). As expected (see above), the GGA functionals also give too small g-tensor
anisotropies (spin densities from natural population analysis, NPA, confirm exaggerated
delocalization onto the ligands at these levels; see Table 4.5).
Standard hybrid functionals such as B3LYP and PBE0 provide substantial improvements
for both g- and HFC A-tensors (Figure 4.3). However, whereas deviations from the “best
EXX admixtures” are small for the g-tensors (in fact, ∆g⊥is obtained somewhat more
accurately at the PBE0 level than with higher EXX values, cf. Figure 4.3), there is consid-
erable room for improvement left for the HFCs. In this case, enhanced EXX admixtures
reduce the average percentage deviations to below 8 %. It may be thus already concluded
that a) the dependence of the HFCs on the EXX admixture is more pronounced compared
to the g-tensors and b) it is easier to reach small relative errors for the HFCs. This is
61
4. 4-component EPR validation studies
Table 4.2.: Comparison of experimental and computed electronic ∆g- (ppt) and
metal HFC tensor (MHz) principal components at the recommended 4c-
mDKS/PBE0-40HF/Dyall(TZ)/IGLO-III level for the benchmark set.
∆giso ∆g∥∆g⊥Aiso A∥A⊥ref. addl. ref.
[MoNCl4]2−calcd. -52 -108 -24 205 293 161
expt. -44 -96 -18 186
[MoOF4]−calcd. -81 -104 -70 175 272 126
expt.e-87 -108 -77 268 187
[MoOCl4]−calcd. -48 -26 -58 140 223 98
expt. -49 -37 -56 145 227 103 187 188,189
[MoOF5]2−calcd. -110 -113 -109 183 278 135
expt. -104 -128 -91 183 279 135 190 191,192
[MoOBr5]2−calcd. -14 92 -67 132 200 98
expt.e-9 87 -57 128 184 99a191,192
[TcNF4]−calcd. -47 -91 -25 -765 -1153 -571
expt. -44 -107 -12 -734 -1129 -537 193 194
[TcNCl4]−calcd. 217 -6 -610 -930 -450
expt. 06 -2 -561 -878 -402 195 196
[TcNBr4]−calcd. 73 171 23 -548 -801 -421
expt. 69 145 32 -488 -743 -360 195 197
[WOCl4]−calcd. -209 -200 -213 -223 -347 -161
expt. -229 -209 -239 198
[WOF5]2−calcd. -391 -464 -354 -329 -473 -257
expt. -368 -443 -330 -331 -469 -262 191,192
[WOBr5]2−calcd. -201 -111 -246 -198 -313 -141
expt.e-172 -99b-206b-105 191,192
[ReNF4]−calcd. -198 -351 -121 -2076 -3054 -1587
expt. -206 -353 -132 -2117 -3079 -1637 199
[ReNCl4]−calcd. -86 -99 -79 -1475 -2265 -1081
expt. -78 -87 -73 -1544 -2263 -1184 200 201–203
[ReNBr4]−calcd. -3 82 -46 -1249 -1915 -917
expt. 367 -29 -1340 -1994 -1013 203 204
[ReOBr4] calcd. -42 237 -182 -865 -1343 -626
expt. -98 171 -232 205
[ReOF5]−calcd. -350 -326 -362 -1809 -2682 -1372
expt. -269 -282 -262 -1959 -2878 -1499 206 207
[OsOF5] calcd. -299 -178 -360 -603 -911 -448
expt. -324 -197 -387c-627 -935d-480 206
aExperimental value for the perpendicular component obtained as A⊥=(3Aiso−A∥)
2.
bNote that numerical data for ∆g∥and ∆g⊥of [WOBr5]2−had been exchanged in refs.
191 and 192, as evident from the experimental giso value and also from our calculations.
cValue averaged over two close g-tensor components.
dNote that two digits in the A∥value for [OsOF5] in ref. 206 had been exchanged (the
−132 ·10−4cm−1should be −312 ·10−4cm−1).
eExperimental HFC values were given in Gauss or Oersted and transformed using ge·µB
h.
62
4.3. Benchmark study
Table 4.3.: Average total deviations for the benchmark set of 4d1and 5d1complexes.
∆g∥∆g⊥A∥A⊥
Effect of the functionala
BP86 38.1 23.6 334.2 306.4
PBE 37.1 24.0 321.8 292.6
B3LYP 28.4 21.0 177.6 181.5
B3LYP-40HF 21.1 21.9 34.2 47.2
PBE0 24.3 20.1 139.0 131.9
PBE0-30HF 21.0 20.5 99.0 97.7
PBE0-35HF 18.5 21.1 61.8 67.0
PBE0-40HF 17.6 21.7 37.6 46.0
PBE0-50HF 19.4 22.8 89.8 68.8
Effect of the basis setb
Dyall(TZ)/IGLO-III 17.6 21.7 37.6 46
Dyall(DZ)/IGLO-III 17.9 21.6 40.9 48.3
Dyall(VDZ)/IGLO-III 16.5 21.6 42.8 51.7
Dyall(TZ)/IGLO-II 18.5 21.9 40.4 47.1
Hirao/IGLO-III 17.7 21.3 38.1 39.3
Hirao/IGLO-II 18.5 21.2 45.7 34.0
aDyall(TZ) basis sets used on the metal atoms, IGLO-III basis sets on the light ligand
atoms, Dyall(VTZ) on Br.
bPBE0-40HF functional used, along with the Dyall(VTZ) basis set on Br.
63
4. 4-component EPR validation studies
Table 4.4.: Average percentage deviations for the benchmark set of 4d1and 5d1complexes.
Due to their very small g-shift values (2-15 ppt), [TcNCl4]−and [TcNF4]−are
only included in the average deviations for HFCs and not for ∆gvalues.
∆g∥∆g⊥A∥A⊥
Effect of the functional a
BP86 35.8 19.6 26.2 47.7
PBE 34.8 18.9 25.3 45.3
B3LYP 24.9 14.4 13.2 26.0
B3LYP-40HF 15.8 19.8 3.4 9.2
PBE0 20.7 13.9 9.7 18.2
PBE0-30HF 16.9 15.4 6.4 13.5
PBE0-35HF 14.2 17.1 4.0 10.0
PBE0-40HF 13.5 19.4 3.3 8.7
PBE0-50HF 14.8 22.9 9.0 16.9
Effect of the basis set b
Dyall(TZ)/IGLO-III 13.5 19.4 3.3 8.7
Dyall(DZ)/IGLO-III 13.5 18.2 3.4 8.9
Dyall(VDZ)/IGLO-III 12.2 19.1 3.6 9.5
Dyall(TZ)/IGLO-II 14.6 19.5 3.4 9.0
Hirao/IGLO-III 13.5 18.0 3.6 8.9
Hirao/IGLO-II 14.0 18.2 4.3 9.0
aDyall(TZ) basis sets used on the metal atoms, IGLO-III basis sets on the light ligand
atoms, and Dyall(VTZ) on Br.
bPBE0-40HF functional used, Dyall(VTZ) on Br.
64
4.3. Benchmark study
Figure 4.4.: Effect of basis-set combinations on average percentage deviations of computed
data from experiment for the test set of d1transition-metal complexes (cf.
Tables A.4 in appendix A and Table 4.4 for numerical data; PBE0-40HF
values).
in part due to the fact that SO effects play a smaller relative role for the HFCs than
for the g-tensor.[178] Moreover, the g-tensor is a valence property and thus more likely
to be influenced by environmental effects, which were neglected here. EXX admixtures
of about 30-40 % appear to provide very reasonable core-shell spin polarizations for the
HFCs, but they also perform reasonably well for the g-tensors (in particular, for the ∆g∥
component). Variation of the “pure DFT ingredients” (e.g., for PBE vs BP86 GGAs,
or for PBE0- vs B3LYP-based hybrids with the same amount of EXX admixture) is less
important than the percentage of EXX alone.
Comparing the results obtained with different basis sets (see Table A.4 in appendix A for
numerical data and Figure 4.4 for percentage deviations for the entire set of d1complexes)
suggests a slight preference for the Dyall(TZ)/IGLO-III combination of basis sets in the
case of the 4d complexes, whereas differences are small for the 5d series (except for A⊥
of some rhenium complexes, where Hirao/IGLO-II performs somewhat better, likely due
to error compensations). The smaller Hirao/IGLO-II basis set combination may thus be
a useful alternative if computational efficiency is important (see below).
Perturbational 1c-DKH calculations were used for a few complexes to analyze the impor-
tance of the SOO term neglected in the 4c calculations of the g-tensor (see Table 4.6),
65
4. 4-component EPR validation studies
Table 4.5.: NPA atomic spin densities and charges in selected d1complexes as a function
of the exchange-correlation functional.a
NPA spin densities NPA atomic charges
M O F q(M) q(O) q(F)
[MoOF5]2−PBE 0.893 -0.067 0.035 1.871 -0.658 -0.642
PBE0 0.949 -0.092 0.028 2.084 -0.703 -0.676
PBE0-40HF 0.983 -0.109 0.025 2.2 -0.732 -0.693
[WOF5]2−PBE 0.913 -0.057 0.029 2.07 -0.779 -0.658
PBE0 0.953 -0.071 0.023 2.266 -0.829 -0.687
PBE0-40HF 0.974 -0.079 0.021 2.372 -0.861 -0.702
[OsOF5] PBE 0.635 -0.068 0.087 2.092 -0.219 -0.375
PBE0 0.745 -0.148 0.082 2.287 -0.224 -0.412
PBE0-40HF 0.874 -0.281 0.081 2.399 -0.222 -0.435
aCalculations done using the built-in NBO subroutines of the Gaussian09 program;
def2-TZVP basis sets in conjunction with scalar-relativistic ECPs at the metal center.
as removal of the SOO term is possible for the AMFI approximation used. Adding only
the separately computed one-electron and SSO terms provides a reference value, and the
SOO contributions may be expressed as a percentage of that sum. Results are ca. 12 %
for the 3d1complex [CrOF5]2−, ca. 4 % for the 4d1system [MoOF5]2−and ca. 2 %
for the 5d1complexes [WOF5]2−and [OsOF5]. Results for the 3d and 4d systems are
consistent with earlier analyses[56] and confirm a) the decreasing relative role of SOO con-
tributions as one descends the Periodic Table, and b) that complexes from the same row
tend to exhibit very similar percentages. It is thus confirmed that the neglected SOO term
is of minor importance compared to other inherent errors (DFT functionals, neglect of
environmental, and counterion effects) in the computations on the heavy-metal complexes.
We are now in a position to select a recommended computational protocol that combines
4c-mDKS calculations with a suitable functional and basis set(s). From Figure 4.3, it
is clear that hybrid functionals are superior to GGA functionals for both g- and HFC
A-tensors. Whereas the g-tensor components depend somewhat less on the EXX admix-
ture than the HFCs, both are reasonably well reproduced by elevated values of xin the
range of 30–40 % (noting that we arrive at somewhat smaller percentage deviations for
g-tensors than for the HFCs; cf. Figure 4.3). We find 30–35 % of EXX admixture (cf.
Table A.3 in appendix A) to be somewhat better than 40 % for the 4d1systems and the
66
4.3. Benchmark study
Table 4.6.: Principal components of the ∆g-tensor calculated using different AMFI spin-
orbit operators (PBE0-40HF/Dyall(TZ)/IGLO-III results at the 1c-DKH
level).a
∆giso ∆g∥∆g⊥
[CrOF5]2−1-el. only -85 -74 -91
1-el. only + SSO -48 -39 -53
1-el. + 2-el. -43 -35 -47
SOO contribution 11 % 10 % 11 %
[MoOF5]2−1-el. only -135 -134 -136
1-el. only + SSO -98 -96 -99
1-el. + 2-el. -94 -93 -94
SOO contribution 4 % 4 % 4 %
[WOF5]2−1-el. only -383 -402 -374
1-el. only + SSO -329 -346 -320
1-el. + 2-el. -322 -340 -313
SOO contribution 2 % 2 % 2 %
[OsOF5] 1-el. only -213 0 -320
1-el. only + SSO -183 -1 -274
1-el. + 2-el. -180 -2 -269
SOO contribution 2 % - 2 %
aThe percentage contribution of the spin-other-orbit term (SOO) is estimated from the
difference between g-shifts computed using a full AMFI approximation (including one-
and two-electron terms) and those including only the one-electron contribution and the
spin-same-orbit (SSO) term.
67
4. 4-component EPR validation studies
reverse for the 5d1complexes. However, the differences for the 4d1complexes are too
small to warrant different protocols for the two transition-metal series. Therefore, hy-
brid functionals with roughly 40 % of Hartree-Fock exchange are recommended as a good
compromise for the entire test set, and for both g- and A-tensors. The mDKS/PBE0-
40HF/Dyall(TZ)/IGLO-III level (or its B3LYP-40HF analogue) based on good-quality
structures should thus provide excellent predictive power for the EPR parameters of 4d
and 5d complexes. Results for the benchmark set at this recommended level are reported
in Table 4.2. Further below it will be investigated if this computational protocol is also
accurate for rather different types of larger complexes.
However, let us first analyze the importance of scalar relativistic (SR) and spin-orbit (SO)
effects on the computed EPR parameters of the smaller d1complexes. To this end the 4c-
mDKS data is compared a) with those obtained by applying the corresponding Breit-Pauli
operators to nonrelativistic (NR) Kohn-Sham wave functions and b) with those calculated
within the second-order perturbation 1c-DKH framework, using identical basis sets and
exchange-correlation potentials. Figures 4.5 and 4.6 provide graphical comparisons for
some selected systems (cf. Tables A.5 and A.6 in appendix A for detailed numerical data
on all compounds). It is obvious from table A.5 that scalar relativistic effects play a
rather minor role for g-tensors of 4d complexes (SR effects are usually only a few ppt,
up to ca. 14 ppt for [TcNBr4]−), whereas they have a sizable negative contribution to
the g-shift components of 5d complexes, in particular for ∆g∥(several tens of ppt up to
-104 ppt in [ReNBr4]−, which corresponds to a decrease of the ∆g∥value by roughly 50 %;
cf. Figure 4.5). Whereas the 1c-DKH ∆g⊥values for the 4d complexes reproduce the
experimental values very well, the computed “parallel” g-tensor component at this level is
insufficiently negative (cf. Table A.5 in appendix A). Here, a variational inclusion of SO
coupling is necessary, as also demonstrated by our previous studies at the 2-component
DKH level.[67,110] Higher-order SO (HOSO) contributions to the g-tensor (beyond leading
order in perturbation theory) become even more vital for the 5d complexes, where these
effects are roughly an order of magnitude larger than for the 4d complexes and contribute
to the ∆g⊥component as well. For instance, HOSO contributions amount up to -180 ppt
(-95 ppt) for ∆g∥(∆g⊥) in the case of [OsOF5] (Figure 4.5).
In contrast, metal hyperfine couplings are significantly affected by SR effects even for the
4d complexes, with enhancements {NR →1c-DKH(SR)}of about 10–15 % and 25–35 %
for A∥and A⊥, respectively. As expected, this enhancement is even more pronounced
for the 5d complexes (ca. 22–61 % and 58–136 % for A∥and A⊥, respectively). Inclu-
sion of leading-order SO corrections increases the absolute value of the HFCs further,
68
4.3. Benchmark study
Figure 4.5.: ∆g∥and ∆g⊥computed for [ReNBr4]−and [OsOF5] within the 1-component
perturbation approach (1c-DKH) and at the 4-component relativistic level
(4c-mDKS) (cf. secs. 3 and 4.2) in comparison with experimental data
(PBE0-40HF/Dyall(TZ)/IGLO-III results; cf. Table A.5 in appendix A for
numerical values). The results for nonrelativistic wave functions with the
application of Breit-Pauli SO operators (denoted as “NR+SO”) are given as
well.
Figure 4.6.: A∥and A⊥components computed for [MOF5]2−(M = Mo, W) at the 1-
component perturbational level (1c-DKH), with and without the inclusion of
second-order SO corrections, and at the 4-component relativistic level (4c-
mDKS) (cf. secs. 3 and 4.2) in comparison with experimental data and
nonrelativistic results (PBE0-40HF/Dyall(TZ)/IGLO-III data; see also Table
A.6 in appendix A).
69
4. 4-component EPR validation studies
more so for the 5d than for the 4d complexes. Interestingly, whereas the perturbational
1c-DKH(SR)+SO A∥values are already close to experiment, the corresponding A⊥data
overshoot appreciably (Figure 4.6). Both A-tensor components are reproduced well at
the 4c-mDKS level, indicating the importance of HOSO effects also for HFCs (more so
for the 5d than for the 4d complexes).
Further qualitative insight is obtained by scaling the speed of light, and thus also the
SO integrals, in the 4c-mDKS calculations with different factors. For the present 4d and
5d systems, the resulting curves are clearly nonlinear, which confirms the influence of
HOSO effects.[110] As illustrative example, Figure 4.7 shows the curves for both ∆g- and
A-tensor components for the two 5d1complexes [ReNBr4]−and [OsOF5]. The nonlinear
behavior is particularly obvious for the g-tensors, where the rhenium complex even ex-
hibits a nonmonotonous trend. It can also be seen that the SO effects may go in either a
positive or a negative direction, explaining the partly strange shapes of the curves. Some-
what smaller deviations from linearity are found for the HFCs, which indicates overall
smaller HOSO effects, corroborating previous analyses at the 2-component level by Verma
and Autschbach.[178]
4.4. Larger iridum(II) and platinum(III) complexes
As an independent test and application of the selected PBE0-40HF/Dyall(TZ)/IGLO-III
computational protocol, a set of larger Ir(II) and Pt(III) complexes with 5d7(S=1
2)
configuration, for which experimental EPR data are available, and which exhibit large
g-tensor anisotropies has been chosen (structures in Figure 4.1).[181] These larger com-
plexes also demonstrate the efficiency of the 4c-mDKS approach, as they contain up to
133 atoms and 607 electrons in the case of [PtI2(IPr)2]+. For this complex, the compu-
tational effort has in fact been reduced by using the somewhat smaller Hirao/IGLO-II
basis set combination (leading to 2960 Cartesian 1-component GTOs). For comparative
purposes, also a slightly truncated complex, where the isopropyl substituents of the “IPr”
ligandawere replaced by a methyl group, has been computed ([PtI2(IPr’)2]+). In this
case, we still compare with data for the larger Dyall(TZ)/IGLO-III combination and find
only rather minor differences between the results obtained with these two basis sets (Ta-
ble 4.7). Truncation of [PtI2(IPr)2]+to [PtI2(IPr’)2]+affects the results for both the ∆g-
aIPR = 1,3-bis(2,6-diisopropylphenyl)imidazole-2-ylidine
70
4.4. Larger iridum(II) and platinum(III) complexes
Figure 4.7.: “Speed of light scaling” analyses (4c-mDKS level) for principal ∆g-
and A-tensor components for [ReNBr4]−and [OsOF5] (PBE0-40HF /
Dyall(TZ)/IGLO-III results).
71
4. 4-component EPR validation studies
and HFC A-tensor relatively little, consistent with the predominantly metal-centered spin
density (see discussion below). In view of the good performances of the Hirao/IGLO-II
basis sets for the small d1complexes (see above), also PBE0-40HF/Hirao/IGLO-II values
for the other systems are included. They differ moderately from the Dyall(TZ)/IGLO-III
data (Table 4.7).
Table 4.7.: Experimental and calculated principal components of the ∆g- (ppt) and hy-
perfine coupling A-tensors (MHz) in larger Ir(II) and Pt(III) complexes. Cal-
culations done at the 4c-mDKS level using different functionals and basis sets.
complex method ∆giso ∆g11 ∆g22 ∆g33 MAiso(M)A11(M)A22(M)A33(M)
trans-[Ir{η2-OC(CF3)2PtBu2}2] PBE/Dyall(TZ)/IGLO-III 269 -140 206 741 193Ir 5 -21 -9 47
PBE0/Dyall(TZ)/IGLO-III 318 -184 211 927 193Ir -3 -49 -26 66
PBE0-40HF/Dyall(TZ)/IGLO-III 336 -235 188 1055 193Ir -9 -67 -40 79
PBE0-40HF/Hirao/IGLO-II 335 -246 168 1084 193Ir -12 -70 -44 79
expt.[208] 358 -202 218 1058
[Ir(Me3tpa)(η2-ethene)]2+ PBE/Dyall(TZ)/IGLO-III 200 -46 190 457 193Ir 136 90 147 171
PBE0/Dyall(TZ)/IGLO-III 238 -41 223 530 193Ir 131 86 144 163
PBE0-40HF/Dyall(TZ)/IGLO-III 258 -38 240 573 193Ir 127 82 140 159
PBE0-40HF/Hirao/IGLO-II 261 -36 240 579 193Ir 123 79 136 155
expt.[180] 258 -27 263 538 193Ir 138
[Ir(C6Cl5)2(cod)] PBE/Dyall(TZ)/IGLO-III 351 -147 488 713 193Ir 463 438 470 481
PBE0/Dyall(TZ)/IGLO-III 433 -143 639 802 193Ir 447 409 460 470
PBE0-40HF/Dyall(TZ)/IGLO-III 462 -128 664 850 193Ir 424 384 437 451
PBE0-40HF/Hirao/IGLO-II 454 -126 647 842 193Ir 414 374 427 440
expt.[181] 545 -149 788 998
[Pt(C6Cl5)4]−PBE/Dyall(TZ)/IGLO-III 206 -511 52 1078 195Pt 7018 6315 6902 7838
PBE0/Dyall(TZ)/IGLO-III 489 -344 826 986 195Pt 7887 7029 8272 8360
PBE0-40HF/Dyall(TZ)/IGLO-III 543 -330 927 1031 195Pt 7507 6600 7940 7981
PBE0-40HF/Hirao/IGLO-II 548 -323 931 1036 195Pt 7345 6445 7773 7816
expt.[209] 594 -400 1005 1177 195Pt 7322 6375 7735 7855
[PtI2(IPr’)2]+PBE/Dyall(TZ)/IGLO-III -220 -1244 -1020 1604 195Pt 431 182 585 524
127Ia250 18 -53 786
PBE0/Dyall(TZ)/IGLO-III -85 -1064 -767 1575 195Pt 472 173 713 531
127Ia252 57 -49 748
PBE0-40HF/Dyall(TZ)/IGLO-III 7-940 -533 1493 195Pt 520 152 872 536
127Ia244 91 -42 685
PBE0-40HF/Hirao/IGLO-II -1 -960 -553 1511 195Pt 476 131 805 491
127Ib256 92 -40 718
expt.[210]c-15 -933 -722 1610 195Pt 500
127I802
[PtI2(IPr)2]+PBE0-40HF/Hirao/IGLO-II19 -932 -508 1497 195Pt 494 136 840 507
127Ib252 99 -41 697
expt.[210] -15 -933 -722 1610 195Pt 500
127I802
aDyall(TZ) basis set used on iodine.
bHirao basis set used on iodine.
cExperimental values for [PtI2(IPr)2]+.
72
4.4. Larger iridum(II) and platinum(III) complexes
As a further test of the optimal EXX admixture, Table 4.7 compares principal compo-
nents of the ∆g- and A-tensors for PBE, PBE0, and PBE0-40HF. Even for the very large
g-tensor anisotropies of some of these systems, the dependence on the functional is not too
pronounced (with the apparent exception of the ∆g22 component in Pt(III) complexes).
Hybrid functionals are better than PBE for g-tensors, and the PBE0-40HF-based pro-
tocol appears to perform overall well (PBE0 appears to be in slightly better agreement
only with the experimental ∆g22 an ∆g33 components for [PtI2(IPr)2]+, but not for the
isotropic g-shift values). Overall, the very large g-tensor anisotropies are reproduced well.
Similarly, a surprisingly small dependence of the HFC components on the EXX admixture
is seen for these complexes, with even PBE reproducing the available tensor components
reasonably well.
An interesting nonmonotonic dependence of the A-tensor components on the EXX admix-
ture is found for [Pt(C6Cl5)4]−(Table 4.7), with PBE <PBE0 >PBE0-40HF. This trend
is seen already at the DKH scalar relativistic level (Table 4.8). Whereas the overall metal
spin density increases with larger EXX admixture (as expected), analyses of NPA atomic
spin densities reveal that the hybridization between 6s and 5d AO contributions is more
involved: the Pt 5d spin density increases monotonously, but the 6s spin density shows a
small peak for PBE0 before decreasing for PBE0-40HF. Even very small changes in these
6s-type spin populations can affect the “direct” SOMO contributions to the metal HFC
significantly, thus explaining the unexpected nonmonotonous trend (which also extends
to the ∆g33 component; see Table 4.7).
In view of the extremely large g-tensor anisotropies for several of the complexes in Ta-
ble 4.7, assessment of the importance of HOSO effects for these tensors is of particular
interest. Table 4.9 compares 4c-mDKS and 1c-DKH results (using identical functionals
and basis sets) and estimates the HOSO contributions from the difference. First of all,
note that the HOSO contributions amount to several hundreds of ppt for the three irid-
ium complexes and to thousands of ppt for the two Pt complexes, and they thus exceed
by far the dependence on the functional (cf. Table 4.7). Even changes of the sign can
be seen for some tensor components (for instance, the ∆g11 component is systematically
overestimated at the 1c-DKH level) and overall fundamental modifications of the entire
tensor. In all five test cases, the 4c-mDKS results exhibit significantly better agreement
with experiment than the 1c-DKH data.
73
4. 4-component EPR validation studies
Table 4.8.: Analyses of NPA atomic spin densities and metal isotropic hyperfine couplings
in selected Ir(II) and Pt(III) complexes as a function of the exact-exchange
admixture.
complex functional metal spin densityaAiso(metal)b
6s 5d total [MHz]
trans-[Ir{η2-OC(CF3)2PtBu2}2] PBE 0.003 0.692 0.696 -36
PBE0 0.004 0.783 0.786 -61
PBE0-40HF 0.005 0.826 0.831 -77
[Ir(C6Cl5)2(cod)] PBE 0.069 0.687 0.756 487
PBE0 0.067 0.797 0.863 455
PBE0-40HF 0.065 0.835 0.9 421
[Pt(C6Cl5)4]−PBE 0.077 0.531 0.608 6827
PBE0 0.098 0.723 0.821 8703
PBE0-40HF 0.095 0.772 0.866 8057
[PtI2(IPr’)2]+PBE 0 0.278 0.278 -240
PBE0 0 0.319 0.319 -395
PBE0-40HF 0 0.351 0.352 -498
a1c-DKH2 results from Gaussian09 using the built-in NBO subroutines; Hirao basis
sets on the metal atoms and iodine, and IGLO-II basis sets on the light ligand atoms.
bMetal isotropic HFCs (193Ir and 195Pt, respectively) computed at the 1-component
scalar-relativistic DKH level, without SO corrections, in MAG.
74
4.4. Larger iridum(II) and platinum(III) complexes
Further insight into the HOSO effects may again be obtained from a “cscaling” analysis
(Figures 4.8 and 4.9), for both g- and HFC A-tensor components. All plots are clearly
nonlinear and confirm the need to include relativistic effects variationally to reproduce
the correct sign and magnitude of EPR parameters in these complexes. We also note
that SO effects may change the sign for a given HFC tensor component, as seen for A33
of trans-[Ir{η2-OC(CF)2PtBu2}2] (Figure 4.8) and A11 of [Ir(Me3tpa)(η2-ethene)]2+ (Fig-
ure 4.9). Similarly, scalar relativistic 1c-DKH calculations are not able to reproduce the
positive sign for all 195Pt A-tensor components of [PtI2(IPr)2]+, and inclusion of only
leading-order SO corrections overshoots the A33 value by more than 100 % (cf. Table A.2
in appendix A).
Figure 4.8.: “Speed of light scaling” analyses of principal g- and A-tensor components
at the 4c-mDKS level for trans-[Ir{η2-OC(CF3)2PtBu2}2] and [Pt(C6Cl5)4]−
(PBE0-40HF/Dyall(TZ)/IGLO-III results).
Finally, it should be noted that the present implementation is able to handle heavy-
metal complexes with more than 100 atoms and 2000 (scalar 1-component) basis func-
tions in affordable time. For instance, the 4c-mDKS calculations for the largest complex
75
4. 4-component EPR validation studies
Figure 4.9.: “Speed of light scaling” analyses in the 4c-mDKS framework for
[Ir(Me3tpa)(η2-ethene)]2+ (‘Ir-ethene’) and [Ir(C6Cl5)2(cod)] (‘Ir-cod’)
(PBE0-40HF/Dyall(TZ)/IGLO-III results).
76
4.4. Larger iridum(II) and platinum(III) complexes
Table 4.9.: Comparison of ∆gcomponents computed at the 1c-DKH and 4c-mDKS rela-
tivistic level, respectively. Results obtained at the PBE0-40HF/Hirao/IGLO-
II level.a
∆giso ∆g11 ∆g22 ∆g33
trans-[Ir{η2-OC(CF3)2PtBu2}2] 1c-DKH 744 343 410 1477
4c-mDKS 335 -246 168 1084
expt.[208] 358 -202 218 1058
HOSO -409 -589 -242 -393
[Ir(Me3tpa)(η2-ethene)]2+ 1c-DKH 377 47 450 634
4c-mDKS 261 -36 240 579
expt.[180] 258 -27 263 538
HOSO -116 -83 -210 -55
[Ir(C6Cl5)2(cod)] 1c-DKH 705 23 976 1116
4c-mDKS 454 -126 647 842
expt.[181] 545 -149 788 998
HOSO -251 -149 -329 -274
[Pt(C6Cl5)4]−1c-DKH 1427 13 2051 2217
4c-mDKS 548 -323 931 1036
expt.[209] 594 -400 1005 1177
HOSO -879 -336 -1120 -1181
[PtI2(IPr’)2]+1c-DKH 1708 219 1170 3736
4c-mDKS -1 -960 -553 1511
expt.[210]b-15 -933 -722 1610
HOSO -1709 -1179 -1723 -2225
aHigher-order spin-orbit (HOSO) effects estimated from the difference between
4c-mDKS and 1c-DKH data.
bExperimental values for [PtI2(IPr)2]+.
77
4. 4-component EPR validation studies
[PtI2(IPr)2]+with 133 atoms and 2960 basis functions required ca. 14 days on 24 CPUs,
Intel Xeon 2.67 GHz, with the three spin-unrestricted SCF calculations done in parallel
(each SCF running on 8 CPUs).
4.5. Conclusions
First applications of the implementation for global hybrid functionals to the 4-component
relativistic calculations of electronic g- and hyperfine coupling A-tensors in ReSpect
have been reported. The efficiency of the implementation allows computations for rather
large complexes and thus makes the method available for interesting applications in a
wide range of fields.
Systematic benchmarking of hybrid functionals and basis sets on a series of 17 small
4d1and 5d1complexes suggested a computational protocol (4c-mDKS/PBE0-40HF/
Dyall(TZ)/IGLO-III) that performed well for both g- and HFC A-tensors. The smaller
Hirao/IGLO-II basis set combination can be considered as a useful alternative if com-
putational efficiency is important. In general, the need for appreciable exact-exchange
admixture in hybrid functionals was apparent, in particular for the HFCs. Application of
this protocol to larger Ir(II) and Pt(III) complexes with very large g-tensor anisotropies
confirmed its applicability and demonstrated the importance of spin-orbit effects beyond
leading order in perturbation theory. This holds particularly true for extreme g-tensor
anisotropies, where the higher-order SO effects can easily amount to several hundreds or
even thousands of ppt and change the appearance of the tensor fundamentally.
78
A. Additional tables for the benchmark
study
Table A.1.: ∆g-tensor components computed for [Pt(C6Cl5)4]−and [PrI2(IPr’)2]+using
different relativistic approaches. PBE0-40HF/Hirao/IGLO-II results.
Complex ∆giso ∆g11 ∆g22 ∆g33
[Pt(C6Cl5)4]−NR + BP(SO)a798 6 1119 1269
1c-DKH 1427 13 2051 2217
4c-mDKS 548 -323 931 1036
expt.[209] 594 -400 1005 1177
[PtI2(IPr’)2]+NR + BP(SO)a2379 71 536 6530
1c-DKH 1708 219 1170 3736
4c-mDKS -1 -960 -553 1511
expt.[210] -15 -933 -722 1610
aBreit-Pauli SO calculations on the nonrelativistic (NR) wavefunction.
Table A.2.: 195Pt A-tensor components computed for [Pt(C6Cl5)4]−and [PrI2(IPr’)2]+us-
ing different relativistic approaches. PBE0-40HF/Hirao/IGLO-II results.
Complex Aiso A11 A22 A33
[Pt(C6Cl5)4]−NR + BP(SO)a4794 3926 5194 5261
1c-DKH 8706 7050 9503 9566
4c-mDKS 7345 6445 7773 7816
expt.[209] 7322 6375 7735 7855
[PtI2(IPr’)2]+NR + BP(SO)a670 -31 1826 214
1c-DKH 1110 113 1928 1288
4c-mDKS 476 131 805 491
expt.[210] - - - 500
aBreit-Pauli SO calculations on the nonrelativistic (NR) wavefunction.
79
A. Additional tables for the benchmark study
Table A.3.: Effect of the exchange-correlation functional on computed principal compo-
nents of electronic ∆g- and metal hyperfine coupling A-tensors. 4c-mDKS
results using Dyall(TZ) basis sets on the metal, IGLO-III basis sets on light
ligand atoms and a Dyall(VTZ) basis set on Br. Negative signs of the exper-
imental HFC components are based on the computations.
Complex Method ∆giso ∆g∥∆g⊥Aiso A∥A⊥
[MoNCl4]2−BP86 -31 -61 -15 131 213 90
PBE -31 -62 -16 133 215 92
B3LYP -42 -87 -20 160 248 116
B3LYP-40HF -53 -111 -24 203 294 158
PBE0 -43 -90 -20 176 262 133
PBE0-30HF -46 -96 -21 185 272 142
PBE0-35HF -49 -102 -22 195 283 151
PBE0-40HF -52 -108 -24 205 293 161
PBE0-50HF -58 -120 -27 227 317 183
expt.[186] -44 -96 -18 - - -
[MoOF4]−BP86 -69 -82 -63 128 219 83
PBE -70 -82 -64 130 220 85
B3LYP -77 -97 -68 148 244 99
B3LYP-40HF -83 -108 -70 175 275 125
PBE0 -77 -96 -67 158 253 111
PBE0-30HF -78 -99 -68 164 260 116
PBE0-35HF -80 -102 -69 169 266 121
PBE0-40HF -81 -104 -70 175 272 126
PBE0-50HF -85 -111 -72 186 285 136
expt.[187] -87 -108 -77 - 268 -
80
Complex Method ∆giso ∆g∥∆g⊥Aiso A∥A⊥
[MoOCl4]−BP86 -35 -4 -50 94 169 57
PBE -35 -4 -50 96 170 59
B3LYP -42 -15 -55 115 196 75
B3LYP-40HF -48 -27 -59 141 226 98
PBE0 -42 -17 -55 123 203 83
PBE0-30HF -44 -20 -56 129 210 88
PBE0-35HF -46 -23 -57 134 217 93
PBE0-40HF -48 -26 -58 140 223 98
PBE0-50HF -52 -33 -61 151 236 108
expt.[187] -49 -37 -56 145 227 103
[MoOF5]2−BP86 -101 -89 -107 135 225 90
PBE -102 -89 -109 136 226 92
B3LYP -108 -105 -110 155 250 107
B3LYP-40HF -110 -116 -107 183 281 133
PBE0 -106 -104 -108 166 259 119
PBE0-30HF -108 -107 -108 171 265 124
PBE0-35HF -109 -110 -108 177 272 130
PBE0-40HF -110 -113 -109 183 278 135
PBE0-50HF -113 -120 -110 194 291 146
expt.[190] -104 -128 -91 183 279 135
[MoOBr5]2−BP86 2 112 -53 82 140 54
PBE 1 111 -53 85 143 56
B3LYP -3 114 -61 105 170 73
B3LYP-40HF -12 98 -67 132 202 97
PBE0 -7 104 -63 114 178 82
PBE0-30HF -9 101 -64 120 185 87
PBE0-35HF -12 96 -66 126 192 92
PBE0-40HF -14 92 -67 132 200 98
PBE0-50HF -19 82 -69 144 214 109
expt.[191] -9 87 -57 128 184 99
81
A. Additional tables for the benchmark study
Complex Method ∆giso ∆g∥∆g⊥Aiso A∥A⊥
[TcNF4]−BP86 -34 -60 -21 -493 -854 -312
PBE -34 -60 -21 -493 -854 -312
B3LYP -41 -78 -23 -600 -986 -407
B3LYP-40HF -49 -95 -26 -768 -1164 -570
PBE0 -41 -78 -23 -650 -1031 -460
PBE0-30HF -43 -82 -23 -686 -1069 -494
PBE0-35HF -45 -87 -24 -724 -1110 -531
PBE0-40HF -47 -91 -25 -765 -1153 -571
PBE0-50HF -52 -100 -27 -860 -1249 -666
expt.[193] -44 -107 -12 -734 -1129 -537
[TcNCl4]−BP86 13 38 0 -351 -635 -208
PBE 12 37 0 -354 -638 -211
B3LYP 9 32 -3 -455 -766 -300
B3LYP-40HF 2 18 -6 -612 -937 -449
PBE0 7 27 -3 -500 -810 -345
PBE0-30HF 5 24 -4 -534 -848 -377
PBE0-35HF 4 21 -5 -570 -888 -412
PBE0-40HF 2 17 -5 -610 -930 -450
PBE0-50HF -2 8 -7 -705 -1028 -544
expt.[195] 0 6 -2 -561 -878 -402
[TcNBr4]−BP86 92 172 52 -287 -498 -181
PBE 91 171 51 -292 -504 -186
B3LYP 88 189 37 -392 -627 -275
B3LYP-40HF 76 180 24 -548 -803 -421
PBE0 81 180 31 -437 -676 -318
PBE0-30HF 78 178 28 -472 -715 -350
PBE0-35HF 76 175 26 -508 -757 -384
PBE0-40HF 73 171 23 -548 -801 -421
PBE0-50HF 67 160 21 -641 -901 -511
expt.[195] 69 145 32 -488 -743 -360
82
Complex Method ∆giso ∆g∥∆g⊥Aiso A∥A⊥
[WOCl4]−BP86 -182 -144 -201 -137 -258 -76
PBE -182 -145 -201 -144 -265 -83
B3LYP -203 -181 -214 -184 -311 -120
B3LYP-40HF -216 -211 -218 -227 -355 -163
PBE0 -198 -179 -208 -193 -316 -131
PBE0-30HF -202 -186 -210 -203 -326 -141
PBE0-35HF -205 -193 -211 -213 -337 -151
PBE0-40HF -209 -200 -213 -223 -347 -161
PBE0-50HF -216 -215 -217 -244 -369 -181
expt.[198] -229 -209 -239 - - -
[WOF5]2−BP86 -387 -434 -364 -228 -376 -154
PBE -398 -449 -373 -234 -382 -161
B3LYP -402 -469 -369 -278 -431 -201
B3LYP-40HF -399 -481 -358 -331 -481 -256
PBE0 -388 -451 -356 -293 -439 -221
PBE0-30HF -388 -455 -355 -305 -450 -233
PBE0-35HF -389 -460 -354 -317 -461 -245
PBE0-40HF -391 -464 -354 -329 -473 -257
PBE0-50HF -393 -474 -352 -353 -496 -281
expt.[191] -368 -443 -330 -331 -469 -262
[WOBr5]2−BP86 -165 -37 -229 -108 -216 -54
PBE -165 -39 -229 -117 -225 -63
B3LYP -188 -73 -245 -160 -276 -103
B3LYP-40HF -206 -115 -251 -203 -322 -144
PBE0 -187 -81 -240 -167 -279 -111
PBE0-30HF -192 -90 -242 -177 -290 -121
PBE0-35HF -196 -100 -244 -188 -301 -131
PBE0-40HF -201 -111 -246 -198 -313 -141
PBE0-50HF -211 -132 -250 -220 -336 -162
expt.[191] -172 -99 -206 - - -105
83
A. Additional tables for the benchmark study
Complex Method ∆giso ∆g∥∆g⊥Aiso A∥A⊥
[ReNF4]−BP86 -170 -287 -112 -1322 -2269 -849
PBE -171 -287 -113 -1355 -2300 -883
B3LYP -190 -332 -119 -1661 -2654 -1164
B3LYP-40HF -205 -365 -125 -2081 -3085 -1580
PBE0 -186 -326 -116 -1788 -2753 -1306
PBE0-30HF -190 -334 -117 -1882 -2850 -1397
PBE0-35HF -193 -342 -119 -1977 -2951 -1491
PBE0-40HF -198 -351 -121 -2076 -3054 -1587
PBE0-50HF -206 -368 -126 -2286 -3274 -1792
expt.[199] -206 -353 -132 -2117 -3079 -1637
[ReNCl4]−BP86 -57 -34 -69 -831 -1568 -463
PBE -58 -36 -70 -868 -1604 -500
B3LYP -73 -66 -77 -1130 -1914 -738
B3LYP-40HF -90 -104 -83 -1477 -2283 -1075
PBE0 -74 -72 -74 -1225 -1994 -841
PBE0-30HF -78 -81 -76 -1305 -2081 -917
PBE0-35HF -82 -90 -77 -1388 -2171 -997
PBE0-40HF -86 -99 -79 -1475 -2265 -1081
PBE0-50HF -96 -119 -84 -1665 -2468 -1264
expt.[200] -78 -87 -73 -1544 -2263 -1184
[ReNBr4]−BP86 31 137 -22 -629 -1224 -332
PBE 29 134 -23 -672 -1267 -375
B3LYP 17 123 -35 -926 -1569 -605
B3LYP-40HF -3 86 -48 -1254 -1929 -917
PBE0 11 109 -38 -1009 -1646 -691
PBE0-30HF 7 101 -40 -1085 -1732 -762
PBE0-35HF 2 92 -43 -1165 -1821 -837
PBE0-40HF -3 82 -46 -1249 -1915 -917
PBE0-50HF -14 60 -52 -1433 -2119 -1090
expt.[203] 3 67 -29 -1340 -1994 -1013
84
Complex Method ∆giso ∆g∥∆g⊥Aiso A∥A⊥
[ReOBr4] BP86 -36 206 -158 -492 -945 -265
PBE -38 203 -158 -529 -983 -302
B3LYP -35 236 -171 -694 -1163 -459
B3LYP-40HF -39 249 -184 -876 -1355 -636
PBE0 -37 231 -172 -733 -1197 -501
PBE0-30HF -39 234 -175 -777 -1244 -543
PBE0-35HF -40 236 -178 -821 -1293 -584
PBE0-40HF -42 237 -182 -865 -1343 -626
PBE0-50HF -47 237 -189 -956 -1448 -710
expt.[205] -98 171 -232 - - -
[ReOF5]−BP86 -352 -303 -376 -1250 -2113 -819
PBE -353 -304 -377 -1276 -2137 -846
B3LYP -361 -327 -378 -1520 -2420 -1070
B3LYP-40HF -357 -339 -366 -1824 -2728 -1372
PBE0 -348 -314 -365 -1600 -2466 -1166
PBE0-30HF -349 -318 -364 -1668 -2536 -1234
PBE0-35HF -349 -322 -363 -1738 -2608 -1302
PBE0-40HF -350 -326 -362 -1809 -2682 -1372
PBE0-50HF -352 -335 -360 -1956 -2836 -1516
expt.[206] -269 -282 -262 -1959 -2878 -1499
[OsOF5] BP86 -333 -200 -400 -404 -691 -260
PBE -334 -200 -401 -411 -698 -268
B3LYP -324 -194 -389 -490 -790 -339
B3LYP-40HF -305 -185 -365 -608 -925 -450
PBE0 -309 -180 -374 -517 -811 -370
PBE0-30HF -306 -179 -369 -543 -840 -394
PBE0-35HF -302 -178 -364 -571 -872 -420
PBE0-40HF -299 -178 -360 -603 -911 -448
PBE0-50HF -292 -178 -349 -685 -1013 -521
expt.[206] -324 -197 -387 -627 -935 -480
85
A. Additional tables for the benchmark study
Table A.4.: Basis-set effects on the computed principal components of electronic ∆g- and
metal hyperfine coupling A-tensors. 4c-mDKS results using the PBE0-40HF
exchange-correlation functional for small d1complexes.
Complex Method ∆giso ∆g∥∆g⊥Aiso A∥A⊥
[MoNCl4]2−Dyall(TZ)/IGLO-III -52 -108 -24 205 293 161
Dyall(DZ)/IGLO-III -50 -106 -23 207 295 162
Dyall(VDZ)/IGLO-III -51 -106 -23 208 295 165
Dyall(TZ)/IGLO-II -53 -111 -24 205 294 161
Hirao/IGLO-III -51 -107 -23 205 294 161
Hirao/IGLO-II -53 -111 -24 212 300 168
[MoOF4]−Dyall(TZ)/IGLO-III -81 -104 -70 175 272 126
Dyall(DZ)/IGLO-III -81 -104 -69 175 273 127
Dyall(VDZ)/IGLO-III -81 -105 -68 178 274 130
Dyall(TZ)/IGLO-II -81 -100 -71 172 269 123
Hirao/IGLO-III -81 -105 -70 175 272 126
Hirao/IGLO-II -81 -101 -71 176 273 127
[MoOCl4]−Dyall(TZ)/IGLO-III -48 -26 -58 140 223 98
Dyall(DZ)/IGLO-III -47 -26 -58 142 226 101
Dyall(VDZ)/IGLO-III -48 -28 -58 144 227 103
Dyall(TZ)/IGLO-II -48 -26 -59 139 223 97
Hirao/IGLO-III -48 -26 -58 141 225 100
Hirao/IGLO-II -48 -27 -58 148 232 107
[MoOF5]2−Dyall(TZ)/IGLO-III -110 -113 -109 183 278 135
Dyall(DZ)/IGLO-III -109 -113 -107 183 278 135
Dyall(VDZ)/IGLO-III -108 -113 -105 184 279 137
Dyall(TZ)/IGLO-II -110 -110 -110 180 274 132
Hirao/IGLO-III -110 -114 -108 182 278 135
Hirao/IGLO-II -110 -111 -110 183 278 136
86
Complex Method ∆giso ∆g∥∆g⊥Aiso A∥A⊥
[TcNF4]−Dyall(TZ)/IGLO-III -47 -91 -25 -765 -1153 -571
Dyall(DZ)/IGLO-III -46 -90 -25 -766 -1154 -573
Dyall(VDZ)/IGLO-III -48 -92 -25 -775 -1158 -583
Dyall(TZ)/IGLO-II -47 -87 -26 -752 -1139 -559
Hirao/IGLO-III -47 -90 -25 -767 -1155 -573
Hirao/IGLO-II -46 -87 -26 -772 -1159 -579
[TcNCl4]−Dyall(TZ)/IGLO-III 2 17 -5 -610 -930 -450
Dyall(DZ)/IGLO-III 3 19 -5 -614 -932 -454
Dyall(VDZ)/IGLO-III 1 15 -6 -623 -940 -464
Dyall(TZ)/IGLO-II 1 17 -6 -610 -930 -450
Hirao/IGLO-III 2 17 -5 -621 -941 -460
Hirao/IGLO-II 2 18 -6 -640 -959 -481
[TcNBr4]−Dyall(TZ)/IGLO-III 73 171 23 -548 -801 -421
Dyall(DZ)/IGLO-III 73 172 24 -550 -802 -424
Dyall(VDZ)/IGLO-III 70 166 21 -559 -811 -433
Dyall(TZ)/IGLO-II 73 171 23 -548 -801 -421
Hirao/IGLO-III 73 172 24 -566 -819 -439
Hirao/IGLO-II 73 172 24 -566 -819 -439
[WOCl4]−Dyall(TZ)/IGLO-III -209 -200 -213 -223 -347 -161
Dyall(DZ)/IGLO-III -204 -195 -209 -225 -349 -163
Dyall(VDZ)/IGLO-III -205 -196 -210 -223 -347 -161
Dyall(TZ)/IGLO-II -214 -210 -216 -225 -350 -163
Hirao/IGLO-III -205 -195 -210 -230 -354 -168
Hirao/IGLO-II -204 -197 -207 -247 -371 -186
[WOF5]2−Dyall(TZ)/IGLO-III -391 -464 -354 -329 -473 -257
Dyall(DZ)/IGLO-III -386 -458 -350 -327 -471 -256
Dyall(VDZ)/IGLO-III -386 -458 -349 -327 -470 -255
Dyall(TZ)/IGLO-II -395 -464 -360 -326 -470 -254
Hirao/IGLO-III -385 -458 -349 -332 -475 -260
Hirao/IGLO-II -383 -457 -345 -340 -483 -269
87
A. Additional tables for the benchmark study
Complex Method ∆giso ∆g∥∆g⊥Aiso A∥A⊥
[ReNF4]−Dyall(TZ)/IGLO-III -198 -351 -121 -2076 -3054 -1587
Dyall(DZ)/IGLO-III -195 -346 -119 -2070 -3045 -1583
Dyall(VDZ)/IGLO-III -195 -346 -119 -2059 -3033 -1571
Dyall(TZ)/IGLO-II -203 -354 -127 -2064 -3044 -1575
Hirao/IGLO-III -195 -348 -119 -2093 -3068 -1606
Hirao/IGLO-II -198 -348 -122 -2141 -3111 -1655
[ReNCl4]−Dyall(TZ)/IGLO-III -86 -99 -79 -1475 -2265 -1081
Dyall(DZ)/IGLO-III -81 -93 -75 -1473 -2256 -1081
Dyall(VDZ)/IGLO-III -82 -94 -76 -1471 -2257 -1078
Dyall(TZ)/IGLO-II -90 -108 -81 -1489 -2283 -1092
Hirao/IGLO-III -83 -95 -76 -1515 -2299 -1122
Hirao/IGLO-II -82 -95 -75 -1602 -2384 -1212
[ReNBr4]−Dyall(TZ)/IGLO-III -3 82 -46 -1249 -1915 -917
Dyall(DZ)/IGLO-III -1 85 -44 -1246 -1908 -915
Dyall(VDZ)/IGLO-III -3 82 -46 -1249 -1915 -916
Dyall(TZ)/IGLO-II -3 82 -46 -1249 -1915 -917
Hirao/IGLO-III -2 84 -44 -1290 -1952 -959
Hirao/IGLO-II -2 84 -44 -1290 -1952 -959
[ReOBr4] Dyall(TZ)/IGLO-III -42 237 -182 -865 -1343 -626
Dyall(DZ)/IGLO-III -39 241 -178 -857 -1331 -620
Dyall(VDZ)/IGLO-III -41 238 -181 -860 -1338 -621
Dyall(TZ)/IGLO-II -42 237 -182 -865 -1343 -626
Hirao/IGLO-III -39 241 -179 -936 -1407 -700
Hirao/IGLO-II -39 241 -179 -936 -1407 -700
88
Complex Method ∆giso ∆g∥∆g⊥Aiso A∥A⊥
[ReOF5]−Dyall(TZ)/IGLO-III -350 -326 -362 -1809 -2682 -1372
Dyall(DZ)/IGLO-III -347 -322 -359 -1796 -2667 -1361
Dyall(VDZ)/IGLO-III -347 -322 -359 -1795 -2667 -1360
Dyall(TZ)/IGLO-II -354 -326 -367 -1794 -2665 -1358
Hirao/IGLO-III -348 -323 -360 -1821 -2691 -1386
Hirao/IGLO-II -349 -324 -361 -1864 -2728 -1431
[OsOF5] Dyall(TZ)/IGLO-III -299 -178 -360 -603 -911 -448
Dyall(DZ)/IGLO-III -296 -174 -357 -596 -903 -442
Dyall(VDZ)/IGLO-III -297 -175 -358 -597 -904 -443
Dyall(TZ)/IGLO-II -302 -177 -364 -598 -904 -444
Hirao/IGLO-III -298 -176 -359 -606 -913 -452
Hirao/IGLO-II -299 -174 -361 -614 -917 -463
89
A. Additional tables for the benchmark study
Table A.5.: ∆gvalues (in ppt) computed at the 1c-DKH and 4c-mDKS levels for small d1
complexes (PBE0-40HF/Dyall(TZ)/IGLO-III results). For comparison, also
g-shift values obtained by using the Breit-Pauli (BP) spin-orbit (SO) coupling
operator on nonrelativistic (NR) wavefunctions are reported. Higher-order
spin-orbit (HOSO) effects are estimated from the difference between 4c-mDKS
and 1c-DKH data.
Complex Method ∆giso ∆g∥∆g⊥
[MoNCl4]2−NR + SO(BP) -43 -97 -17
1c-DKH -43 -98 -16
4c-mDKS -52 -108 -24
expt.[186] -44 -96 -18
HOSO -9 -10 -8
[MoOF4]−NR + SO(BP) -71 -89 -62
1c-DKH -70 -89 -61
4c-mDKS -81 -104 -70
expt.[187] -87 -108 -77
HOSO -11 -15 -9
[MoOCl4]−NR + SO(BP) -39 -14 -52
1c-DKH -40 -19 -50
4c-mDKS -48 -26 -58
expt.[187] -49 -37 -56
HOSO -8 -7 -8
[MoOF5]2−NR + SO(BP) -96 -90 -99
1c-DKH -93 -90 -95
4c-mDKS -110 -113 -109
expt.[187] -104 -128 -91
HOSO -17 -23 -14
90
Complex Method ∆giso ∆g∥∆g⊥
[MoOBr5]2−NR + SO(BP) -2 113 -59
1c-DKH -4 104 -58
4c-mDKS -14 92 -67
expt.[191] -9 87 -57
HOSO -10 -12 -9
[TcNF4]−NR + SO(BP) -37 -78 -17
1c-DKH -38 -81 -17
4c-mDKS -47 -91 -25
expt.[193] -44 -107 -12
HOSO -9 -10 -8
[TcNCl4]−NR + SO(BP) 11 29 2
1c-DKH 7 20 1
4c-mDKS 2 17 -5
expt.[195] 0 6 -2
HOSO -5 -3 -6
[TcNBr4]−NR + SO(BP) 87 194 34
1c-DKH 80 180 30
4c-mDKS 73 171 23
expt.[195] 69 145 32
HOSO -7 -9 -7
[WOCl4]−NR + SO(BP) -155 -117 -174
1c-DKH -163 -152 -168
4c-mDKS -209 -200 -213
expt.[198] -229 -209 -239
HOSO -46 -48 -45
91
A. Additional tables for the benchmark study
Complex Method ∆giso ∆g∥∆g⊥
[WOF5]2−NR + SO(BP) -322 -316 -325
1c-DKH -316 -327 -311
4c-mDKS -391 -464 -354
expt.[191] -368 -443 -330
HOSO -75 -137 -43
[WOBr5]2−NR + SO(BP) -130 28 -209
1c-DKH -150 -48 -200
4c-mDKS -201 -111 -246
expt.[191] -172 -99 -206
HOSO -51 -63 -46
[ReNF4]−NR + SO(BP) -136 -280 -64
1c-DKH -150 -311 -69
4c-mDKS -198 -351 -121
expt.[199] -206 -353 -132
HOSO -48 -40 -52
[ReNCl4]−NR + SO(BP) -17 -6 -23
1c-DKH -45 -73 -31
4c-mDKS -86 -99 -79
expt.[200] -78 -87 -73
HOSO -41 -26 -48
[ReNBr4]−NR + SO(BP) 84 216 19
1c-DKH 38 112 2
4c-mDKS -3 82 -46
expt.[203] 3 67 -29
HOSO -41 -30 -48
92
Complex Method ∆giso ∆g∥∆g⊥
[ReOBr4] NR + SO(BP) 48 362 -109
1c-DKH 6 281 -132
4c-mDKS -42 237 -182
expt.[205] -98 171 -232
HOSO -48 -44 -50
[ReOF5]−NR + SO(BP) -252 -140 -308
1c-DKH -261 -175 -304
4c-mDKS -350 -326 -362
expt.[206] -269 -282 -262
HOSO -89 -151 -58
[OsOF5] NR + SO(BP) -142 58 -242
1c-DKH -176 2 -265
4c-mDKS -299 -178 -360
expt.[206] -324 -197 -387
HOSO -123 -180 -95
93
A. Additional tables for the benchmark study
Table A.6.: Metal hyperfine couplings (in MHz) computed at different relativistic levels
for small d1complexes; PBE0-40HF/Dyall(TZ)/IGLO-III results. NR de-
notes nonrelativistic calculations; 1c-DKH (SR) denotes 1-component scalar
relativistic DKH calculations; A(SO) signifies second-order SO corrections to
the hyperfine coupling tensor, and 4c-mDKS corresponds to the 4-component
relativistic treatment within the matrix Dirac-Kohn-Sham method. Negative
signs of the experimental HFC components are based on the computations.
Complex Method Aiso A∥A⊥
[MoNCl4]2−NR 164 232 131
1c-DKH (SR) 198 264 165
1c-DKH (SR) + A(SO) 211 298 168
4c-mDKS 205 293 161
expt.[186] ---
[MoOF4]−NR 131 211 90
1c-DKH (SR) 162 240 123
1c-DKH (SR) + A(SO) 180 271 135
4c-mDKS 175 272 126
expt.[187] - 268 -
[MoOCl4]−NR 109 181 72
1c-DKH (SR) 130 201 95
1c-DKH (SR) + A(SO) 144 222 105
4c-mDKS 140 223 98
expt.[187] 145 227 103
[MoOF5]2−NR 134 212 94
1c-DKH (SR) 166 242 127
1c-DKH (SR) + A(SO) 189 274 145
4c-mDKS 183 278 135
expt.[187] 183 279 135
94
Complex Method Aiso A∥A⊥
[MoOBr5]2−NR 108 177 73
1c-DKH (SR) 126 194 92
1c-DKH (SR) + A(SO) 135 196 105
4c-mDKS 132 200 98
expt.[191] 128 184 99
[TcNF4]−NR -584 -883 -435
1c-DKH (SR) -730 -1025 -582
1c-DKH (SR) + A(SO) -791 -1165 -604
4c-mDKS -765 -1153 -571
expt.[193] -734 -1129 -537
[TcNCl4]−NR -489 -758 -354
1c-DKH (SR) -596 -862 -463
1c-DKH (SR) + A(SO) -631 -940 -476
4c-mDKS -610 -930 -450
expt.[195] -561 -878 -402
[TcNBr4]−NR -476 -743 -342
1c-DKH (SR) -572 -837 -440
1c-DKH (SR) + A(SO) -566 -806 -446
4c-mDKS -548 -801 -421
expt.[195] -488 -743 -360
[WOCl4]−NR -126 -203 -88
1c-DKH (SR) -200 -270 -164
1c-DKH (SR) + A(SO) -254 -363 -198
4c-mDKS -223 -347 -161
expt.[198] ---
95
A. Additional tables for the benchmark study
Complex Method Aiso A∥A⊥
[WOF5]2−NR -159 -243 -117
1c-DKH (SR) -296 -374 -257
1c-DKH (SR) + A(SO) -380 -496 -323
4c-mDKS -329 -473 -257
expt.[191] -331 -469 -262
[WOBr5]2−NR -123 -196 -87
1c-DKH (SR) -172 -239 -138
1c-DKH (SR) + A(SO) -226 -317 -180
4c-mDKS -198 -313 -141
expt.[191] - - -105
[ReNF4]−NR -1066 -1567 -815
1c-DKH (SR) -2022 -2505 -1780
1c-DKH (SR) + A(SO) -2365 -3292 -1901
4c-mDKS -2076 -3054 -1587
expt.[199] -2117 -3079 -1637
[ReNCl4]−NR -859 -1302 -637
1c-DKH (SR) -1436 -1864 -1222
1c-DKH (SR) + A(SO) -1668 -2405 -1299
4c-mDKS -1475 -2265 -1081
expt.[200] -1544 -2263 -1184
[ReNBr4]−NR -821 -1254 -605
1c-DKH (SR) -1261 -1680 -1052
1c-DKH (SR) + A(SO) -1407 -1996 -1113
4c-mDKS -1249 -1915 -917
expt.[203] -1340 -1994 -1013
96
Complex Method Aiso A∥A⊥
[ReOBr4] NR -558 -990 -342
1c-DKH (SR) -796 -1218 -585
1c-DKH (SR) + A(SO) -969 -1331 -788
4c-mDKS -865 -1343 -626
expt.[205] ---
[ReOF5]−NR -814 -1322 -560
1c-DKH (SR) -1571 -2061 -1325
1c-DKH (SR) + A(SO) -2077 -2741 -1744
4c-mDKS -1809 -2682 -1372
expt.[206] -1959 -2878 -1499
[OsOF5] NR -267 -428 -186
1c-DKH (SR) -523 -690 -439
1c-DKH (SR) + A(SO) -689 -918 -575
4c-mDKS -603 -911 -448
expt.[206] -627 -935 -480
97
5. Tungstoenzymes
“The worthwhile problems are the ones you can
really solve or help solve, the ones you can
really contribute something to. [...] No problem
is too small or too trivial if we can really do
something about it.”
— Richard Feynman
Copyright notice: Section 5.3, appendix B, and parts of the introduction and
computational details sections with all tables and graphics therein are reproduced with
permission according to the “ACS Policy on Theses and Dissertations” (21/06/2017)
from S. Gohr, P. Hrob´arik, and M. Kaupp, J. Phys. Chem. A,2017, 121(47), 9106-9117
(DOI: 10.1021/acs.jpca.7b08768).
5.1. Introduction
Molybdenum and tungsten are the only 4d and 5d elements, respectively, with known
natural functions in biological systems. They are thus both unique as the heaviest known
elements with well-defined biological roles. Molybdoenzymes have been known for sev-
eral decades and were found in prokaryotes as well as eukaryotes.[10,211] In contrast, the
first tungstoenzyme was purified in 1983.[212] Leaving molybdenum-containing nitrogenase
aside, both molybdenum and tungsten enzymes are oxidoreductases, and the single metal
site is able to cycle between the three oxidation states +IV, +V, and +VI in a relatively
narrow potential range.[213,214] While molybdoenzymes can be found in all kinds of cells,
tungstoenzymes have only been identified in prokaryots, especially in hyperthermophilic
archaea.[10] The mechanistic details for this tungsten specificity are, however, still under
investigation.[215] Later findings showed that tungstoenzymes have a higher temperature
stability than their molybdenum analogues, which could be a possible explanation for the
specificity towards tungsten,[7] possibly related to its earlier evolutionary role on a still
99
5. Tungstoenzymes
hotter planet earth.[32] It is interesting to note that the higher temperature stability and
lower reduction potential of tungsten is due to the larger relativistic expansion of its 5d
valence orbitals compared to molybdenum’s 4d ones.[35,216]
Molybdoenzymes have been classified into three main categories: sulfite oxidases; xanthine
oxidases (XO), and dimethyl sulfoxide reductases (DMSOR). All active site structures of
these enzymes contain at least one bispterin cofactor (‘molybdopterin’ - MPT), which is
shown in Figure 5.1. In the cases of sulfite oxidases and xanthine oxidases, the molyb-
denum atom is coordinated by one MPT, one oxido, and one hydroxy group as well as
a sulfido group. For sulfite oxidases, the sulfur atom belongs to a coordinating cysteine
amino acid. In the case of DMSOR, Mo is coordinated by two molybdopterin cofactors
instead of just one. The latter is also the case in all known tungstoenzymes, most likely
since tungsten complexes have in general lower reduction potentials due to relativistic
effects and adding sulfur instead of oxygen will increase the reduction potential of the
transition metal.[10] The πorbitals on the sulfur atom of MPT are partially delocalized
into the unsaturated backbone and are therefore particularly weak πdonors in contrast to
pure sulfido ligands.[217] In addition, adjustments to the remaining ligands (oxygen, sulfur
or selenium) are seen to be used in nature to adjust the redox potentials and therefore
catalyze different reactions.[7,218]
Figure 5.1.: The molybdopterin cofactor (MPT). The name can be seen as a heritage from
its discovery in molybdoenzymes since the cofactor itself does not contain
molybdenum and is also found in combination with tungsten.
While the reactions catalyzed by the different tungstoenzymes are known, the reaction
mechanisms are still under debate.[7–11,215,219–231] One of the most frequently used meth-
ods to investigate the d1M(V) intermediate states (S=1
2)during the catalytic cycles
are EPR measurements. The specificity of EPR for the active-site composition is partic-
100
5.2. Additional computational details
ularly important in the field, since those enzymes also contain iron-sulfur clusters in close
proximity (around 10 ˚
A) with relatively strong optical absorptions.[214]
5.2. Additional computational details
Structures: All model structures in sec. 5.3 were optimized with Turbomole[134] using
the PBE0[139,140] hybrid functional, whereas the models in sec. 5.4 were optimized us-
ing the BP86[136–138] functional in conjunction with the resolution-of-the-identity (RI)
approximation[232,233] – if not stated otherwise. A quasi-relativistic energy-consistent
small-core pseudopotential (ECP)[141] was used for tungsten with a (8s7p6d1f)/[6s4p3d1f]
Gaussian-type orbital valence basis set, and all-electron def2-TZVP[142] basis sets for
the ligand atoms. In addition, Grimme’s atom-pairwise D3 dispersion correction[143]
with Becke-Johnson damping[144] (BJ) has been added, “-D3(BJ)”. The experimental
structures are derived from X-ray crystal data, but since the EPR measurements have
been conducted with frozen-solution samples, in some cases the conductor-like screening
model (COSMO)[145] has been used to take bulk solvent effects on the structure into
account. The following solvents with their corresponding dielectric constants have been
considered: dimethylformamide (DMF) ε= 37.219, acetonitrile (CH3CN) ε= 35.688,
dichloromethane (CH2Cl2)ε= 8.93, and toluene (tol) ε= 2.38. An averaged dielectric
constant was used to model the CH2Cl2/tol solvent mixture as ε= 5.65 for [W(mdt)3]−,
assuming a 1:1 ratio in the absence of more detailed information.[234] For [WO(bdt)2]−,
a DMF/CH3CN mixture with ε= 36.4 is used. A variety of possible solvents (CH2Cl2,
DMF, THF) were mentioned for the EPR measurements of [Tp*WO(OPh)2] without
specifying the one actually used. Therefore, just DMF was chosen as a test case. How-
ever, note that COSMO was used solely for the structure optimizations and not in the
subsequent property calculations.
A static dielectric constant set to ε= 4.0 is applied for the enzyme structures as often
done in the case of proteins.[235]
EPR Parameter Calculations: The all-electron Hirao[149] basis set together with fully
uncontracted Huzinaga-Kutzelnigg-type IGLO-II[157] basis sets for the ligand atoms, or
all-electron Dyall basis sets of triple-ζquality[153] for tungsten together with fully uncon-
tracted Huzinaga-Kutzelnigg-type IGLO-III[157] basis sets for the light ligand atoms will
be tested in sec. 5.3. PBE0-xHF functionals with variable amounts xof exact-exchange
admixture are applied.
101
5. Tungstoenzymes
g-shifts are reported in ppt as deviations from the free-electron value (ge= 2.002319):
∆g= (g−ge)·1000. All reported HFC values are in MHz and refer to 183W(V). Exper-
imental values in 10−4cm−1were converted to MHz as ν=λ·cby a factor 2.99792458.
HFC values in Oe, G or mT were converted by the factor g·µB
h, where gis the measured
gfactor appropriate for the given HFC component (use of geinstead changes the con-
verted HFCs by 4–5 MHz for [WO(bdt)2]−). Negative signs of all HFC components are
based on the computations. RMSD values between computed and experimental structures
have been determined using the superposition functionality of Maestro as included in
the Schr¨
odinger 2015.3 package.[163]
5.3. Validation on model complexes for tungstoenzymes
5.3.1. Introduction
A 5d element like tungsten requires special care regarding the incorporation of both scalar
relativistic and spin-orbit (SO) effects for magnetic property calculations: due to special
relativity, the 5d orbitals are much more destabilized and expanded than the respective
4d orbitals. This will enhance bond ionicities and stabilize the higher W(V) and W(VI)
oxidation states. Therefore, before embarking on a detailed study of W(V) active sites in
tungsten enzymes, it will be necessary to close the gap between the previous benchmark
study in chapter 4 and the work on tungsten-containing enzyme sites. Since none of the
previous systems contained a tungsten-sulfur bond, here a set of seven different tung-
sten complexes is chosen to extend the method screening, as represented in Figure 5.2.
[WOCl5]2−is included additionally to provide comparison against a small non-sulfur
containing system. [Tp*WO(OPh)2] (Tp* = hydrotris(3,5-dimethylpyrazol-1-yl)borate)
represents a nitrogen-oxygen combination of ligands. [WO(SPh)4]−has four thiophenol
groups. [WO(bdt)2]−(bdt = benzene-1,2-dithiolate) and [WO(edt)2]−(edt = ethane-1,2-
dithiolate) have two dithiolene ligands and thus are most closely related to the enzyme
active sites. [W(mdt)3]−(mdt = 1,2-dimethyl-1,2-dithiolene) and [W(pdt)3]−(pdt = 1,2-
diphenyl-1,2-dithiolate) provide finally homoleptic tris-dithiolene examples.
102
5.3. Validation on model complexes for tungstoenzymes
Figure 5.2.: Overview on the investigated tungsten(V) complexes, based on PBE0-
D3(BJ)/def2-TZVP optimizations.
103
5. Tungstoenzymes
5.3.2. Results and discussion
5.3.2.1. Structures
In line with our previous study and with the findings of ref. 236, the chosen PBE0-
D3(BJ)/def2-TZVP level was expected to provide reliable structures for our EPR pa-
rameter calculations. This is confirmed in the comparison with available experimental
data in Table 5.1. Except for d(W–Cl)eq /d(W–Cl)ax in the dianionic [WOCl5]2−and
d(W=O) in [WO(SPh)4]−all deviations from experiment are below ca. 0.05 ˚
A. The ex-
perimental d(W=O) value for the latter complex seems to be unusually small compared
to related systems and might be an artifact. Larger RMSD values in three cases are re-
lated to the orientation of some ligand planes rather than to the direct metal coordination
environment.
5.3.2.2. Effect of the structure on EPR parameters
Given the relatively good agreement between optimized and experimental structures (see
above), moderate differences between the EPR parameters obtained at these structures
can be expected. While this is largely the case, the effects are nevertheless explored in
Table 5.2 (PBE0-40HF/Hirao/IGLO-II level), probing also the effects of COSMO solvent
modeling in the structure optimizations.
Generally good agreement with experiment is found for [WOCl5]2−. Only for g33 the exper-
imental structures give slightly larger values and thus closer agreement with experiment.
Particularly small structural effects are seen for the relatively rigid [Tp*WO(OPh)2],
where excellent agreement with experiment is found both for the g-tensor and Aiso, in
fact even somewhat better than one could expect at the DFT level used. This superior
performance could also be attributed to the neutral charge of these systems, where a sim-
ulation of possible counterion effects is not required as compared to other systems within
the benchmark series.
While [WO(SPh)4]−features saturated thiolate ligands rather than dithiolenes, its square
pyramidal coordination with four basal sulfur atoms and an apical oxo ligand resembles
already a number of structural aspects of molybdo- and tungstoenzymes. Solvent effects
on the structure and thus on the computed EPR parameters are small, and the overall
agreement with experiment is good, albeit g33 is too close to the free-electron value.
Here, a test of adjusting specific geometrical parameters shows an appreciable impact of
the W–S bond length on g33 but not on the other two g-tensor components (cf. Table B.1
in appendix B). Visualization of the g-tensor (Figure B.1 in appendix B) shows that g33
is oriented along the W=O bond and roughly perpendicular to the W–S bonds. This
104
5.3. Validation on model complexes for tungstoenzymes
Table 5.1.: PBE0-D3(BJ)/def2-TZVP optimized structural parameters compared to ex-
periment. a
[WOCl5]2−d(W=O) (˚
A) d(W–Cl)ax (˚
A) d(W–Cl)eq (˚
A) RMSD (˚
A)
X-ray[237] b1.724 2.565 2.390 -
X-ray[238] c1.669 2.664 2.372 -
PBE0-D3(BJ) 1.696 2.556 2.432 0.0557/0.0725
[Tp*WO(OPh)2]d(W=O) (˚
A) d(W–OPh) (˚
A) d(W–N) (˚
A) RMSD (˚
A)
X-ray[239] 1.705 1.941 2.200 -
PBE0-D3(BJ) 1.704 1.954 2.229 0.4247d
+ COSMO(DMF) 1.711 1.960 2.223 0.4334d
[WO(SPh)4]−d(W=O) (˚
A) d(W–S) (˚
A) RMSD (˚
A)
X-ray[240] 1.589 2.456 -
PBE0-D3(BJ) 1.705 2.403 2.679e
+ COSMO(CH3CN) 1.708 2.399 2.560e
[WO(bdt)2]−d(W=O) (˚
A) d(W–S) (˚
A) RMSD (˚
A)
X-ray[241] 1.689 2.366 -
PBE0-D3(BJ) 1.700 2.396 0.1403
+ COSMO(DMF/CH3CN) 1.712 2.387 0.1385
[WO(edt)2]−d(W=O) (˚
A) d(W–S) (˚
A) RMSD (˚
A)
PBE0-D3(BJ) 1.703 2.416 -
[W(mdt)3]−d(W–S) (˚
A) RMSD (˚
A)
X-ray[234] 2.394 -
PBE0-D3(BJ) 2.382 0.2082
+ COSMO(CH2Cl2/tol) 2.380 0.2047
[W(pdt)3]−d(W–S) (˚
A) RMSD (˚
A)
X-ray[234] 2.380 -
PBE0-D3(BJ) 2.375 0.7193f
aIn case of the X-Ray structures (especially for [W(mdt)3]−and [W(pdt)3]−), not all
bonds of one type are of the exact same length, and only their average values are given.
bCambridge structural database (CSD) ID: FORSUA. cCSD-ID: ZAPFUR. dThe high
RMSD values are attributed mostly to the phenol rings. Including only the W, B, O,
and N atoms, the RMSD values are 0.0518 and 0.0519 ˚
A, respectively. eIncluding only
the W, O, and S atoms, the RMSD values reduce to 0.1575 ˚
A and 0.1611 ˚
A,
respectively. fIncluding only W and S, the RMSD value reduces to 0.3066 ˚
A.
105
5. Tungstoenzymes
Table 5.2.: Effect of the structure on computed EPR parameters (PBE0-
40HF/Hirao/IGLO-II level; hyperfine couplings in MHz).
expt. data / structure giso g11 g22 g33 Aiso A11 A22 A33
[WOCl5]2−expt. EPR[242] 1.773 1.758 1.758 1.804 -381
X-ray struct.[237] 1.745 1.726 1.726 1.781 -257 -196 -196 -378
X-ray struct.[238] 1.780 1.762 1.767 1.812 -250 -189 -190 -369
PBE0-D3(BJ) 1.748 1.744 1.744 1.758 -260 -196 -196 -388
[Tp*WO(OPh)2]expt. EPR[239] 1.785 1.707 1.802 1.847 -255
X-ray struct.[239] 1.785 1.714 1.795 1.847 -246 -189 -193 -354
PBE0-D3(BJ) 1.784 1.701 1.804 1.847 -253 -195 -198 -365
+ COSMO(DMF) 1.784 1.702 1.802 1.848 -252 -194 -197 -364
[WO(SPh)4]−expt. EPR[243] 1.936 1.903 1.903 2.018 -165 -133 -133 -234
X-ray struct.[240] 1.928 1.892 1.909 1.982 -144 -98 -100 -234
PBE0-D3(BJ) 1.926 1.888 1.889 2.001 -142 -99 -99 -227
+ COSMO(CH3CN) 1.928 1.892 1.892 2.001 -142 -99 -99 -227
[WO(bdt)2]−expt. EPR[244] 1.962 1.911 1.931 2.044 -153 -111 -119 -235
X-ray struct.[244] 1.962 1.907 1.929 2.049 -136 -94 -98 -215
PBE0-D3(BJ) 1.947 1.901 1.923 2.018 -137 -93 -97 -219
+ COSMO(DMF/CH3CN) 1.948 1.901 1.923 2.020 -137 -94 -98 -218
[W(mdt)3]−expt. EPR[234] 1.994 1.988 2.001 2.009 -82 -33 -102 -96
X-ray struct.[234] a2.036 2.034 2.037 2.038 -2 13 -10 -10
PBE0-D3(BJ) 1.990 1.981 1.995 1.995 -81 -25 -109 -109
+ COSMO(CH2Cl2/tol) 1.993 1.982 1.998 1.998 -82 -27 -109 -110
[W(pdt)3]−expt. EPR[234] 1.993 1.991 2.002 2.008 -81 -36 -93 -90
X-ray struct.[234] 1.982 1.954 1.994 1.998 -118 -54 -155 -146
PBE0-D3(BJ) 1.991 1.987 1.992 1.993 -70 -19 -97 -96
aThe experimental structure was obtained from [W(mdt)3]2−, while the EPR data refer
to [W(mdt)3]−.
106
5.3. Validation on model complexes for tungstoenzymes
has already been found for related 5d transition metal complexes and was attributed to
contributions of ‘in-plane’ d orbitals.[185]
Similar observations are made for the bis-dithiolene oxo complex [WO(bdt)2]−, which is
an even closer structural model for tungstoenzyme active sites. Again g33 is particularly
sensitive to the W–S bond lengths (Table B.2 in appendix B and Figure 5.3), and as
a result of its shorter distances, the experimental structure provides better agreement
with the experimental EPR parameters than the optimized ones. The structural effect
on the HFC values is minor, significantly below the effect of the functional, as will be
shown in the next section. Effects of the solvent model on the structure optimization are
also small. Unfortunately, closer experimental details such as the solvent or structural
parameters for the close analogue [WO(edt)2]−could not be found. However, it can be
seen that – compared to [WO(bdt)2]−– the change in d(W=O) is negligible, while d(W–S)
differs by 0.02 ˚
A, probably due to the missing stabilization by phenyl substituents.
Figure 5.3.: Directions of g-tensor principal components computed for [WO(bdt)2]−at the
4-component PBE0-D3(BJ)/def2-TZVP//PBE0-40HF/Dyall(TZ)/IGLO-III
level.
Next, the homoleptic tris-dithiolene complex [W(mdt)3]−exhibits a different type of struc-
ture dependence. Unfortunately, only the dianion has been characterized crystallographi-
cally. While this structure has been denoted as “X-ray struct.” in Table 5.2, its dithiolene
fold angles in the mdt ligands are almost non-existent. The resulting EPR data, espe-
cially the HFC values, do not match the experimental data. Structure optimization of
the monoanion introduces a dithiolene fold angle of approximately 15◦(cf. Figure B.2 in
107
5. Tungstoenzymes
appendix B and ref. 234) and provides EPR parameters in much better agreement with
experiment (but with a somewhat too low g33). Axial symmetry of the tensors can be ex-
pected for this almost C3symmetric structure where small deviations in the experimental
data are probably attributable to solvent and/or packing effects in the frozen solutions,
which are absent in the (gas-phase) optimizations. On the other hand, for the closely re-
lated [W(pdt)3]−molecule, the experimental structure refers correctly to the monoanion.
In contrast to [W(mdt)3]−, it possesses twist angles of roughly 23◦and dithiolene fold
angles in a range from 6 to 15◦.[234] Better agreement with the experimental EPR data
is nevertheless found again for the optimized structure, in particular for the HFC tensor
and g11.
It is noted in passing that 1-component DKH calculations, which either neglect SO effects
(on the HFCs) or include them only to leading order in perturbation theory (HFCs and
g-tensors), cannot compete with the 4-component relativistic methods with variational
treatment of spin-orbit coupling (cf. Table B.4 in appendix B), consistent with the pre-
vious findings in chapter 4.
5.3.2.3. Effect of the functional
Based on the results of the previous benchmark studies in chapter 4, the comparison will
be restricted to PBE0-based global hybrid functionals with only three different amounts
of EXX admixture (25, 40, and 50 %). The results for the PBE0-D3(BJ) optimized struc-
tures are collected in Table 5.3 and visualized in Figure 5.4. It appears that the 25 %
EXX admixture of the PBE0 functional is already sufficient to provide good agreement
with experiment for the g-tensors. This contradicts somewhat earlier studies with linear-
response methods, but those neglected higher-order SO effects and thus underestimated
the g-tensors for different reasons.[45,58,184,245] The previous evaluation of 4-component
methods suggested somewhat larger optimum EXX admixtures for a number of 5d sys-
tems, but again PBE0 gave already rather good agreement for the g-tensors.
In contrast, larger EXX admixtures are clearly required for the HFC tensors in the present
test set: best agreement is found variously for PBE0-40HF or PBE0-50HF (Table 5.3,
Figure 5.4). The higher EXX admixture required for the metal HFCs is related to the
description of spin polarization for core-shell s orbitals, which is enhanced (improved) by
larger EXX admixture in the core region,[127,128] as recently confirmed also by preliminary
evaluations with local hybrid functionals.[131]
108
5.3. Validation on model complexes for tungstoenzymes
Given the rather high demands on the accuracy for the application to tungsten enzymes,
and since local hybrid functionals – which might remedy this discrepancy – currently
are not yet available in the 4-component code, this may force the adoption of a rather
pragmatic protocol at this point in time: PBE0 may be used for g-tensor calculations but
PBE0-40HF or PBE0-50HF for the metal HFCs (likely also for ligand HFCs, which tend
to also benefit from larger EXX admixtures).
Figure 5.4.: Effect of EXX admixture (PBE0 vs PBE0-40HF vs PBE0-50HF) on devia-
tions from the experimental ∆g- (ppt) and HFC- (MHz) tensor components.
Hirao/IGLO-II based calculations at PBE0-D3(BJ)/def2-TZVP structures.
5.3.2.4. Basis-set effects
Given the relatively large size of models required to describe enzyme active sites (typically
above 100 atoms), use of moderately-sized basis sets might be desirable. In chapter 4,
the somewhat smaller relativistic Hirao/IGLO-II basis-set combination agreed well with
the larger Dyall(TZ)/IGLO-III combination and provided partially even better agree-
ment with experiment. This performance is confirmed here also for the present W(V)
test set (Table 5.4): remarkably, the Hirao/IGLO-II results again agree better with ex-
109
5. Tungstoenzymes
Table 5.3.: Dependence of computed EPR parameters on the EXX admixture in the func-
tional (PBE0 vs PBE0-40HF vs PBE0-50HF). a
∆giso ∆g11 ∆g22 ∆g33 Aiso A11 A22 A33
[WOCl5]2−expt.[242] -229 -244 -244 -198 -381
PBE0 -241 -253 -253 -217 -226 -163 -163 -353
PBE0-40HF -254 -258 -258 -245 -260 -196 -196 -388
PBE0-50HF -262 -263 -262 -262 -283 -219 -219 -412
[Tp*WO(OPh)2]expt.[239] -217 -295 -200 -155 -255
PBE0 -205 -282 -189 -144 -218 -160 -163 -329
PBE0-40HF -218 -301 -198 -155 -253 -195 -198 -365
PBE0-50HF -227 -313 -205 -162 -276 -218 -221 -389
[WO(SPh)4]−expt.[243] -66 -99 -99 16 -165 -133 -133 -234
PBE0 -62 -102 -101 16 -116 -75 -75 -198
PBE0-40HF -76 -114 -114 -1 -142 -99 -99 -227
PBE0-50HF -86 -122 -122 -14 -159 -115 -115 -247
[WO(bdt)2]−expt.[244] -40 -91 -71 42 -153 -111 -119 -235
PBE0 -40 -90 -69 38 -109 -66 -72 -188
PBE0-40HF -55 -101 -80 16 -137 -93 -97 -219
PBE0-50HF -65 -108 -87 1 -155 -112 -115 -240
[WO(edt)2]−expt.[246] -21 -96 -81 114 -153 -119 -122 -217
PBE0 -32 -102 -84 90 -111 -65 -74 -193
PBE0-40HF -49 -114 -96 62 -140 -93 -101 -225
PBE0-50HF -61 -122 -103 43 -159 -112 -118 -246
[W(mdt)3]−expt.[234] -8 -15 -1 7 -82 -33 -102 -96
PBE0 -8 -17 -3 -3 -60 -11 -85 -85
PBE0-40HF -12 -21 -8 -8 -81 -25 -109 -109
PBE0-50HF -17 -26 -13 -13 -100 -39 -131 -131
[W(pdt)3]−expt.[234] -9 -12 -1 5 -81 -36 -93 -90
PBE0 -9 -14 -7 -7 -52 -8 -75 -74
PBE0-40HF -12 -16 -10 -10 -70 -19 -97 -96
PBE0-50HF -15 -19 -14 -13 -87 -30 -116 -116
aResults with the Hirao/IGLO-II basis and PBE0-D3(BJ)/def2-TZVP structures.
Hyperfine couplings in MHz, ∆gvalues in ppt.
110
5.3. Validation on model complexes for tungstoenzymes
periment in almost all cases (exceptions are: [WO(bdt)2]−∆g33, [W(mdt)3]−A22/A33,
and [WO(SPh)4]−A33). Given that the Dyall(TZ) basis is clearly larger and optimized
within the 4-component realm (the Hirao basis originates from DKH3 calculations), this
excellent performance of the smaller basis appears to be due to some favorable error com-
pensation.
Table 5.4.: Basis-set effects on computed EPR parameters at the PBE0-40HF level. a
∆giso ∆g11 ∆g22 ∆g33 Aiso A11 A22 A33
[WOCl5]2−expt.[242] -229 -244 -244 -198 -381
Hirao / IGLO-II -254 -258 -258 -245 -260 -196 -196 -388
Dyall / IGLO-III -262 -267 -267 -251 -235 -171 -171 -364
[Tp*WO(OPh)2]expt.[239] -217 -295 -200 -155 -255
Hirao / IGLO-II -218 -301 -198 -155 -253 -195 -198 -365
Dyall / IGLO-III -218 -300 -200 -154 -250 -192 -195 -362
[WO(SPh)4]−expt.[243] -66 -99 -99 16 -165 -133 -133 -234
Hirao / IGLO-II -76 -114 -114 -1 -142 -99 -99 -227
Dyall / IGLO-III -82 -119 -119 -7 -142 -98 -99 -229
[WO(bdt)2]−expt.[244] -40 -91 -71 42 -153 -111 -119 -235
Hirao / IGLO-II -55 -101 -80 16 -137 -93 -97 -219
Dyall / IGLO-III -56 -104 -84 20 -114 -72 -75 -196
[WO(edt)2]−expt.[246] -21 -96 -81 114 -153 -119 -122 -217
Hirao / IGLO-II -49 -114 -96 62 -140 -93 -101 -225
Dyall / IGLO-III -51 -117 -99 65 -119 -73 -80 -205
[W(mdt)3]−expt.[234] -8 -15 -1 7 -82 -33 -102 -96
Hirao / IGLO-II -12 -21 -8 -8 -81 -25 -109 -109
Dyall / IGLO-III -14 -26 -8 -8 -73 -15 -102 -102
[W(pdt)3]−expt.[234] -9 -12 -1 5 -81 -36 -93 -90
Hirao / IGLO-II -12 -16 -10 -10 -70 -19 -97 -96
Dyall / IGLO-III -14 -22 -11 -11 -67 -12 -94 -93
aPBE0-D3(BJ)/def2-TZVP structures used. Hyperfine couplings in MHz, ∆gvalues in
ppt.
111
5. Tungstoenzymes
5.3.3. Effect of sulfido vs. oxo substitution in [WX(bdt)2]−and
[WX(edt)2]−, X = O, S
One possible question for tungstoenzyme active sites may be if oxo or sulfido ligands
are present.[247] It is therefore of interest if such a substitution can be identified by EPR
parameters alone (in the absence of 17O labeling). We have thus optimized also the
sulfido analogues [WS(bdt)2]−and [WS(edt)2]−. PBE0-D3(BJ)/def2-TZVP structures
(Table 5.5) reveal the expected longer W=S bond (by ca. 0.44 ˚
A) and small further
structural changes compared to the oxo analogues (cf. Table 5.1).
Table 5.5.: Comparison of PBE0-D3(BJ)/def2-TZVP optimized structural parameters for
[WX(bdt)2]−and [WX(edt)2]−(X = O, S).
PBE0-D3(BJ) d(W=S/O)ax (˚
A) d(W–S)eq (˚
A)
[WO(bdt)2]−1.700 2.396
[WS(bdt)2]−2.136 2.380
[WO(edt)2]−1.703 2.416
[WS(edt)2]−2.146 2.384
PBE0/Hirao/IGLO-II 4-component g-tensor results for the two sets of complexes are com-
pared in Table 5.6 together with PBE0-50HF HFC results (PBE0-40HF/Hirao/IGLO-II
results are available from Table B.3 in appendix B). The changes of both g- and HFC ten-
sors are at best moderate, and based on the benchmarking reported above, it is currently
unclear if these small differences can be distinguished computationally beyond any doubt.
However, trends may be easier to detect for a series of complexes, or the combination
with further spectroscopic techniques (e.g. EXAFS, IR, pNMR) may help to distinguish
such species.
5.3.4. Conclusions
Based on the previous benchmarks for a wider set of 4d and 5d complexes (chapter 4),
the 4-component relativistic DFT methodology for g-tensors and metal hyperfine coupling
tensors has been fine-tuned for a series of synthetic W(V) complexes closer to the tungsten
active site targets, including a number of dithiolene and thiolate complexes. W–S bond
lengths were found to be important for the computed EPR data, requiring thus a careful
structure optimization.
112
5.3. Validation on model complexes for tungstoenzymes
Table 5.6.: Effect of oxo vs. sulfido substitution on the EPR parameters of [WX(bdt)2]−
and [WX(edt)2]−(X = O, S). a
∆giso ∆g11 ∆g22 ∆g33 Aiso A11 A22 A33
[WO(bdt)2]−-40 -90 -69 38 -155 -112 -115 -240
[WS(bdt)2]−-43 -90 -70 31 -173 -128 -140 -251
[WO(edt)2]−-32 -102 -84 90 -159 -112 -118 -246
[WS(edt)2]−-36 -108 -87 86 -174 -129 -140 -253
aPBE0 results for the g-tensor, PBE0-50HF data for the HFC tensor, using
Hirao/IGLO-II basis sets and PBE0-D3(BJ)/def2-TZVP structures. Hyperfine
couplings in MHz, ∆gvalues in ppt.
The best agreement with experimental g-tensor data was achieved with the 25 % exact-
exchange admixture included in the PBE0 hybrid functional, whereas larger admixtures
of about 40 % – 50 % provided the best metal hyperfine tensors. Notably, these choices
are based on the inclusion of higher-order spin-orbit contributions within the 4-component
relativistic framework. Until better electronic structure methods become routinely appli-
cable to large transition-metal complexes, such a combined protocol may currently offer
the best compromise. The somewhat more affordable Hirao/IGLO-II basis set combina-
tion was found to perform as well as the larger Dyall(TZ)/IGLO-III combination and is
therefore a reasonable choice for applications to the larger-sized tungstoenzyme centers.
Differences in the EPR parameters between W=O and W=S analogues are predicted here
to be rather small, which may render identification of this substitution by EPR parame-
ters alone difficult.
113
5. Tungstoenzymes
5.4. Identifying the W(V) state of tungsten
oxidoreductase enzymes by computational EPR
investigations
5.4.1. Introduction
One of the best studied tungstoenzyme-containing archaea is Pyrococcus furiosus (Pf ) for
which a total of five different tungsten-containing enzymes were identified,[248] all belong-
ing to the aldehyde oxidoreductase (AOR) family: AOR,[249] formaldehyde oxidoreductase
(FOR),[250] Glyceraldehyde-3-phosphate (GAPOR),[251] and the not yet more closely spec-
ified tungsten-containing oxidoreductases WOR4[252] and WOR5.[224] Furthermore, from
the formate dehydrogenase (FDH) family, two enzyme types have been characterized (not
from Pf ): FDH[221,253] and formylmethanofuran dehydrogenase (FMDH).[222,254]
All these tungstoenzymes have in common that they contain one or more (identical) ac-
tive sites in which tungsten is ligated by two MPT cofactors as shown in Figure 5.5. So
far, the depicted magnesium-bridge has only been identified in the crystal structure of
FOR,[250] but not for AOR.[249] In case of the F(M)DH enzymes, an additional guanine
monophosphate is attached to the phosphate group of MPT.[255]
Figure 5.5.: Structure of the active center of AOR/FOR as found in Pyrococcus furiosus.
Many questions on these enzymes are still open. Three of the most intriguing ones are:
a) How do the archaea ensure their tungsten-specificity even in the presence of very high
molybdenum excess?[7,10,215] b) The specific physiological substrates are only known for
114
5.4. Identifying the W(V) state of tungsten oxidoreductase enzymes
GAPOR (Glyceraldehyde-3-phosphate), whereas for the remaining enzymes it is only
known what kind of reaction they catalyze. AOR and FOR are responsible for the oxi-
dation of aldehydes to carboxylates. While AOR oxidizes a broad variety of aldehydes,
FOR seems to be restricted to small aldehydes with C1 – C3 chain lengths.[247] WOR5 is
assumed to catalyze similar reactions as AOR,[224] and WOR4 catalyzes the production
of hydrogen sulfide from sulfur and hydrogen.[223] c) The catalytic cycles themselves have
been subject to various experimental and quantum chemical investigations but are not
completely understood.[7–11,214,215,219–231,246,247,249,250,252,255–264]
Herein, the focus will be on the latter issue. Electron paramagnetic resonance (EPR)
spectroscopy of the paramagnetic W(V) state of the enzyme has become crucial to such
mechanistic studies due to iron-sulfur clusters in close proximity (around 10 ˚
A) with rela-
tively strong optical absorptions.[214] Tungsten(V) is a 5d1system, which gives well-defined
S=1
2spectra without zero-field splitting. A variety of experimental EPR investigations
has been conducted for the AOR and FOR enzymes over the past 20 years: Hensgens et
al. isolated AOR from Desulfovibrio gigas (Dg) and obtained the EPR spectra for an ‘as
isolated’ form in 1995.[256] The quite complex experimental EPR spectrum was simulated
using a sum of four full sets of EPR parameters, thereby assuming a mixture of different
active-site configurations contributing to the final EPR signal. In 1996, a very detailed
investigation for the Pf AOR by Koehler et al.[246] identified seven distinct W(V) signals
and suggested that the isolated enzyme represents a mixture where all three tungsten
oxidation states (+IV/+V/+VI) are accessible in different species of the same enzyme
type. They identified those species by first reducing the mixture to W(IV) using sodium
dithionite. In this ‘dithionite-reduced’ form, a ‘spin-coupled’ signal arose from weak spin-
spin interactions between a W(V) S=1
2state and a S=3
2[4Fe-4S]+cluster (they
assumed that after the reduction, some of the W(IV) reoxidizes to W(V) by transferring
one electron to the iron-sulfur cluster [4Fe-4S]2+ →[4Fe-4S]+).[257] Redox titrations were
performed on the dithionite-reduced form using potassium ferricyanide to subsequently
transform the distinct species into their EPR-active W(V) states. First, a ‘low-potential’
species was identified with two sequential one-electron oxidations {W(IV) →W(V) and
W(V) →W(VI)}, both around -400 mV. In the overall mixture, this low-potential species
accounted for 20 to 30 %. It should be noted that the same low-potential form was also
obtained by incubation with formaldehyde. Further increasing the potential yielded a
‘mid-potential’ species that only accounted for a few percent in the mixture and is there-
fore only of minor importance. However, it is noteworthy that this species did not seem to
cycle to a W(VI) state. The next species appeared at non-physiological potentials above
+100 mV and was termed ‘high-potential’. It corresponded to approximately 30 % of
115
5. Tungstoenzymes
the total amount of tungsten in the mixture and is therefore another major component.
Like the ‘mid-potential’ species, it did not seem to be able to redox-cycle and is therefore
most likely not a catalytically active species. In fact, oxidizing the sample to +300 mV
and subsequently reducing it down to -190 mV produced two entirely new EPR signals,
not seen before this aggressive oxidation (termed ‘re-reduced 1’ and ‘re-reduced 2’). This
procedure involved a complete loss of catalytic activity. On the other hand, oxidizing and
re-reducing without going above 0 mV did not change the catalytic abilities of the sample.
This suggested that the ‘low-potential’ species is responsible for the catalytic activity of
the enzyme. Finally, another species was observed that appeared after the incubation
with glycol (or glycerol) and resulted in an irreversible loss of catalytic activity. This
‘diol-inhibited’ version was also identified by its distinct EPR signal.
In 2000, the same authors published their findings for the Thermococcus litoralis (Tl)
FOR.[247] The Tl FOR is tetrameric in contrast to the previously investigated dimeric
Pf AOR. Remarkably, Tl FOR was not inhibited by glycol or glycerol, even though
both have a very high sequence homology. Furthermore, no W(V) state was obtained for
Tl FOR upon incubation with formaldehyde, possibly due to higher potentials for the
W(IV)/W(V) transition (Pf AOR -436 mV vs Tl FOR -335 mV). However, the existence
of the ‘spin-coupled’, ‘low-’, ‘mid-’, and ‘high-potential’ species was confirmed for this
Tl FOR, based on (very) similar EPR parameters in comparison to the previous Pf AOR
case. This may suggest related active-site structures. In addition to the redox titrations,
the samples were also incubated with an excess of sodium sulfide and sodium dithionate
in order to obtain ‘sulfide-activated’ forms. This produced EPR signals that were very
similar to the mid- and high-potential versions. The signal in the low-potential range
vanished. The procedure was termed sulfide-activation, because of an observed 8-fold ac-
tivity enhancement after the treatment. Due to the enhancement in activity and the fact
that Tl was found in sulfur-rich environments, it was assumed that the sulfide-activated
species represent either partially or completely the physiologically active form of Tl FOR.
This would also suggest that the Tl FOR low-potential W(V) species corresponds to a
desulfo form of the enzyme. Even though the sulfide-activated mid- and high-potential
forms are most likely not physiologically relevant (see above), four possible structures
have been suggested. The proposals are based on the assumption that their existence is
an artifact of a ligand-based oxidation of an already oxidized W(VI) species. In order to
arrive at a net one-electron oxidation for the mid-potential structures, it was proposed
that W(VI) is reduced to W(V) and a two-electron oxidation of one of the MPT ligands
results in a loosely attached MPT. For the high-potential structures, it is proposed that
the one-electron reduction to W(V) would be compensated by a terminal SH−or S2−
116
5.4. Identifying the W(V) state of tungsten oxidoreductase enzymes
ligand that undergoes a two-electron oxidation and insertion into one of the W–S MPT
bonds to provide a net one-electron oxidation. Both proposals were already mentioned in
the aforementioned ref. 246 as possible explanations for the high-potential structure. An
overview on all proposed W(V) structures and those considered in this work is provided
in Figure 5.6.
In 2006, Bol and coworkers published an EPR study on Pf FOR,[214] comparing it to
the aforementioned Tl FOR study. Also these two enzymes have a very high sequence
homology and would therefore be expected to behave similarly. However, in contrast to
Tl FOR, several differences have been found. Pf FOR did not show a high-potential struc-
ture and a mid-potential signal was only identified in one of the samples. A low-potential
species, on the other hand, was detected again after incubation with formaldehyde as
already seen for Pf AOR. It was not found during redox titrations, which was attributed
to the fact that they only decreased the potential to -400 mV and the formaldehyde re-
duction appears most likely below this point. In addition, only a small enhancement (by
a factor of 1.4) after excess incubation with sulfide/dithionite was observed. All these
results were rationalized by the assumption that Pf FOR was already present in the more
stable sulfide-activated form, while the Tl FOR from ref. 247 was found in a desulfo form.
While the solid-state structures for Pf AOR and FOR have been determined crystallo-
graphically,[249,250] the W(V) coordination sphere for the low-potential (or formaldehyde
reduced) variant is to the best of my knowledge still unspecified. According to the FOR
X-ray data, the W–O distance amounts to 2.10 ˚
A, which would point to a W–O single
bond and therefore either a W–OH or W–OH2unit. However, as these data correspond
to a snapshot with ill-defined tungsten oxidation states, possibly a mixture, the distance
might very well be an artifact: it could therefore also correspond to a double bond W=O
(d(W-O) around 1.7 ˚
A). Together with the mid- and high-potential forms and the respec-
tive desulfo or sulfide-activated forms, this gives a wealth of possibilities for the S=1
2
W(V) states as summarized in Figure 5.6.
To identify the best candidate(s), first all the suggestions are structure-optimized based
on the FOR crystal structure.[250] The calculated EPR parameters are then compared to
experimental data to help identify those structures that most likely represent the observed
W(V) states.
117
5. Tungstoenzymes
Figure 5.6.: Overview on structural proposals for different EPR-active W(V) species for
AOR/FOR tungstoenzymes studied in this work. Only the direct tungsten
coordination is shown.
118
5.4. Identifying the W(V) state of tungsten oxidoreductase enzymes
5.4.2. Results and discussion
5.4.2.1. Model building for the structures
Truncation to the structure models
The ‘1B25’ protein structure for FOR[250] taken from the protein database[265] is used as
starting point. The protein framework was cut at the peptide bonds outside a ca. 5 ˚
A
radius from the metal center, adding hydrogen atoms in such a way that the carboxyl
groups from threonine (THR), aspartic acid (ASP), and glutamic acid (GLU) are satu-
rated, and the histidine (HIS) residue is assumed to be protonated.aThis model, in which
one oxygen ligand complements the two MPT ligands, is shown in Figure 5.7. The C-α
atoms of the amino acids were kept frozen during the optimizations. Since 13 different
structures were considered by using 4-component relativistic methods, computational ef-
ficiency has been important. The previous work in chapter 4 suggests to aim at less than
2000 basis functions. Therefore, several approximations for simplification were tested:
a) remove the entire magnesium phosphate bridge (in the following short “Mg-bridge”)
together with the phosphate groups and freeze the remaining terminal carbon atoms
instead. In addition, cut the MPTs to just leave the furan ring and saturate it with
hydrogen atoms
b) add to a) two NH2groups at the respective positions of the furan rings
c) as a), but only cleave the 2-amino-4-oxo-pyrimidine residue of the MPTs
d) as a), but keep the Mg-bridge and do not freeze the carbon atoms associated with
the bridge
e) use d) for the structure optimization but remove the Mg-bridge for the EPR pa-
rameter calculations
f) as e) but remove also all surrounding amino acid residues and water molecules after
the optimization
Option a) resulted in appreciable bending of one of the MPT ligands (cf. Figure C.2 in
appendix C), which seems unreasonable and rules out this option. This preliminary struc-
ture was nevertheless usedbto test the influences of options b) and c): the entire structure
aNote that the entire system gains an additional positive charge compared to the charges given in
Figure 5.6.
bFurther structures were not yet available at this point and I experienced severe convergence problems
with the experimental structure in the 4-component calculations.
119
5.4. Identifying the W(V) state of tungsten oxidoreductase enzymes
was kept frozen and only the added NH2groups or pyrazine rings were reoptimized after
their addition, so that changes to the EPR parameters will be clearly attributable to
these alterations alone (cf. Figure 5.8). The EPR results are collected in Table 5.7. The
effect of pyrazine compared to only NH2is negligible while removing the NH2group has
effects of up to 10 % on the ∆g33 component. This is not unexpected, since the preceding
investigations on the synthetic model complexes have shown that ∆g33 (which usually
points along the direction of the W=O bond; cf. Figures C.3 to C.7 in appendix C) is
most sensitive to structural changes in the dithiolene ligands. However, note that ∆g33 is
generally small, and thus small absolute changes will translate into large relative changes.
Unfortunately, even the inclusion of only the (four) NH2groups requires more than
2100 basis functions. This does not seem to be justified by the overall moderate ad-
ditional accuracy.
Table 5.7.: Effects of different truncations of the MPT cofactor on the EPR parameters.a
‘WO{THR}’ modelb∆giso ∆g11 ∆g22 ∆g33 Aiso A11 A22 A33 #atoms # b.f.c
truncated MPT with pyrazine -100 -161 -111 -27 -100 -74 -53 -173 109 2302
truncated MPT with NH2-100 -162 -111 -27 -100 -74 -53 -173 101 2150
fully truncated MPT -98 -161 -109 -24 -105 -78 -58 -179 93 1998
amDKS-PBE0-40HF/Hirao/IGLO-II results referring to options a), b), and c) in the
text. RI-BP86-D3(BJ)+COSMO/def2-TZVP optimized structures (without the
Mg-bridge).
bCf. Figure 5.9.
cNumber of (Cartesian GTO) basis functions.
Abandoning options a), b), and c), options d) and e) are the next step: Table 5.8 shows
that including the Mg-bridge during the structure optimization but removing it for the
EPR parameter calculations has almost negligible influences on both the g-tensor and
the HFCs (cf. Figure 5.8). A comparison of the last rows in Tables 5.7 and 5.8 reveals,
however, a strong influence on all parameters (except ∆g22) due to the structural changes
in the optimization. That is, the bridge has an important structural role during the
optimizations but no direct electronic influence on the EPR parameters. Removal of the
Mg-bridge allows a considerable reduction of computational effort in the 4-component
EPR calculations (giving less than 2000 basis functions).
Consequently, the next logical step f) would be a removal of all surroundings except the
tungsten-oxo core itself (cf. Figure 5.8). In contrast to the previous truncations, this ap-
proach results in significant influences especially on the HFCs (cf. Table 5.9). In addition,
in some of the model structures, the amino acid residues add – besides hydrogen-bond
interactions – even weakly coordinating oxygen atoms to the tungsten-oxo center (cf. Fig-
121
5. Tungstoenzymes
Figure 5.8.: Overview on the structure truncations under consideration (top to bottom,
respectively). Left: a) – c) Different truncations of the MPT cofactor. Middle:
d) & e) Removal of the Mg-bridge. Right: e) & f) Removal of all surrounding
amino acid residues and water.
122
5.4. Identifying the W(V) state of tungsten oxidoreductase enzymes
Table 5.8.: Effect of the Mg-bridge on EPR parameters.a
‘WO{THR}’ modelb∆giso ∆g11 ∆g22 ∆g33 Aiso A11 A22 A33 #atoms # b.f.c
with Mg-Bridge -99 -182 -108 -8 -113 -83 -71 -186 116 2559
without Mg-Bridge -100 -184 -109 -7 -114 -84 -72 -187 93 1998
amDKS-PBE0-40HF/Hirao/IGLO-II results referring to options d) and e) in the text.
RI-BP86-D3(BJ)+COSMO/def2-TZVP optimized structures (including the Mg-bridge
and fully truncated MPT ligands).
bCf. Figure 5.9.
cNumber of (Cartesian GTO) basis functions.
ure 5.9). Therefore, and in order to keep all model systems comparable, it is advisable to
keep the amino acid residues in all cases.
Table 5.9.: Effect of the surrounding amino acid residues on EPR parameters.a
‘WO{HIS}’ modelb∆giso ∆g11 ∆g22 ∆g33 Aiso A11 A22 A33 #atoms # b.f.c
with AA residues -97 -170 -98 -22 -116 -85 -74 -188 92 1990
without AA residuesc-100 -166 -102 -34 -130 -98 -85 -208 36 1036
aRI-BP86-D3(BJ)+COSMO/def2-TZVP optimized structures including the Mg-bridge,
which has been removed for the EPR calculations. mDKS-PBE0-40HF/Hirao/IGLO-II
results with fully truncated MPT ligands, referring to option f) in the text.
bCf. Figure 5.9.
cNumber of (Cartesian GTO) basis functions.
Evaluating functionals for the structure optimizations
In the preceding study on synthetic tungsten model complexes (sec. 5.3), PBE0-D3(BJ)/
def2-TZVP optimized structures were used. However, as the computationally much more
expedient RI-BP86-D3(BJ) approach is more feasible for the larger protein models,cit
is necessary to establish possible deviations due to this modification of the protocol.
Table 5.10 shows the resulting differences in the EPR parameters for three of the models.
The use of RI-BP86-D3(BJ) instead of PBE0-D3(BJ) structures influences mostly ∆g33,
albeit moderately so on an absolute scale. Nevertheless, it is also remarkable, that the
∆g11 component is in all cases noticeably influenced, even though the structure changes
seem to be negligible in the cases of ‘WO{THR}’ and ‘hp WO{HIS}’ (cf. Figures C.8
to C.10 in appendix C). However, we have already seen in our previous studies that
cPrimarily, PBE0 optimized structures were pursued but during the course of the optimizations it be-
came clear that some of those structure optimizations would take too long to allow a timely completion
of the project.
123
5. Tungstoenzymes
Table 5.10.: Effects of different functionals used for the structure optimizations.a
∆giso ∆g11 ∆g22 ∆g33 Aiso A11 A22 A33 RMSD RMSD
(WOS4C4only)
‘WO{HIS}’ (PBE0) -92 -156 -98 -23 -113 -84 -70 -187 - -
‘WO{HIS}’ (BP86) -96 -168 -99 -23 -116 -86 -73 -189 1.5388 0.0247
‘WO{THR}’ (PBE0) -108 -193 -112 -17 -119 -87 -76 -194 - -
‘WO{THR}’ (BP86) -99 -182 -108 -8 -113 -83 -71 -186 0.2329 0.0585
‘hp WO{HIS}’ (PBE0) -61 -114 -103 34 -136 -100 -92 -216 - -
‘hp WO{HIS}’ (BP86) -65 -122 -105 31 -137 -102 -93 -217 0.2359 0.0571
adef2-TZVP basis results with D3(BJ) correction and COSMO (ε= 4.0). EPR
calculations at the mDKS-PBE0-40HF/Hirao/IGLO-II level including the Mg-bridge
but fully truncated MPT ligands.
bCf. Figure 5.9.
the g11 value is strongly influenced by the W–O bond length (cf. Tables B.1 and B.2 in
appendix B), which is in all cases around 0.03 ˚
A longer for the BP86-optimized structures.
In conclusion, the above evaluations suggest the following steps: first, keep the Mg-bridge
in the structure optimizations for its structural role but remove it for the 4-component
EPR parameter calculations. Second, the inclusion of NH2groups at the truncated MPT
residues would be desirable but the increased accuracy does not balance the additional
computational efforts. Third, the influence of RI-BP86 instead of PBE0 for the structure
optimizations does not follow a specific trend but it is most distinct for the g11 component
and amounts there only to around 10 ppt (i.e. well below 10 %), which is justifiable given
the saved computational time (but should nevertheless be kept in mind).
5.4.2.2. Comparison of structures
As all systems contain either the two MPT ligands or one of the original MPTs together
with one persulfide variant (Figure 5.6), we may use the nature of the direct tungsten
coordination (after the structure optimizations) as basis of our shorthand notation for the
various W(V) species studied. Figure 5.9 provides an overview of the optimized structures
(respective stereo representations are available in Figures C.11 to C.19 in appendix C).
For clarity, the Mg-bridge has already been removed in the representations. The neutrald
dWithout the additional positive charge from the protonated histidine residue, the model would be
monoanionic as stated in Figure 5.6.
124
5.4. Identifying the W(V) state of tungsten oxidoreductase enzymes
‘WO{HIS}’ model is characterized by a hydrogen bond between the tungsten-oxo group
and the histidine residue, the monoanionic ‘WOOH’ model adds a hydroxy group, and
so on. The ‘sulfide-activated’ versions, where W=O is replaced by W=S, are structurally
very similar to their W=O analogues. Therefore, only the differing ‘hp WS’ is shown
explicitly, but not the ‘WS{HIS}’, ‘WS{THR}’, ‘WSH’, ‘WOSH’, and ‘WSSH’ structures.
While the original X-ray data would suggest a somewhat distorted coordination, the
optimizations of the ‘WO{HIS}’ or ‘WS{HIS}’ models generally give a rather regular
square pyramidal structure with apical W=O (or W=S), similar to synthetic analogues.
This may indicate artifacts in the crystallographically found structure from ill-defined
oxidation states. Six-coordinate species, such as ‘WOOH’, provide almost 90◦dihedral
angles between the two MPT ligands (see Table C.1 in appendix C for a collection of
selected bond lengths and MPT dihedrals). A structure with a smaller dihedral angle
and more irregular W–O coordination, closer to the original crystal data, is found for
‘hp WO{H2O}’ and ‘WOH’. In both cases, a second oxygen coordinates weakly to tung-
sten, causing the distortion compared to the ‘WO{HIS}’ and ‘WO{THR}’ models. For
‘hp WO{H2O}’, this second oxygen ligand is a water molecule (2.29 ˚
A distance to W),
formed through proton abstraction from HIS by the hydroxy group. An analogous behav-
ior is also found for the low-potential model when the hydroxy group is added at the HIS
site (‘WO{H2O}’). Adding the hydroxy group at the THR/ASP site results instead in
the octahedral ‘WOOH’ model. However, ‘WO{H2O}’ is ca. 100 kJ/mol lower in energy
(BP86/def2-TZVP), suggesting the proton abstraction from HIS to be far more favorable.
For the ‘WOH’ model, which originates from a W–OH2starting structure, the coordinat-
ing water ligand transfers one of its protons to a nearby water to form H3O+, which in
turn is stabilized by hydrogen bonding from THR and ASP. The resulting W–OH group
exhibits a W–O bond length of 1.98 ˚
A. This allows the additional weak coordination to
tungsten by the hydroxy group of GLU at 2.37 ˚
A. The ‘WO{THR}’ model also transfers
a proton from the W-OH ligand to a water molecule, forming H3O+and a W=O ligand.
In fact, the ‘WO{HIS}’ and ‘WO{THR}’ structures differ primarily in the orientations
of the amino acid residues (cf. Figure 5.10).
In general, the models for the high-potential species do not differ in any characteristic
way from the low-potential models (except for the changed MPT ligand).
It is noteworthy, that two optimizations (‘WO{HIS}’/‘WS{HIS}’, and ‘hp WS’) returned
an awkward 70 – 80◦angle around the dithiolene bonds in one of the MPT ligands. To
ensure that this does not reflect some unrealistic local minima, the angle was manually
adjusted and the structures reoptimized, with the same result.
125
5.4. Identifying the W(V) state of tungsten oxidoreductase enzymes
Figure 5.10.: Comparison of the optimized ‘WO{HIS}’ (green) and ‘WO{THR}’ (red)
structures (BP86/def2-TZVP).
For the diol-inhibited model, the smaller (ethylene) glycol was used instead of glycerol,
since the experimental EPR data are very similar for glycol and glycerol. Therefore,
only small structural differences in the closer surroundings of tungsten are assumed. The
‘Wdiol’ model exhibits relatively long W–O distances (1.98 and 2.02 ˚
A) due to hydrogen
bonding to those oxygen atoms by the nearby H2O and HIS (d(H-O) = 1.741 and 1.563 ˚
A,
respectively).
5.4.2.3. Comparison of computed and experimental EPR data
The comparison between theory and experiment will concentrate on experimental g-
tensors and 183W HFCs for various signals from FOR/AOR of Pf and Tl from refs. 214,
246, and 247.
Expected error margins
The expected accuracy of the calculations is derived from the aforementioned approxi-
mations and the previous results in Table 5.3, with a focus on the most closely related
systems, i.e.[WO(SPh)4]−, [WO(bdt)2]−, and [WO(edt)2]−. These considerations (cf. Ta-
127
5. Tungstoenzymes
ble 5.11) lead to expected error margins of ca. ±5 ppt for ∆g, -10 MHz for A11 and
A22, and +15 MHz for A33. However, possible uncertainties for g33 are larger than for the
other g-tensor components.
Table 5.11.: Derivation of expected error ranges based on previous validations (upper
part) and different approximations for the modeling of enzyme sites (lower
part).
∆g11 a∆g22 ∆g33 A11 bA22 A33
[WO(SPh)4]−c3 2 0 -18 -18 13
[WO(bdt)2]−c-1 -2 4 1 -4 5
[WO(edt)2]−c6 3 24 -7 -4 29
error (worst case) ±3 (+6) ±2 (±3) +4 (+24) -15 (-20) -15 (-20) +10 (+30)
MPT truncation d0 -2 -3 4 5 6
removed Mg-bridge e2 1 1 1 -1 1
RI-BP86 optimization f±10 ±4±10 ±4±3±5
∑error (worst case) ±5 (±15) ±4 (±7) ±4 (+30) -10 (+5/-20) -10 (+5/-20) +15 (+40)
a∆gErrors in ppt. b183W HFC errors in MHz.
cCf. Table 5.3; dcf. Table 5.7; ecf. Table 5.8; fcf. Table 5.10
Diol-inhibited form
One might expect the diol-inhibited states (from Pf AOR[246]) to provide a particularly
well-defined structure, suitable as an additional internal calibration of the methodology.
Unfortunately, larger deviations than expected are found for the g-tensors (Table 5.12):
even at the PBE0 level, the g-shifts are already somewhat too negative, and the devi-
ations grow further with larger EXX admixture. It is currently unclear if this reflects
inaccuracies in the model (e.g. the g-tensor is likely influenced by the W–O bonds, which
are weakened and lengthened by hydrogen bonding). The HFCs are better but still not
in perfect overall agreement with the experimental values. This is unexpected and raises
questions about the quality of the ‘Wdiol’ model structure used here. The glycol molecule
is rather large in contrast to all other models discussed here. This could require the con-
sideration of additional amino acid residues to properly model structural effects, which
might, e.g., indirectly weaken the hydrogen bonding to the diol oxygen atoms. A possi-
bly crucial importance of further surroundings is also indicated by experimental studies,
which could not identify an inhibition of Tl FOR by glycerol.[247] Since the active centers
are very similar,[249,250] the mechanisms of diol inhibition might involve more than just
the narrowest surroundings of the active centers. Furthermore, it is possible that the
histidine residue is unprotonated, which would certainly result in structural influences
due to the missing hydrogen bond. It is also noteworthy, that the HFC values are clearly
128
5.4. Identifying the W(V) state of tungsten oxidoreductase enzymes
distinct from those of the other experimentally reported species (low-, mid-, and high-
potential) and all other model structures from Figure 5.6, even the ‘WOOH’ model, which
also features a six-coordinated tungsten center (cf. Table 5.13). That is, comparison for
the diol-inhibited species leaves a few questions open regarding the proper modeling of
intermolecular interaction effects and therefore cannot be used as an internal calibration
here.
Table 5.12.: Collection of experimental and calculated values for the diol-inhibited model
(‘Wdiol’).
∆giso ∆g11 ∆g22 ∆g33 Aiso A11 A22 A33
glycerol-inhibited (ES-4 AOR)[246] -75 -121 -62 -41 - - - -
glycerol-inhibited (Pf AOR)[246] -72 -118 -61 -37 -183 -237 -111 -201
glycol-inhibited (Pf AOR)[246] -74 -115 -64 -44 -180 -237 -105 -198
calculateda-98 -149 -80 -65 -182 -253 -111 -182
aCalculations done at the mDKS-PBE0-xHF/Hirao/IGLO-II level, with PBE0 for the
g-tensor and PBE0-50HF for HFCs. RI-BP86-D3(BJ)+COSMO/def2-TZVP structures.
Low- and high-potential forms
Figure 5.11 provides a graphical overview on the calculated gand HFC components to-
gether with the range of experimental values for the low-potential variant. Corresponding
absolute values (also for the high-potential variant) are given in Table 5.13. Especially the
experimental values for ∆g22 and A22 lie in a narrow range and are therefore specifically
important in the identification of suitable structure proposals.
First, we will focus on the low-potential version: it is noticeable, that no structure satisfies
all six required parameters. Many of the investigated structures can in fact be excluded
readily (Table 5.14 provides an overview on the (dis)agreements), i.e. ‘WOOH’, ‘WSSH’,
‘hp WO{H2O}’, ‘hp WS’, ‘hp WO{HIS}’, ‘WOH’, ‘WSH’, ‘WOSH’, and ‘WO{H2O}’.
‘WOOH’ and ‘WSSH’ show almost no agreement at all and can be considered as unsuit-
able proposals. Also the three high-potential models are consistently outside the range
of experimental values, especially for the A11 component. ‘WOH’ and ‘WSH’ fail partic-
ularly for ∆g11 and A33. ‘WO{H2O}’ and ‘WOSH’ show reasonable agreements only for
∆g33,A11, and ∆g11, rendering them unlikely as well. Notably, these structure proposals
possess the highest dihedral angles between the MPT ligands in the test set (cf. Table C.1
in appendix C: ‘WOOH’ 72◦, ‘WOH’ 49◦, ‘WO{H2O}’ 44◦, ‘hp WO{H2O}’ 38◦), which
leads us to the conclusion that large MPT dihedral angles above 30◦seem to be unlikely.
129
5. Tungstoenzymes
Figure 5.11.: Graphical overview on the agreement with experiment for the low-potential
forms with all calculated EPR values at the PBE0 (g) and PBE0-
50HF/Hirao/IGLO-II (HFC) level. The error bars refer to Table 5.11 (gray
= worst case). The green lines signify the available range (min. and max.)
of expt. values (gray dashed lines refer to expt. HFC values from ref. 214,
which were only partially observed). Cf. Table 5.13 for absolute values.
130
5.4. Identifying the W(V) state of tungsten oxidoreductase enzymes
Table 5.13.: Collection of the experimental and calculated EPR parameters for all inves-
tigated model structures except ‘Wdiol’.a
∆giso ∆g11b∆g22 ∆g33 AisocA11 A22 A33 Ref.
low-pot (ES-4 AOR) - -142 -102 - - - - - 246
low-pot (Pf AOR) -84 -139 -101 -13 -126 -138 -81 -156 246
low-pot (Tl FOR) -96 -159 -104 -25 -135 -150d-84 -171 247
low-pot (Pf FOR) (dithionite) -96 -155 -101 -33 - - -93e- 214
low-pot (Pf FOR) (formaldehyde) -92 -151 -100 -26 - -117e-88e- 214
WO{HIS}-86 -159 -89 -11 -133 -102 -90 -208
WO{THR}-88 -171 -98 4 -131 -99 -88 -205
WOH -153 -307 -111 -40 -157 -146 -91 -233
WOOH -201 -272 -223 -108 -198 -158 -146 -291
WO{H2O}-83 -139 -81 -28 -172 -136 -116 -264
WS{HIS}-92 -169 -91 -17 -142 -126 -90 -210
WS{THR}-85 -173 -81 -1 -135 -116 -89 -199
WSH -115 -234 -102 -10 -217 -217 -151 -282
WOSH -124 -179 -173 -20 -165 -132 -120 -242
WSSH -204 -278 -242 -91 -179 -157 -123 -258
high-pot (ES-4 AOR) -57 -112 -42 -16 - - - - 246
high-pot (Pf AOR) -53 -110 -40 -10 -153 -186 -129 -144 246
high-pot (Tl FOR) -62 -119 -46 -21 -144 -177 -120 -135 247
sulfide-activated high-pot (Tl FOR) -60 -107 -50 -21 - - - - 247
hp WO{HIS}-49 -108 -91 51 -154 -117 -109 -236
hp WO{H2O}-106 -149 -121 -49 -204 -164 -145 -303
hp WS -49 -100 -79 31 -160 -127 -116 -239
re-reduced 1 (Pf AOR) -13 -45 -14 19 - - - - 246
re-reduced 2 (Pf AOR) -97 -152 -86 -53 - - - - 246
aThe ∆gvalues (in ppt) were obtained using the PBE0 functional, while PBE0-50HF
was employed to obtain the hyperfine coupling values (in MHz).
bThe ∆gvalues are collected in ascending order and the HFC values are linked to ∆g
according to the expt. references and the calculations, respectively.
cHFC signs are assumed in accordance with the calculations.
dA large linewidth hindered the precise determination of this HFC value in ref. 247.
eSplittings were only partially observed in ref. 214 and might therefore not be allocated
correctly.
131
5. Tungstoenzymes
Table 5.14.: Overview on the structures in agreement with the expt. values under consid-
eration of the error margins with respect to Figure 5.11.
in agreement range? ∆g11 ∆g22 ∆g33 A11 A22 A33 ∑(max = 6)a
WO{HIS}! % ! ?!?3
WO{THR}?! % ?!?2.5
WOH %?! ! ! % 1.5
WOOH % % % ?% % 0
WO{H2O}! % ! ! % % 0
hp WO{HIS}% % % % % % 0
hp WO{H2O}! % ?%%! 0
hp WS % % % % % % 0
WS{HIS}? ? ! ! ! ?4.5
WS{THR}?% % ?!?0.5
WSH % ! ! % % % 0
WOSH % % ! ! % % 0
WSSH % % % ?% % 0
aSums using the following values (to a minimum of 0): != 1; ?= 0.5; %= -1.
The remarkable differences between the ‘WOOH’, ‘WOSH’ and ‘WSSH’ models are most
likely related to the sequential weakening of the hydrogen bonds from GLU and HIS.
Differences between the ‘WO{X}’ and ‘WS{X}’ structures (X= HIS, THR) are proba-
bly also related to the weakening of a hydrogen bond – this time from THR. The sulfur
substition influences particularly the A11 value (the principal axis of A11 points roughly
along the W=O axis, cf. Figure C.3 in appendix C), which is elevated (in absolute values)
by around 15 to 20 MHz and thereby brought into closer agreement with the experimental
values.
‘WO{THR}’ and ‘WS{THR}’ agree with the experimental data for all HFC components
and fail only notably for ∆g33 (in case of ‘WS{THR}’ also for ∆g22). However, the
best agreement is found for the structurally similar ‘WO{HIS}’ model and its sulfur-
substituted analogue ‘WS{HIS}’. In fact, from the data one could get the impression of a
slight preference towards the sulfur-substituted version. However, according to ref. 214,
the EPR values in ref. 247 are probably obtained from the desulfo form and, therefore, the
range of experimental values might include data from sulfide-activated as well as desulfo
forms. In addition, we already concluded in sec. 5.3 that a decisive distinction between
the WO and WS forms might not be possible given the accuracy of our calculations.
132
5.4. Identifying the W(V) state of tungsten oxidoreductase enzymes
Finally, none of the tested structure proposals can be a possible candidate for the EPR
data on the high-potential variants (Table 5.13). Hence, this contradicts the proposed
sulfur insertion into one of the MPT cofactors. Further work and different proposals are
therefore required to (better) understand the nature of the high-potential enzyme state.
5.4.3. Conclusions
13 different proposed model structures for the W(V) low- and high-potential forms of the
AOR/FOR tungstoenzyme active sites have been investigated. A detailed structure that
is in all aspects consistent with the variety of experimental data could not be determined.
However, a promising model has been identified, and together with the exclusion of several
proposed model structures, it is possible to deduce distinct properties for the catalytically
active low-potential W(V) structure:
1. The MPT ligands are unaltered at the coordination site.
2. The MPT ligands are slightly twisted towards each other but probably not as much
as found in the crystal structure, excluding the coordination of a second oxygen
ligand to the tungsten center. In comparison to the crystal structure, the low-
potential version is likely to possess a smaller dihedral angle between the two MPT
ligands, most likely below 30◦.
3. A protonation of the tungsten-oxo center to W–OH or W–OH2is unlikely.
4. The closer surroundings, especially the amino acid residues, have measurable effects
on the EPR components and should not be excluded from future considerations and
calculations.
5. A distinction between “W=S” and “W=O” analogues is at the limits of our available
accuracy and therefore not reliably possible.
Moreover, it could be demonstrated that neither the models for the high-potential vari-
ant nor any other of the investigated models is able to account fully for the respective
experimental EPR data. The insertion of an additional sulfur atom into one of the MPT
ligands seems at least not to be the sole alteration – if it is present at all. In addition,
also the diol-inhibited form does not seem to be as easily describable as one might assume
on first glance. Further modeling work – probably including another layer of amino acid
residues – will be necessary to identify the (structural) nature of these species.
133
B. Additional tables and figures for the
study of tungsten model complexes
Table B.1.: Effects of artificial structure modifications on the EPR parameters of
[WO(SPh)4]−.a
[WO(SPh)4]−giso g11 g22 g33 Aiso A11 A22 A33
expt.[243] 1.936 1.903 1.903 2.018 -165 -133 -133 -234
PBE0-D3(BJ) + COSMO 1.928 1.891 1.891 2.002 -133 -90 -90 -218
W-O 1.708 ˚
A→1.758 ˚
A 1.921 1.881 1.881 2.000 -132 -89 -89 -217
W-O 1.708 ˚
A→1.658 ˚
A 1.935 1.901 1.901 2.003 -134 -91 -91 -220
W-S 2.399 ˚
A→2.299 ˚
A 1.956 1.917 1.917 2.034 -123 -86 -86 -197
a4-component
PBE0-40HF/Dyall(TZ)/IGLO-III//PBE0-D3(BJ)+COSMO(DMF)/def2-TZVP results.
183W HFC values in MHz.
Table B.2.: Effects of artificial structure modifications on the EPR parameters of
[WO(bdt)2]−.a
[WO(bdt)2]−giso g11 g22 g33 Aiso A11 A22 A33
expt.[244] 1.962 1.911 1.931 2.044 -153 -111 -119 -235
B3LYP/def2-TZVP 1.948 1.904 1.919 2.020 -97 -52 -56 -184
WO 1.71 ˚
A→1.61 ˚
A 1.957 1.917 1.930 2.023 -99 -54 -57 -187
WO 1.71 ˚
A→1.81 ˚
A 1.938 1.889 1.907 2.019 -97 -52 -56 -182
WS 2.43 ˚
A→2.36 ˚
A 1.969 1.923 1.929 2.057 -108 -68 -69 -188
WS 2.43 ˚
A→2.40 ˚
A 1.957 1.912 1.923 2.036 -105 -61 -64 -189
WS 2.43 ˚
A→2.50 ˚
A 1.922 1.882 1.906 1.979 -65 -15 -21 -158
WXSY 156◦→150◦1.940 1.899 1.917 2.003 -104 -59 -62 -192
WXSY 156◦→160◦1.952 1.907 1.919 2.029 -89 -42 -50 -174
a4-component PBE0-30HF/Dyall(TZ)/IGLO-III//B3LYP/def2-TZVP results. 183W
HFC values in MHz.
135
B. Additional tables and figures for the study of tungsten model complexes
Figure B.1.: Directions of g-tensor principal components computed for [WO(SPh)4]−at
the 4-component PBE0-40HF/Hirao/IGLO-II//PBE0-D3(BJ)/def2-TZVP
level.
Figure B.2.: Overlay of the X-ray structure[234] for [W(mdt)3]2−(green) and the PBE0-
D3(BJ)/def2-TZVP optimized structure for [W(mdt)3]−(red); created in
Maestro.[163]
136
Table B.3.: Effect of oxo vs. sulfido substitution on the EPR parameters of [WX(bdt)2]−
and [WX(edt)2]−(X = O, S). a
∆giso ∆g11 ∆g22 ∆g33 Aiso A11 A22 A33
[WO(bdt)2]−-55 -101 -80 16 -137 -93 -97 -219
[WS(bdt)2]−-60 -106 -85 12 -153 -109 -121 -230
[WO(edt)2]−-49 -114 -96 62 -140 -93 -101 -225
[WS(edt)2]−-56 -127 -104 62 -155 -110 -122 -233
a4-component PBE0-40HF/Hirao/IGLO-II//PBE0-D3(BJ)/def2-TZVP results. 183W
HFC values in MHz, ∆gvalues in ppt.
Table B.4.: Comparison of leading-order perturbation and full variational SO effects on
EPR parameters. a
∆giso ∆g11 ∆g22 ∆g33 Aiso A11 A22 A33
[WOCl5]2−expt.[242] -229 -244 -244 -198 -381
PBE0-40HF 1c-DKH (wo SO) 00 0 0 -228 -296 -194 -194
PBE0-40HF 1c-DKH -190 -204 -204 -162 -295 -405 -240 -240
PBE0-40HF 4c-mDKS -254 -258 -258 -245 -260 -196 -196 -388
[Tp*WO(OPh)2]expt.[239] -217 -295 -200 -155 -255
PBE0-40HF 1c-DKH (wo SO) 00 0 0 -230 -295 -197 -197
PBE0-40HF 1c-DKH -172 -254 -162 -101 -284 -386 -235 -231
PBE0-40HF 4c-mDKS -218 -301 -198 -155 -253 -195 -198 -365
[WO(SPh)4]−expt.[243] -66 -99 -99 16 -165 -133 -133 -234
PBE0-40HF 1c-DKH (wo SO) 00 0 0 -127 -186 -98 -98
PBE0-40HF 1c-DKH -49 -85 -84 22 -163 -243 -123 -123
PBE0-40HF 4c-mDKS -76 -114 -114 -1 -142 -99 -99 -227
[WO(bdt)2]−expt.[244] -40 -91 -71 42 -153 -111 -119 -235
PBE0-40HF 1c-DKH (wo SO) 00 0 0 -125 -183 -97 -95
PBE0-40HF 1c-DKH -33 -75 -55 32 -158 -238 -121 -114
PBE0-40HF 4c-mDKS -55 -101 -80 16 -137 -93 -97 -219
[WO(edt)2]−expt.[246] -21 -96 -81 114 -153 -119 -122 -217
PBE0-40HF 1c-DKH (wo SO) 00 0 0 -127 -187 -98 -95
PBE0-40HF 1c-DKH -26 -86 -69 78 -161 -243 -125 -116
PBE0-40HF 4c-mDKS -49 -114 -96 62 -140 -93 -101 -225
a1c-DKH calculations with or without (“wo”) perturbational contributions of SO effects
(SOMF meanfield operators) performed with ORCA.[148] All results obtained at the
PBE0-40HF/Hirao/IGLO-II//PBE0-D3(BJ)/def2-TZVP level. ∆gvalues in ppt, HFC
values in MHz.
137
C. Additional tables and figures for the
study on the W(V) state of
tungsten oxidoreductase
Table C.1.: Selected bond-lengths in ˚
A for the presented structure models after their
optimization (RI-BP86/def2-TZVP) with the Mg-bridge.
avg. d(W–S) d(W–O)/(W–S) next-nearest d(W–O) d(P–P)aMPT ‘dihedral angle’ (◦)b
Wdiol 2.40 1.98 2.02 6.72 46
WO{HIS}2.40 1.74 >4 6.49 15
WO{THR}2.39 1.76 >4 6.58 13
WOH 2.36 1.98 2.37 6.06 49
WOOH 2.51 1.81 1.93 6.51 72
WO{H2O}2.47 1.74 2.27 6.48 44
hp WO{HIS}2.40 1.73 >4 6.45 11
hp WO{H2O}2.48 1.73 2.29 6.36 38
hp WS 2.39 2.13 3.96 6.28 6
expt.[250] 2.49 2.09 >4 6.57 33
ad(P–P) can be seen as a measure for the length of the Mg-bridge.
bNote that the ‘dihedral angle’ is approximated as the angle between the vectors that
are spanned between the two sulfur atoms (bound to W) of each MPT.
139
C. Additional tables and figures for the study on the W(V) state of tungsten oxidoreductase
Figure C.1.: Stereo representation of the active center of FOR as found from its crystal
structure in ref. 250 with added hydrogen atoms (Histidin is assumed to be
present in its protonated form).
Figure C.2.: Optimized structure (without further alterations to the crystal structure)
with frozen terminal carbon atoms. Due to the apparent bending of the
truncated MPT ligand on the left, this model was abandoned.
140
Figure C.3.: Principal axes of the g- and HFC-tensors for the ‘WO{HIS}’ model.
BP86/def2-TZVP optimized structure, PBE0-40HF/Hirao/IGLO-II EPR
calculation.
Figure C.4.: Principal axes of the g-tensor for the ‘Wdiol’ model. BP86/def2-TZVP op-
timized structure, PBE0-40HF/Hirao/IGLO-II EPR calculation.
141
C. Additional tables and figures for the study on the W(V) state of tungsten oxidoreductase
Figure C.5.: Principal axes of the g-tensor for the ‘WOOH’ model. BP86/def2-TZVP
optimized structure, PBE0-40HF/Hirao/IGLO-II EPR calculation.
Figure C.6.: Principal axes of the g-tensor for the ‘WO{H2O}’ model. BP86/def2-TZVP
optimized structure, PBE0-40HF/Hirao/IGLO-II EPR calculation.
142
Figure C.7.: Principal axes of the g-tensor for the ‘WOH’ model. BP86/def2-TZVP opti-
mized structure, PBE0-40HF/Hirao/IGLO-II EPR calculation.
Figure C.8.: Comparison of PBE0/def2-TZVP (green) and RI-BP86/def2-TZVP (red) op-
timized structures for the ‘WO{HIS}’ model. The two structures have been
superimposed with respect to the tungsten center, the ligating sulfur atoms,
and the W=O oxygen.
143
C. Additional tables and figures for the study on the W(V) state of tungsten oxidoreductase
Figure C.9.: Comparison of PBE0/def2-TZVP (green) and RI-BP86/def2-TZVP (red) op-
timized structures for the ‘WO{THR}’ model. The two structures have been
superimposed with respect to the tungsten center, the ligating sulfur atoms,
and the W=O oxygen.
Figure C.10.: Comparison of PBE0/def2-TZVP (green) and RI-BP86/def2-TZVP (red)
optimized structures for the ‘hp WO{HIS}’ model. The two structures
have been superimposed with respect to the tungsten center, the ligating
sulfur atoms, and the W=O oxygen.
144
Figure C.11.: Stereo representation of the optimized structure for the ‘WO{HIS}’ model.
The Mg-bridge has only been removed after the optimization to allow a
better graphical representation.
Figure C.12.: Stereo representation of the optimized structure for the ‘WO{THR}’ model.
The Mg-bridge has been removed after the optimization to allow a better
graphical representation.
145
C. Additional tables and figures for the study on the W(V) state of tungsten oxidoreductase
Figure C.13.: Stereo representation of the optimized structure for the ‘WOH’ model. The
Mg-bridge has been removed after the optimization to allow a better graph-
ical representation.
Figure C.14.: Stereo representation of the optimized structure for the ‘WOOH’ model.
The Mg-bridge has been removed after the optimization to allow a better
graphical representation.
146
Figure C.15.: Stereo representation of the optimized structure for the ‘WO{H2O}’ model.
The Mg-bridge has been removed after the optimization to allow a better
graphical representation.
Figure C.16.: Stereo representation of the optimized structure for the ‘hp WO{HIS}’
model. The Mg-bridge has been removed after the optimization to allow a
better graphical representation.
147
C. Additional tables and figures for the study on the W(V) state of tungsten oxidoreductase
Figure C.17.: Stereo representation of the optimized structure for the ‘hp WO{H2O}’
model. The Mg-bridge has been removed after the optimization to allow a
better graphical representation.
Figure C.18.: Stereo representation of the optimized structure for the ‘hp WS’ model.
The Mg-bridge has been removed after the optimization to allow a better
graphical representation.
148
Figure C.19.: Stereo representation of the optimized structure for the ‘Wdiol’ model. The
Mg-bridge has been removed after the optimization to allow a better graph-
ical representation.
149
6. NMR shifts of paramagnetic
heavy-metal complexes
“If there is no complete agreement between the
results of one’s work and the experiment, one
should not allow oneself to be too discouraged.”
— Paul Dirac
Up to now, our considerations were focussed on electron paramagnetic resonance, which is
a rather specific spectroscopic technique with a naturally limited applicability. NMR spec-
trometers are more commonly available in laboratories than EPR spectrometers, due to
the overall larger scope of NMR. In principle, NMR on paramagnetic substances (pNMR)
combines the worlds of information from EPR and diamagnetic NMR, especially if the
interpretation is supported with quantum-chemical calculations.[2] However, as a result
of the magnetic fields generated by the unpaired electron(s), pNMR exhibits additional
complications compared to NMR spectra of diamagnetic compounds.[23] This can, e.g.,
shift the signals considerably. A second problem arises from paramagnetic relaxation pro-
cesses, which can cause a significant line broadening, particularly for nuclei close to the
paramagnetic center. Therefore, even experienced experimentalists might not always be
able to find all signals, since they might either lie outside the scanning range or are so
severely broadened that they become unobservable.
Despite ongoing developments[2,3,28,29] quantum-chemical interpretations of NMR spectra
for paramagnetic compounds still lack behind the success on diamagnetic analogues where
calculations have already become a standard tool to help with the evaluations.[1] This
is largely caused by the more complicated description of the additional hyperfine shift
contributions (cf. sec. 2.10). Initial applications of the ReSpect program for pNMR shift
calculations have been reported in 2013 by Komorovsk´y et al.[109] on small- to medium-
sized doublet systems (around 46 atoms). In this work, the investigations are extended
to larger systems (65 to 133 atoms) with up to two heavy-atom centers and S≥1. That
is, zero-field splitting (ZFS) needs to be included as well. The ZFS will be approximated
151
6. NMR shifts of paramagnetic heavy-metal complexes
here as the energy differenceabetween SO-split states with different projections of the
total magnetization J(cf. sec. 2.7.5). This neglects direct dipolar spin-spin couplings,
which, however, are considered to be negligible for heavy-atom systems.[80]
During the writing process of this manuscript, Rouf et al. published[266] a related study
considering mainly 3d metallocenes but including also rhodocene (4d) and iridocene (5d).
gand Awere obtained at the NEVPT2/DKH-TZV level in ORCA including DKH to
account for scalar relativistic effects and a perturbational treatment of SOC. In addition,
they calculated the HFC values at the mDKS-PBE0/DKH-TZV level in ReSpect. While
this is beyond doubt a sophisticated approach, variable amounts of EXX admixture for
the HFC calculations were not considered and the study focused on scalar relativistic
effects, which were found to be significant only for iridocene.
Since the previous investigations in this work already provided considerable insight into
the calculation of EPR parameters within the 4-component framework (cf. sec. 4), this
part will be organized as follows: a) validation of the computational protocol previously
determined for EPR also for pNMR on doublet systems, b) inclusion of ZFS, and c) ex-
tension to diruthenium complexes.
In a), the same iridium and platinum complexes as already discussed in the EPR context
in sec. 4 will be analyzed. Furthermore, a previously not considered iridium pincer com-
plex is added to the set (sec. 6.3). A similar ruthenium pincer complex is then used for
b), and finally two closely related diruthenium complexes are investigated (all in sec. 6.4).
6.1. Additional computational details I
The applied calculations follow closely the considerations that are laid out in section 2.10
and use the 2004 theory by Moon and Patchkovskii.[30] Structure optimizations are carried
out at the PBE0/def2-TZVP level in Turbomole and all pNMR values are obtained
using EPR and NMR parameters calculated with ReSpect. GIAOs are used for the
orbital shift calculations, but some restrictions apply due to the implementation stage in
ReSpect (version 3.4.2):[267]
•Only pure GGA functionals are supported, not hybrid functionals. Therefore, the
PBE/Hirao/IGLO-II level was used for the 4-component orbital shift calculations.
•Only the numerical DFT kernel is stable and has been used.
aADvalue of 1 cm−1corresponds to an energy difference of around 2.2·10−7a.u.[80]
152
6.1. Additional computational details I
Analyses will be given by separating the chemical shifts into three contributing terms,
the orbital shielding σorb, the Fermi-contact shielding σFC (not to be mistaken with the
Fermi-contact hyperfine coupling AFC), and the pseudocontact shielding σPC. The shift
δis then calculated as:
δ=δorb +δFC +δPC =σref −σorb
δorb
−σFC −σPC (6.1)
σorb is analogous to the NMR chemical shielding of diamagnetic compounds and σFC+σPC
can therefore be interpreted as the hyperfine contributions to the shielding (σhf =σFC +
σPC;δhf =−σhf). With respect to eqs. (2.106) and (2.107) in sec. 2.10, the individual
contributions are defined as
σFC =−µB
γMkT ·1
3Tr (giso · ⟨SS⟩ · Aiso)
σPC =−µB
γMkT ·1
3Tr (g· ⟨SS⟩ · Aaniso +ganiso · ⟨SS⟩ · Aiso)
σtot =σorb +σFC +σPC .
(6.2)
The isotropic parts of (symmetric) tensors K(s)are defined as
Kiso =1
3Tr (K(s))
Kiso =Kiso1
Kaniso =K−Kiso =K−1
3Tr (K(s))1.
(6.3)
Comparing this separation with Table 2.2 on page 46, we can conclude that the FC con-
tributions depend on the contact interactions (or more generally on the isotropic part) of
the HFC and are therefore a good indicator for the spin density at the respective nuclei.
The PC contribution, on the other hand, can be seen as an indicator for the importance
of anisotropic interactions.
Also “4-component spin densities” will be considered in this chapter. In the nonrelativistic
case, spin densities Q(r) are defined as[105]
Q(r) = ρα(r)−ρβ(r)
=N∫Ψ(r, s1,x2, ..., xN)σz(s1) Ψ(r, s1,x2, ..., xN) ds1dx2...dxN,(6.4)
153
6. NMR shifts of paramagnetic heavy-metal complexes
i.e. the expectation value of the Pauli spin matrix can be linked to the spin density[94]
⟨σz⟩=∫Q(r) dr.(6.5)
Therefore, in close analogy, we define the “4-component spin density” as[268]
⟨ψ(Jz)
4c |Σz|ψ(Jz)
4c ⟩=∫Q4c(r) dr(6.6)
with
Σz=(σz0
0σz).(6.7)
As mentioned in the main computational details, section 3, it is important for the
4-component calculations of gto first obtain a good guess on the orientation of the prin-
cipal axes since the applied computational scheme relies on their (pre-)alignment with
the Cartesian axes. However, the HFC principal axes of the ligand atoms are often not
collinear with the principal axes system for g, justifiably raising questions on their relia-
bility in pNMR calculations. Therefore, the influence of correctly aligned HFC principal
axes will be tested below. A correct alignment is achieved by a subsequent 4-component
calculation where the molecule is rotated such that the Cartesian axes are aligned with
(one of) the HFC principal axes from the previous 4-component calculation. This, of
course, doubles the computational cost.
All chemical shifts δ=σref −σare given in ppm (∆gin ppt, HFC in MHz, Din cm−1).
In general, temperatures are used in accordance with the available information in the
respective literature. Unless noted otherwise, room temperature (298 K) is assumed.
Calculated shifts that refer to a group of nuclei are given as averaged values over the
individual shifts of all nuclei in this group.
6.2. Reference shieldings
In accordance with the above mentioned restrictions to GGA functionals for orbital shield-
ings of open-shell systems, the absolute shieldings for the reference systems have been
obtained at the PBE/IGLO-II level from PBE0-D3(BJ)/def2-TZVP optimized structures
154
6.2. Reference shieldings
as reproduced in Table 6.1.
Table 6.1.: Reference orbital shieldings σref in ppm.
1H (TMS) 13C (TMS) 19F (CFCl3)a31P (H3PO4)aqb
PBE / IGLO-IIc31.8 187.4 169.5 344.5
CCSD(T)d35.6e193.4f207.9g358.2h
aCalculated using CF3H as secondary standard (σ(CF3H, calcd.) = 248.47). See text
for details.
bCalculated using PH3as secondary standard (σ(PH3, calcd.) = 610.57). See text for
details.
c4-component results from this work. Structures optimized at the
PBE0-D3(BJ)/def2-TZVP level.
dValues based on CCSD(T) σresults and experimental shifts δ, calculated using
secondary standards with σref =σ+δ
eσ(H2O) = 30.9 ppm[269] and δ= 4.65 ppm[270]
fσ(CH3CN) = 192.1 ppm[271] and δ= 1.3 ppm[270]
gσ(CF3H) = 286.9 ppm[272] and δ=−79 ppm[270]
hσ(PH3) = 624.31 ppm[273] and δ=−266.1 ppm[274]
Since the reference substance for 31P is an 85 % aqueous solution of phosphoric acid,
it would be difficult to directly obtain a theoretical absolute shielding. The problem is
circumvented by using gas-phase PH3as secondary standard. Thus, σref for H3PO4is
obtained as
σref(H3PO4) = σ(PH3)−266.1 ppm ,(6.8)
where −266.1 ppm represent the experimental chemical shift of gaseous PH3relative to
(H3PO4)aq.[274] Similarly, CF3H has been used as secondary standard for 19F.[270]
σref(CFCl3) = σ(CF3H) −79 ppm (6.9)
A comparison of the 4-component PBE data with external data obtained from (nonrel-
ativistic) coupled-cluster calculations shows small underestimations of 4 to 39 ppm, i.e.
3 % to 20 % (Table 6.1). The largest discrepancy occurs for the 19F reference shielding.
However, in view of possible error compensations in the shift calculations (δ=σref −σ),
we will use the PBE data in the following. Further discussions follow below.
155
6. NMR shifts of paramagnetic heavy-metal complexes
6.3. Iridium and platinum complexes
Besides the trans-Ir[η2-OC(CF3)2PtBu]2system, for which we have studied the EPR pa-
rameters in section 4, also the [IrClN(CHCHPtBu2)2] complex, which has not been con-
sidered before due to missing experimental EPR data, is investigated.
6.3.1. [Ir(II)ClN(CHCHPtBu2)2]
An X-ray structure of this iridium(II) PNP pincer complex (Figure 6.1) is available to-
gether with experimental giso and 1H NMR shift values.[275] To complement Table 4.7,
Table D.1 in appendix D contains the calculated EPR data. Table 6.2 gives the chemical
shifts for the hydrogen atoms marked in Figure 6.1.
Figure 6.1.: Structure of [IrClN(CHCHPtBu2)2] with positions of the distinct sets of hy-
drogen atoms marked.
Considering first the results obtained with the full g,A, and σorb matrices (i.e. the
‘standard’ approach), we find a good (‘PNPout’) to very good (‘PNPmid’ and ‘isobutyl’)
agreement with experiment (Table 6.2).
The separation of different shielding contributions in Figure 6.2 shows a large FC term
for the ‘PNPout’ group. This is supported by a closer look at the spin densities, which
have sizable 1H atom contributions only for the ‘PNPout’ group (cf. Figure 6.3), leading
to a large hyperfine shielding that contributes more than 50 % to the total σ. In general,
the magnitude of the hyperfine shielding is much smaller for the ‘PNPmid’ and ‘isobutyl’
156
6.3. Iridium and platinum complexes
Table 6.2.: 1H Chemical shifts in ppm for the paramagnetic iridium complex
[IrClN(CHCHPtBu2)2]. a
‘standard’ corrected HFC orientation bsolvent model cexpt.[275] #d
PNPmid -8.53 -8.49 1.23 -6.80 2
PNPout -98.85 -98.81 -112.29 -138.20 2
isobutyl 11.59 11.61 12.72 10.50 36
aBased on the X-ray structure,[275] calcd. at the mDKS-PBE/Hirao/IGLO-II level for
σorb, and at the mDKS-PBE0-40HF/Hirao/IGLO-II level for gand A.
bAfter additional ReSpect calculations with the principal axes of the HFC tensor from
hydrogen atom 15 (cf. Figure 6.1) as guess orientation (cf. computational details in
secs. 3 and 6.1) (gnot recalculated).
cg- and A-tensors obtained using the PCM method for ethanol in ReSpect.
dNumber of (magnetically equivalent) nuclei.
Figure 6.2.: Separation of the contributions to the total 1H nuclear shieldings (cf. eq.
(6.2)) of [IrClN(CHCHPtBu2)2] at the mDKS-PBE0-40HF/Hirao/IGLO-II
level.
157
6. NMR shifts of paramagnetic heavy-metal complexes
groups, but still sizable with respect to σorb (around 25 % of σtot).
Figure 6.3.: 4-Component spin density for [IrClN(CHCHPtBu2)2]. Isosurface values are
-0.000157 (red) and +0.000157 (blue). mDKS-PBE0-40HF/Hirao/IGLO-II
results. The molecule is rotated by 180◦w.r.t. Figure 6.1.
Application of the PCM solvent model for ethanol has a moderate effect on the ob-
tained shifts for the ‘PNPmid’ and ‘PNPout’ groups (Table 6.2). The sign change of δ
for ‘PNPmid’ can be traced back to a corresponding change in the HFC value, which is
very small and hence sensitive (‘PNPmid’ Aiso:−0.191 →+0.133 MHz). No clearcut
improvement of agreement with experiment is found. However, it should be noted that
this might change if different solvent models, that take hydrogen bonds specifically into
account, become available in ReSpect.
As already mentioned in section 2.7.1, the present approach with “only” three 4-component
SCF calculations requires an appropriate a priori orientation of the molecule (or better of
the principal axes of the tensors). Since we are now interested in the shifts of atoms that
might be very far away from the heavy-atom center and whose principal HFC tensor axes
usually differ considerably from the g-tensor axes, this represents an additional source of
possible error(s). In principle, it can be circumvented by a second set of calculations with
correct alignments of the axes. However, it is obviously not feasible to obtain the HFC
tensors for every hydrogen atom with perfectly suitable principal axes (for the molecule
at hand, this would, e.g., result in 16 ·3 = 48 additional 4-component ReSpect calcula-
tions). Therefore, in order to estimate the effect of the necessary approximation, a second
calculation was performed using the principal axes of the HFC tensor at hydrogen atom
‘H15’ of the ‘PNPmid’ group (see Figure 6.1). In fact, this scheme did not improve the
158
6.3. Iridium and platinum complexes
results but rather slightly increased the differences between calculations and experiment
(Table 6.2). Since this occurs for all shifts, not just ‘PNPmid’, it might be linked to a
poorer description of all other HFCs. While we may already conclude that the effects of
the reorientation of the axes are small, and anyway not computationally feasible for all
nuclei, we will test possible influences again for the second iridium complex.
6.3.2. trans-Ir(II)[η2-OC(CF3)2PtBu]2
trans-Ir[η2-OC(CF3)2PtBu]2represents an S=1
2system for which not only experimental
1H but also 19F shifts are available.[208] The calculated and experimental data are listed
in Table 6.3 where the labeling refers to Figure 6.4.
Table 6.3.: Chemical shifts in ppm for the paramagnetic iridium complex trans-Ir[η2-
OC(CF3)2PtBu]2.a
‘standard’ corrected HFC orientationbsolvent modelcexpt.[208] #d
19F-67.44 -67.01 -65.16 -58.05 12
1H(methylene, avgd.) 24.50 24.55 27.76 24.10 4
1H(tert-butyl, avgd.) -2.39 -2.35 -1.87 1.20 36
aBased on the X-ray structure;[208] calcd. at the mDKS-PBE/Hirao/IGLO-II level for
σorb, and at the mDKS-PBE0-40HF/Hirao/IGLO-II level for gand A.
bAfter additional ReSpect calculations with the principal axes of the HFC tensor from
fluorine atom 5 (cf. Figure 6.4) as guess orientation (cf. computational details in secs. 3
and 6.1) (gnot recalculated).
cg- and A-tensors obtained using the PCM method for THF in ReSpect.
dNumber of (magnetically equivalent) nuclei.
Results obtained with the full g- and A-tensors are in very good agreement for the 1H
‘methylene’ shift and acceptable for the 19F and the 1H ‘tert-butyl’ shifts. Separations
of different shielding contributions are provided in Figure 6.5. Since σFC and σPC cancel
to a large degree, the 19F shift is almost exclusively determined by the σorb part, which
is independent of EPR parameters. Using σref = 207.9 ppm, as obtained from CCSD(T)
calculations (Table 6.1), the 19F shift would be underestimated by around 30 ppm.
While also the 1H shift of the ‘tert-butyl’ group does not differ much from those in a
diamagnetic analogue, the ‘methylene’ shift experiences a strong hyperfine shift of more
than 20 ppm. This is not too surprising since the ‘methylene’ groups are positioned closer
to the heavy-atom center and therefore exhibit appreciable spin density (cf. Figure 6.6).
159
6. NMR shifts of paramagnetic heavy-metal complexes
Figure 6.4.: Structure of trans-Ir[η2-OC(CF3)2PtBu]2with positions of the distinguishable
hydrogen atoms marked (fluorine atoms are shown in purple).
Figure 6.5.: Separation of the contributions to the total nuclear shieldings of trans-Ir[η2-
OC(CF3)2PtBu]2at the PBE0-40HF/Hirao/IGLO-II level, cf. eq. (6.2).
160
6.3. Iridium and platinum complexes
Figure 6.6.: 4-Component spin density for trans-Ir[η2-OC(CF3)2PtBu]2. Isosurface values
are -0.000157 (red) and +0.000157 (blue). mDKS-PBE0-40HF/Hirao/IGLO-
II results.
An application of the PCM method for the THF solvent does not improve the results sig-
nificantly and is therefore not necessary. In addition, the dependence of the HFC tensors
on a proper orientation of their principal axes (along the Cartesian axes) was tested for
one of the fluorine atoms. As before, we can conclude that the resulting effects are not
sufficiently significant to justify the additional computational effort. Therefore, for the
remainder of this work, and as recommendation for future investigations, all HFC tensors
will be calculated using g-tensor principal axes to properly (pre-)align the axis system for
the property computations.
6.3.3. [Pt(III)I2(IPr)2]+
As a third test, pNMR shifts are investigated for the largest molecule from the previ-
ous EPR benchmark set, [PtI2(IPr)2]+. The overall numbers are given in Table 6.4 (cf.
Figure 6.7 for labels). The general agreement with experiment is good, even though the
‘isopropyl CH3outward’, ‘phenyl H (para)’ and ‘phenyl H (meta)’ shifts would be difficult
to assign a priori to experimental peaks without taking the number of contributing nuclei
and thus the peak integrals into account.
161
6. NMR shifts of paramagnetic heavy-metal complexes
Figure 6.7.: Structure of [PtI2(IPr)2]+with positions of the several distinct groups of 1H
nuclei marked.
Table 6.4.: 1H Chemical shifts in ppm for the paramagnetic platinum complex
[PtI2(IPr)2]+.a
‘standard’ expt.[210] #b
isopropyl CH3inward 6.15 7.92 24
imidazol H 1.98 4.13 4
phenyl H (meta) -3.07 2.08 8
phenyl H (para) 0.83 1.24 4
isopropyl CH3outward -5.68 -0.6 24
isopropyl H -9.82 -7.27 8
aBased on the X-ray structure,[210] calcd. at the mDKS-PBE/Hirao/IGLO-II level for
σorb, and at the mDKS-PBE0-40HF/Hirao/IGLO-II level for gand A.
bNumber of (magnetically equivalent) nuclei.
162
6.3. Iridium and platinum complexes
While spin-density distributions have not been specifically determined here,bthe sepa-
rated shielding contributions allow us to draw some conclusions (Figure 6.8): sizable FC
contributions are seen for the ‘isopropyl CH3inward’, ‘phenyl H (meta)’ and the ‘iso-
propyl H’ group. Hence, significant portions of spin density from the paramagnetic center
seem to be delocalized into the π-systems of the ligands. However, in comparison with
the ‘phenyl H (para)’ groups, the noticeable underestimation of the 1H shift for ‘phenyl H
(meta)’ might hint to an overestimation of the spin polarization effect by the PBE0-40HF
functional. A better description of δfor the ‘phenyl H (meta)’ group could provide a
correct trend for all calculated shifts.
Figure 6.8.: Separation of the contributions to the total 1H nuclear shieldings of
[PtI2(IPr)2]+at the PBE0-40HF/Hirao/IGLO-II level, cf. eq. (6.2).
bThe calculations were performed in an older version of ReSpect.
163
6. NMR shifts of paramagnetic heavy-metal complexes
6.4. Ruthenium complexes
We will now extend the computational protocol validated above to include the effects of
zero-field splitting. [Ru(PNPtBu)Cl] represents a triplet system (S= 1) with a PNPtBu
pincer ligand,crelated to the aforementioned iridium complex.[276] The second system,[277]
[Ru2(O2C-p-tolyl)4-(THF)2]0/+, whose structure is reminiscent of a paddlewheel, includes
a diruthenium group and is investigated in its neutral triplet state as well as in the quartet
state (S=3
2)of the monocation.
6.4.1. Additional computational details II
X-ray structural data are available for all three molecules. Their hydrogen atom positions
have been optimized at the same level as the full structure optimizations: in the case
of [Ru(PNPtBu)Cl] this refers to PBE0-D3(BJ)+COSMO(toluene)/def2-TZVP (toluene
ε= 2.38) and for [Ru2(O2C-p-tolyl)4-(THF)2]0/+to PBE0-D3(BJ)+COSMO(methanol/
THF)/def2-TZVP, both in Turbomole.[134] In the latter case, no closer specification
on the solvent mixture could be found in ref. 277, therefore a 1:1 mixture is assumed
(methanol ε= 32.6 and THF ε= 7.4 leading to a methanol/THF combination with
ε= 20.0). In accordance with the knowledge gained in the previous chapters, all EPR and
NMR calculations are performed using the computationally more expedient Hirao/IGLO-
II basis-set combination. PBE0-type functionals are applied with variable amounts of
exact exchange. As mentioned before, the (open-shell) orbital shieldings have to be cal-
culated with a pure GGA functional, therefore PBE has been used. Two partially different
approaches will be applied for the pNMR calculations: a) the S(S+1)
3generalized form of
the 2004 theory by Moon and Patchkovskii (“without ZFS”), and b) the 2015 theory by
Vaara et al. that includes ZFS specifically (“with ZFS”). Both are described in more
detail in sec. 2.10.
6.4.2. Expected accuracy for ZFS
Even though the ruthenium systems are not fully axially symmetric (cf. Tables D.2, D.3,
and D.4 in appendix D), zero-field splittings will be calculated as outlined in section 2.7.5,
since – for the time being – no better 4-component approach is available in ReSpect.
cPNPtBu = (tBuPCH2SiMe2)2N
164
6.4. Ruthenium complexes
This additional approximation seems to be sufficient here compared to previous calcu-
lations on transition metal systems with around 20 % deviation,[278,279] as we see for
[Ru(PNPtBu)Cl]: D= ca. 220 cm−1calcd. vs 273 cm−1expt. (19 % deviation).
6.4.3. [Ru(II)(PNPtBu)Cl]
In 2003, [Ru(PNPtBu)Cl] was reported as the first monomeric, paramagnetic (S= 1),
4-coordinate Ru(II) complex.[276] Up to then, the known divalent ruthenium complexes
were mostly 6-coordinate (octahedral), while some 5-coordinate square pyramidal com-
plexes were also known, but no 4-coordinate species.[280] Figure 6.9 shows the complex
and denotes all NMR-relevant nuclei. Besides 1H, also 13C and 31P chemical shifts are
available. Experiments were done at 20 ◦C (T= 293 K).
Figure 6.9.: Structure of [Ru(PNPtBu)Cl] with positions of the distinct 1H, 13C, and 31P
atoms/groups marked.
First, it is noticeable that the 31P shifts are remarkably large, providing a clear expla-
nation why they have not been found in experiment where only a range of ±500 ppm
was scanned. This seems a bit odd, since a high spin density can be expected at these
nuclei, and the experimentalists were well aware of the paramagnetism in this molecule.
Hence, it might be possible that ±500 ppm reflect the limitations of their spectrometer.
Without paramagnetism, the inspected range would have been reasonable, given that 31P
shifts of diamagnetic systems are usually found in this range.[270] Due to the large magni-
tude of the shift and its strong dependence on the exact-exchange (EXX) admixture, the
aforementioned error in the reference shielding becomes negligible.
165
6. NMR shifts of paramagnetic heavy-metal complexes
Table 6.5.: pNMR shifts in ppm for [Ru(PNPtBu)Cl] at different computational levels.
1H13C31P
structure levelaCH2Si-CH3CMe3CH2Si-CH3CMe3CMe3P
expt. -42 29 -11 -152 468 -271 147 >±500 ppm
#b4 12 36 2 4 4 12 2
X-ray / without ZFScPBE -22 13 14 -116 808 -299 192 -3450
PBE0 -30 10 5 -182 527 -314 146 -4884
PBE0-40HF -33 9 3 -180 420 -331 134 -5884
PBE0-50HF -32 10 0 -173 366 -336 125 -6469
X-ray / with ZFS PBE -30 22 5 -154 783 -288 167 -3263
PBE0 -36 19 -1 -187 530 -323 133 -4796
PBE0-40HF -40 19 -5 -186 426 -340 119 -5764
PBE0-50HF -40 22 -9 -180 375 -346 109 -6342
opt. / without ZFS PBE -17 20 10 -136 822 -278 184 -3269
PBE0 -23 17 3 -167 566 -311 147 -4764
PBE0-40HF -24 17 0 -165 457 -323 133 -5671
PBE0-50HF -25 17 -2 -159 405 -332 125 -6293
opt. / with ZFS PBE -23 28 3 -140 813 -285 168 -3194
PBE0 -29 26 -3 -173 570 -320 134 -4690
PBE0-40HF -31 26 -7 -171 464 -334 119 -5582
PBE0-50HF -33 28 -10 -166 414 -343 110 -6190
aAll calculations at the mDKS-PBE/Hirao/IGLO-II level for σorb.
mDKS/Hirao/IGLO-II level for g,D, and Aas stated. T = 293 K.
bNumber of (magnetically equivalent) nuclei.
c“X-ray” = based on the X-ray structure.[276] “opt.” = based on the
PBE0-D3(BJ)+COSMO(toluene)/def2-TZVP optimized structure.
166
6.4. Ruthenium complexes
The 13C shifts show an overall good agreement and correct trends {δ(CMe3)< δ(CH2)<
δ(CMe3)< δ(Si-CH3)}. Differences between the use of the X-ray and optimized structures
are significant only for the ‘CH2’ and ‘Si-CH3’ groups. A visual comparison of both
structures (cf. Figure D.2 in appendix D) does not provide any evidence of distinct
structural differences. Also the respective EPR parameters (Table D.2 in appendix D)
show only small differences.
The overall agreement is best with the PBE0-40HF functional. The need of 40 % EXX ad-
mixture agrees well with our previous recommendations for a 4d transition metal system
and global hybrid functionals are clearly superior to pure GGA functionals. Interestingly,
the results are slightly better without the inclusion of ZFS, whereas the 1H chemical shifts
depend much more on it. The Dvalue is in good agreement with experiment (cf. Ta-
ble D.2 in appendix D) and – based on our calculations – negative. With the application
of the 2015 theory by Vaara et al., the 1H chemical shifts are all in (very) good agree-
ment with experiment. Here, values obtained with 50 % of EXX admixture provide even
somewhat better agreement than those with 40 %, emphasizing again the importance of
increased amounts of EXX admixture.
Separations of the contributing terms are provided in Figures 6.10 and 6.11. The 13C
shieldings and especially the 31P shieldings are predominantly influenced by the FC con-
tributions. This is also reflected by the spin-density distributions in Figure 6.12, revealing
considerable spin-density concentrations at phosphorus and at some of the carbon atoms.
These large spin densities explain the large dependence on the exact-exchange admixture,
which influences the Aiso value and thus the FC contribution. Structural changes between
the X-ray and the optimized structure affect mostly the orbital shielding (Figure 6.11).
This explains the noticeable influences of the structure even though the changes in the
EPR parameters are not equally pronounced (Table D.2 in appendix D).
In summary, we can conclude that a precise reproduction of the chemical shifts consti-
tutes a challenge, but the orders of magnitude and trends are all well reproduced at the
mDKS-PBE0-40HF/Hirao/IGLO-II level. Furthermore, in subsequent pNMR measure-
ments, the 31P shift can probably be expected in a scanning range of -5000 to -6500 ppm.
167
6. NMR shifts of paramagnetic heavy-metal complexes
Figure 6.10.: Separation of the contributions to the total nuclear shieldings of
[Ru(PNPtBu)Cl] at the PBE0-40HF/Hirao/IGLO-II level (X-ray structure,
with ZFS, T = 293 K), cf. eq. (6.2).
Figure 6.11.: Separation of the contributions to the total 1H nuclear shieldings of
[Ru(PNPtBu)Cl] at the PBE0-40HF/Hirao/IGLO-II level (X-ray struc-
ture, with ZFS, T = 293 K), X-ray structure (left) and PBE0-
D3(BJ)+COSMO(toluene)/def2-TZVP optimized structure (right), cf. eq.
(6.2).
168
6.4. Ruthenium complexes
Figure 6.12.: 4-Component spin density for [Ru(PNPtBu)Cl]. Isosurface values are -0.0002
(red) and +0.0002 (blue). mDKS-PBE0-40HF/Hirao/IGLO-II//PBE0-
D3(BJ)+COSMO/def2-TZVP results.
6.4.4. [Ru(II)
2(O2C-p-tolyl)4-(THF)2]0
The most noteworthy feature of the paddlewheel-like tetracarboxylate [Ru2(O2C-p-tolyl)4-
(THF)2]0complex is its diruthenium center. The neutral molecule is shown in Figure 6.13.
The electronic structure of this complex can be rationalized as a metal-metal quadruple
bonded compound formed by overlaps of the d orbitals in a distorted X4M-MX4arrange-
ment (Figure 6.14).[281] Therefore, the neutral Ru(II)
2system is assumed to be present in
the triplet (S= 1) state.
Explicit NMR shifts for the THF-related 1H atoms were not reported in ref. 277. Never-
theless, they are calculated here and shown in Table 6.6 together with the proton shifts
from the toluene carboxylate ligands. In addition, specific temperature information was
only provided for experiments on a closely related compound at -58 ◦C, but not for the
systems targeted here. A temperature of 298 K is therefore assumed but calculations at
215 K are additionally included in Table 6.6 to cover this possibility. EPR parameters
are provided in appendix D, Table D.3.
The agreement with experiment is almost perfect for the ‘o-H’ group upon inclusion of
the full D-tensor at the PBE0-40HF level (X-ray structure / with ZFS / 298 K). While
the ‘m-H’ and ‘p-CH3’ shifts are underestimated by around 3 ppm, all three calculated
values can be clearly assigned. They show only small dependencies on the differences
169
6. NMR shifts of paramagnetic heavy-metal complexes
Figure 6.13.: Structure of [Ru2(O2C-p-tolyl)4-(THF)2]0with positions of the several dis-
tinct groups of 1H nuclei marked.
170
6. NMR shifts of paramagnetic heavy-metal complexes
Table 6.6.: 1H pNMR shifts in ppm for [Ru2(O2C-p-tolyl)4-(THF)2]0at different compu-
tational levels.
structure levelao-H m-H p-CH32,5-THF 3,4-THF
expt. 16 9 3 - -
#b8 8 12 4 4
X-ray / without ZFS / 298 K cPBE 17 7 -2 14 0
PBE0 17 7 0 3 -3
PBE0-40HF 20 7 1 -7 -8
PBE0-50HF not converged
X-ray / with ZFS / 298 K PBE 13 5 -2 25 5
PBE0 13 5 -1 15 2
PBE0-40HF 16 6 0 3 -5
PBE0-50HF not converged
X-ray / with ZFS / 215 K PBE 11 4 -4 39 9
PBE0 12 4 -2 25 6
PBE0-40HF 16 5 -1 8 -5
PBE0-50HF not converged
opt. / without ZFS / 298 K PBE 19 7 -2 -5 0
PBE0-40HF 22 7 0 -19 -6
opt. / with ZFS / 298 K PBE 15 5 -3 8 6
PBE0-40HF 19 6 0 -10 -2
opt. / with ZFS / 215 K PBE 14 4 -5 18 11
PBE0-40HF 20 5 -1 -8 -1
aAll calculations at the mDKS-PBE/Hirao/IGLO-II level for σorb.
mDKS/Hirao/IGLO-II level for g,D, and Aas stated.
bNumber of (magnetically equivalent) nuclei.
c“X-ray” = based on the X-ray structure.[277] “opt.” = based on the
PBE0-D3(BJ)+COSMO(methanol/THF)/def2-TZVP optimized structure.
172
6.4. Ruthenium complexes
between the X-ray and the optimized structure (cf. Figure D.3 in appendix D). Based on
the comparison of computed and experimental shifts, the assumption of T = 298 K seems
to be better justified than 215 K.
Here, it can be also concluded that the PBE0-D3(BJ)+COSMO/def2-TZVP optimization
level provides a good starting structure for pNMR shift calculations if no X-ray structure
is available. The influence of ZFS is considerable but overall not crucial for the toluene
carboxylate related pNMR shifts. The THF-related shifts show a stronger dependence on
the structure and ZFS effects. Further NMR data for the THF ligands are clearly desir-
able. For the moment, an analysis of different contributions, as displayed in Figure 6.15,
provides additional insights.
Figure 6.15.: Separation of the contributions to the total 1H nuclear shieldings of
[Ru2(O2C-p-tolyl)4-(THF)2]0at the PBE0-40HF/Hirao/IGLO-II level (X-
ray structure, with ZFS, T = 298 K), cf. eq. (6.2).
In contrast to the [Ru(PNPtBu)Cl] complex, the total shieldings are now mainly driven
by the orbital contributions except for ‘o-H’ where the FC term amounts to ca. 25 %.
Therefore, the small values for the ‘p-CH3’ group result from the similarity of the orbital
shieldings of the ‘p-CH3’ and the methyl group in TMS. The larger magnitude of the ‘m-H’
shift originates primarily from the difference in the orbital shielding of the aromatic proton
with respect to TMS.
Ref. 277 provides THF-related chemical shift data at 215 K for a similar complex
(Ru2(O2C(CH2)6CH3)4-THF2]): -19 and -15 ppm for ‘2,5-THF’ and ‘3,4-THF’, respec-
173
6. NMR shifts of paramagnetic heavy-metal complexes
tively. Especially with regard to the optimized structure, values in this range are only
obtained through the inclusion of exact exchange.
6.4.5. [Ru(II/III)
2(O2C-p-tolyl)4-(THF)2]+
Changing from [Ru2(O2C-p-tolyl)4-(THF)2]0(S= 1) to [Ru2(O2C-p-tolyl)4-(THF)2]+
(S=3
2)has only small structural consequences: the Ru-Ru bond length was found not
to be affected.[277] However, the Ru-O and Ru-O(THF) bond lengths decrease – as ex-
pected – for the cationic version (by around 0.05 and 0.1 ˚
A, respectively). The latter
is accompanied by a reorientation of the THF ligands to a more perpendicular position
with respect to the paddlewheel structure (cf. Figures 6.16 and 6.13). Chemical shifts
are collected in Table 6.7, while Figure 6.17 provides the usual separation of the shielding
contributions.
In contrast to the neutral variant, the experimentally determined ‘o-H’ shift is now twice
as large, and the ‘p-CH3’ shift has the opposite sign. This can be readily explained by the
increased spin density in the π-system of the toluene carboxylate ligands (cf. Figure 6.18)
and the hence increased hyperfine contributions (cf. Figures 6.17 and 6.15).
The overall agreement of the calculated values with the experimental data is very good at
the mDKS-PBE0-40HF/Hirao/IGLO-II level with ZFS. The correct trend is reproduced
at all levels, allowing an unambiguous assignment of the shifts. Again, the 215 K data are
in worse agreement with experiment than the 298 K data. Therefore, it can be assumed
that the reported measurements have been conducted at room temperature.
In contrast to most of the preceding compounds, it is striking that a change of the EXX
admixture does not seem to provide a defined trend for the pNMR shifts. This is already
the case for the EPR parameters (Table D.4 in appendix D). A spin-contamination effect
seems to be unlikely given the ⟨S2⟩values obtained at the 1c-DKH level (Table D.4),
and the spin densities also do not provide obvious differences to explain this behavior
(Figure D.5 in appendix D). A comparison of orbitals at the DKH level (Table D.6) re-
veals that the increased EXX admixture stabilizes the occupied π∗and δ∗orbitals (cf.
Figure D.1) compared to the PBE0 level. This does not occur for the neutral complex
(Table D.5) and is in general not expected. Therefore, the pNMR shift values obtained
with large EXX admixtures should be viewed with caution.
174
6.4. Ruthenium complexes
Figure 6.16.: Structure of [Ru2(O2C-p-tolyl)4-(THF)2]+with positions of the several dis-
tinct groups of 1H nuclei marked.
175
6. NMR shifts of paramagnetic heavy-metal complexes
Table 6.7.: 1H pNMR values in ppm for [Ru2(O2C-p-tolyl)4-(THF)2]+at different compu-
tational levels.
structure levelao-H m-H p-CH32,5-THF 3,4-THF
expt. 32 7 -3 - -
#b8 8 12 4 4
X-ray / without ZFS / 298 K cPBE 19 4 2 50 2
PBE0 28 2 -9 24 -5
PBE0-40HF not converged
PBE0-50HF 32 -8 -16 26 1
X-ray / with ZFS / 298 K PBE 19 4 1 46 0
PBE0 29 2 -8 15 -8
PBE0-40HF not converged
PBE0-50HF 31 -8 -16 33 3
X-ray / with ZFS / 215 K PBE 24 3 0 56 -2
PBE0 37 1 -11 9 -14
PBE0-40HF not converged
PBE0-50HF 38 -13 -22 42 4
opt. / without ZFS / 298 K PBE 19 4 -6 43 2
PBE0 28 1 -11 20 -3
PBE0-40HF 35 2 -8 -6 -15
PBE0-50HF 32 -9 -18 24 3
opt. / with ZFS / 298 K PBE 20 4 -6 39 1
PBE0 29 1 -10 13 -5
PBE0-40HF 36 5 -3 -36 -22
PBE0-50HF 31 -9 -18 29 5
opt. / with ZFS / 215 K PBE 24 3 -8 47 -1
PBE0 37 0 -14 9 -10
PBE0-40HF 45 6 -4 -61 -33
PBE0-50HF 38 -15 -25 36 6
aAll calculations at the mDKS-PBE/Hirao/IGLO-II level for σorb.
mDKS/Hirao/IGLO-II level for g,D, and Aas stated.
bNumber of (magnetically equivalent) nuclei.
c“X-ray” = based on the X-ray structure.[277] “opt.” = based on the
PBE0-D3(BJ)+COSMO(methanol/THF)/def2-TZVP optimized structure.
176
6.4. Ruthenium complexes
Figure 6.17.: Separation of the contributions to the total 1H nuclear shieldings
of [Ru2(O2C-p-tolyl)4-(THF)2]+at the PBE0-40HF/Hirao/IGLO-II level
(PBE0-D3(BJ)+COSMO(methanol/THF)/def2-TZVP structure, with ZFS,
T = 298 K), cf. eq. (6.2).
Inclusion of the ZFS D-tensor (at the PBE/PBE0 level) has only small influences of
around 1 ppm on the toluene carboxylate pNMR shifts in this system. Admittedly, in
contrast to the previous two ruthenium complexes, the D-value is rather small (PBE:
-55 cm−1compared to -180 and -220 cm−1, respectively. Cf. Tables D.2 to D.4 in ap-
pendix D).
177
6. NMR shifts of paramagnetic heavy-metal complexes
Figure 6.18.: 4-Component spin density for [Ru2(O2C-p-tolyl)4-(THF)2]0(left) and
[Ru2(O2C-p-tolyl)4-(THF)2]+(right). Isosurface values are -0.0002
(red) and +0.0002 (blue). mDKS-PBE0-40HF/Hirao/IGLO-II//PBE0-
D3(BJ)+COSMO/def2-TZVP results.
178
6.5. Conclusions
6.5. Conclusions
In general, the previously recommended protocol (mDKS-PBE0-40HF/Hirao/IGLO-II)
for EPR parameter calculations represents a suitable choice also for pNMR shifts. The
increased exact-exchange admixture is again vital for g,A, and the ZFS Dparameters.
Since (4-component) orbital shieldings are at the moment only available with pure GGA
functionals, the influence of the functional on this term remains to be examined. It is
still advisable to use the best structure information available, i.e. X-ray structures if
possible. These deductions extend also to non-doublet systems with S≥1 where the in-
clusion of the calculated D-tensor becomes very important, specifically for larger Dvalues
(≫50 cm−1), whereas the approximation without ZFS seems to be sufficient for small
magnitudes of D.
Finally, we can conclude that the overall agreement with experimental data is very good.
Comparing the obtained accuracy with previous approaches,[2,28,109,266,278] it is in fact ex-
cellent. Two key factors hoped for with computational approaches – to allow assignments
and predictions – are achieved. Only the 31P shift prediction for [Ru(PNPtBu)Cl] remains
to be confirmed although paramagnetic relaxation enhancement may possibly hamper
detection.
179
D. Additional tables and figures for the
pNMR studies
Table D.1.: EPR parameters for [IrClN(CHCHPtBu2)2] at the 4c-mDKS level using dif-
ferent functionals and basis-set combinations.a
method∆gisob∆g11 ∆g22 ∆g33 AisocA11 A22 A33
expt.[275] 408 “large anisotropy”
PBE0 / Dyall(TZ)/IGLO-III 230 -676 -154 1520 20 -87 -58 205
PBE0-40HF / Dyall(TZ)/IGLO-III 176 -914 -403 1847 15 -118 -87 250
PBE0-40HF / Hirao/IGLO-II 182 -907 -406 1860 12 -118 -87 242
aCrystal structure from ref. 275 used.
b∆gvalues in ppt. cHFC values for 193Ir in MHz.
Table D.2.: EPR parameters for [Ru(PNPtBu)Cl] at the 4c-mDKS level using different
functionals with the Hirao/IGLO-II basis-set combination.
structure method ∆gisoa∆g11 ∆g22 ∆g33 AisobA11 A22 A33 D(cm−1)⟨S2⟩c
expt.[276] 87 273 2.00
X-ray PBE 109 -30 68 289 -137 -168 -158 -85 -221 2.01
PBE0 165 -41 92 445 -98 -149 -112 -31 -204 2.01
PBE0-40HF 201 -31 87 548 -83 -143 -94 -11 -219 2.01
PBE0-50HF 233 -6 66 639 -83 -143 -95 -11 -236 2.01
opt. dPBE 104 -45 78 279 -126 -170 -141 -68 -226 2.01
PBE0 161 -43 93 432 -100 -150 -116 -35 -203 2.01
PBE0-40HF 195 -34 89 531 -85 -143 -97 -15 -217 2.01
PBE0-50HF 225 -3 63 616 -84 -143 -95 -13 -233 2.01
a∆gvalues in ppt. bHFC values for 101Ru in MHz.
cFrom a 1c-DKH calculation in MAG.
dPBE0-D3(BJ)/def2-TZVP optimized structure.
181
D. Additional tables and figures for the pNMR studies
Table D.3.: EPR parameters for [Ru2(O2C-p-tolyl)4-(THF)2]0at the 4c-mDKS level using
different functionals with the Hirao/IGLO-II basis-set combination.
structure method ∆gisoa∆g11 ∆g22 ∆g33 Aiso A11bA22 A33 D(cm−1)⟨S2⟩c
X-ray[277] PBE 76 -28 122 134 30 14 38 39 -178 2.01
PBE0 156 -10 224 255 37 25 43 43 -174 2.04
PBE0-40HF 258 -147 395 526 36 31 37 40 -133 2.05
PBE0-50HF not converged
opt. dPBE 78 -12 116 130 31 15 37 39 -173 2.01
PBE0-40HF 326 -11 486 502 34 32 34 37 -104 2.05
a∆gvalues in ppt. bHFC values for 101Ru in MHz.
cFrom a 1c-DKH calculation in MAG.
dPBE0-D3(BJ)/def2-TZVP optimized structure.
Table D.4.: EPR parameters for [Ru2(O2C-p-tolyl)4-(THF)2]+at the 4c-mDKS level using
different functionals with the Hirao/IGLO-II basis-set combination.
structure method ∆gisoa∆g11 ∆g22 ∆g33 Aiso A11bA22 A33 D(cm−1)⟨S2⟩c
X-ray[277] PBE 71 -32 120 124 35 30 31 44 -55 3.76
PBE0 280 -41 426 454 32 17 20 59 -133 3.78
PBE0-40HF not converged
PBE0-50HF 45 -55 85 105 67 62 64 76 103 3.78
opt. dPBE 65 -31 113 115 35 31 32 44 -54 3.76
PBE0 227 -40 352 370 37 24 26 60 -108 3.78
PBE0-40HF 576 -46 884 890 14 -14 -11 67 -640 3.79
PBE0-50HF -16 -53 -5 12 72 69 71 75 98 3.77
a∆gvalues in ppt. bHFC values for 101Ru in MHz.
cFrom a 1c-DKH calculation in MAG.
dPBE0-D3(BJ)/def2-TZVP optimized structure.
182
Figure D.1.: MO level diagram for the highest molecular orbitals in [Ru2(O2C-p-tolyl)4-
(THF)2]0(Figure 6.13). Gaussian09 unrestricted PBE0-xHF-D3(BJ)/def2-
TZVP results. Cf. also Figure 6.14. Molecular orbitals visualized in Chem-
Craft (https://www.chemcraftprog.com).
Table D.5.: [Ru2(O2C-p-tolyl)4-(THF)2]0: Orbital numbers for the Ru-Ru bond with re-
spect to Figures 6.14 and D.1.a,b
PBE0 PBE0-40HF PBE0-50HF
α β α β α β
σ∗204 208 208 208 208 208
π∗197/198 198/199 198/199 196/199 198/199 196/199
δ∗199 197 197 198 197 198
δ196 196 185 197 186 197
π193/194 194/195 186/187 193/195 185/187 187/195
σ194 193 183 194 196 194
aGaussian09 unrestricted PBE0-xHF-D3(BJ)/def2-TZVP results.
bOccupied orbitals are marked. HOMO: 199 (α) and 197 (β).
183
D. Additional tables and figures for the pNMR studies
Table D.6.: [Ru2(O2C-p-tolyl)4-(THF)2]+: Orbital numbers for the Ru-Ru bond with re-
spect to Figure 6.14.a,b
[Ru2(O2C-p-tolyl)4-(THF)2]+
PBE0 PBE0-40HF PBE0-50HF
α β α β α β
σ∗205 205 200 204 200 204
π∗198/199 198/199 189/191 198/199 189/191 198/199
δ∗197 197 190 197 190 197
δ186 196 185 196 185 196
π172/173 186/192 169/170 186/188 169/170 186/188
σ188 191 188 185 188 185
aGaussian09 unrestricted PBE0-xHF-D3(BJ)/def2-TZVP results.
bOccupied orbitals are marked. HOMO: 199 (α) and 196 (β).
Figure D.2.: Comparison of the X-ray[276] (green) and optimized struc-
tures (red) for [Ru(PNPtBu)Cl]. Optimization at the PBE0-
D3(BJ)+COSMO(toluene)/def2-TZVP level. RMSD: 0.1436 (value
obtained using the superposition function of Maestro[163]).
184
Figure D.3.: Comparison of the X-ray[277] (green) and optimized (red) struc-
tures for [Ru2(O2C-p-tolyl)4-(THF)2]0. Optimization at the PBE0-
D3(BJ)+COSMO(toluene)/def2-TZVP level. RMSD: 0.5925 (value obtained
using the superposition function of Maestro[163]).
Figure D.4.: Comparison of the X-ray[277] (green) and optimized (red) struc-
tures for [Ru2(O2C-p-tolyl)4-(THF)2]+. Optimization at the PBE0-
D3(BJ)+COSMO(toluene)/def2-TZVP level. RMSD: 0.2775 (value obtained
using the superposition function of Maestro[163]).
185
D. Additional tables and figures for the pNMR studies
Figure D.5.: 4-Component spin densities for [Ru2(O2C-p-tolyl)4-(THF)2]+. Isosurface val-
ues are -0.0002 (red) and +0.0002 (blue). mDKS-PBE0-xHF/Hirao/IGLO-
II//PBE0-D3(BJ)+COSMO/def2-TZVP results (x= 0,25,40,50 % EXX
admixture; “PBE0-0HF” = PBE).
186
7. Final conclusions and outlook
“Mankind’s greatest achievements have come
about by talking, and its greatest failures by not
talking.”
— Stephen Hawking
Throughout this work, the extraordinary importance of relativistic effects for magnetic
resonance properties of 4d and 5d transition metal systems has been investigated in the
framework of “fully relativistic” 4-component matrix Dirac-Kohn-Sham density functional
theory (4c-mDKS-DFT). This approach includes spin-orbit (SO) effects – which are often
essential for a proper description of (very) heavy elements – in a variational framework,
i.e. to all orders. The usually applied 1-component approach includes SO effects only
as first-order perturbation, which is sufficient for light-atom systems, such as organic
radicals, but can already fail for 3d transition-metal complexes. The g-shift in electron
paramagnetic resonance (EPR) is, e.g., almost exclusively due to SO effects.
All 4c-mDKS-DFT calculations were performed with the ReSpect program package (ver-
sions 3.3.0(beta) to 3.4.2).
First, a comprehensive benchmark study on 17 small 4d1and 5d1transition metal com-
plexes (S=1
2)has been conducted in chapter 4 for EPR g- and hyperfine coupling ten-
sors. Based on this study, the inclusion of exact-exchange (EXX) admixture through
hybrid functionals was found to be vital. Hyperfine coupling constants (HFCCs) are
particularly influenced by the improved description of metal s-type core-shell spin po-
larization. A general computational protocol with increased EXX admixture (mDKS-
PBE0-40HF/Dyall(TZ)/IGLO-III) is recommended for future EPR calculations at this
level. Besides the Dyall(TZ)/IGLO-III basis-set combination for the heavy element and
the ligand atoms, respectively, also the smaller Hirao/IGLO-II combination performed
remarkably well and can be recommended for larger systems or in general when computa-
tional efficiency is important. The modified PBE0 functional with 40 % EXX admixture
187
7. Final conclusions and outlook
represents a compromise between the application to 4d systems, where a somewhat lower
EXX admixture (around 35 %) seems to perform better, and 5d systems, which sometimes
require around 45 %. In addition, gvalues were in some cases reproduced slightly better
with an unmodified PBE0 functional (25 % EXX). However, these subtle differences did
not justify separate recommendations with respect to the inherent additional computa-
tional effort. Despite the increased spin polarization, spin contamination problems were
not observed for these d1systems.
Using the recommended approach, all calculated values were in good to very good agree-
ment with experimental data. To ensure the validity of the data, additional systems have
been included, where possible, to minimize errors from the measurements. The protocol
was subsequently tested on six larger Ir(II) and Pt(III) complexes (43 - 133 atoms) with
5d7(S=1
2)configuration and some very large g-tensor anisotropies. Here, the appli-
cability was confirmed, especially the importance of higher-order spin-orbit effects was
demonstrated by scaling the speed of light and by a comparison with calculations at the
1-component level. Higher-order SO effects accumulated to thousands of ppt in some
cases and thereby changed the appearance of the g-tensor fundamentally. Finally, also
the computational efficiency of the 4c-mDKS implementation in ReSpect was proven by
its successful application to a complex with 133 atoms (2960 basis functions).
Applications to more specific tungsten complexes and investigations on the active-site
structure of tungstoenzymes have been presented in chapter 5. Tungsten oxidoreductase
enzymes were chosen due to tungsten being the heaviest known element with a natu-
ral function in biological systems and heavy enough to bring in large (higher-order) SO
effects. In addition, detailed EPR studies are available for these systems. Thus, such
enzymes represent an attractive case to test the developed computational protocol in an
actual application. Since the active sites involve tungsten-sulfur bonds, which were not
part of the benchmark study of chapter 4, an additional validation study has been per-
formed in section 5.3. Due to the size of the targeted active sites, the Hirao/IGLO-II
basis-set combination was used. While PBE0-40HF performed well overall, specific fine-
tuning revealed that PBE0 for g-tensors and PBE0-50HF for HFCCs provides even better
results in these cases. Differences between the W=O and W=S analogues were found to
be small, and therefore it will be rather difficult to identify such a substitution based on
EPR investigations alone.
188
Several structure proposals exist for the S=1
2EPR-active W(V) state of tungsten oxi-
doreductase obtained from potentiometric titration. However, despite various experimen-
tal efforts (including EPR spectroscopy), the actual structure of the W(V) state could not
be determined. Hence, 13 of the proposed structures have been optimized in this work
and their calculated EPR parameters were compared to the available set of experimental
data. The active-site models were cut from an available enzyme X-ray structure of the
EPR-inactive state. Possible ramifications due to the approximations made for the mod-
els were carefully evaluated and considered in the final comparisons. Whilst none of the
chosen models was able to satisfactorily explain all experimental EPR parameters, it was
possible to exclude many of them and thereby draw specific conclusions on the nature of
the W(V) state: a) The two surrounding molybdopterin (MPT) cofactors are unaltered
at the coordination site. b) The dihedral angle between these MPT ligands should be
somewhere around 30 degree. c) Only a single oxygen atom is bound to the tungsten cen-
ter as W=O, i.e. the tungsten-oxo group is not protonated. d) The surrounding amino
acid residues have measurable effects on the EPR parameters and will therefore also be
important in future calculations.
In the third and final project of this work, the calculations were extended to nuclear
magnetic resonance (NMR) shifts of paramagnetic compounds (pNMR). pNMR chemical
shifts contain orbital shieldings that can be approximated analogously to shieldings of
diamagnetic systems and the hyperfine shift contributions that can be described with
the help of EPR Spin Hamiltonian parameters. The ‘standard’ approach recommended
above was successfully validated on a few Ir(II) and Pt(III) complexes. Furthermore,
also non-doublet ruthenium compounds have been considered, testing again the effect of
EXX admixture, with particular attention to the additionally introduced zero-field split-
ting (ZFS) contributions. Increased amounts of EXX admixture were again found to be
vital for a satisfactory reproduction of the measured chemical shifts. However, it should
be noted that orbital shieldings are only available at the pure GGA level in ReSpect.
Nevertheless, even at the current development stage, the calculations presented here are
sufficient to a) help with the assignment of peaks in experimental spectra and to b) pre-
dict shift ranges. Especially the latter might be very helpful, since pNMR signals can be
significantly shifted due to the magnetic fields generated by the unpaired electron(s).
Based on all of the above considerations, future applications to EPR- or pNMR-related
problems in 4d and 5d transition-metal chemistry can be performed without the explicit
need of additional benchmarks. Even large systems with more than 100 atoms can be
189
7. Final conclusions and outlook
targeted. However, several problems still remain for magnetic properties of open-shell
systems, not just on the theoretical side. First of all, due to the nature of least-squares
fitting schemes and the amount of parameters, an unambiguous fit of the Spin Hamilto-
nian (SH) might not always be possible in the case of complex spectra.[256,282]
Use of the SH is based on the inherent assumption that spin is (approximately) a good
quantum number, which is usually not the case anymore if SO effects are large. Also the
pNMR theory used in this thesis is only valid for relatively small SO effects and should
not be used without special caution for applications on lanthanides or actinides.[283] Fur-
ther problems often arise for such systems due to low-lying excited states, which require
a multideterminantal approach and prohibit therefore the herein used DFT methods. On
the other hand, multireference approaches such as CASPT2 (“complete active space with
second-order perturbation theory”) are not feasible for larger molecules that often require
large active spaces. Recently, a 4-component DMRG (“density matrix renormalization
group”) method[284] has been reported where already a state-interaction approach and
initial g-tensor calculations on neptonyl and plutonyl are promising.[285] Nevertheless, the
general feasibility for large-scale molecules remains to be seen for this approach.
These are open challenges, which might even require to either adjust or improve the SH
or to abandon the concept of the Spin Hamiltonian and directly calculate the spectra
instead. Strategies to overcome some of the aforementioned difficulties with usually very
sophisticated levels of theory have been published[278,285–288] but did not yet solve all chal-
lenges to a satisfying degree. Irrespective of the nature of future approaches, they will
certainly need highly motivated people on both sides (experiment and theory) with com-
bined efforts.
In summary, the work in this thesis has clearly shown that the 4-component mDKS-
DFT approach provides good accuracy to reproduce EPR and NMR Spin Hamiltonian
parameters for open-shell 4d and 5d systems by properly taking the significant higher-
order spin-orbit effects into account.
190
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“Everything that has a beginning comes to an
end.”
— Quintilian
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