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Electronic structure of strongly correlated systems: method development and applications to molecular magnets

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Electronic structure of strongly correlated systems: method development and applications to molecular magnets

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Content ELECTRONIC STRUCTURE OF STRONGLY CORRELATED SYSTEMS:
METHOD DEVELOPMENT AND APPLICATIONS TO MOLECULAR MAGNETS
by
Saikiran Kotaru
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CHEMISTRY)
August 2023
Copyright 2023 Saikiran Kotaru
Acknowledgements
First and foremost, I would like to express my deepest appreciation to my research adviser, Pro-
fessor Anna I. Krylov. She has been incredibly supportive and a constant source of inspiration
throughout my Ph.D. Her unwavering guidance helped me in my personal development and to
become a better researcher.
I would like to extend my sincere thanks to Professors Oleg Prezhdo, Curt Wittig, Mark Thomp-
son, Stephan Haas, Susumu Takahashi, Andrey Vilesov, and Jahan Dawlaty for their invaluable
discussions.
I would like to thank members of Prof. Krylov’s group for creating an intellectually stimulat-
ing and friendly environment in the lab: Dr. Maristella Alessio, Sourav Dey, Ronit Sarangi, Dr.
Sven K¨ ahler, Dr. Kaushik Nanda, Dr. Pavel Pokhilko, Dr. Yongbin Kim, Goran Giudetti, Mad-
hubani Mukherjee, Pawel Wojcik, Dr. Florian Hampe, Dr. Tirthendu Sen, Dr. Sahil Gulania, Dr.
Sarai Dery Folkestad, Nayanthara Jayadev, Kyle Tanovitz, and George Baffour Pipim. Numerous
discussions with Drs. Maristella Alessio and Sven K¨ ahler assisted me in my research profoundly.
I am also grateful to Dr. Pavel Pokhilko for helpful advice in debugging the spin-orbit coupling
code. I would like to acknowledge the assistance of Drs. Evgeny Epifanovsky and Xintian Feng at
Q-Chem, Inc. with navigating through the Q-Chem software during my initial explorations.
I would like to extend my gratitude to friends for their support: Dr. Swetha Erukala, Shivalee
Dey, Anwesha Maitra, Sraddha Agrawal, Akhil Valsalan, Devansh Shukla, Yogesh Todarwal, Car-
los Mora Perez, Shreyans Jain, Goutham Sekharamantri, and Dr. Pratyusha Das. I am indebted to
Vishnu Papisetty for his kindness and for being my guardian in Los Angeles.
ii
I am extremely grateful to Naveena and Raghava Sigireddy for generously hosting me during
Spring of 2022. I also wish to thank Uma Devi and SVR Prasad for their valuable advice during
my Ph.D. Finally, I would like to thank my parents, brother, sister-in-law, and grandparents for
their love and support over the years. This thesis would not have been possible without their
encouragement and guidance.
iii
Table of Contents
Acknowledgements ii
List of Tables vi
List of Figures ix
Abstract xiv
Chapter 1: Introduction and overview 1
1.1 Single-molecule magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The spin-flip approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Chapter 2: Magnetic Exchange Interactions in Binuclear and Tetranuclear Iron(III)
Complexes Described by Spin-Flip DFT and Heisenberg Effective Hamilto-
nians 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.1 Natural orbital analysis of mono- and bi-nuclear complexes . . . . . . . . . 21
2.4.2 Benchmark calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.3 Tetranuclear Fe(III) complexes . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Appendix A: Multiple-center molecular magnets: Mayhall’s approach . . . . . . . 36
2.7 Appendix B: Structures of mononuclear Fe(III) systems . . . . . . . . . . . . . . . 42
2.8 Appendix C: Natural orbitals in mononuclear Fe(III) systems . . . . . . . . . . . . 43
2.9 Appendix D: Natural orbitals in binuclear Fe(III) systems . . . . . . . . . . . . . . 47
2.10 Appendix E: Magneto-structural correlations . . . . . . . . . . . . . . . . . . . . . 50
Chapter 2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Chapter 3: Spin–orbit couplings within spin-conserving and spin-flipping time-dependent
density functional theory: Implementation and benchmark calculations 61
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2.1 TD-DFT/TDA and SF-TD-DFT/TDA . . . . . . . . . . . . . . . . . . . . 63
iv
3.2.2 Spin–orbit Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.1 Spin–orbit couplings in selected organic molecules . . . . . . . . . . . . . 69
3.4.1.1 Formaldehyde and acetone . . . . . . . . . . . . . . . . . . . . . 69
3.4.1.2 Biacetyl (BIA) and (2Z)-2-buten-2-ol (BOL) . . . . . . . . . . . 73
3.4.1.3 Psoralens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4.2 Spin–orbit couplings calculated with SF-TD-DFT . . . . . . . . . . . . . . 76
3.4.2.1 CH
2
, NH
+
2
, SiH
2
, and PH
+
2
. . . . . . . . . . . . . . . . . . . . . 76
3.4.2.2 BH, AlH, HSiF, HSiCl, HSiBr . . . . . . . . . . . . . . . . . . . 78
3.4.2.3 Spin reversal energy barrier in Fe(III) SMM . . . . . . . . . . . . 79
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.6 Appendix A: NTO analysis and energies of excited states . . . . . . . . . . . . . . 83
3.7 Appendix B: One electron SOCCs . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Chapter 3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Chapter 4: Origin of Magnetic Anisotropy in Nickelocene Molecular Magnet and Re-
silience of its Magnetic Behavior 94
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3 Computational details and model systems . . . . . . . . . . . . . . . . . . . . . . 102
4.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.6 Appendix A: Wave function analysis . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.7 Appendix B: Basis-set effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.8 Appendix C: Molar susceptibility within the Curie law . . . . . . . . . . . . . . . 126
4.9 Appendix D: Analysis of orbital angular momentum and spin–orbit matrix elements 127
4.10 Appendix E: Calculation of orbital angular momentum and spin–orbit coupling . . 128
4.11 Appendix F: Nickelocene on the MgO surface . . . . . . . . . . . . . . . . . . . . 135
Chapter 4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Chapter 5:Future work 149
5.1 Implementation of mixed-reference non-collinear SF-TDDFT . . . . . . . . . . . . 149
5.2 Magnetic properties of a spin-frustrated trinuclear copper complex . . . . . . . . . 150
Chapter 5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
v
List of Tables
2.1 Energy gaps
a
(E, eV),hS
2
i values
b
, and density-based analysis of spin-flip states
(denoted as “SF
n
”) obtained from the high-spin hextet reference state (denoted as
“ref.”) in model mononuclear Fe(III) complexes
c
. . . . . . . . . . . . . . . . . . . 21
2.2 Occupations of frontier natural orbitals ( n
i
) in the lowest hextet (idealhS
2
i is 8.75)
and quartet (idealhS
2
i is 3.75) state in mononuclear Fe(III) complexes. . . . . . . 22
2.3 Frontier natural orbital occupations ( n
i
) andn
u;nl
of the lowestS = 4 state showing
an increase of ionic character with the absolute value of J (cm
1
) in binuclear
Fe(III) complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 J-couplings (cm
1
) for 12 binuclear Fe(III) SMMs computed using NC-SF-TDDFT
with selected functionals and the 6-31G(d,p) basis set. . . . . . . . . . . . . . . . 26
2.5 Basis-set dependence ofJ-couplings (cm
1
) for the 12 binuclear complexes com-
puted with NC-SF-TDDFT with!PBEh. . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Frontier natural orbital occupations ( n
i
) and n
u;nl
of the lowest S = 9 state in
tetranuclear Fe(III) complexes. The first five (from NO
1
to NO
5
) and the last
five (from NO
16
to NO
20
) natural orbitals with lowest and biggest occupations,
respectively, are reported. Intermediate orbitals have occupations n
i
of one. . . . . 32
2.7 J-couplings (cm
1
) for tetranuclear iron (III) complexes computed with NC-SF-
TDDFT/!PBEh/6-31G(d,p). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.8 Structural (from experiments of Refs. 69–84) and magnetic data of this work
(LRC-!PBEh/6-31G(d,p)) for the 16 complexes under study. . . . . . . . . . . . . 50
3.1 SOMF SOCCs in formaldehyde and acetone computed with TD-DFT/TDA (B3LYP,
PBE0, !PBEh, !B97X-D, and !B97M-V) and EOM-EE-CCSD using cc-pVTZ
compared with values from Ref. 28. . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 SOMF SOCCs in formaldehyde and acetone computed with TD-DFT/TDA (B3LYP,
PBE0, !PBEh, !B97X-D, and !B97M-V) and EOM-EE-CCSD using aug-cc-
pVTZ compared with values from Ref. 28. . . . . . . . . . . . . . . . . . . . . . 72
3.3 SOCC in BIA and BOL computed with TD-DFT/TDA (B3LYP/cc-pVTZ) com-
pared with the EOM-CCSD and RASCI values. . . . . . . . . . . . . . . . . . . . 74
3.4 SOCCs in psoralen and its thio derivatives computed with B3LYP/cc-pVDZ and
compared with previous calculations. . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5 SOCCs (cm
1
) between
3
B
2
and
1
A
1
states in CH
2
, NH
+
2
, SiH
2
, and PH
+
2
computed
using SOMF with NC-SF-TD-DFT. . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.6 SOCCs (cm
1
) in BH, AlH, HSiF, HSiCl, and HSiBr computed using SOMF with
NC-SF-TD-DFT (PBE0/!PBEh; cc-pcVTZ). . . . . . . . . . . . . . . . . . . . . 79
vi
3.7 Energy gaps (E, in cm
1
) and of the target spin-flip states obtained from
the high spin hextet reference state computed with PBE0,!PBEh/cc-pVDZ. . . . . 80
3.8 Energy barrierU (cm
1
) computed using 2, 3, and 5 lowest SF states in trigonal
bipyramidal (PMe
3
)Fe(III)Cl
3
complex with NC-SF-DFT/PBE0/cc-pVDZ. . . . . 81
3.9 Vertical excitation energies (in eV) and dominant excitation character (Exc) of the
lowest singlet and triplet states in formaldehyde and acetone computed with TD-
DFT and EOM-EE-CCSD using cc-pVTZ. . . . . . . . . . . . . . . . . . . . . . 83
3.10 Vertical excitation energies (in eV) and dominant excitation character (Exc) of low-
est singlet and triplet states in formaldehyde and acetone computed with TD-DFT
and EOM-EE-CCSD using aug-cc-pVTZ. . . . . . . . . . . . . . . . . . . . . . . 83
3.11 Vertical excitation energies E (in eV), dominant excitation character (Exc), PR
NTO
,
and their weights (
2
) in BIA and BOL; B3LYP/cc-pVTZ. . . . . . . . . . . . . . 84
3.12 Vertical excitation energies (E, in eV), and dominant excitation character (Exc)
of psoralen molecules; B3LYP/cc-pVDZ. . . . . . . . . . . . . . . . . . . . . . . 84
3.13 NTO descriptors of excited states in psoralen molecules; B3LYP/cc-pVDZ. . . . . 85
3.14 One-electron SOCCs in formaldehyde and acetone computed with TD-DFT/TDA
(B3LYP, PBE0, !PBEh, !B97X-D, and !B97M-V) and EOM-EE-CCSD using
cc-pVTZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.1 Energies (in cm
1
) of the four lowest states of the isolated NiCp
2
and of NiCp
2
on
top of MgO(001), computed using EOM-SF-CCSD/cc-pVTZ and SF-TD-DFT/cc-
pVTZ with PBE0. “NiCp
2
/(Mg)O” and “NiCp
2
/Mg(O)” stand for the adsorption
complexes of NiCp
2
/MgO(001) with the Ni atom on-top of Mg
2+
and O
2
ad-
sorption sites, respectively (cluster model: NiCp
2
/Mg
25
O
25
). Differences between
NiCp
2
and NiCp
2
/MgO with SF-PBE0 are reported in parenthesis. . . . . . . . . . 114
4.2 Wave function properties of the target states of NiCp
2
; EOM-SF-CCSD/cc-pVTZ. 118
4.3 Wave function properties of the target states of the six ring-substituted NiCp
2
com-
pounds; EOM-SF-CCSD/cc-pVTZ (unless specified otherwise). . . . . . . . . . . 123
4.4 Wave function properties of the target states of NiCp
2
obtained by EOM-SF-CCSD
and SF-PBE0 calculations using the cc-pVDZ and cc-pVTZ basis sets. . . . . . . 124
4.5 CalculatedD parameter of NiCp
2
as extracted from the spin-orbit splitting of the
triplet ground state (S = 1,U =jDjS
2
=D) using both the EOM-SF-CCSD and
SF-PBE0 methods and the cc-pVDZ and cc-pVTZ basis sets. . . . . . . . . . . . . 125
4.6 Values of (4/g
2
)T as a function of the spin. . . . . . . . . . . . . . . . . . . . . . 126
4.7 Spin-orbit mean-field reduced matrix elements of NiCp
2
. Only A(lowest energy)
! B(higher energy) transition is shown. values are the singular values. The
sum of the contribution from the two leading NTO pairs recovers with accuracy
the reduced spin-orbit matrix elements (full EOM-SF-CCSD values).
a
. . . . . . . 127
4.8 Adsorption energies (E) and adsorption complex distances (R(Ni–Mg/O)) of
NiCp
2
on the MgO(001) surface. Model system: QM cluster (NiCp
2
/Mg
49
O
49
)
embedded into an array of point charges. Method: PBE0/6-31G

. “NiCp
2
/(Mg)O”
and “NiCp
2
/Mg(O)” stand for the adsorption complexes of NiCp
2
/MgO(001) with
the Ni atom on-top of Mg
2+
and O
2
adsorption sites, respectively. . . . . . . . . 136
vii
4.9 Energies (in cm
1
) of the target states of NiCp
2
in different environments: isolated
and on MgO(001). State energies are computed for the NiCp
2
/Mg
25
O
25
cluster us-
ing cc-pVTZ basis set and EOM-SF-CCSD, EOM-SF-MP2, and SF-TDDFT with
both PBE0 and LRC-!PBEh methods. “NiCp
2
/(Mg)O” and “NiCp
2
/Mg(O)” stand
for the adsorption complexes of NiCp
2
/MgO(001) with the Ni atom on-top of Mg
2+
and O
2
adsorption sites, respectively.S
2
values are in parenthesis. . . . . . . . . . 138
4.10 Wave function properties of the target states of the isolated NiCp
2
and of NiCp
2
/Mg
25
O
25
adsorption complex obtained by EOM-SF-CCSD and SF-PBE0 calculations using
cc-pVDZ basis set. The NiCp
2
/Mg(O) adsorption model is considered. . . . . . . 139
4.11 Spin-orbit mean-field reduced matrix elements of the isolated NiCp
2
(full EOM-
SF-CCSD and full SF-PBE0 values with cc-pVDZ and cc-pVTZ basis sets) and
of NiCp
2
/Mg
25
O
25
adsorption complex (SF-PBE0/cc-pVDZ values only). The
NiCp
2
/Mg(O) adsorption model is considered. . . . . . . . . . . . . . . . . . . . 140
viii
List of Figures
1 Illustration showing the equivalence of single spin-flip (SF) and multiple spin-flips
in the description of electronic structure of single-molecule magnets. . . . . . . . xiv
1.1 (a) Structure of Fe
4
-based SMM, Fe
III
4
(acac)
6
(Br-mp)
2
with iron ions depicted as
large orange spheres, where acac is acetyl acetonate, and Br-mp
3
is the anion
of 2-(bromomethyl)-2-(hydroxymethyl)-1,3-propanediol (Br-mpH
3
). The arrows
denote the relative orientations of the magnetic moments of each iron ion in the S
= 5 ground state. (b) Double-well potential energy diagram depicting the energy
barrier between spin-up and spindown. (c) Resonant photon–spin interaction (Rabi
cycle) between magnetic sublevels. Reproduced with permission from Ref. 4.
Copyright 2008 American Physical Society. . . . . . . . . . . . . . . . . . . . . . 1
1.2 Spin–orbit (dark blue) and Zeeman (light blue) splitting of a doubly degenerate
S = 2 ground state. Reproduced with permission from Ref. 12. Copyright 2021
American Chemical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Two-electrons-in-three-orbitals example. The spin-flip approach employs a high-
spin reference and spin-flipping operators to describe multi-configurational low-
spin states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Effect of the ligand-field strength on the local ground-state spin configuration of
Fe(III). The orbital splitting pattern corresponds to octahedral coordination. In the
weak-field limit, the low-lying manifold of electronic states is derived from config-
urations with five unpaired electrons, i.e., when eachd-orbital is singly occupied.
In the intermediate and strong field, the configuration of Fe(III) is a low spin in
which (some) electrons are paired. . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Three types of configurations in Fe(III) binuclear SMMs: Neutral Hund-type (with
fixed orbital occupations), locally excited (non-Hund-type with paired electrons),
and ionic (in which the number of electrons on each center is different). . . . . . . 16
2.3 Crystal structures of the binuclear (1-12) and tetranuclear (13-16) complexes with
Fe(III) centers investigated in this study with their Cambridge structural database
names. Color code: Fe — orange, Cl — green, S — yellow, N — blue, C — gray,
and O — red. Hydrogen atoms are not shown. . . . . . . . . . . . . . . . . . . . 19
2.4 Orbital occupations of the lowest state (i.e., SF
1
) in mononuclear Fe(III) complexes. 23
2.5 Mean absolute error (MAE) ofJ-couplings calculated using different functionals
relative to experimental values for 12 binuclear Fe(III) systems. The 6-31G(d,p)
basis set was used for all atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
ix
2.6 Theoretical versus experimentalJ-couplings for functionals with MAE< 10 cm
1
.
The black line shows the perfect match. . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 MAE of theJ-couplings computed using different basis sets with the!PBEh func-
tional for the 12 iron (III) binuclear systems. . . . . . . . . . . . . . . . . . . . . 28
2.8 Iron core in star-type complexes 13, 14 (left) and butterfly-type complexes 15, 16
(right) showing different exchange interactions. The superscriptsw andb denote
wing and body iron atoms, respectively. . . . . . . . . . . . . . . . . . . . . . . . 30
2.9 Flowchart of the protocol.H
HDvV
is the Heisenberg-Dirac-van Vleck Hamiltonian
and
~

ON
I
are the local spin functions. . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.10 Model mononuclear Fe (III) systems: [Fe(Cl)
6
]
3
(left), a monomer unit built from
ABIZOA complex (center), and [Fe(CN)
6
]
3
(right). . . . . . . . . . . . . . . . . 42
2.11 Frontier and natural orbitals of the lowest hextet (left) and quartet state (right)
in [Fe(Cl)
6
]
3
with their occupations. . . . . . . . . . . . . . . . . . . . . . . . . 44
2.12 Frontier and natural orbitals of the lowest hextet (left) and quartet state (right)
in ABI-m with their occupations. . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.13 Frontier and natural orbitals of the lowest hextet (left) and quartet state (right)
in [Fe(CN)
6
]
3
with their occupations. . . . . . . . . . . . . . . . . . . . . . . . 46
2.14 Frontier and natural orbitals of the lowestS = 4 state in complex 2 with their
occupations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.15 Frontier and natural orbitals of the lowestS = 4 state in complex 7 with their
occupations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.16 Frontier and natural orbitals of the lowestS = 4 state in complex 12 with their
occupations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.17 Left: Plot of the Fe–Fe distance vs the exchange interactions (J) for the 16 com-
plexes under study. Right: Plot of the (Fe-O-Fe) bond angle vs the exchange inter-
actions (J) for the 16 complexes under study. . . . . . . . . . . . . . . . . . . . . 51
3.1 Molecules studied in this work. BIA and BOL denote biacetyl and (2Z)-2-buten-
2-ol, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2 Canonical frontier Kohn-Sham molecular orbitals (,n, and

) of formaldehyde
and acetone (the shapes of the NTOs from the excited-state calculations are very
similar); B3LYP/cc-pVTZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3 Canonical Kohn-Sham molecular orbitals (,n,

) of BIA and BOL; B3LYP/cc-
pVTZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4 Canonical Kohn-Sham molecular orbitals (,n,

) of psoralen compounds; B3LYP/cc-
pVDZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5 Energy levels arising from the splitting of the two lowest quartet states induced by
SOC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
x
4.1 a) Structures of the nickelocene with staggered rings (coordinates are from Ref.
66). The bond of the metal with the Cp centroid is shown with blue dash lines.
Color code: Ni — purple, C — gray, and H — white. b) Electronic configurations
of the high-spinM
S
= 1 triplet reference and the low-spin target states: M
S
= 0
triplet (j1i) and singlets (j2i,j3i, andj4i). State energies and character are obtained
from EOM-SF-CCSD calculations. The two closed-shell configurations in states
j3i andj4i have equal weights. Based on natural orbital analysis, the two unpaired
electrons reside on the d
yz
and d
xz
orbitals. c) Four lowest electronic states and
spin-orbit splitting of theS = 1 ground state. Spin-orbit splitting is computed via
a two-step state-interaction scheme. Magnetic anisotropyD is obtained from the
energy splitting between the magnetic sublevels of the triplet ground state. . . . . 103
4.2 Structures of six ring-substituted nickelocene derivatives. In complex 1 two C–H
groups are substituted with two P atoms. In complex 2, 3, and 6, two H atoms are
substituted with methyl, cyano, and aromatic groups, respectively. Complexes 4
and 5 are bent structures taken from Ref. 73 and Ref. 74, respectively. The bond
of the metal with the Cp centroid is shown with blue dash lines. Color code: Ni —
purple, P — orange, N — blue, C — gray, and H — white. . . . . . . . . . . . . . 104
4.3 a) The embedded cluster setup used for structure optimization: the all-electron
QM region (NiCp
2
/Mg
49
O
49
) is treated with PBE0/6-31G

, while the outermost
region contains point charges. The QM region is shown as lifted for clarity. Here,
NiCp
2
is on-top of Mg
2+
adsorption site. b) Top and side views of the embed-
ded NiCp
2
/Mg
49
O
49
PBE0 region. c) Top and side views of a smaller cut-out
(NiCp
2
/Mg
25
O
25
) used for the SF-TD-DFT calculations. The bond of the metal
with the Cp centroid is shown with blue dash lines. Color code: Ni — purple, Mg
— green, O — red, C — gray, and H — white. . . . . . . . . . . . . . . . . . . . 105
4.4 Hole and particle NTO pairs of the spinless density matrix, giving rise to SOC
within the states 1 and 2 of nickelocene (EOM-SF-CCSD/cc-pVTZ). Singular val-
ues are 1.19 and 1.15, respectively. Red, green, and blue axes indicatex,y, and
z coordinates axes, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.5 Calculated field-dependent magnetizations (on the left) of NiCp
2
at low tempera-
ture (T = 2 K). Magnetization is in Bohr magneton (
b
) units. Calculated
Zm
T ,

Xm
T (
Ym
T ), and
av
T (on the right) of NiCp
2
in the temperature range from 5 to
300 K and under an applied field of 1 T. “av” stands for isotropic powder averaging. 109
4.6 Calculated temperature dependence of the inverse susceptibility (1/
av
) of NiCp
2
in the temperature range from 5 to 250 K (on the left) and from 5 to 80 K (on
the right), and under an applied field of 1 T. Calculated curves including three and
four electronic states are in blue and red, respectively. Experimental susceptibility
data, i.e. black and green curves, are taken from Ref. 67 and Ref. 68, respectively.
Experimental magnetization data are not available. . . . . . . . . . . . . . . . . . 110
xi
4.7 Calculated field-dependent magnetizations (top) of (i) NiCp
2
, (ii) six ring-substituted
NiCp
2
compounds, and (iii) NiCp
2
/Mg
25
O
25
adsorption complex (T = 2 K). Mag-
netization is in Bohr magneton (
b
) units. Calculated
Zm
T ,
Xm
T (
Ym
T ), and

av
T (bottom) in the temperature range from 5 to 300 K and under an applied
field of 1 T. “av” stands for isotropic powder averaging. Properties are obtained
by EOM-SF-CCSD/cc-pVTZ calculations, with the exception of complexes 5 and
6 for which we employed EOM-SF-CCSD/cc-pVDZ and of NiCp
2
/Mg
25
O
25
for
which we used SF-PBE0/cc-pVDZ. . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.8 Hole and particle NTO pairs of the spinless density matrix between states 1 and
2 of the NiCp
2
/Mg
25
O
25
adsorption complex (SF-PBE0/cc-pVTZ). Ni atom is on-
top of O
2
. Singular values are 0.5 and 0.5, respectively. Red, green, and blue
axes indicatex,y, andz coordinates axes, respectively. . . . . . . . . . . . . . . . 115
4.9 Singly occupied natural orbitals (SONOs) of triplet (state 1) and singlet states
(states 2, 3, and 4); EOM-SF-CCSD/cc-pVTZ.n = n

+n

and n = n

n

(in parenthesis) are provided.n

andn

values are obtained from the occupancies
of the and natural orbitals, respectively. Red, green, and blue axes indicatex,
y, andz coordinates axes, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 118
4.10 Hole and particle NTO pairs of the spinless density matrix, contributing to the over-
all SOC between the states 1 and 3 of NiCp
2
(EOM-SF-CCSD/cc-pVTZ). Singular
values are 1.19 (above) and 1.16 (below). Red, green, and blue axes indicatex,
y, andz coordinates axes, respectively. . . . . . . . . . . . . . . . . . . . . . . . 119
4.11 Hole and particle NTO pairs of the spinless density matrix, contributing to the over-
all SOC between the states 1 and 4 of NiCp
2
(EOM-SF-CCSD/cc-pVTZ). Singular
values are 1.17 (above) and 1.11 (below). Red, green, and blue axes indicatex,
y, andz coordinates axes, respectively. . . . . . . . . . . . . . . . . . . . . . . . 120
4.12 Hole and particle NTO pairs of the spinless density matrix, contributing to the over-
all SOC between the states 1 and 2 of complexes 1, 2, and 3 (EOM-SF-CCSD/cc-
pVTZ). Color code: Ni — purple, P — orange, N — blue, C — gray, and H —
white. Red, green, and blue axes indicatex,y, andz coordinates axes, respectively. 121
4.13 Hole and particle NTO pairs of the spinless density matrix, contributing to the over-
all SOC between the states 1 and 2 of complexes 4, 5, and 6 (EOM-SF-CCSD/cc-
pVTZ, with the exception of complexes 5 and 6 for which cc-pVDZ basis set has
been used). Color code: Ni — purple, C — gray, and H — white. . . . . . . . . . 122
4.14 Calculated magnetization (left) and susceptibility (right) plots of NiCp
2
obtained
with the cc-pVDZ and cc-pVTZ basis sets. Both EOM-SF-CCSD and SF-PBE0
methods are used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.15 Electronic configurations of the high-spinM
S
= 1 triplet reference and the low-
spin target states: M
S
= 0 triplet (i.e.,j1i) and singlets (i.e.,j2i,j3i, andj4i).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
xii
4.16 From the 4x4 supercell of MgO(001) (on the left) to the NiCp
2
/MgO(001) adsorp-
tion complex (on the right) consisting in a QM cluster (NiCp
2
/Mg
25
O
25
) embedded
in point charges. The QM region is lifted for clarity. Here, NiCp
2
is on-top of Mg
2+
adsorption site. The number of point charges is 2206, corresponding to a 6x6 su-
percell of the original MgO(001) slab model. The bond of the metal with the Cp
centroid is shown with blue dash lines. Color code: Ni — purple, Mg — green, O
— red, C — gray, and H — white. . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.17 Adsorption complex distances (R(Ni–Mg)) of NiCp
2
on the MgO(001) surface.
Here, NiCp
2
is on-top of Mg
2+
adsorption site. Model system: QM cluster (NiCp
2
/Mg
25
O
25
).
Blue dash line indicates R(Ni–Mg) distance. Color code: Ni — purple, Mg —
green, O — red, C — gray, and H — white. . . . . . . . . . . . . . . . . . . . . . 136
4.18 Calculated spin density of a) isolated NiCp
2
, b) adsorbed NiCp
2
on (Mg)O, and c)
adsorbed NiCp
2
on Mg(O). Color code: Ni — purple, Mg — green, O — red, C
— gray, and H — white. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.1 MR-SF-TDDFT employs two references shown in the upper panel by black and red
arrows. In the lower panel, electronic configurations generated by mixed reference
spin-flip operator are shown in blue, black, and red arrows. The blue ones are
generated by both references, while the black and red ones are generated fromM
s
=
+1 andM
s
= -1 references respectively. Gray arrows correspond to configurations
that cannot even be obtained in MR-SF-TDDFT. Reproduced with permission from
Ref. 6; Copyright 2022 American Chemical Society. . . . . . . . . . . . . . . . . 150
5.2 Molecular structure for the trinuclear copper complex. Hydrogen atoms are omit-
ted for clarity. Reproduced from Ref. 11 with permission from the Royal Society
of Chemistry; Copyright 2018 Royal Society of Chemistry. . . . . . . . . . . . . . 151
xiii
Abstract
Shut yourself up in a room, and have with you an unbiased 25-cent coin. Keep tossing the coin.
Suppose, on one day, you get tails as the first outcome and heads every other time you toss the
coin. Amazed by the asymmetry of this configuration, you follow this experiment the next day
to have heads as the first outcome and tails every other time. These different configurations on
day 1 and 2 are merely different microstates available to the system, and can be obtained from
an all-heads configuration with a single flip and multiple flips. It turns out that the low-energy
Figure 1: Illustration showing the equivalence of single spin-flip (SF) and multiple spin-flips
in the description of electronic structure of single-molecule magnets.
spectra of a system of interacting spins can be described by just a single spin-flip from a high-spin
configuration and is equivalent to a configuration produced by multiple spin-flips in terms of its
xiv
physical properties. The work presented here elucidates electronic structure of single-molecule
magnets (SMMs) within a single spin-flip approach.
This dissertation begins with the description of SMMs and theoretical framework of spin-flip
approach in Chapter 1. In Chapter 2, we extract exchange coupling constants for a set of iron (III)
binuclear and tetranuclear SMMs from all-electron calculations using non-collinear spin-flip time-
dependent density functional theory (NC-SF-TD-DFT). We investigate the range of applicability
of the Heisenberg model by analyzing bonding patterns in these Fe(III) complexes using natural
orbitals, their occupations, and the number of effectively unpaired electrons.
In Chapter 3, we present a new implementation for computing spin–orbit couplings within
TD-DFT framework in the standard spin-conserving and the spin-flip variants (SF-TD-DFT). This
approach employs the Breit–Pauli Hamiltonian and Wigner–Eckart’s theorem applied to the re-
duced one-particle transition density matrices, together with the spin–orbit mean-field treatment of
the two-electron contributions. We use state-interaction procedure and compute the SOC matrix
elements using zero-order non-relativistic states and provide benchmark calculations on several
closed-shell organic molecules, diradicals, and a Fe(III) SMM.
In Chapter 4, we investigate the magnetic behavior of nickelocene (NiCp
2
, Cp = cyclopen-
tadienyl) molecular magnet using the spin-flip variant of the equation-of-motion coupled-cluster
(EOM-SF-CC) method and use the EOM-SF-CC results to benchmark SF-TD-DFT. We also con-
sidered a set of six ring-substituted NiCp
2
derivatives and a model system of the NiCp
2
/ MgO(001)
adsorption complex, for which we used SF-TD-DFT method. To gain insight into the electronic
structure of these systems, we analyze spinless transition density matrices and their natural transi-
tion orbitals (NTOs).
In Chapter 5, I present the future directions of the work presented in this dissertation. The
results presented in this dissertation have been published in the following papers:
1. Kotaru, S.; K¨ ahler, S.; Alessio, M.; Krylov, A. I. Magnetic exchange interactions in binuclear
and tetranuclear iron(III) complexes described by spin-flip DFT and Heisenberg effective
Hamiltonians J. Comput. Chem. 2023, 44, 367–380 [Chapter 2]
xv
2. Kotaru, S.; Pokhilko, P.; Krylov, A. I. Spin–orbit couplings within spin-conserving and spin-
flipping time-dependent density functional theory: Implementation and benchmark calcula-
tions J. Chem. Phys. 2022, 157, 224110 [Chapter 3]
3. Alessio, M.; Kotaru, S.; Giudetti, G.; Krylov, A. I. Origin of magnetic anisotropy in nick-
elocene molecular magnet and resilience of its magnetic behavior J. Phys. Chem. C 2023,
127, 3647–3659 [Chapter 4]
xvi
Chapter 1: Introduction and overview
1.1 Single-molecule magnets
Single-molecule magnets (SMMs) exhibit magnetic properties such as magnetic hysteresis of a
purely molecular origin and quantum tunneling of magnetization.
1–3
They have attracted consider-
able attention recently due to potential applications as basic units in information storage devices or
quantum computing.
4–7
The first described and widely explored SMM is Mn
12
acetate, which has
a magnetic blocking temperature of 3 K.
8
Significant research efforts have been focused on finding
molecules with higher blocking temperatures.
Figure 1.1: (a) Structure of Fe
4
-based SMM, Fe
III
4
(acac)
6
(Br-mp)
2
with iron ions depicted
as large orange spheres, where acac is acetyl acetonate, and Br-mp
3
is the anion of 2-
(bromomethyl)-2-(hydroxymethyl)-1,3-propanediol (Br-mpH
3
). The arrows denote the rel-
ative orientations of the magnetic moments of each iron ion in the S = 5 ground state. (b)
Double-well potential energy diagram depicting the energy barrier between spin-up and
spindown. (c) Resonant photon–spin interaction (Rabi cycle) between magnetic sublevels.
Reproduced with permission from Ref. 4. Copyright 2008 American Physical Society.
SMMs commonly include high-spin transition metals, such as iron (III), iron (II), and man-
ganese (III) as magnetic centers.
9–11
For example, a Fe
4
-based SMM (shown in Fig. 1.1) withS
1
= 5 spin ground state shows magnetic bistability for as long as hundreds of nanoseconds, and was
proposed as a qubit candidate.
4
A key quantity that determines the magnetic relaxation time in the
SMMs is the effective magnetic relaxation energy barrierU for spin inversion (also known as spin-
reversal barrier), which arises due to the zero-field splitting of the 2S+1 degenerate components of
the spinS ground state.
12
This is illustrated in Fig. 1.2 for a doubly degenerate quintet state. The
spin-reversal barrier arises from the spin–orbit coupling (SOC) of the ground state with low-lying
excited states, and leads to anisotropy of the magnetic properties. The height of this barrier depends
on the square of the total spin of the molecule (in the ground state) and its magnetic anisotropy
(or zero-field splitting parameter,D). Another important interaction is the exchange coupling (J)
between the localized effective spins, which determines the energy separations between the ground
and excited states, and thereby the magnetic relaxation rate.
13, 14
Figure 1.2: Spin–orbit (dark blue) and Zeeman (light blue) splitting of a doubly degenerate
S = 2 ground state. Reproduced with permission from Ref. 12. Copyright 2021 American
Chemical Society.
Theoretical studies of SMMs play an essential in interpreting experiments and designing
novel magnetic materials.
15, 16
Spin-Hamiltonians are commonly used to interpret experimen-
tally observed magnetic behavior and to provide a link between theory and experiment.
16
Spin-
Hamiltonian parameters — such as exchange-coupling constantJ and magnetic anisotropy param-
eterD — can be determined from the energies and wave functions from ab initio calculations, and
then compared with the respective quantities obtained by fitting experimental magnetization and
susceptibility data.
2
SMMs are strongly correlated systems with multiple nearly degenerate states and their wave
functions are multi-configurational. Consequently, the ab initio description of SMMs is challeng-
ing because standard quantum-chemistry methods, such as Kohn–Sham DFT or single reference
methods based on perturbation theory (PT) or coupled-cluster (CC) formalisms fail when wave
functions are dominated by more than one Slater determinant. We overcome this challenge with
the spin-flip (SF) approach, which is explained in detail in Section 1.2.
1.2 The spin-flip approach
Within the spin-flip approach, a high-spin reference, which is accurately described by a single-
reference wave function, is used to describe the target problematic low-spin states as

S;S1
Ms=S1
=
^
R
Ms=1

S
Ms=S
: (1.1)
Here, the spin-flip operator
^
R
Ms=1
generates all possible singly excited determinants in which the
spin of one electron is flipped with respect to the high-spin reference.
17–23
For example, a spin-flip
operator on a high-spin triplet reference ofM
s
= 1 generates a set ofM
s
= 0 states, which are
used to describe low-spin triplet and singlet states. Similarly, a high-spin reference state ofS = 5
can be used to describe multiconfigurationalS = 5 andS = 4 states in a system with 10 unpaired
electrons, such as Fe(III) binuclear complexes.
Figure 1.3: Two-electrons-in-three-orbitals example. The spin-flip approach employs a high-
spin reference and spin-flipping operators to describe multi-configurational low-spin states.
3
We note that spin-flip solutions are not pure spin eigenfunctions — for example, only those
configurations corresponding to two-electrons-in-two-orbitals (i.e, the first 4 configurations shown
in Fig. 1.3) form a spin-complete set, whereas the configurations obtained by excitations that
change the number of electrons in the open-shell subspace are missing their spin-complements.
23–25
This may result in a slight spin-contamination of the proper spin-flip states.
In Chapter 2, we calculate exchange coupling constants in binuclear and tetranuclear Fe(III)
SMMs using spin-flip method and Heisenberg spin Hamiltonian.
26
The binuclear Fe (III) SMMs
have 10 unpaired electrons, whereas the tetranuclear SMMs have 20 unpaired electrons. It is
straightforward to extract exchange parameters in systems with two radical centers through the use
of Land´ e interval rule.
27
However, in systems with multiple radical centers, in order to construct
an effective Hamiltonian, one needs to use energies and wave functions from single spin-flip cal-
culations to parameterize the Heisenberg Hamiltonian.
28, 29
The key question here is the validity of
the Heisenberg model for systems under study. We use density-based analysis and natural orbitals
(NOs) to characterize the electronic structure and assess the reliability of the Heisenberg model for
these systems.
SOC plays a crucial role in determining the magnetic properties of SMMs, it leads to mag-
netic anisotropy, thereby affecting the spin-reversal energy barrier and magnetic relaxation.
30–32
In
Chapter 3, we present a new implementation for computing SOCs within time-dependent density-
functional theory framework (TD-DFT) and in the spin-flip variant (SF-TD-DFT).
33
We employ
the Breit–Pauli Hamiltonian and use an effective one-electron spin–orbit mean-field (SOMF) ap-
proximation to reduce the computational cost of the two-electron contribution.
34–36
We present
benchmark results for molecules featuring different types of electronic structure and evaluate spin-
reversal energy barrier in a Fe(III) SMM using SF-TD-DFT.
In Chapter 4, we characterize magnetic properties of the nickelocene (NiCp
2
, Cp = cyclopen-
tadienyl) molecular magnet with EOM-SF-CC and SF-TD-DFT using ezMagnet and the Q-Chem
software.
12, 37, 38
We investigate the sensitivity of NiCp
2
’s electronic structure and magnetic behav-
ior to its coordination environment by modifying its local ligand field. To this end, we consider
4
isolated NiCp
2
, six differently ring-substituted NiCp
2
compounds, and the adsorption complex of
a single NiCp
2
molecule on a model magnesium oxide surface.
5
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8
Chapter 2: Magnetic Exchange
Interactions in Binuclear and Tetranuclear
Iron(III) Complexes Described by Spin-Flip
DFT and Heisenberg Effective
Hamiltonians
2.1 Introduction
Single-molecule magnets (SMMs) are of interest due to their potential use in high-density infor-
mation storage, spintronics, and quantum computing.
1–5
Rational tuning of their ground-state spin,
magnetic anisotropy, and magnetic exchange couplings is essential in the design of SMMs with
desired magnetic behavior (i.e., ferromagnetic or antiferromagnetic, switchable, etc.)
6
Magnetic
properties of SMMs arise due to their spin-ordering, which in turn is governed by the exchange
interactions between the localized effective spins. The magnitude of the exchange coupling also
controls the magnetic relaxation rate, as high exchange results in higher energy separations be-
tween the ground and excited states, thereby suppressing magnetic relaxation via close-lying spin
states.
7, 8
High-spin transition-metal compounds including multiple magnetic centers, such as iron
(III), iron (II), and manganese (III), are well suited for the design of SMMs.
9–12
Among the best
9
examples, a complex of 4 Fe(III) with S = 5 spin ground state, which shows magnetic bistabil-
ity for as long as hundreds of nanoseconds, was proposed as a candidate for implementing qubits
in molecule-based quantum computers.
9
Similar magnetic frameworks anchored to a conducting
surface have been investigated in the context of molecular spintronics.
13
Despite the observation
of magnetic hysteresis in such multi-nuclear complexes, their low blocking temperature precludes
their incorporation into a real quantum device, and calls for a joint effort by experiment and theory
to tweak the SMMs magnetic properties through optimization of their structures.
First-principles calculations of exchange interactions in transition-metal complexes are chal-
lenging because small energy gaps between the spin states require accurate treatment of both
static and dynamic correlation effects. Traditionally, such systems have been described by
multi-reference methods, such as complete active space self-consistent field (CASSCF) methods
augmented with perturbative corrections. For example, CASPT2
14
and NEVPT2
15
(n-electron
valence-state PT) are frequently used in studies of magnetic molecules, providing accurate esti-
mates of their magnetic properties.
16, 17
Due to the high cost of such calculations, more afford-
able DFT-based approaches are often employed, such as the broken symmetry (BS) methods.
18–20
Notwithstanding its wide use, the limitations of BS-DFT are well known—it is based on unphysical
broken-symmetry solutions and its application relies on a somewhat arbitrary choice of projectors.
Although BS-DFT was shown to provide reasonable energies and charge density, spin densities
are qualitatively wrong, which undermines the reliability of computed spin-dependent properties.
Furthermore, BS-DFT does not scale well with the number of radical sites, as the number of BS
solutions grows rapidly.
We follow an alternative strategy based on the spin-flip approach. Spin-flip (SF) methods
21–27
offer a balanced treatment of multi-configurational states of polyradicals within a single-reference
formalism, making the SF-protocol suitable for applications to SMMs.
28–33
The SF approach does
not rely on scrambled spin-states and does not require choosing an active space. When combined
with DFT,
24, 25
the spin-flip approach provides a computationally more efficient yet reliable ap-
proach to tackle extended SMMs.
10
Following previous SF studies of SMMs (see Ref. 28 and references therein), here we use
non-collinear spin-flip time-dependent density functional theory (NC-SF-TDDFT)
25, 34
to calculate
exchange interactions in binuclear and tetranuclear Fe(III) complexes. In these molecules, each
Fe(III) center has a d
5
electronic configuration, which, depending on the strength of the ligand
field, can result in hextet, quartet, or doublet states, as illustrated in Fig. 2.1. In a weak ligand field,
the splitting between thed-orbitals is small, so the electrons first occupy available vacant orbitals,
before pairing up, as prescribed by the Hund rule. This gives rise to high-spin configurations (left
panel of Fig. 2.1). In contrast, in a strong ligand field, the energy separation betweend-orbitals is
larger than the electron pairing energy, so the electrons first occupy the lowest energy orbitals and
pair, before moving to orbitals higher in energy, as prescribed by the Aufbau principle. This gives
rise to low-spin configurations, such as those shown in the middle and right panels of Fig. 2.1.
Figure 2.1: Effect of the ligand-field strength on the local ground-state spin configuration of
Fe(III). The orbital splitting pattern corresponds to octahedral coordination. In the weak-
field limit, the low-lying manifold of electronic states is derived from configurations with
five unpaired electrons, i.e., when eachd-orbital is singly occupied. In the intermediate and
strong field, the configuration of Fe(III) is a low spin in which (some) electrons are paired.
Magnetic properties of SMMs can be modeled using phenomenological spin Hamiltonians of
varying complexity. These spin Hamiltonians contain parameters that can be determined either by
fitting to experimental measurements or by using ab initio calculations.
35
Thus, spin Hamiltonians
establish a connection between theory and experiment. The Heisenberg-Dirac-Van Vleck (HDvV)
Hamiltonian is often employed to describe the inter-site effective exchange interactions. This
model is expected to be suitable for treating exchange interactions in multi-center Fe(III) SMMs
within the weak-field limit, when the non-Hund configurations (such as those in the middle and
right panels of Fig. 2.1) are well separated from the configurations in which the local electronic
configuration has five unpaired electrons.
11
The focus of this work is on magnetic exchange interactions in Fe(III) SMMs. We consider
12 binuclear Fe(III) complexes (referred to as the benchmark set below),
36–38
4 tetranuclear com-
plexes, and 3 mono-nuclear model systems. Our specific goals are:
1. Assess the applicability of the HDvV model for binuclear Fe(III) complexes from the bench-
mark set using density-based analysis and natural orbitals (NOs). We first consider weak-
and strong-field regimes in three mononuclear Fe(III) model systems with octahedral sym-
metry. We characterize resulting high- and low-spin states using NOs and corresponding
electronic occupations (ranging from 2 for doubly occupied to 1 for singly occupied and
0 for unoccupied orbitals) as appropriate descriptors of their electronic structure. We then
employ such quantities to investigate the contribution of non-Hund configurations in the
low-energy states in the SMMs from the benchmark set and thus assess the reliability of the
HDvV model for these systems.
2. Investigate the functional and basis-set dependence of exchange coupling constants for the
benchmark set.
36–38
This study extends previous SF-TDDFT benchmarking work on bin-
uclear Cu(II) systems
28
to more challenging Fe(III) binuclear systems with a much larger
number of unpaired electrons.
3. Extend the analysis of exchange interactions to four tetranuclear Fe(III) SMMs. To treat
these challenging systems, we rely on the best performing DFT functional (as determined by
the benchmark calculations) and parameterize the Heisenberg Hamiltonian by performing
only a single SF calculation from the highest-multiplicity state, following the approach of
Mayhall and Head-Gordon.
39
This SF-based approach combined with the analysis of NOs
and their occupations affords a general treatment of all exchange interactions in multi-center
SMMs, without making simplifying assumptions (as commonly done in experimental stud-
ies). Hence, it allows us to assess the quality of the HDvV model and to validate simplified
forms of HDvV Hamiltonians.
12
The chapter is organized as follows. Section 2.2 presents the spin-flip formalism, the theory of
the HDvV Hamiltonians, the Mahyall’s approach for extracting exchange couplings of multi-center
SMMs from SF calculations, and the density-based analysis. Section 2.3 provides the computa-
tional details. Section 2.4.1 presents the NO analysis of electronic states in mono- and binuclear
complexes, followed by benchmark results for binuclear complexes in Section 2.4.2, and calcula-
tions on tetranuclear SMMs in Section 2.4.3. Our concluding remarks are given in Section 2.5.
2.2 Theoretical Framework
To describe spin states of the Fe(III) complexes, we rely on the SF method. SF methods have shown
robust performance in treating diradicals,
25, 40–43
triradicals,
44, 44, 45, 45–47
conical intersections,
48–52
and metal-containing SMMs.
28–30, 32, 33, 36
In the SF approach,
21, 22, 24, 53
a high-spin state is used
as a reference from which multi-configurational lower-spin states are obtained by spin-flipping
excitations. The high-spin state does not exhibit strong correlation and thus its wave function can
be accurately described by any single-reference method, including standard Kohn-Sham DFT, and
then used as a reference to access the lower-spin manifold as a manifold of “excited” states in
the spin-flipped sector of the Fock space. Within the SF approach, the target low-spin states are
described as

S;S1
Ms=S1
=
^
R
Ms=1

S
Ms=S
; (2.1)
where the spin-flip operator
^
R
Ms=1
generates all possible singly excited determinants in which the
spin of one electron is flipped with respect to the high-spin reference (e.g., Kohn-Sham reference
in the SF-TDDFT formulation
24, 53
). Here, we employ non-collinear formulation of SF-TDDFT
(NC-SF-TDDFT). For large molecules including multiple metal centers (as the tetranuclear Fe(III)
complexes under study), using SF-TDDFT is attractive owing to its low computational cost. The
NC-SF-TDDFT is a general and robust implementation of the SF approach, which extends appli-
cability of the original (collinear) SF-TDDFT, available for hybrid functionals only, to any type
13
of DFT functional. The performance of SF-TDDFT has been tested against experiments for a set
of binuclear Cu(II) SMMs, illustrating ability of the NC kernel to improve the accuracy of the
computed exchange interactions.
28
Here we extend such benchmarking study to binuclear Fe(III)
systems and employ the resulting best-performing protocol to model tetranuclear Fe(III) SMMs.
Phenomenological spin Hamiltonians are commonly used to interpret magnetic measurements,
serving as a bridge between theory and experiment.
35, 54
The low-energy spectrum of a system of
interacting spins can be described by the HDvV Hamiltonian,
55–57
^
H
HDvV
=2
X
A 0) or antiferromagnetic (J
AB
< 0). This model spin Hamiltonian is
valid when the low-energy states are dominated by neutral configurations and the non-Hund and
ionic configurations are much higher in energy. Fig. 2.2 shows neutral (following Hund’s rule),
non-Hund, and ionic configurations for a generic binuclear Fe(III) system. Here neutral config-
urations characterize electronic states in which the 10 unpaired electrons are evenly distributed
between the two Fe(III) centers, whereas non-Hund and ionic configurations involve electron pair-
ing (the ionic configurations result from electron transfer between the centers).
Experimentally, theJ coupling constants are obtained by fitting temperature-dependent mag-
netic susceptibility to the HDvV model.
58
Theoretically, a full-electron quantum-chemistry cal-
culation can, in principle, provide all electronic states of the system. By analyzing the resulting
energies and wave functions, one can then assess the validity of the spin Hamiltonian and determine
the exchange interactions between the spin centers.
35, 54
However, such brute-force calculations of
the entire manifold of low-lying states come with an exponential computational cost. This prob-
lem can be circumvented by an elegant approach developed by Mayhall and Head-Gordon,
39
who
demonstrated that one can construct and parameterize the HDvV Hamiltonian of Eq. (2.2) for
14
an arbitrary number of unpaired electrons and metallic centers from the two highest-multiplicity
states, the high-spinjSi and adjacentjS 1i spin states of theM
s
=S 1 manifold. These states
can be computed by any spin-flip method with a single spin-flip from the highest-multiplicity refer-
ence state. This effective Hamiltonian approach provides a full set of exchange couplings derived
from all-electron ab initio calculations, thus enabling the validation of simplified spin Hamiltoni-
ans. Such an approach can even be applied an infinite number of spins, as was recently illustrated
for spin chains in solid copper oxalate
33
and for extended conjugated carbon-based molecules with
polyradical character.
59
In binuclear systems, the procedure of extractingJ-values is simplified by employing the Land´ e
interval rule:
60
E(S)E(S 1) =2SJ
AB
; (2.3)
which is derived by solving the HDvV Hamiltonian analytically. Hence, a single SF calculation
(1SF) is sufficient to compute theJ-coupling between the two metallic centers. Mayhall and Head-
Gordon validated this approach by comparing the 1SF results against the full SF calculations (i.e.,
with multiple spin-flips) that directly compute all components of the multiplet.
61
In contrast, for multi-nuclear SMMs, energies alone are not sufficient, and the procedure of pa-
rameterizing the HDvV Hamiltonian requires additional steps based on the effective Hamiltonian
theory, grounded in the Bloch formalism and described in detail elsewhere.
31, 35, 39
Within May-
hall’s approach,
39
effective HamiltonianH
eff
for a system withM metal centers is constructed from
theM lowest-energy eigenvalues (E) and the corresponding orthonormalized local spin function
vectors (j
~

ON
i) produced by a 1SF calculation from the highest spin statejS >
^
H
eff
=
X
I
j
~

ON
I
iE
I
h
~

ON
I
j: (2.4)
The full set ofJ parameters is then derived from the off-diagonal matrix elements of this effec-
tive Hamiltonian. This approach assumes the validity of the HDvV Hamiltonian by restricting the
15
model space to the neutral configurations only. Importantly, it provides all possible pairwise cou-
plings between the centers—eachJ
AB
parameter is obtained from a distinct effective Hamiltonian
matrix element. The diagonalization of
^
H
HDvV
constructed from these parameters yields the entire
manifold of the Heisenberg states (S;S 1;S 2;:::).
This approach, in which a spin Hamiltonian of reduced dimensionality is parameterized by
using ab initio calculations, can be described as a coarse-grained treatment of strong correlation.
Because it only requires a single SF calculation, it scales linearly rather than exponentially with the
number of strongly correlated electrons. Mayhall and Head-Gordon
39
validated this approach by
comparing the results against full SF calculations (i.e., flipping all spins) as well as experimental
values for several SMMs. The extraction of the couplings from the SF-DFT calculations is imple-
mented in a post-processing Python script within the ezMagnet module.
32
Further details can be
found in the appendices.
Here, we exploit Mayhall’s approach to investigate exchange interactions in tetranuclear Fe(III)
systems in which each metal center has five unpaired electrons, giving rise to 20 strongly correlated
electrons. A brute-force ab initio calculation of the ground state would be prohibitively expensive,
i.e., it would require 10 spin-flips or a CASSCF calculation with 20 active electrons. However,
following Mayhall’s approach, we obtain the J-couplings from 1SF calculations from the high-
spin (S = 10) reference that computes only the four lowest targetS = 9 states.
Figure 2.2: Three types of configurations in Fe(III) binuclear SMMs: Neutral Hund-type
(with fixed orbital occupations), locally excited (non-Hund-type with paired electrons), and
ionic (in which the number of electrons on each center is different).
In the context of applications, the key question is the domain of applicability of Mayhall’s ap-
proach, as it is formulated assuming that the Heisenberg picture is valid for a system under study.
16
The bilinear exchange interactions of the HDvV model of Eq. (2.2) can be derived from the more-
general Hubbard model using second-order quasidegenerate perturbation theory.
35
The derivation
assumes a large energy gap between the manifold of states with fixed orbital occupations, such
as all possible determinants with five unpaired electrons for Fe(III), and states with doubly oc-
cupied orbitals, such as locally excited non-Hund and ionic configurations in which the number
of magnetic electrons on each center changes. These different types of configurations are shown
in Fig. 2.2 for a generic binuclear Fe(III) system. When present with large weights, these non-
Hund and ionic configurations result in a non-Heisenberg spectral pattern, leading to discrepancies
between theoretically and experimentally obtainedJ-couplings. In such cases, the HDvV model
breaks down and models incorporating non-Hund configurations explicitly should be considered
instead.
35
To analyze the deviations from the Heisenberg behavior, we employ density-based analysis
and use NOs and their occupations to gain insight into the nature of electronic states. Similar
analysis has been carried out for diradicals and triradicals, elucidating bonding patterns between
the formally unpaired electrons.
28, 62
Here, we extend it to much more complex Fe(III) binuclear
and tetranuclear systems containing 10 and 20 unpaired electrons, respectively. The one-electron
spin density matrix of the electronic stateI is defined as

II;
pq
=h
I
ja
y
p
a
q
j
I
i; (2.5)
where stands for either or spin anda
y
p
anda
q
are the creation and annihilation operators as-
sociated with
p
and
q
spin orbitals. NOs, which are eigenfunctions of the one-particle density
matrix,
63–66
offer a compact description of the correlated wave functions. For each NO, the cor-
responding eigenvaluen
i
can be interpreted as orbital occupation and used to quantify the radical
17
character of the respective electronic state. As in our previous work,
28, 62
we use Head-Gordon’s
index
67
to compute the number of effectively unpaired electrons,
n
u;nl
=
X
i
n
2
i
(2 n
i
)
2
; (2.6)
where n
i
is the occupation number of spin-free natural orbital i and the sum runs over all NOs.
The occupation numbers are the eigenvalues of the spin-free one-electron density matrix (the sum
the and blocks of
). For a system withm unpaired electrons, high-spin states shown
u;nl
exactly equal to the number of singly occupied frontier NOs (i.e., n
i
= 1, thus n
u;nl
= m). In
contrast, low-spin states that result from the pairing of a few electrons, giving rise to a certain
number of doubly occupied ( n
i
= 2) and unoccupied ( n
i
= 0) NOs, and a smaller number of
effectively unpaired electrons. We use the number of unpaired electronsn
u;nl
and occupations n
i
of frontiers NOs as descriptors to differentiate between strong and weak ligand-field regimes in
three model mononuclear Fe(III) systems, and to estimate the contributions of non-Hund and ionic
configurations in binuclear and tetranuclear Fe(III) complexes. We note that similar assessments
of bonding patterns can be done using charge cumulants within Greens function theory.
68
For Fe(III) mononuclear complexes, high-spinS =
5
2
states are expected to have five unpaired
electrons (n
u;nl
= 5) and five singly occupied ( n
i
= 1) NOs. In contrast, their low-spin S =
3
2
states would have one doubly occupied ( n
i
= 2) and one unoccupied ( n
i
= 0) NO. If the HDvV
model is valid in binuclear Fe(III) complexes, their low-lying spin states should have n
u;nl
= 10
and the occupations n
i
of 10 frontier NOs should be close to 1. Locally excited non-Hund and
ionic configurations appear as a result of electron pairing, which would manifest itself in n
u;nl
significantly lower than 10 for the binuclear Fe(III) complexes and occupancies of the frontier
NOs much different from 1. In the tetranuclear Fe(III) complexes, the Heisenberg physics would
result inn
u;nl
= 20 and 20 singly occupied NOs.
18
2.3 Computational Details
Figure 2.3: Crystal structures of the binuclear (1-12) and tetranuclear (13-16) complexes with
Fe(III) centers investigated in this study with their Cambridge structural database names.
Color code: Fe — orange, Cl — green, S — yellow, N — blue, C — gray, and O — red.
Hydrogen atoms are not shown.
Fig. 2.3 shows structures of 12 binuclear Fe(III) SMMs (the benchmark set) and 4 tetranuclear
Fe(III) SMMs.
69–84
Below we refer to the individual molecules by the numbers shown in Fig.
2.3. Each Fe(III) metal center has a d
5
configuration and is embedded in a distorted octahedral
coordinational environment, with the exception of complex 5, which has penta-coordinated Fe(III)
centers with a trigonal bipiramidal symmetry. In binuclear complexes 1-12, the Fe atoms are
19
connected by a dibridged oxo unit (complexes 1-4), single oxo unit (complexes 5-7), oxo and
acetate bridges (complexes 8-12). In the tetranuclear complexes 13 and 14, the Fe atoms are
bridged by oxymethyl groups whereas 15 and 16 have both oxo and acetate bridges. We note that
several of these binuclear complexes have been studied using BS-DFT by Joshi et al.
38
In addition to these multi-nuclear Fe(III) compounds, we also examined three model mononu-
clear Fe(III) systems of varying crystal-field strengths: [Fe(Cl)
6
]
3
(weak field), a monomer unit
built from ABIZOA complex, which we call ABI-m, and [Fe(CN)
6
]
3
(strong field). The ABI-m
compound has been cut out from the ABIZOA complex (complex 4 in Fig. 2.3) by saturating
dangling bonds with hydrogen atoms; this model system represents the local coordinational envi-
ronment and ligand-field strength of metal centers in binuclear and tetranuclear complexes. For
[Fe(Cl)
6
]
3
and [Fe(CN)
6
]
3
, we used PCM solvent to stabilize the multiply charged anions. The
corresponding structures are given in Appendix B. We use these model systems to examine elec-
tronic structure patterns that emerge in the weak- and strong-field regimes in terms of NO occu-
pations and the number of effectively unpaired electrons, with a goal to use these descriptors to
assess the performance of the HDvV model in the binuclear and tetranuclear complexes.
We used high-spin reference states of S =
5
2
, 5, and 10 to compute spin-flip states in the
mononuclear, binuclear, and tetranuclear Fe(III) complexes, respectively. For the mononuclear
Fe(III) systems, single spin-flip excitations from the high-spin hextet (S =
5
2
) reference state lead
to target low-spin states, i.e., quartet (S =
3
2
) and hextetd
5
configurations of Fe(III). For binuclear
and tetranuclear Fe(III) SMMs (Fig. 2.3), we extractJ-couplings using the Land´ e interval rule and
Mayhall’s approach,
39
respectively. To apply the Land´ e interval rule of Eq. (2.3), we compute two
adjacent spin states (S = 5 andS = 4). For tetranuclear complexes, the target states are four spin
states ofS = 9, generated by spin-flip excitations from high-spin reference states ofS = 10. These
target states are then used to build the effective spin Hamiltonian of Eq. (2.4), as prescribed by
Mayhall and Head-Gordon.
39
We employ the non-collinear formulation of SF-TDDFT.
25
To investigate the functional depen-
dence, we used the following:
20
• Hybrid functionals: B3LYP,
85
B5050LYP,
24
PBE0,
86
and PBE50.
25
• Minnesota functionals:
87
meta-GGA M06-L, and hybrid meta-GGA M06-2X.
• Range-separated hybrid functionals: CAM-B3LYP,
88
!PBE,
89
!PBEh,
90
!B97X-D,
91
and
!B97M-V.
92
We also evaluated basis-set effects (cc-pVDZ, def2-TZVP, and 6-31G(d,p)) and the performance
of effective core potentials (CRENBL, SBKJC, SRSC, and LANL2DZ).
All calculations were carried out using theQ-Chem electronic structure package.
93, 94
2.4 Results and discussion
2.4.1 Natural orbital analysis of mono- and bi-nuclear complexes
Table 2.1: Energy gaps
a
(E, eV),hS
2
i values
b
, and density-based analysis of spin-flip states
(denoted as “SF
n
”) obtained from the high-spin hextet reference state (denoted as “ref.”) in
model mononuclear Fe(III) complexes
c
.
Complex ref. SF
1
SF
2
SF
3
SF
4
SF
5
SF
6
SF
7
[Fe(Cl)
6
]
3
hS
2
i 8.75 8.76 3.81 3.81 3.81 3.82 3.82 3.82
E 0.00 1.61 1.61 1.61 1.73 1.74 1.74
n
u;nl
5.00 3.16 3.19 3.04 3.03 3.09 3.08
ABI-m
hS
2
i 8.77 8.80 3.85 3.84 3.85 3.83 3.83 3.84
E 0.00 0.35 0.63 0.74 0.99 1.40 1.54
n
u;nl
5.00 3.02 3.03 3.03 3.02 3.04 3.04
[Fe(CN)
6
]
3
hS
2
i 8.76 3.83 3.82 3.82 3.82 3.82 3.82 8.76
E 0.00 0.04 0.04 0.18 0.19 0.19 1.59
n
u;nl
3.01 3.01 3.01 3.01 3.05 3.05 5.00
a
The energy gaps are computed with respect to the lowest spin-flip state (SF
1
).
b
The idealhS
2
i
values of high-spin (S =
5
2
) and low-spin (S =
3
2
) states are 8.75 and 3.75, respectively.
c
!PBEh/6-31G(d,p).
In this section, we use NO-based analysis (NO occupations and the number of effectively
unpaired electrons; see Section 2.2) to estimate the contributions of neutral (Hund), non-Hund,
21
Table 2.2: Occupations of frontier natural orbitals ( n
i
) in the lowest hextet (idealhS
2
i is 8.75)
and quartet (idealhS
2
i is 3.75) state in mononuclear Fe(III) complexes.
Complex hS
2
i NO
1
NO
2
NO
3
NO
4
NO
5
[Fe(Cl)
6
]
3
8.76 1.00 1.00 1.00 1.00 1.00
3.81 0.20 0.86 1.00 1.14 1.82
ABI-m
8.80 1.00 1.00 1.00 1.00 1.00
3.85 0.05 1.00 1.00 1.00 1.96
[Fe(CN)
6
]
3
3.83 0.02 1.00 1.00 1.00 1.97
8.76 1.00 1.00 1.00 1.00 1.00
and ionic configurations in mono- and binuclear Fe(III) systems in order to verify the applicability
of the HDvV model for the SMMs from our benchmark set.
To investigate the electronic structure pattern of the Fe(III) ion in a ligand field of various
strength, we begin by considering three model single-center Fe(III) systems: [Fe(Cl)
6
]
3
, ABI-m,
and [Fe(CN)
6
]
3
. [Fe(Cl)
6
]
3
and [Fe(CN)
6
]
3
represent weak- and strong-field cases, respec-
tively. Starting with a high-spin hextet reference state, we perform SF-TDDFT calculations (using
!PBEh/6-31G(d,p)) and analyze the manifold of the computedS 1 electronic states. The choice
of the level of theory (functional/basis set) is justified by our benchmark calculations described
below (Section 2.4.2). Here, our aim is to verify the applicability of our SF protocol to describe
the electronic structure of the Fe(III) complexes within the strong field regime, where non-Hund
configurations appear at low energy, spoiling the HDvV model, and to determine, on the basis of
the NO analysis, whether the ABIZOA complex (and other complexes shown in Fig 2.3) fall in the
weak field or the strong field category.
Table 2.1 shows energy gaps (E), the number of effectively unpaired electrons (n
u;nl
), and
the expectation value of theS
2
operator (hS
2
i) for the spin-flip states of each model system (energy
gaps are reported with respect to the lowest SF state). The occupations of the five frontier NOs ( n
i
)
for the lowest hextet and quartet states are reported in Table 2.2. The respective NOs are shown in
Figs. 2.11 - 2.13 in the Appendix C (we show spin-orbitals). [Fe(Cl)
6
]
3
and [Fe(CN)
6
]
3
exhibit
22
two threefold degeneracies of the quartet states due to O
h
symmetry. As expected from the crystal-
field theory, the hextet state with five unpaired electrons is the lowest SF state in [Fe(Cl)
6
]
3
. In
contrast, in [Fe(CN)
6
]
3
the quartet state appears 1.6 eV below the hextet state. These patterns
represent weak- and strong-field cases, respectively. The ABI-m model system clearly shows a
weak-field pattern, with the hextet state being the lowest. Due to lower local symmetry, the orbital
and state degeneracies are lifted.
Figure 2.4: Orbital occupations of the lowest state (i.e., SF
1
) in mononuclear Fe(III) com-
plexes.
We use NO occupations ( n
i
) to determine the character of the lowest spin states in these com-
plexes. For non-Hund configurations in mononuclear Fe(III) compounds, we expect (at least) one
frontier orbital to be doubly occupied ( n
i
= 2) and one frontier orbital to be unoccupied ( n
i
= 0)
as a result of electron pairing. This is reflected in a lowering of the number of effectively unpaired
electrons from 5 to 3. We observe that the high-spin hextet states have occupations of all 5 frontier
orbitals close to 1, whereas in low-spin quartet states, the electrons become partially paired forming
non-Hund configurations and the occupations of two NOs differ considerably from 1, one being
almost 0 (unoccupied) and the other one nearly 2 (doubly occupied), see Table 2.2. Whereas the
high-spin hextet state appears as the lowest state in ABI-m and [Fe(Cl)
6
]
3
, the non-Hund quartet
is the lowest state in [Fe(CN)
6
]
3
. Hence, from the density-based analysis of the states and state
ordering in these mononuclear systems, we characterize weak- and strong-field limits and identify
that the Fe(III) center in ABI-m is in a weak-field environment. Thus, Fe(III) centers in the binu-
clear ABIZOA compound (built from two ABI-m units) and in the other multi-nuclear complexes
under study (based on similar nature of ligands surrounding each Fe(III) center) can be described
as having a high-spind
5
configuration.
23
Table 2.3: Frontier natural orbital occupations ( n
i
) andn
u;nl
of the lowestS = 4 state show-
ing an increase of ionic character with the absolute value of J (cm
1
) in binuclear Fe(III)
complexes.
ComplexJ-exp
a
J-theo
b
n
u;nl
NO
1
NO
2
NO
3
NO
4
NO
5
NO
6
NO
7
NO
8
NO
9
NO
10
1 -6.4 -7.1 9.99 0.96 0.98 0.98 0.99 1.00 1.00 1.01 1.01 1.02 1.04
2 -13.6 -16.6 9.98 0.94 0.97 0.98 0.98 0.99 1.01 1.02 1.02 1.03 1.06
3 -13.7 -15.7 9.98 0.94 0.97 0.97 0.98 0.99 1.01 1.02 1.03 1.03 1.06
4 -21.3 -20.4 9.98 0.94 0.96 0.98 0.98 0.99 1.01 1.02 1.02 1.03 1.06
5 -100
c
-106.8 9.91 0.90 0.91 0.93 1.00 1.00 1.00 1.00 1.07 1.09 1.10
6 -87.5 -91.9 9.92 0.89 0.93 0.93 1.00 1.00 1.00 1.00 1.07 1.07 1.10
7 -98 -100.7 9.91 0.89 0.93 0.93 1.00 1.00 1.00 1.00 1.07 1.07 1.11
8 -108 -105.4 9.90 0.89 0.92 0.93 1.00 1.00 1.00 1.00 1.07 1.07 1.11
9 -119 -109.8 9.90 0.89 0.90 0.93 0.99 1.00 1.01 1.01 1.07 1.09 1.11
10 -121 -124.7 9.88 0.88 0.90 0.92 0.99 0.99 1.01 1.01 1.08 1.10 1.12
11 -130 -123.0 9.88 0.88 0.91 0.92 0.99 0.99 1.01 1.01 1.08 1.09 1.12
12 -132 -119.4 9.88 0.88 0.91 0.92 0.99 0.99 1.01 1.01 1.08 1.09 1.12
a
Experimental exchange coupling from Refs.
69–80 b
Computed exchange coupling using
!PBEh/6-31G(d,p) and the Land´ e interval rule.
c
The exchange coupling is taken to be -100 cm
1
due to discrepancies in reported experimental values.
For multi-center Fe(III) compounds, the appearance of non-Hund and ionic configurations re-
sults in non-Heisenberg behavior, which requires mapping to biquadratic Heisenberg and Hubbard
models to capture their magnetic behavior. Below we extend such NO-based analysis to the bin-
uclear complexes to determine the contribution of non-Hund configurations in their low-energy
spectra, thus, assessing whether the fit of magnetic susceptibility measurements to the HDvV
model is appropriate for these systems. For the binuclear Fe(III) SMMs, SF-TDDFT calcula-
tions using high-spin reference ofS = 5 yieldS = 5 andS = 4 states ofM
S
= 4 components from
which the exchange couplings are computed using Eq. (2.3). Table 2.3 shows the NO analysis
for the benchmark set (see Fig. 2.3 for their structures). For complexes 2 (dibridged oxo unit), 7
(single oxo unit), and 12 (oxo and acetate bridges) of Fig. 2.3, the associated 10 frontier NOs of the
lowestS = 4 state are reported in Figs. 2.14 - 2.16 respectively in the Appendix D. Visualization
of the NOs shows that the unpaired electrons are localized on thed-orbitals of iron, with almost
negligible contribution from the ligands. In all complexes the high-spinS = 5 states have orbital
occupations of 1 for all 10 frontier natural orbitals, with expectedn
u;nl
= 10. In contrast, the lower
24
S = 4 states (reported in Table 2.3) show slight deviations of orbital occupations from 1, and this
deviation increases in the higher exchange coupling regime. Whereas some NOs have occupations
that become smaller than 1 with an increase in magnitude of the exchange coupling (e.g., NO
1
low-
ers from 0.96 to 0.88), others have occupations that become greater than 1 (e.g., NO
10
increases
from 1.04 to 1.12). Similarly, the number of unpaired electrons decreases from 9.99 to 9.88 upon
an increase in exchange interactions. This is indicative of an electron pairing and mixing of the
ionic or non-Hund configurations with the neutral configurations. The consequences of a large
exchange coupling can be compared to the effect of strong ligand field, as seen in [Fe(CN)
6
]
3
where a large crystal field splitting stabilizes the non-Hund states in the lowest SF state (Fig. 2.4).
However, deviation ofn
u;nl
from 10 (less than 2%) and of NO occupations from 1 (within 12 %) is
small, meaning that contributions of non-Hund configurations are small, so that the HDvV model
can be considered valid for the binuclear Fe(III) compounds under study. Therefore, experimen-
tally derived exchange couplings (see Refs. 69–80) employing the HDvV model are reliable and
can be directly compared with the theoretically calculated exchange couplings of this work. Over-
all, the discrepancy between experimentally derivedJ-values and the ones extracted from ab initio
calculation using Eq. (2.3) is small with a mean absolute error (MAE) smaller than 5 cm
1
.
2.4.2 Benchmark calculations
Having established a procedure to validate the HDvV model, we now proceed to benchmarking
the functional and basis-set choice against experimentally derived values. We examine the perfor-
mance of 11 DFT functionals, 3 different basis sets, and 4 effective core potentials for describing
magnetic exchange interactions in SMMs from the benchmark set (Fig. 2.3) for which experi-
mentally derived exchange couplings exist. ThehS
2
i-values for the spin-flipped S = 5 and S =
4 states are close to 30 and 20 values respectively, with negligible spin contamination (between
0.01 and 0.06). We compute statistical measures of errors such as mean absolute error (MAE),
mean error (ME), and standard deviation error (SDE) of the computedJ-values with respect to the
experimentally derived ones.
25
Table 2.4: J-couplings (cm
1
) for 12 binuclear Fe(III) SMMs computed using NC-SF-
TDDFT with selected functionals and the 6-31G(d,p) basis set.
Complex Exp.
a
B3LYP B5050 PBE0 PBE50 M06- M06- CAM- !PBE !PBEh !B97X- !B97M-
LYP L 2X B3LYP D V
1 -6.4 -10.9 -3.6 -6.9 -2.7 -11.5 -5.3 -7.7 -8.2 -7.1 -7.3 -7.7
2 -13.6 -24.7 -8.5 -16.9 -7.2 -25.5 -11.9 -19 -22.8 -16.6 -18.1 -18.4
3 -13.7 -23.5 -6.3 -15.2 -5 -28.5 -10.8 -17.6 -23.6 -15.7 -13.4 -17.4
4 -21.3 -27.3 -11.7 -20 -10 -31.9 -6.5 -21.1 -27.8 -20.4 -22.3 -23
5 -100.0 -134.2 -70.2 -101.9 -61.8 -129.9 -163.4 -119.5 -139.5 -106.8 -116.1 -115.5
6 -87.5 -115.7 -59.4 -87.4 -52.7 -125.3 -44.2 -103.2 -122.4 -91.9 -99.9 -101.5
7 -98.0 -124.1 -66.6 -95.6 -58.8 -119.6 -90.8 -113.1 -133.2 -100.7 -109.3 -111
8 -108.4 -140.3 -64.7 -102.4 -57.3 -147.4 -104 -118 -142 -105.4 -114.6 -112.1
9 -119.0 -147.3 -67 -106.9 -59 -155.2 -96.5 -123.5 -148.8 -109.8 -119.3 -118
10 -121.0 -164.7 -74.8 -119.9 -66.1 -172.8 -56.9 -140.4 -171.2 -124.7 -136.5 -132.1
11 -130.5 -166.3 -77.1 -120.1 -66.9 -170.3 -173.2 -139.2 -166.1 -123.1 -134.4 -131.9
12 -132.0 -155.9 -74 -114.5 -65 -163.7 -87.4 -134.7 -162 -119.4 -130.5 -130.6
MAE 23.7 30.6 4.9 36.6 27.5 26.0 8.8 26.4 4.7 6.1 6.0
ME -23.7 30.6 3.6 36.6 -27.5 8.4 -8.8 -26.4 0.8 -5.8 -5.6
SDE 12.8 20.4 6.5 23.7 28.13 35.4 7.0 15.4 6.0 11.48 6.0
a
Experimental exchange coupling from Refs. 69–80.
Figure 2.5: Mean absolute error (MAE) ofJ-couplings calculated using different functionals
relative to experimental values for 12 binuclear Fe(III) systems. The 6-31G(d,p) basis set was
used for all atoms.
Table 2.4 collects the results for selected functionals using the 6-31G(d,p) basis set on all
atoms; the respective MAEs are shown graphically in Fig. 2.5. Range-separated hybrid functionals
with short-range Hartree–Fock exchange such as!PBEh,!B97M-V ,!B97X-D, and CAM-B3LYP
provide a good estimate of exchange interactions (with MAEs around 5 cm
1
). PBE0 also performs
26
Figure 2.6: Theoretical versus experimental J-couplings for functionals with MAE < 10
cm
1
. The black line shows the perfect match.
well. In contrast, M06-L, !PBE, B3LYP, CAM-B3LYP, !B97X-D, and !B97M-V functionals
underestimate the couplings whereas the other functionals overestimate them. Hybrid functionals
PBE50, B5050LYP, B3LYP, and Minnesota-06 functionals have MAEs greater than 20 cm
1
, to
be compared with MAEs below 10 cm
1
for range-separated functionals. Fig. 2.6 provides a more
detailed view of this trend by showing the theoretical and experimental couplings as a scatter plot
for the five best-performing functionals (!PBEh,!B97M-V ,!B97X-D, CAM-B3LYP, and PBE0).
Exchange couplings for complexes 1, 2, 4, 7, 8 and 12 of Fig. 2.3 are also available from
BS-DFT calculations.
38
One of the best performing functional of this BS-DFT work is the hybrid
range-separated HSE functional (also known as HSE06), which gives MAE of 5.4 cm
1
, to be
compared with MAE of 3.8 cm
1
obtained with our NC-SFTDDFT/!PBEh protocol.
All DFT functionals used in this work reproduce the main experimental observation: the con-
nectivity between the iron atoms plays a crucial rule in tuning the exchange interaction. Dibridged
oxo units feature low couplings, whereas a single oxo bridge or oxo bridge paired with acetate
bridges result in higherJ values. We attempted to correlateJ values of the binuclear SMMs with
Fe–Fe distances and Fe-O-Fe bridging angles; the results are given in Section 2.10. For the diiron
compounds including a single oxo bridge (complexes 5-7) and a single oxo bridge combined with
27
acetate bridges (complexes 8-12), we note that an increase of the Fe–Fe distance and of the Fe-O-
Fe bond angle corresponds to a decrease of the antiferromagnetic coupling strength (from -125 to
-92 cm
1
for!PBEh/6-31G(d,p)). In contrast, in complexes 1-4 with dibridged oxo units, the ex-
change couplings are much smaller (between -7 and -20 cm
1
for!PBEh/6-31G(d,p)) and do not
correlate with the Fe–Fe distance, suggesting that the smaller Fe-O-Fe bridging angle, in between
103

and 107

for all these systems, is the key parameter controlling the magnitude of the antifer-
romagnetic coupling. Empirical or semi-empirical models
95–97
for the interpretation of exchange
interactions in multi-nuclear compounds typically correlate theJ’s with the orbital energies and
the overlap between the localized “magnetic orbitals” of the paramagnetic centers. Microscopic
interpretations of such magneto-structural correlation in bi- and multi-nuclear complexes are chal-
lenging and still scarce due to the large number of unpaired electrons and magnetic orbitals, and
thus, multiple factors contributing to the total coupling. Such investigations require further detailed
studies on simplified model systems and would entail systematic scans of the computed magnetic
properties over a large range of bond distances and angles (e.g., see Ref. 98), which is not the
scope of the current work.
Figure 2.7: MAE of the J-couplings computed using different basis sets with the !PBEh
functional for the 12 iron (III) binuclear systems.
28
Table 2.5: Basis-set dependence ofJ-couplings (cm
1
) for the 12 binuclear complexes com-
puted with NC-SF-TDDFT with!PBEh.
Complex Exp.
a
cc-pVDZ def2-TZVP 6-31G(d,p) CRENBL SBKC SRSC LANL2DZ
1 -6.4 -5.9 -5.8 -7.1 -5.3 -5.5 -6.0 -7.2
2 -13.6 -16.1 -15.9 -16.6 -14.5 -15.0 -15.2 -19.0
3 -13.7 -15.1 -14.5 -15.7 -13.9 -14.2 -14.4 -17.5
4 -21.3 -20.2 -19.4 -20.4 -18.9 -18.8 -18.9 -21.9
5 -100.0 -108.8 -108.2 -106.8 -101.0 -99.4 -101.8 -115.2
6 -87.5 -91.4 -92.3 -91.9 -83.6 -84.2 -84.1 -96.5
7 -98.0 -103.3 -104.6 -100.7 -94.8 -93.6 -96.8 -109.8
8 -108.4 -104.7 -105.4 -105.4 -93.9 -92.2 -126.3 -146.6
9 -119.0 -109.4 -109.9 -109.8 -96.3 -95.5 -97.7 -111.7
10 -121.0 -122.0 -123.2 -124.7 -109.1 -107.2 -109.9 -127.2
11 -130.5 -122.3 -123.2 -123.1 -109.2 -108.8 -109.8 -127.1
12 -132.0 -118.3 -119.4 -119.4 -106.4 -104.8 -140.3 -160.9
MAE 5.0 4.9 4.7 9.1 9.7 7.6 10.9
ME 1.2 0.8 0.8 8.7 9.4 2.5 -9.1
SDE 6.6 6.4 6.0 10.0 10.5 11.1 13.1
a
Experimental exchange coupling from Refs. 69–80.
To determine optimal computational settings to be used for binuclear and then tetranuclear
Fe(III) systems, we investigated the effects of the basis set and ECPs when combined with one
of the best performing functionals of this study, i.e., the !PBEh functional. Fig. 2.7 shows the
results obtained using the 6-31G(d,p), cc-pVDZ, and def2-TZVP basis sets (see also Table 2.5).
The calculatedJ values are not sensitive to the basis set, and agree well with experimental values
with MAE of less than 5 cm
1
. In contrast, ECPs perform poorly with MAE of around 10 cm
1
and high standard deviations (see Fig. 2.7).
We conclude that the !PBEh/6-31G(d,p) computational setting affords the best performance
for computing exchange interactions in the present benchmark set. This result is consistent
with findings by Orms and Krylov for binuclear copper SMMs
28
that showed that hybrid and
range-separated hybrid functionals, such as PBE0 and!PBEh, outperform other GGA-type func-
tionals. The present benchmarking study confirms the robust performance of NC-SF-TDDFT
with!PBEh/6-31G(d,p) for treating magnetic exchange interactions in SMMs featuring different
29
transition-metal centers. In the next section, we employ this protocol to describe more complex
SMMs with multiple Fe(III) centers and larger numbers of unpaired electrons.
2.4.3 Tetranuclear Fe(III) complexes
Figure 2.8: Iron core in star-type complexes 13, 14 (left) and butterfly-type complexes 15, 16
(right) showing different exchange interactions. The superscriptsw andb denote wing and
body iron atoms, respectively.
Having validated our SF-based computational protocol, we proceed to investigate tetranuclear
Fe(III) SMMs (Fig. 2.3). Computational treatment of these multi-center SMMs is challenging for
any electronic structure method, owing to their large system size, a large number of unpaired elec-
trons in nearly degenerate orbitals, and multi-configurational character of the resulting spin states.
For these reasons, accurate determination of exchange constants is important, because small differ-
ences can change the ground state spin of the molecule.
99
The combination of a reliable SF-DFT
method with a low-cost post-processing tool for extracting all J-values allows us to investigate
the multi-configurational character of the spin states and to capture the nature of the magnetic
exchange interactions between the metal centers.
As described in Section 2.3, we follow the framework developed by Mayhall and Head-
Gordon.
39
First, we perform a single SF calculation (with!PBEh/6-31G(d,p)) using a high-spin
reference (S = 10). The calculation yieldsS = 9 states whosehS
2
i values are close to 90 showing
small spin contamination (between 0.02 and 0.38). Second, we project the eigenstates onto the
neutral determinant basis and construct an effective Hamiltonian from the 4 lowest single spin-
flipped states and their energies. Third, we extract exchange constants by mapping the effective
30
Hamiltonian to the HDvV model. Additional details on Mayhall’s approach (translated into a post-
processing Python script) and a sample input for running Q-Chem calculations when combined
with the parameterization of the HDvV Hamiltonian are given in the Section 2.6.
Complexes 13 and 14 have star-like structures (left panel of Fig. 2.8) and the experimental
fits
81, 82
show a predominant exchange interaction between the central and three peripheral irons.
Complexes 15 and 16 feature a butterfly-like core (right panel of Fig. 2.8) with four dominant
wing-body interactions (J
wb
) and two much smaller interactions—one body-body (J
bb
) and one
wing-wing (J
ww
).
83, 84
In the fitting of the experimental susceptibility data to the HDvV model,
simplified spin Hamiltonians were adopted in which all dominant exchange interactions were con-
strained to the same value. For example, for butterfly-like SMMs, J
ww
-couplings were assumed
to be zero and allJ
wb
were constrained to have the same value.
83, 84
In contrast, our procedure en-
ables the determination of all individual exchange interactions without assuming simplified HDvV
models.
For the lowest target S = 9 states, the occupations of the frontier NOs is close to 1 and the
deviation of the number of unpaired electrons (n
u;nl
) from 20 is small, i.e., less than 0.04, consistent
with the low exchange interaction regime for which the HDvV model is a good approximation.
These examples illustrate the utility of such calculations and the NOs analysis in validating the
simplified HDvV models used for fitting macroscopic properties of multi-nuclear transition-metal
complexes.
It is cumbersome to visualize and analyze the large number of frontier natural orbitals (20
and 20 NOs) involved in these systems. Thus, we use NO occupancies and number of unpaired
electrons as compact descriptors of the character of states. Table 2.6 shows the n
i
and n
u;nl
for
the lowest S = 9 state. Similar to the case of binuclear iron complexes, n
u;nl
is smaller than
the expected value of 20. The deviation is larger for complexes 15 and 16 with higher magnetic
exchange interaction (i.e.,n
u;nl
= 19.6) than the one for complexes 13 and 14 (i.e.,n
u;nl
= 19.8)
exhibiting lower exchange couplings. A similar trend is observed for the NO occupations, whose
deviation from 1 increases in the higher exchange coupling regime. While the occupation of the
31
frontier natural orbital NO
20
increases from 1.05 to 1.07, the NO
1
occupation decreases from
0.95 to 0.93 as the magnitude of exchange interaction increases, indicating pairing of the active
electrons. However, the deviation ofn
u;nl
from 20 (less than 2%) and of NO occupations from 1
(within 7 %) is small. In addition, the remaining targetS = 9 states (states 2, 3, and 4 higher in
energy) have occupations n
i
of the frontier NOs close to 1 and the number of effectively unpaired
electrons as given byn
u;nl
is exactly 20. On the basis of the NO-based analysis, we conclude that
each Fe(III) is locally a high-spin hextet and the contribution of non-Hund or ionic configurations
to theS = 9 spin states is negligible. These results support the validity of the HDvV model and
justify direct comparison between theory and experiment.
Table 2.6: Frontier natural orbital occupations ( n
i
) and n
u;nl
of the lowest S = 9 state in
tetranuclear Fe(III) complexes. The first five (from NO
1
to NO
5
) and the last five (from NO
16
to NO
20
) natural orbitals with lowest and biggest occupations, respectively, are reported.
Intermediate orbitals have occupations n
i
of one.
Complexn
u;nl
NO
1
NO
2
NO
3
NO
4
NO
5
NO
16
NO
17
NO
18
NO
19
NO
20
13 19.98 0.95 0.96 0.97 0.97 0.98 1.02 1.03 1.03 1.04 1.05
14 19.98 0.95 0.96 0.97 0.97 0.98 1.02 1.03 1.03 1.04 1.05
15 19.96 0.94 0.95 0.95 0.97 0.97 1.02 1.03 1.03 1.05 1.06
16 19.96 0.93 0.96 0.96 0.97 0.97 1.02 1.03 1.04 1.04 1.07
Table 2.7: J-couplings (cm
1
) for tetranuclear iron (III) complexes computed with NC-SF-
TDDFT/!PBEh/6-31G(d,p).
Complex Exp.
a
!PBEh/6-31G(d,p) aveJ
b
13 -8.3 -10.1; -11.4; -11.2 -10.9
14 -8.5 -10.6; -11.7; -11.7 -11.3
15 -45.5; -8.9 -43.7; -42.3; -43.3; -44.5; -3.5; 5.1
c
-43.5; 0.8
16 -46; 0.0 -27.8; -30.1; -45.7; -58.6; -1.0; -1.9
c
-40.6; -1.5
a
Experimental exchange coupling from Refs.
81–84 b
“Ave” stands for the average ofJ values
extracted using!PBEh/6-31G(d,p).
c
The biggerJ-values refer to the four wing-body couplings,
while the smallerJ-values are the body-body and wing-wing ones.
Table 2.7 shows the calculated and experimentally-derivedJ couplings. For complexes 13 and
14, experimental fitting procedures
81, 82
employ a simplified HDvV model providing one coupling
constant only—J of -8.3 and -8.5 cm
1
, respectively. For these two systems, our calculations yield
32
three differentJ-couplings, all within a narrow range (less than 2 cm
1
). For complex 13,J values
range from -10.1 to -11.4 cm
1
, whereas for complex 14J couplings are in between -10.6 and -
11.7 cm
1
. Therefore, from the comparison between experimentally and computationally derived
couplings, we conclude that the simplified HDvV models adopted in the experimental study are
justified for complexes 13 and 14. This is also the case for complex 15. In contrast, in complex 16,
a simple two-J model (which was assumed in the experimental study
84
) appears to be insufficient:
the experimental fitting considered all wing-body interactions to be the same, giving rise to only
oneJ
wb
of -46.0 cm
1
,
84
whereas our calculations show that the 4 wing-body interactions can differ
significantly, by up to 30 cm
1
. However, the calculated average J’s matched the experimental
coupling.
Overall, the SF protocol for computing J-couplings combined with the NO-based analysis
enables validation of the HDvV model used for fitting macroscopic properties of multi-nuclear
transition-metal complexes and reproduces the dominant interactions between the iron centers with
an error of less than 6 cm
1
.
Finally, we remark on the magneto-structural correlation for these SMMs. As noted in Section
2.4.2, for the binuclear complexes the presence of oxo and acetate bridging units correlates with a
larger exchange coupling constant. Compounds 15 and 16 in which the Fe(III) ions are linked via
oxo and acetate bridges exhibit stronger antiferromagnetic couplings than the ones in complexes
13 and 14, which have oxymethyl groups as bridging units. We also consider correlation of the
magnetic data in Table 2.7 with Fe–Fe distances and Fe-O-Fe bond angles (see Section. 2.10) and
observe an increase ofJ coupling with the Fe–Fe distances and Fe-O-Fe bond angles, opposite to
the trend observed for the binuclear complexes (Section 2.4.2). However, considering the small
number of the tetranuclear SMMs under study and that two complexes (13 and 14) out of four
have Fe-O-Fe bridging angles (103.8

and 102.0

, respectively) as small as the ones for complexes
1-4 (outlier cases in the trend of binuclear compounds), we cannot establish any relation between
structural parameters and the value of the coupling constant for the tetranuclear SMMs.
33
2.5 Conclusions
We investigated magnetic properties of 12 binuclear and 4 tetranuclear Fe(III) SMMs. We em-
ployed the NC-SF-TDDFT method to parameterize HDvV Hamiltonians. The validity of this ap-
proach was assessed using NO analysis, which shows that the target SF states in all complexes are
within the HDvV domain, corresponding to the weak-field regime. On the basis of this analysis,
we suggest usingn
u;nl
(the number of effectively unpaired electrons) and n
i
(NO occupations) as
descriptors of the radical character of relevant spin states in multi-nuclear complexes; whenn
u;nl
deviates significantly from the expected number of unpaired electrons and a certain number of
NOs shows occupation n
i
either much larger than 1 or much smaller than 1, electrons start to pair
up and non-Hund or ionic configurations cannot be neglected. Spin-contamination is negligible in
all these complexes. We employed several representative functionals (hybrid, Minnesota, range-
separated hybrids), basis sets, and ECPs. In agreement with earlier studies of Cu(II) SMMs,
28
range-separated hybrid functionals with short-range Hartree–Fock exchange agree well with exper-
iments (MAE< 10 cm
1
). We recommend!PBEh/6-31G(d,p) for computingJ-couplings using
NC-SF-TDDFT. In multi-nuclear SMMs, the couplings extracted from the ab initio-parameterized
effective Hamiltonians reliably describe interactions between all metal centers, without introduc-
ing any simplifying assumptions. Thus, this type of calculations can be used for the a priori
determination of the validity of simplified HDvV models, which are often adopted in experimental
studies. The effective Hamiltonian approach combined with SF-TDDFT can be extended to study
even larger SMMs.
100, 101
Our results underscore the role of the connectivity between the metal
centers in tuning the exchange interactions in SMMs. Among the 16 Fe(III) complexes, the largest
exchange couplings are the ones between Fe (III) centers linked by oxo units and acetate bridges.
For the binuclear compounds including oxo units and acetate bridges, by correlating magnetic data
to selected bond distances and angles, we identify stronger antiferromagnetic couplings among the
ones with shorter Fe–Fe distances and the Fe-O-Fe bond angles that deviate from 180

. These
34
results highlight the need for further theoretical investigations of magneto-structural correlations
in multi-nuclear complexes using the techniques described in this study.
35
2.6 Appendix A: Multiple-center molecular magnets: May-
hall’s approach
For strongly correlated systems that are well described by the Heisenberg model, Mayhall and
Head-Gordon proposed to parameterize the Heisenberg-Dirac-van Vleck (HDvV) Hamiltonian us-
ing single spin-flip (1SF) methods in which a single spin-flipping excitation operator from the
highest spin state generates a manifold of adjacent spin states.
39, 61
For systems with two radical
centers, the Land´ e interval rule can be used to obtain theJ parameter from a 1SF calculation us-
ing energies alone. For systems with multiple radical centers, J parameters can be obtained by
fitting of energy levels. However, without considering the wave functions it is difficult to judge
whether the fit correctly represents the underlying interactions. Mayhall and Head-Gordon intro-
duced an effective Hamiltonian-based approach to parameterize the HDvV spin Hamiltonian for
systems with an arbitrary number of radical centers and unpaired electrons using energies and wave
functions from 1SF calculations.
Effective Hamiltonian theory provides an effective tool to parameterize phenomenological
models like the HDvV Hamiltonian by mapping states of the ab initio electronic Hamiltonian
onto the states of the model Hamiltonian. To do so, one computes a set of energies and the corre-
sponding eigenfunctions of the many-body electronic Hamiltonian, which correspond to electronic
states whose properties the model is to capture. The effective Hamiltonian is then defined such that
the computed electronic states are mapped onto the states of a model Hamiltonian that have the
same energies as the full Hamiltonian. Via this connection between the model states and electronic
wave functions, matrix elements of the effective Hamiltonian can be interpreted as the interactions
between effective local spins.
The HDvV Hamiltonian describes the low-energy part of the spectrum under the assumption
that the respective states are dominated by the neutral configurations. For a system withM radical
centers, one computes theM lowest energy 1SF excitations starting from the reference state with
maximum spin projectionS at each site thus an overall spin projection ofSM. By localizing
36
all partially occupied molecular orbitals and transforming the excited-state solution vectors to this
local orbital basis, one can verify that the obtained solutions are predominantly neutral and assign
each partially occupied orbital to a specific radical site.
If the 1SF states are indeed not ionic, the flipped spin of that state is located at a specific
radical site, which can be interpreted as a localS 1 spin. Determinants of the partially occupied
orbitals at that site I are used to define a local S 1 spin functionj
S1
I
i. If the M lowest
energy SF excitations each result in a local SF at a different radical site, one obtains a localS 1
spin functions at each site. Thej
S1
i functions are then used as the basis for representing the
effective Hamiltonian. Following the approach of des Cloizeaux,
102
the local spin functions are
orthonormalized to form the orthonormalized spin functionsj
~

ON
I
i. One then defines the effective
Hamiltonian via the spectral representation:
^
H
eff
=
X
I
j
~

ON
I
iE
I
h
~

ON
I
j; (2.7)
where E
I
refers to the energy of the single SF excitation that resulted in (S 1) spin at site I.
Each of these states thus maps onto one specific vector of model spin states in which all but one
site have maximum spin projectionS and one site has the projection (S 1). By this construction,
the energy differences between two 1SF excited states result from the coupling between their local
spin functions and the off-diagonal matrix elements of the effective Hamiltonian represented in
the basis of local spin functions provide the exchange coupling parameters J
IJ
for the HDvV
Hamiltonian.
Figure 2.9: Flowchart of the protocol. H
HDvV
is the Heisenberg-Dirac-van Vleck Hamilto-
nian and
~

ON
I
are the local spin functions.
37
Fig. 2.9 shows a flowchart of the computational protocol we apply to compute J couplings
of tetranuclear iron (III) compounds. The first step requires a simple 1SF calculation from
the high-spin reference state. The next step is parameterization of the HDvV Hamiltonian, as
prescribed by Mayhall and Head-Gordon;
39
it requires eigenstates and energies, and provides
all J-values. Mayhall’s approach is implemented as a post-processing Python script (available
within the ezMagnet module
32
). The script reads the raw data from the Q-Chem output (en-
ergies and projected eigenvectors) and provides the J-values. Therefore, two separate calcu-
lations need to be performed: 1) 1SF calculation using Q-Chem and 2) construct the effective
Hamiltonian by executing the Python script in which Mayhall’s approach of Ref. 39 is imple-
mented. For additional details about Mayhall’s approach, see Refs. 39, 61 and ezMagnet manual
athttp://iopenshell.usc.edu/downloads/ezmagnet/.
The following example illustrates the assignment of exchange parametersJ
AB
from the effec-
tive Hamiltonian for the (Fe)
4
YAYPOD complex. The section of theQ-Chem input below contains
the keywords used to compute the four lowest energy SF-TDDFT states and request the partially
occupied orbitals of the respective eigenvectors to be localized; the molecular geometry is given in
the supplementary information of the manuscipt.
103
Any ionic determinants, i.e. those that move
charges between radical centers, are projected out and only the neutral determinant coefficients are
kept.
$comment
Spin-flip calculation for building effective spin models
$end
$rem
BASIS = 6-31G(d,p)
CIS_N_ROOTS = 4
EXCHANGE = LRC-wPBEh
WANG_ZIEGLER_KERNEL = TRUE
MEM_TOTAL 230000
SCF_MAX_CYCLES = 500
MAX_CIS_CYCLES = 150
SPIN_FLIP = 1
SCF_CONVERGENCE = 7
SCF_ALGORITHM = diis_gdm
$end
38
$development
1sf_heis_projection 1
$end
The combined weight of the neutral coefficients for each eigenvector is listed, which can be
used as an additional check for the assumptions that ionic configurations can be neglected. The
weights for our example YAYPOD complex is printed below
[ 9.731619E-01 9.803481E-01 9.845902E-01 9.986758E-01]
as well as the energies in Hartrees
[-4.902195E-03 -3.549901E-03 -2.509316E-03 -1.136423E-03]
and the projected eigenvector coefficients
[[-2.357448E-01 -1.451038E-01 2.445182E-01 2.432505E-01]
[-2.560876E-01 -1.563791E-01 2.499574E-01 2.444646E-01]
[-1.726807E-01 2.247644E-01 -2.558274E-01 2.249857E-01]
[-2.440463E-01 -1.407701E-01 2.491572E-01 2.370230E-01]
[-2.360271E-01 -1.446474E-01 2.399208E-01 2.374273E-01]
[ 1.745107E-01 -2.828061E-01 -2.063682E-01 2.085861E-01]
[-2.318909E-01 -1.395355E-01 2.423395E-01 2.378598E-01]
[ 1.675363E-01 -2.831508E-01 -1.967223E-01 2.074008E-01]
[-1.762714E-01 2.407006E-01 -2.558516E-01 2.299831E-01]
[ 2.718992E-01 1.930640E-01 1.773049E-01 2.169889E-01]
[ 2.849877E-01 2.030972E-01 1.832275E-01 2.211781E-01]
[ 1.677904E-01 -2.830680E-01 -1.962313E-01 2.069361E-01]
[ 2.693438E-01 1.902356E-01 1.799501E-01 2.171512E-01]
[ 1.756897E-01 -2.925299E-01 -2.025575E-01 2.074907E-01]
[-1.692216E-01 2.316783E-01 -2.494809E-01 2.261656E-01]
[-1.671659E-01 2.278674E-01 -2.468704E-01 2.235684E-01]
[-1.707318E-01 2.301026E-01 -2.507412E-01 2.253798E-01]
[ 2.750819E-01 1.895552E-01 1.866389E-01 2.193897E-01]
[ 1.786666E-01 -3.009937E-01 -2.013876E-01 2.103869E-01]
[ 2.889924E-01 2.013276E-01 1.827070E-01 2.173048E-01]]
In addition to the projected vectors, the calculation lists for each of the neutral determinants which
radical center the localization procedure has assigned it to. For our example this list reads
2
2
39
3
2
2
0
2
0
1
3
1
3
1
0
0
3
3
1
0
1
with labels 0-3 indicating one of the four radical centers. The projected eigenvectors are then
sorted according to the radical center and the vectors are orthonormalized to yield the orthonor-
malized projected eigenvectors
~
b. A determinant-wise effective Hamiltonian
^
H
eff
det
is constructed as
an intermediate for defining local spin functions. The determinant-wise effective Hamiltonian is
defined via the spectral representation
H
eff
det
=
M
X
i
~
b
i
E
i
~
b
T
i
; (2.8)
where
~
b
i
are the orthonormalized projected eigenvector and theE
i
are the energies of the excitation
localized at radical centeri. The resulting effective Hamiltonian matrix then reads
[[-2.001038E-01 -1.966752E-01 -1.964967E-01 -2.021006E-01 -2.054433E-01
-9.373975E-04 -1.304710E-05 -7.746025E-04 -1.062858E-03 -5.464950E-04
1.040648E-03 2.322353E-03 6.060696E-03 1.471196E-03 2.626110E-03
-4.626224E-03 -4.155277E-04 -2.158070E-03 -2.539972E-03 -2.443947E-03]
[-1.966752E-01 -1.933928E-01 -1.932178E-01 -1.987054E-01 -2.020376E-01
-4.010136E-04 5.676767E-04 -2.833044E-04 -5.935006E-04 6.298311E-05
-2.816039E-03 -1.736351E-03 2.092956E-03 -2.349120E-03 -1.185951E-03
-3.018491E-03 1.177874E-03 -5.839896E-04 -9.810514E-04 -8.698211E-04]
[-1.964967E-01 -1.932178E-01 -1.930434E-01 -1.985288E-01 -2.018603E-01
-4.765316E-04 4.868534E-04 -3.591546E-04 -6.730988E-04 -2.028387E-05
-2.750589E-03 -1.666742E-03 2.155002E-03 -2.284049E-03 -1.123410E-03
-2.759880E-03 1.438404E-03 -3.294987E-04 -7.292871E-04 -6.145350E-04]
[-2.021006E-01 -1.987054E-01 -1.985288E-01 -2.042113E-01 -2.076430E-01
2.045231E-04 1.194402E-03 3.322930E-04 6.953314E-06 6.362585E-04
40
-7.909018E-04 3.838455E-04 4.257559E-03 -3.441321E-04 8.476892E-04
-9.489562E-04 3.430026E-03 1.560497E-03 1.125308E-03 1.264799E-03]
[-2.054433E-01 -2.020376E-01 -2.018603E-01 -2.076430E-01 -2.111669E-01
2.042289E-04 1.221684E-03 3.117807E-04 -5.138784E-05 6.477844E-04
-2.315739E-03 -1.186094E-03 2.808236E-03 -1.841882E-03 -6.226146E-04
1.174347E-03 5.687757E-03 3.722124E-03 3.252114E-03 3.421183E-03]
[-9.373975E-04 -4.010136E-04 -4.765316E-04 2.045231E-04 2.042289E-04
-1.935601E-01 -2.011935E-01 -1.927978E-01 -1.959667E-01 -2.010310E-01
-2.231787E-03 4.453537E-03 -2.491262E-04 -1.476652E-04 -2.849404E-03
1.349143E-03 -1.877397E-03 -2.391640E-03 -2.122112E-03 -1.258197E-03]
[-1.304710E-05 5.676767E-04 4.868534E-04 1.194402E-03 1.221684E-03
-2.011935E-01 -2.091600E-01 -2.003843E-01 -2.036735E-01 -2.090140E-01
-5.467537E-05 6.983128E-03 1.993541E-03 2.080725E-03 -7.457944E-04
1.976064E-03 -1.386488E-03 -1.925742E-03 -1.649725E-03 -7.448187E-04]
[-7.746025E-04 -2.833044E-04 -3.591546E-04 3.322930E-04 3.117807E-04
-1.927978E-01 -2.003843E-01 -1.920576E-01 -1.952316E-01 -2.002143E-01
-3.969282E-03 2.615042E-03 -2.007412E-03 -1.871275E-03 -4.553912E-03
2.192632E-03 -9.981130E-04 -1.536300E-03 -1.278937E-03 -4.078753E-04]
[-1.062858E-03 -5.935006E-04 -6.730988E-04 6.953314E-06 -5.138784E-05
-1.959667E-01 -2.036735E-01 -1.952316E-01 -1.984907E-01 -2.035162E-01
-4.881284E-03 1.775992E-03 -2.884964E-03 -2.736987E-03 -5.458033E-03
4.604758E-03 1.435112E-03 8.142564E-04 1.044908E-03 1.960875E-03]
[-5.464950E-04 6.298311E-05 -2.028387E-05 6.362585E-04 6.477844E-04
-2.010310E-01 -2.090140E-01 -2.002143E-01 -2.035162E-01 -2.089031E-01
1.732442E-03 8.841561E-03 3.801115E-03 3.844770E-03 1.013372E-03
3.834072E-03 5.346037E-04 -6.155376E-05 1.903724E-04 1.119037E-03]
[ 1.040648E-03 -2.816039E-03 -2.750589E-03 -7.909018E-04 -2.315739E-03
-2.231787E-03 -5.467537E-05 -3.969282E-03 -4.881284E-03 1.732442E-03
-1.989524E-01 -2.072206E-01 -1.999658E-01 -1.963908E-01 -1.953341E-01
-8.182138E-05 1.892052E-04 -8.165408E-05 -1.603088E-04 -1.775712E-04]
[ 2.322353E-03 -1.736351E-03 -1.666742E-03 3.838455E-04 -1.186094E-03
4.453537E-03 6.983128E-03 2.615042E-03 1.775992E-03 8.841561E-03
-2.072206E-01 -2.160771E-01 -2.083761E-01 -2.046271E-01 -2.034385E-01
1.548046E-05 3.887539E-04 1.314907E-04 4.089293E-05 -6.361604E-06]
[ 6.060696E-03 2.092956E-03 2.155002E-03 4.257559E-03 2.808236E-03
-2.491262E-04 1.993541E-03 -2.007412E-03 -2.884964E-03 3.801115E-03
-1.999658E-01 -2.083761E-01 -2.011301E-01 -1.974232E-01 -1.963625E-01
5.222942E-05 2.530613E-04 2.869076E-05 -4.396447E-05 -7.209462E-05]
[ 1.471196E-03 -2.349120E-03 -2.284049E-03 -3.441321E-04 -1.841882E-03
-1.476652E-04 2.080725E-03 -1.871275E-03 -2.736987E-03 3.844770E-03
-1.963908E-01 -2.046271E-01 -1.974232E-01 -1.938850E-01 -1.928156E-01
-1.183409E-04 1.734013E-04 -8.378418E-05 -1.630464E-04 -1.898894E-04]
[ 2.626110E-03 -1.185951E-03 -1.123410E-03 8.476892E-04 -6.226146E-04
-2.849404E-03 -7.457944E-04 -4.553912E-03 -5.458033E-03 1.013372E-03
-1.953341E-01 -2.034385E-01 -1.963625E-01 -1.928156E-01 -1.917967E-01
-2.251579E-05 1.990746E-04 -5.507758E-05 -1.285017E-04 -1.430428E-04]
[-4.626224E-03 -3.018491E-03 -2.759880E-03 -9.489562E-04 1.174347E-03
1.349143E-03 1.976064E-03 2.192632E-03 4.604758E-03 3.834072E-03
-8.182138E-05 1.548046E-05 5.222942E-05 -1.183409E-04 -2.251579E-05
-1.995102E-01 -2.049386E-01 -1.990967E-01 -1.965925E-01 -1.991549E-01]
[-4.155277E-04 1.177874E-03 1.438404E-03 3.430026E-03 5.687757E-03
-1.877397E-03 -1.386488E-03 -9.981130E-04 1.435112E-03 5.346037E-04
1.892052E-04 3.887539E-04 2.530613E-04 1.734013E-04 1.990746E-04
-2.049386E-01 -2.106650E-01 -2.046309E-01 -2.020440E-01 -2.046650E-01]
[-2.158070E-03 -5.839896E-04 -3.294987E-04 1.560497E-03 3.722124E-03
-2.391640E-03 -1.925742E-03 -1.536300E-03 8.142564E-04 -6.155376E-05
-8.165408E-05 1.314907E-04 2.869076E-05 -8.378418E-05 -5.507758E-05
-1.990967E-01 -2.046309E-01 -1.987870E-01 -1.962770E-01 -1.988195E-01]
[-2.539972E-03 -9.810514E-04 -7.292871E-04 1.125308E-03 3.252114E-03
-2.122112E-03 -1.649725E-03 -1.278937E-03 1.044908E-03 1.903724E-04
-1.603088E-04 4.089293E-05 -4.396447E-05 -1.630464E-04 -1.285017E-04
-1.965925E-01 -2.020440E-01 -1.962770E-01 -1.937998E-01 -1.963111E-01]
[-2.443947E-03 -8.698211E-04 -6.145350E-04 1.264799E-03 3.421183E-03
-1.258197E-03 -7.448187E-04 -4.078753E-04 1.960875E-03 1.119037E-03
-1.775712E-04 -6.361604E-06 -7.209462E-05 -1.898894E-04 -1.430428E-04
-1.991549E-01 -2.046650E-01 -1.988195E-01 -1.963111E-01 -1.988592E-01]]
41
Each of the 55 sub-block matrices that describe effective interactions within a specific radical
site is then diagonalized. Since only one SF excited state is located on each site and the matrix is
thus rank deficient each diagonal block only yields one non-zero eigenvalue. The resulting eigen-
vectors of this block-diagonalization are a linear combination of local SF excited determinants that
can be interpreted a local spin function. Only the local spin functions that correspond to non-zero
eigenvalues are kept and all others projected out. The remaining eigenvectors are orthonormalized
yet again to restore orthonormality after the projection. The final effective Hamiltonian is then con-
structed from the spectral representation in the basis of the orthonormalized local spin functions
j
~

ON
I
i as defined in equation 2.7. The resulting effective Hamiltonian matrix is printed below:
[[-1.001885E+00 2.152256E-05 6.327713E-04 1.040924E-03]
[ 2.152256E-05 -1.002200E+00 1.335355E-03 6.852854E-04]
[ 6.327713E-04 1.335355E-03 -1.001802E+00 4.339833E-05]
[ 1.040924E-03 6.852854E-04 4.339833E-05 -1.001665E+00]]
The exchange coupling parameters are then obtained from the off-diagonal matrix element that
corresponds to the pair of radical sites via
J
AB
=
H
eff
AB
2
p
S
A
S
B
: (2.9)
2.7 Appendix B: Structures of mononuclear Fe(III) systems
Figure 2.10: Model mononuclear Fe (III) systems: [Fe(Cl)
6
]
3
(left), a monomer unit built
from ABIZOA complex (center), and [Fe(CN)
6
]
3
(right).
42
2.8 Appendix C: Natural orbitals in mononuclear Fe(III) sys-
tems
43
Figure 2.11: Frontier and natural orbitals of the lowest hextet (left) and quartet state
(right) in [Fe(Cl)
6
]
3
with their occupations.
44
Figure 2.12: Frontier and natural orbitals of the lowest hextet (left) and quartet state
(right) in ABI-m with their occupations.
45
Figure 2.13: Frontier and natural orbitals of the lowest hextet (left) and quartet state
(right) in [Fe(CN)
6
]
3
with their occupations.
46
2.9 Appendix D: Natural orbitals in binuclear Fe(III) systems
Figure 2.14: Frontier and natural orbitals of the lowest S = 4 state in complex 2 with
their occupations.
47
Figure 2.15: Frontier and natural orbitals of the lowest S = 4 state in complex 7 with
their occupations.
48
Figure 2.16: Frontier and natural orbitals of the lowestS = 4 state in complex 12 with
their occupations.
49
2.10 Appendix E: Magneto-structural correlations
Table 2.8: Structural (from experiments of Refs. 69–84) and magnetic data of this work
(LRC-!PBEh/6-31G(d,p)) for the 16 complexes under study.
Complex J (cm
1
) Fe–Fe (
˚
A) Fe-O-Fe (

)
1 -7.1 3.34 103.8
2 -16.6 3.17 104.4
3 -15.7 3.09 102.7
4 -20.4 3.20 107.3
5 -106.8 3.45 151.6
6 -91.9 3.45 143.7
7 -100.7 3.62 180.0
8 -105.4 3.20 125.5
9 -109.8 3.11 119.8
10 -124.7 3.15 123.6
11 -123.1 3.11 121.4
12 -119.4 3.15 123.9
13
a
-10.9 3.10 102.6
14
a
-11.3 3.10 102.0
15
a;b
-43.5 3.40 127.9
16
a;b
-40.6 3.40 133.2
a
AverageJ values, average Fe–Fe distances, and average Fe-O-Fe bond angles.
b
Only dominant
wing-body exchange interaction is considered.
50
Figure 2.17: Left: Plot of the Fe–Fe distance vs the exchange interactions (J) for the 16
complexes under study. Right: Plot of the (Fe-O-Fe) bond angle vs the exchange interactions
(J) for the 16 complexes under study.
51
Chapter 2 References
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2 1976, 72, 1441–1446.
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[98] Cauchy, T.; Ruiz, E.; Alvarez, S. Magnetostructural correlations in polynuclear complexes:
The Fe
4
butterflies J. Am. Chem. Soc. 2006, 128, 15722–15727.
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complex Chem. Phys. Lett. 2005, 415,
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[ClO
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60
Chapter 3: Spin–orbit couplings within
spin-conserving and spin-flipping
time-dependent density functional theory:
Implementation and benchmark
calculations
3.1 Introduction
Spin-orbit coupling (SOC) is a relativistic effect arising from the interaction of the orbital angular
momentum of an electron with its intrinsic spin angular momentum. SOC plays a crucial role
in various chemical phenomena. For example, SOC determines magnetic properties of single-
molecule magnets (SMMs) and rates of spin-forbidden processes
1–6
such as phosphorescence, in-
tersystem crossing, and, more generally, nonadiabatic dynamics of molecules and materials.
7–11
In SMMs, it leads to magnetic anisotropy, thereby affecting the spin-reversal energy barrier and
magnetic relaxation. Although SOC is much larger in heavy atoms, it also plays a role in systems
composed of light atoms, such as organic molecules.
12
An accurate quantum-mechanical treatment of SOC is required in many applications.
13–18
SOC
is commonly computed using the state interaction scheme—often called a perturbative approach—
in which a small number of zero-order non-relativistic states are used to compute the matrix ele-
ments of the Breit-Pauli Hamiltonian, followed by the diagonalization of the resulting matrix to
61
yield the spin–orbit coupled states and energies.
19, 20
The Breit-Pauli Hamiltonian contains one-
and two-electron parts, with the latter being about 50 % of the full SOC value in light molecules.
The costs of computing the two-electron contribution can be be significantly reduced using an
effective one-electron spin–orbit mean-field (SOMF) approximation.
21–23
Various implementations employing full and approximate Breit-Pauli SOC operators have
been reported for density-functional theory (DFT),
24–29
density-functional theory/multireference
configuration interaction (DFT/MRCI),
30, 31
coupled-cluster (CC),
32
equation-of-motion CC
(EOM-CC),
33–36
multireference CC (MRCC),
37
complete active-space self-consistent field
(CASSCF),
19, 38, 39
restricted active space self-consistent field (RASSCF),
40
restricted active
space configuration interaction (RASCI),
41
MRCI,
42
and density-matrix renormalization group
(DMRG)
43, 44
methods.
In contrast to matrix elements describing non-relativistic transition properties—such as nona-
diabatic couplings or transition dipole moments—SOCs are tensorial quantities requiring calcu-
lations between all the components of the interacting multiplets. Wigner-Eckart’s theorem
45–47
allows one to circumvent the explicit calculation of all multiplet components by providing a recipe
for generating the full set of the SOC matrix elements from just one spin projection, for example,
the M
s
= 0 component. Using this strategy, Pokhilko et al. developed a framework for com-
puting SOCs by the application of Wigner-Eckart’s theorem to the reduced one-particle density
matrices.
36
The efficiency of this protocol was illustrated by application to the EOM-CC wave-
functions. Formulated in spin-orbital representation, this approach is ansatz-agnostic and can be
applied to any electronic structure method that can provide transition density matrices. Following
this work, SOCs for frozen-core core-valence separated EOM-CCSD (fc-CVS-EOM-CCSD),
48
RASCI, and RAS-spin-flip (RAS-SF)
41, 49
wave functions were implemented.
Here, we extend this algorithm
36
to compute SOCs using time-dependent DFT (TD-DFT)
50
in its standard spin-conserving and spin-flipping (SF-TD-DFT)
51–55
variants within the Tamm-
Dancoff approximation (TDA).
56
Owing to its favorable computational scaling, TD-DFT is often
employed to calculate excited states in extended systems. Implementations of SOCs between the
62
TD-DFT states are available in program packages such as MolSOC
25
and PySOC,
28
however, the
SOCs for SF-TD-DFT have not yet been reported. The SF approach extends Kohn-Sham TD-
DFT to treat certain types of strong correlation, such as bond-breaking, conical intersections, and
systems with two or more unpaired electrons.
55, 57–59
This work describes the implementation of SOCs using TD-DFT and SF-TD-DFT within the
Q-Chem electronic structure package
60, 61
and presents benchmark results for molecules featur-
ing different types of electronic structure: e.g., closed-shell organic molecules, diradicals, and a
molecular magnet. We compare the results obtained with different DFT and wave-function-based
methods and assess the effect of specific density functionals and basis sets on the SOC.
The chapter is organized as follows. Section 3.2 presents the theory of TD-DFT/TDA and SF-
TD-DFT/TDA, and the key equations for the calculation of the Breit-Pauli SOC matrix elements
using Wigner–Eckart’s theorem. The computational details are given in Section 3.3. Sections 3.4.1
and 3.4.2 provide the benchmark results obtained using TD-DFT and SF-TD-DFT, respectively.
Our concluding remarks are given in Section 3.5.
3.2 Theory
3.2.1 TD-DFT/TDA and SF-TD-DFT/TDA
Within TDA, TD-DFT treatment involves solving the following Hermitian eigenvalue equation:
AX =X
; (3.1)
with
A =
ab

ij
(
a

i
) + (iajjb) + (iajf
xc
jjb); (3.2)
where indexesi, j denote the occupied orbitals, a, b denote the virtual orbitals,
a
and
i
are the
orbital energies of the virtual and occupied Kohn-Sham orbitals, respectively,f
xc
is the adiabatic
exchange-correlation kernel,
is a diagonal matrix with excitation energies on the diagonal, X
63
contains the eigen-vectors of A, and the two-electron integrals are given in Mulliken’s notation.
TD-DFT/TDA yields excitation energies very close to the corresponding (linear response) TD-DFT
excitation energies for closed- and open-shell molecules.
56
In SF-TD-DFT, a high-spin reference
is used to describe target multi-configurational lower-spin states by spin-flipping excitations as:

S;S1
Ms=S1
=
^
R
Ms=1

S
Ms=S
; (3.3)
where the spin-flip operator
^
R
Ms=1
generates singly excited determinants in which the spin of
one electron is flipped with respect to the high-spin reference.
55
In the standard collinear formulation, the target spin-flipped determinants can only be cou-
pled by the Hartree-Fock exchange; hence, in the original SF-TD-DFT method functionals with
high fractions of the exact exchange—such as B5050LYP—were employed.
51
This limitation was
overcome by using non-collinear formulation.
52–54
Previous benchmark studies for organic dirad-
icals,
54
binuclear Cu(II)
62
and Fe(III)
58
SMMs illustrated robust performance of the non-collinear
SF-TD-DFT, especially when combined with the functionals from the PBE family. Here, we use
the best performing functionals to assess the performance of the SF-TD-DFT for computing SOCs
in selected diradicals and in a Fe(III) SMM.
3.2.2 Spin–orbit Hamiltonian
Originally derived by Pauli,
63, 64
the Breit-Pauli Hamiltonian describes the relativistic effects. In
particular, it can be used to evaluate spin–orbit matrix elements between non-relativistic electronic
states. In atomic units, the spin–orbit part of the Breit-Pauli Hamiltonian has the following form:
H
SO
BP
=
1
2c
2
"
X
i
X
K
Z
K
(r
i
R
K
)p
i
jr
i
R
K
j
3
s(i)

X
i6=j
(r
i
r
j
)p
i
jr
i
r
j
j
3
(s(i) + 2s(j))
#
(3.4)
64
where c is the speed of light,r
i
andp
i
denote the coordinate and momentum operators of the
ith electron respectively, s(i) is the spin operator, and R
K
and Z
K
are the coordinates and the
charge of theKth nucleus. The first term, the one-electron part of the Breit-Pauli Hamiltonian, is
proportional to the nuclear charge and, therefore, dominates in heavy elements. The second term,
the two-electron part of the Hamiltonian, describes spin-same-orbit and spin-other-orbit interac-
tions; it is significant in molecules composed of light atoms, such as typical organic molecules.
20
A
full calculation of SOC involves computation of one- and two-particle transition density matrices
and contracting them with appropriate spin-orbit integrals.
35
Fortunately, the cost of the evaluation
of the two-electron contribution can be significantly reduced by invoking spin–orbit mean-field
(SOMF) approximation.
21
The SOMF approximation amounts to considering only the contribu-
tions from the separable part of the two-particle density matrix,
35
which captures most of the effect,
yielding insignificant errors.
The symmetry of the one-electron term in Eq. (3.4) is such that one can write down the second-
quantized form of it using triplet excitation operators. Here, we use the irreducible spherical tensor
operators to represent the SOMF Hamiltonian. The triplet excitation operators are given in the
second quantization as:
T
1;1
pq
=a
y
p
a
q
; (3.5)
T
1;0
pq
=
1
p
2

a
y
p
a
q
a
y
p
a
q

; (3.6)
T
1;1
pq
=a
y
p
a
q
; (3.7)
where theT
1;1
are spin-flipping (change the spin-projection) andT
1;0
conserves the spin projec-
tion. Using these operators, the SOMF Hamiltonian can be written as:
H
SOMF
=
1
2
X
pq
h
h
SOMF
L
+
;pq
T
1;1
pq
+h
SOMF
z;pq
T
1;0
pq
+h
SOMF
L

;pq
T
1;1
pq
i
; (3.8)
65
whereh
SOMF
L
+
;L

are constructed using the sum of one-electron and mean-field contributions:
h
SOMF
L
+
=h
SOMF
x
+ih
SOMF
y
; (3.9)
h
SOMF
L

=h
SOMF
x
ih
SOMF
y
: (3.10)
In the above expression, the two-electron spin–orbit integrals are contracted with the density matrix
of the reference state (hence, mean-field).
Through the application of Wigner–Eckart’s theorem to the triplet excitation operators, Eq.
(3.8) can be implemented as:
hI
0
S
0
M
0
jT
1;M
pq
jI
00
S
00
M
00
i =hS
00
M
00
; 1MjS
0
M
0
ihI
0
S
0
jjT
1;
pq
jjI
00
S
00
i; (3.11)
wherejI
0
S
0
M
0
i denotes theI
0
th electronic state with spinS
0
and spin-projectionM
0
respectively,
hS
00
M
00
; 1MjS
0
M
0
i is a Clebsh–Gordan coefficient, andhI
0
S
0
jjT
1;
pq
jjI
00
S
00
i is a spinless triplet tran-
sition density matrix (denoted asu
pq
below).u
pq
can be obtained
36
from the one-particle transition
density matrix between the states with the same spin projection as:
u
pq
hI
0
S
0
jj
^
T
1;
pq
jjI
00
S
00
i =
1
p
2

Ms=0
pq;

Ms=0
pq;

=hS
00
M
0
; 10jS
0
M
0
i; (3.12)
where the transition density matrix
pq
is defined as:

pq
=hI
0
S
0
M
0
ja
y
p
a
q
jI
00
S
00
M
0
i: (3.13)
The SOC matrix elements between any two spin-states can then be obtained as:
hI
0
S
0
M
0
jH
SOMF
jI
00
S
00
M
00
i =
1
2
X
pq
h
h
SOMF
L
+
;pq
hS
00
M
00
; 1 1jS
0
M
0
i +
p
2h
SOMF
z;pq
hS
00
M
00
; 10jS
0
M
0
i
h
SOMF
L

;pq
hS
00
M
00
; 11jS
0
M
0
i
i
u
pq
:
(3.14)
66
We use theu
pq
matrix between spin-multipletsjS
0
M
0
i andjS
00
M
0
i to compute the entire set
of the SOC matrix elements for all pairs of the interacting spin-states: S
0
M
0
S
0
and
S
00
M
00
S
00
. Further details of the theory can be found in Ref. 36.
The key quantity involved in the calculation of inter-system crossing rates and oscillator
strengths is the SOC constant (SOCC). While couplings between different multiplet components
are dependent on spatial orientation, the SOCC is rotationally invariant and can be computed by
summing over all projections as:
SOCC =
s
X
M
0
;M
00
jhS
0
M
0
jH
SO
jS
00
M
00
ij
2
; (3.15)
In this work, we use the SOCC values to benchmark with those previously available in the litera-
ture.
3.3 Computational Details
Fig. 3.1 shows the structures of the molecules used for benchmarking in this work; below, we refer
to the molecules by the letters shown in the figure.
Molecules a-g are representative closed-shell organic molecules. Their excited states and SOCs
are calculated using standard TD-DFT/TDA. Ground-state optimized geometries of formalde-
hyde and acetone (molecules a and b, at !B97X-D/ TZVP) and psoralens (molecules e-g, at
PBE0/TZVP) are taken from Ref. 28, and biacetyl (BIA) and (2Z)-2-buten-2-ol (BOL) (molecules
c and d, at B3LYP/cc-pVDZ) are taken from Ref. 41.
Molecules h-q feature electronic degeneracies of the frontier orbitals and, therefore, cannot be
described by standard Kohn-Sham TD-DFT. To compute relevant spin-states in this set, we use
the non-collinear formulation of SF-TD-DFT/TDA. For diradicals h-k we use equilibrium triplet-
state geometries (
3
B
2
), following the previous studies.
65–68
Experimental structures are used for
molecules l-p, taken from Ref. 32, and the structure of the trigonal bipyramidal Fe(III) SMM
(molecule q) is taken from Ref. 69. High-spin triplet reference was used to compute the target
67
triplet and singlet spin-flip states in molecules h-p. To access the quartet ground state in molecule
q, a high-spin hextet reference was used.
We tested functional dependencies of the SOC for formaldehyde and acetone by considering
B3LYP,
70
PBE0,
71
!PBEh,
72
!B97X-D,
73
and !B97M-V,
74
and basis set dependencies by con-
sidering cc-pVTZ and aug-cc-pVTZ. For organic molecules c, d and e-g we used B3LYP with the
cc-pVTZ and cc-pVDZ bases, respectively. Similarly, for diradicals h-k we tested the effect of
the functional choice on SOCs using the cc-pVTZ basis. We used PBE0 and!PBEh functionals
with the cc-pCVTZ and cc-pVDZ basis sets for molecules l-p and q, respectively. The Cartesian
coordinates of all molecules are provided in the supplementary information of the manuscript.
75
Figure 3.1: Molecules studied in this work. BIA and BOL denote biacetyl and (2Z)-2-buten-
2-ol, respectively.
68
To assign the state characters, we used natural transition orbitals (NTOs), the respective leading
singular values (), and NTO participation ratios (PR
NTO
),
76–78
which define the number of NTO
pairs necessary to describe the excitation.
2
gives the weight of the electronic transition for
each NTO pair. Using dominant excitations, NTOs and descriptors, we carefully compare the
states obtained in this work with those reported in previous studies in order to make meaningful
comparisons of the respective SOCs.
We use the following acronyms for SOCs taken from previous studies: ‘1e-eff’ for SOCs ob-
tained using effective charges with the one-electron part of the Breit-Pauli Hamiltonian, ‘1e’ for
SOCs computed with only the one-electron part of the Breit-Pauli Hamiltonian, ‘SOMF’ for SOCs
obtained using spin–orbit mean-field approximation, ‘full BP’ for SOCs obtained using the full
Breit-Pauli Hamiltonian.
All calculations were performed using the Q-Chem software
60, 61
in which the presented ap-
proach was implemented and released in version 6.0.
3.4 Results and discussion
3.4.1 Spin–orbit couplings in selected organic molecules
3.4.1.1 Formaldehyde and acetone
We first consider carbonyl compounds formaldehyde and acetone and investigate the effect of
the functional and basis sets on the SOC. We use hybrid functionals (B3LYP, PBE0) and range-
separated hybrid functionals (LRC-!PBEh,!B97X-D, and!B97M-V) and the cc-pVTZ and aug-
cc-pVTZ basis sets. The results for formaldehyde and acetone have been reported previously by
Gao et al. (employing the one-electron Breit-Pauli operator with an effective charge approxi-
mation) using TD-DFT and TD-DFTB,
28
by de Carvalho et al. (using one-electron Breit-Pauli
operator) using TD-DFT,
27
by Dinkelbach et al. (using SOMF) using TD-DFT,
29
and Liu et al.
(‘1e-eff’) using semiempirical orthogonalization-corrected methods (OMx) combined with config-
uration interaction with single excitations (CIS) OMx/CIS and LR-TD-DFT.
11
Below, we compare
69
the spin–orbit coupling constants from PySOC by Gao et al. with our TD-DFT/TDA calcula-
tions.
28
We also perform calculations using the highly accurate EOM-EE-CCSD method, which
we use as a reference for this study.
Figure 3.2: Canonical frontier Kohn-Sham molecular orbitals (,n, and

) of formaldehyde
and acetone (the shapes of the NTOs from the excited-state calculations are very similar);
B3LYP/cc-pVTZ.
Fig. 3.2 shows the relevant molecular orbitals involved in the S
1
, T
1
, and T
2
excited states.
With the cc-pVTZ basis, irrespective of the choice of the functional, the lowest singlet state, S
1
is
ofn

character, and triplet states T
1
and T
2
are ofn

and

character, respectively. Tables
3.9 and 3.10 present the vertical excitation energies calculated using different functionals with cc-
pVTZ and aug-cc-pVTZ as well as the EOM-EE-CCSD values. The excitation energies computed
with different functionals are within 0.3 eV , with S
1
lying between T
1
and T
2
states. The choice
of the basis, cc-pVTZ versus aug-cc-pVTZ, has a small effect on the excitation energies with a
difference of less than 0.06 eV .
Tables 3.1 and 3.2 show the SOMF SOCCs computed between the ground state, excited singlet
and triplet states in formaldehyde and acetone using the cc-pVTZ and aug-cc-pVTZ bases with
different functionals. The one-electron SOCC computed with cc-pVTZ is reported in Table 3.14.
In agreement with El-Sayed’s rules,
79, 80
the SOC is large for transitions involving a change of the
orbital type, i.e., SOC is large between the
1
n

/
3

states and negligible between
1
n

/
3
n

states. Both one-electron and SOMF SOCCs are insensitive to the choice of functionals and basis
70
sets, with a variation of less than 2 cm
1
. While the SOCC between the ground state and
3
n

com-
puted using different functionals matches perfectly the EOM-EE-CCSD value, the SOCC between
the
1
n

and
3

states differ by about 12 cm
1
and 8 cm
1
, from the EOM-CCSD values for the
one-electron and SOMF parts. A comparison with 1e-eff values (obtained using effective charges
by Gao et al.
28
) shows a close match with SOMF SOCCs in this study, however, the full Breit-Pauli
SOCC (full BP-B3LYP/DALTON) differs by about 10 cm
1
for the
1
n

/
3

transition.
Table 3.1: SOMF SOCCs in formaldehyde and acetone computed with TD-DFT/TDA
(B3LYP, PBE0,!PBEh,!B97X-D, and!B97M-V) and EOM-EE-CCSD using cc-pVTZ com-
pared with values from Ref. 28.
Transition B3LYP PBE0 !PBEh !B97X-D !B97M-V EOM-EE- B3LYP/ B3LYP/
CCSD PySOC
a
Dalton
b
Formaldehyde
GS/
3
n

62.45 61.63 61.66 62.12 61.07 61.41 - -
GS/
3

0 0 0 0 0 0 - -
1
n

/
3
n

0 0 0 0 0 0 0 0
1
n

/
3

44.68 43.8 43.92 44.13 43.36 51.48 45 54
Acetone
GS/
3
n

59.14 58.5 58.74 59.16 58.41 59.28 - -
GS/
3

0.27 0.26 0.25 0.26 0.31 0.24 - -
1
n

/
3
n

0.03 0.04 0.04 0.04 0.04 0.05 0 0
1
n

/
3

43.75 43.18 43.36 43.47 42.72 50.57 44 54
a
1e-eff SOCC (B3LYP/TZVP);
b
full Breit-Pauli SOCC (B3LYP/cc-pVTZ) using response theory.
71
Table 3.2: SOMF SOCCs in formaldehyde and acetone computed with TD-DFT/TDA
(B3LYP, PBE0,!PBEh,!B97X-D, and!B97M-V) and EOM-EE-CCSD using aug-cc-pVTZ
compared with values from Ref. 28.
Transition B3LYP PBE0 !PBEh !B97X-D !B97M-V EOM-EE- B3LYP/ B3LYP/
CCSD PySOC
a
Dalton
b
Formaldehyde
GS/
3
n

60.62 60.00 60.03 60.67 59.25 59.52 - -
GS/
3

0 0 0 0 0 0 - -
1
n

/
3
n

0 0 0 0 0 0 0 0
1
n

/
3

44. 33 43.47 43.56 43.81 42.93 50.88 45 54
Acetone
GS/
3
n

57.61 57.16 57.40 57.92 56.81 57.63 - -
GS/
3

0.17 0.18 0.18 0.20 0.23 0.15 - -
1
n

/
3
n

0.02 0.01 0.01 0.01 0.01 0.01 0 0
1
n

/
3

43.08 42.71 42.96 43.07 42.16 49.92 44 54
a
1e-eff SOCC (B3LYP/TZVP);
b
full Breit-Pauli SOCC (B3LYP/cc-pVTZ) using response theory.
72
3.4.1.2 Biacetyl (BIA) and (2Z)-2-buten-2-ol (BOL)
Our next benchmark set comprises a diketone (BIA) and conjugated alcohol (BOL) previously
studied by Carreras et al. using the RASCI and EOM-CCSD methods with SOMF approxima-
tion.
41
Fig. 3.3 shows the relevant frontier MOs: ,n, and

. The LUMO in BIA and LUMO+1
in BOL are of

character, whereas the HOMO in BIA is ofn type and in BOL it is of character.
Following this MO energy order, the lowest singlet and triplet states in BIA are ofn

character,
in BOL, they are of

character.
Figure 3.3: Canonical Kohn-Sham molecular orbitals (,n,

) of BIA and BOL; B3LYP/cc-
pVTZ.
Table 3.11 presents the vertical excitation energies, and NTO descriptors of low-lying singlet
and triplet excited states computed using B3LYP/cc-pVTZ in BIA and BOL. Except for

state
(S
9
state) of BIA for which PR
NTO
=2.08, all other singlet and triplet states have PR
NTO
close to
1, meaning that all computed excited states, except for S
9
in BIA, can be described by a single
excitation. Table 3.3 shows the one-electron and SOMF SOCCs computed using B3LYP, compar-
ing the results with EOM-CCSD and RASCI couplings.
41
The trends in SOCCs follow El-Sayed’s
rules,
79, 80
featuring large SOCCs between states of different orbital characters, such as
1

=
3
n

and
1
n

=
3

transitions. In BIA, the SOMF SOCC between the
1

and
3
n

states computed
with B3LYP differs by about 22 cm
1
from those evaluated by the EOM-CCSD and RASCI meth-
ods. This can be explained by the different character of the S
9
state—at the B3LYP level, this state
73
shows configuration mixing whereas the EOM-CCSD and RASCI wave-function retain pure
1

character. All other computed SOMF DFT couplings are within 6 cm
1
from the EOM-CCSD
SOMF couplings. Overall, the SOCs computed with B3LYP agree well with the SOCs computed
with EOM-EE-CCSD. The differences between RASCI and EOM-CCSD couplings observed in
BOL can be attributed to insufficient treatment of dynamic correlation by RASCI.
41
Table 3.3: SOCC in BIA and BOL computed with TD-DFT/TDA (B3LYP/cc-pVTZ) com-
pared with the EOM-CCSD and RASCI values.
Transition B3LYP EOM-CCSD
a
RASCI
b
1e SOMF SOMF SOMF
BIA
GS/
3
n

0.00 0.00 0.00 0.00
GS/
3
n
0

130.31 82.05 85.79 81.25
GS/
3

0.00 0.00 0.00 0.00
1
n

/
3
n

1.01 0.66 0.17 0.16
1
n

/
3
n
0

0.00 0.00 0.00 0.00
1
n

/
3

71.09 45.62 51.84 55.33
1
n
0

/
3
n

0.00 0.00 0.00 0.00
1
n
0

/
3
n
0

0.68 0.43 0.02 0.45
1
n
0

/
3

0.00 0.00 0.00 0.00
1

/
3
n

33.65 21.72 44.19 44.6
1

/
3
n
0

0.00 0.00 0.00 0.00
1

/
3

0.23 0.02 0.13 0.04
BOL
GS/
3

0.67 0.01 0.01 0.00
GS/
3
n

43.34 24.77 23.14 18.57
1

/
3

0.09 0.02 0.02 0.03
1

/
3
n

24.73 16.22 15.43 7.06
1
n

/
3

20.52 13.34 11.07 3.44
1
n

/
3
n

2.43 1.41 0.10 0.02
a
EOM-CCSD/cc-pVTZ; from Ref. 41;
b
RASCI/cc-pVTZ; from Ref. 41.
74
3.4.1.3 Psoralens
Psoralens are photosensitizers used in pharmaceutical applications.
81
They occur naturally in some
plants, such as Heracleum maximum (commonly known as cow parsnip).
82
Contained in the skin of
the plant, psoralens are responsible for the ability of cow parsnip to cause skin rashes and blistering,
initiated by sunlight (phytophotodermatitis). An important aspect of psoralens’ pharmacological
action is that it can react with DNA, and the reaction, initiated by UV light, proceeds in the triplet
state.
83
SOCs in psoralen and its thio-derivatives (psoralenOO, psoralenOS, and psoralenSO) have
been characterized theoretically in previous studies by Tatchen et al. (SOMF) using DFT/MRCI,
84
by Chiodo and Russo (full BP) using TD-DFT,
25
and, more recently, by Gao et al. (1e-eff) using
TD-DFT and TD-DFTB,
28
and by Liu et al. (1e-eff) using OM2/CIS and TD-DFT.
11
Here, we use
B3LYP/cc-pVDZ and compare the results with some of the previously available SOCCs between
low-lying singlet and triplet states. Fig. 3.4 shows the frontier molecular orbitals involved in the
low-lying excited states for psoralen compounds. In all three compounds, the HOMO-2 is of n
type, HOMO-1 and HOMO are of type, and LUMO, LUMO+1, and LUMO+2 are

orbitals.
Figure 3.4: Canonical Kohn-Sham molecular orbitals (, n,

) of psoralen compounds;
B3LYP/cc-pVDZ.
Tables 3.12 and 3.13 show the vertical excitation energies and NTO analysis of the low-lying
singlet and triplet excited states. To make meaningful comparisons with previous studies,
25, 28, 84
75
we carefully analyzed excited-state characters in our calculations and matched the computed states
with those from previous studies by their leading TD-DFT amplitudes. We found that the reported
state numbering does not match ours—apparently, some states were missed in earlier studies. Ta-
bles 3.12 and 3.13 give the state labels from Q-Chem calculations, whereas in Table 3.4 we give
both sets of state labels.
Table 3.4 shows the 1el and SOMF SOCCs for the psoralen compounds. For all three
molecules, most of the computed SOMF SOCCs agree with the 1e-eff-B3LYP values (computed
with PySOC) within 3 cm
1
. Only the S
2
/T
2
couplings in psoralenOO and psoralenOS show the
largest deviation from the 1e-eff treatment, but agree well with the full Breit-Pauli treatment. For
the larger SOCCs (that are greater than 1 cm
1
), DFT/MRCI values are larger than our SOMF
B3LYP values, while the couplings that are less than 1 cm
1
agree within 0.1 cm
1
. While the
couplings between the ground state and excited triplets are typically overestimated in this work
with respect to the full Breit-Pauli treatment (computed withMolSOC), those between singlet and
triplet excited states are usually smaller. Overall, we observe a good qualitative agreement between
our values and the SOCCs reported in previous studies.
3.4.2 Spin–orbit couplings calculated with SF-TD-DFT
3.4.2.1 CH
2
, NH
+
2
, SiH
2
, and PH
+
2
In these molecules, the description of the low-lying states using standard Kohn-Sham DFT and TD-
DFT is inadequate because of the diradical character of the singlet states.
85
This problem can be
circumvented by using SF-TD-DFT with a high-spin triplet reference.
55
In contrast to other types
of organic diradicals, methylene-like diradicals are highly sensitive to the functional employed,
as documented in previous SF-TD-DFT studies.
51, 54
Specifically, only non-collinear SF-TD-DFT
(NC-SF-TD-DFT) can yield accurate results for these species, and only with functionals that do not
use Becke’s exchange. The best results were obtained with the functionals from the PBE family.
Benchmark calculations on other classes of molecules have also shown superior performance of
NC-SF-TD-DFT with PBE0 and !PBEh.
58, 62
Following previous studies,
35, 36, 41
we computed
76
Table 3.4: SOCCs in psoralen and its thio derivatives computed with B3LYP/cc-pVDZ and
compared with previous calculations.
B3LYP DFT/MRCI
b
B3LYP
c
PBE0
d
1el SOMF SOMF 1el-eff full BP
PsoralenOO
GS/T
1
1.60 0.03 - - -
GS/T
2
0.24 0.03 0.05 1 (GS/T
1
) 0.07 (GS/T
1
)
GS/T
3
0.58 0.05 - - -
GS/T
4
66.09 41.85 50.01 43 33.46
S
1
/T
1
0.72 0.04 - - -
S
1
/T
2
0.97 0.00 0.01 1 (S
1
/T
1
) 0.08 (S
1
/T
1
)
S
1
/T
3
0.74 0.08 - - -
S
1
/T
4
10.99 6.70 10.22 8 12.45
S
2
/T
1
28.43 17.78 - - -
S
2
/T
2
6.44 4.22 28.1 19 (S
3
/T
1
) 9.96 (S
2
/T
1
)
S
3
/T
1
0.67 0.02 - - -
PsoralenOS
GS/T
1
1.31 0.01 - - -
GS/T
2
0.25 0.02 0.04 1 (GS/T
1
) 0.05 (GS/T
1
)
GS/T
3
99.15 69.48 78.53 70 (GS/T
4
) 71.45 (GS/T
4
)
GS/T
4
0.69 0.30 - - -
S
1
/T
1
0.46 0.06 - - -
S
1
/T
2
1.01 0.00 0.04 0 (S
1
/T
1
) 0.08 (S
1
/T
1
)
S
1
/T
3
38.13 34.51 35.6 37 (S
1
/T
4
) 45 (S
1
/T
4
)
S
1
/T
4
0.73 0.05 - - -
S
2
/T
1
22.47 22.89 - - -
S
2
/T
2
35.36 27.49 11.13 10 (S
2
/T
1
) 22.81 (S
2
/T
1
)
S
3
/T
1
0.98 0.03 - - -
PsoralenSO
GS/T
1
1.12 0.13 0.04 0 0.03
GS/T
2
1.18 0.22 - - -
GS/T
3
0.74 0.20 - - -
GS/T
4
64.76 40.89 49.44 42 (GS/T
5
) 31.70
S
1
/T
1
1.03 0.03 0.01 1 0.08
S
1
/T
2
0.11 0.03 - - -
S
1
/T
3
0.94 0.07 - - -
S
1
/T
4
6.34 4.12 6.21 4 (S
1
/T
5
) 5.99
S
2
/T
1
0.55 0.00 - - -
S
2
/T
2
1.51 0.01 - - -
S
3
/T
1
28.00 17.93 25.88 16 13.74 (S
2
/T
1
)
b
DFT-MRCI/TZVP; Ref. 84;
c
1e-eff SOCC with TD-DFT/B3LYP/TZVP, Ref. 28; reported state labels are given in parenthesis;
d
full Breit-Pauli SOCC with TD-DFT/PBE0/TZVP, Ref. 25; reported state labels are given in parenthesis.
77
SOCCs between the lowest triplet (
3
B
2
) and singlet (
1
A
1
) states of these diradicals (CH
2
, NH
+
2
,
SiH
2
, and PH
+
2
). We used NC-SF-TD-DFT and considered B3LYP, PBE0,!PBEh,!B97X-D, and
!B97M-V .
Table 3.5: SOCCs (cm
1
) between
3
B
2
and
1
A
1
states in CH
2
, NH
+
2
, SiH
2
, and PH
+
2
computed
using SOMF with NC-SF-TD-DFT.
Method
a
CH
2
NH
+
2
SiH
2
PH
+
2
PBE0 10.36 15.15 68.67 135.38
!PBEh 10.32 15.03 68.93 135.66
B3LYP 12.29 19.83 75.69 150.12
!B97X-D 12.85 18.11 75.20 157.21
!B97M-V 13.16 20.20 79.73 155.90
EOM-SF-CCSD
b
10.86 18.26 56.74 119.97
a
cc-pVTZ basis;
b
From Ref. 36.
Table 3.5 presents the SOMF SOCCs in these diradicals using NC-SF-TD-DFT, comparing
them with SOMF SOCCs obtained with EOM-SF-CCSD.
36
In contrast to TD-DFT calculations
of formaldehyde and acetone discussed above, here we observe a strong functional dependence,
with PBE0 and!PBEh performing similarly and B3LYP,!B97x-D, and!B97M-V showing small
differences in couplings with each other. For CH
2
, SiH
2
, and PH
+
2
, PBE0 and!PBEh compare well
with EOM-SF-CCSD, with differences increasing with an increase in atomic number, 0.5 cm
1
in
the case of CH
+
2
and 16 cm
1
in the case of PH
+
2
. Overall, we confirm the earlier recommendation
54
to use PBE0 and !PBEh in NC-SF-TD-DFT calculations—as these functionals yield accurate
estimates of both energy differences between the states and the respective SOCCs.
3.4.2.2 BH, AlH, HSiF, HSiCl, HSiBr
Next, we consider closed-shell molecules with moderate diradical character, BH, AlH, HSiF,
HSiCl, and HSiBr, previously studied by Christiansen et al. using linear response CCSD (LR-
CCSD, equivalent to EOM-EE-CCSD for energies and slightly different for properties) and Epi-
fanovsky et al. using EOM-SF-CCSD/EOM-EE-CCSD methods.
32, 35
Here, we consider SOCs
between the 1
1

+
and 1
3
states for BH, AlH, and between 1
1
A
0
and 1
3
A
00
states for silylenes
78
HSiX, X = F, Cl, Br. We use NC-SF-TD-DFT with PBE0/cc-pCVTZ and !PBEh/cc-pCVTZ.
Table 3.6 shows the results for these molecules. As one can see, there is an excellent agreement
between the SOCCs computed using PBE0/!PBEh and EOM-SF-CCSD, with differences less than
3 cm
1
(except for HSiBr for which we observe 15 cm
1
and 31 cm
1
difference between PBE0
and!PBEh with EOM-SF-CCSD, respectively). Similarly, there is a good agreement between the
SOCCs computed with LR-CCSD and PBE0/!PBEh.
Table 3.6: SOCCs (cm
1
) in BH, AlH, HSiF, HSiCl, and HSiBr computed using SOMF with
NC-SF-TD-DFT (PBE0/!PBEh; cc-pcVTZ).
Method BH AlH HSiF HSiCl HSiBr
1
1

+
/ 1
3
1
1

+
/ 1
3
1
1
A
0
/1
3
A
00
1
1
A
0
/1
3
A
00
1
1
A
0
/1
3
A
00
PBE0 4.39 34.98 79.52 106.85 267.78
!PBEh 4.41 35.17 79.87 105.47 252.40
EOM-SF-CCSD
a
4.10 32.94 78.28 108.43 283.47
LR-CCSD
b
3.48
c
27.06
c
71.10
d
99.38
d
270.93
d
a
EOM-CCSD/cc-pCVTZ, from Ref. 35;
b
LR-CCSD, from Ref. 32;
c
aug-cc-pVTZ;
d
ANO2 basis.
3.4.2.3 Spin reversal energy barrier in Fe(III) SMM
Finally, we consider a mononuclear Fe(III) SMM, (PMe
3
)Fe(III)Cl
3
, which is reported to have the
highest effective energy barrierU
eff
= 81 cm
1
among Fe(III)-based SMMs.
69
This energy barrier
for spin inversion arises because of the splitting of the ground state due to SOC. The experimental
spin-reversal barrier is U = 100, as computed from the magnetic anisotropy (D =50 cm
1
)
and ground-state spin (S = 3=2), withU =jDj(S
2
1=4).
69
Alessio and Krylov used the EOM-
SF-CCSD treatment to calculate the spin reversal energy barrier and magnetic properties of this
complex.
86
Following this study, here we compute the energy barrier using the spin-orbit coupling
obtained with SOMF NC-SF-TD-DFT treatment.
For this complex, we use PBE0 and!PBEh functionals to access the quartet ground state (S
= 3=2) and other closely lying excited states by spin-flip excitations from a high-spin reference
79
Table 3.7: Energy gaps (E, in cm
1
) and of the target spin-flip states obtained from
the high spin hextet reference state computed with PBE0,!PBEh/cc-pVDZ.
Ref SF
1
SF
2
SF
3
SF
4
SF
5
PBE0 8.76 3.84 3.85 8.74 3.83 3.82
E - 0 398 553 4122 4471
!PBEh 8.76 3.82 3.82 8.77 3.81 3.82
E - 0 344 715 4116 4495
EOM-SF-CCSD
a
8.75 3.81 3.81 8.73 3.83 3.83
E - 0 92 2074 7245 7447
a
EOM-SF-CCSD/cc-pVDZ, from Ref. 86.
Figure 3.5: Energy levels arising from the splitting of the two lowest quartet states induced
by SOC.
of S = 5=2. We obtain three spin-flip states (see Table 3.7) lying within 715 cm
1
; SF
1
and SF
2
are quartet states while SF
3
is a hextet state. This differs from the EOM-SF-CCSD results, where
SF
3
is about 2000 cm
1
above SF
2
.
86
We then computed the spin reversal energy barrier with 2,
3, and 5 low-lying SF states by including the SOC effects using the state-interaction procedure.
36
The inclusion of only two states is insufficient to characterize the energy barrier with SF-TD-
DFT (see Table 3.8), however, the energy barrier calculated with 3 states gives a value of 97
(PBE0) cm
1
and 100 (!PBEh) cm
1
, in excellent agreement with the experimental estimate of
100 cm
1
. The energy barrier computed with EOM-SF-CCSD converges to 130 cm
1
with the
inclusion of 5 states. To assess the effects of energy gaps of the SF states, we computed the energy
80
barrier using PBE0 SOCs and EOM-SF-CCSD energy gaps (denoted as EOM/PBE0 in Table 3.8).
This combined EOM/PBE0 calculation gives energy barriers close to the EOM-SF-CCSD results.
Therefore, we conclude that the differences in energy barriers computed using PBE0/!PBEh and
EOM-SF-CCSD result from the different energy gaps of the SF states predicted by these methods.
Table 3.8: Energy barrierU (cm
1
) computed using 2, 3, and 5 lowest SF states in trigonal
bipyramidal (PMe
3
)Fe(III)Cl
3
complex with NC-SF-DFT/PBE0/cc-pVDZ.
No. of states PBE0 !PBEh EOM-SF-CCSD
b
EOM/PBE0
c
Exp-U
d
2 52 59 103 99
3 97 100 130 119
5 94 92 128 117 100
b
EOM-SF-CCSD, Ref. 86;
c
Energy barrier calculated with EOM-SF-CCSD energies and PBE0 SOCs;
d
Experimental value, Ref. 69.
3.5 Conclusions
We presented the implementation of SOCs with TD-DFT and SF-TD-DFT and benchmark calcula-
tions for several organic molecules of a closed-shell character as well as diradicals and one SMM.
The algorithm is based on evaluating matrix elements of the Breit-Pauli operator by the application
of Wigner–Eckart’s theorem to the reduced one-particle density matrices. We used NTO analysis
to characterize the nature of the interacting states. We tested functional and basis-set dependencies
for formaldehyde and acetone with TD-DFT using B3LYP, PBE0, LRC-!PBEh, !B97X-D, and
!B97M-V functionals and the cc-pVTZ and aug-cc-pVTZ basis sets. The results demonstrate that
SOCCs are rather insensitive to the choice of functionals and basis sets, with a variation of less
than 2 cm
1
. In agreement with El-Sayed’s rules, SOCs are large for transitions involving a change
of the orbital type. We validated our SOMF SOCC results by comparisons with the reference val-
ues from TD-DFT, EOM-EE-CCSD, RASCI, DFT/MRCI, EOM-SF-CCSD, and LR-CCSD stud-
ies. Calculations for diradicals with NC-SF-TD-DFT show strong functional dependencies, with
81
PBE0 and!PBEh performing similarly to EOM-SF-CCSD, and B3LYP,!B97X-D, and!B97M-
V performing similarly to each other; this is in agreement with the previous benchmark studies in
which energy gaps and state characters were considered.
54, 87
Using the state-interaction approach,
we computed the spin-reversal energy barrier in Fe(III) SMM with PBE0 and LRC-!PBEh, which
matches the experimental estimate when the three lowest SF states are included in the calculation.
This new implementation extends the scope of computational tools for modeling spin-forbidden
processes in large molecular systems, as illustrated by our recent study in which we applied this SF-
TD-DFT SOC code to describe the magnetic behavior of nickelocene molecular magnet adsorbed
on the MgO(001) surfaces using the state-interaction scheme.
59
82
3.6 Appendix A: NTO analysis and energies of excited states
Table 3.9: Vertical excitation energies (in eV) and dominant excitation character (Exc) of the
lowest singlet and triplet states in formaldehyde and acetone computed with TD-DFT and
EOM-EE-CCSD using cc-pVTZ.
State Exc B3LYP PBE0 !PBEh !B97X-D !B97M-V EOM-EE-CCSD
Formaldehyde
S
1
n

4.07 4.08 4.05 4.09 4.05 4.14
T
1
n

3.37 3.32 3.29 3.44 3.51 3.67
T
2

6.02 5.91 5.94 6.11 6.23 6.16
Acetone
S
1
n

4.51 4.54 4.54 4.57 4.56 4.62
T
1
n

3.90 3.88 3.88 3.98 4.09 4.22
T
2

6.06 5.99 6.02 6.16 6.29 6.3
Table 3.10: Vertical excitation energies (in eV) and dominant excitation character (Exc) of
lowest singlet and triplet states in formaldehyde and acetone computed with TD-DFT and
EOM-EE-CCSD using aug-cc-pVTZ.
Exc B3LYP PBE0 !PBEh !B97X-D !B97M-V EOM-EE-CCSD
Formaldehyde
n

4.02 4.03 4.01 4.06 4.00 4.09
n

3.34 3.30 3.27 3.42 3.48 3.64

6.00 5.90 5.93 6.09 6.21 6.16
Acetone
n

4.48 4.52 4.52 4.55 4.54 4.59
n

3.90 3.88 3.88 3.99 4.09 4.21

6.06 5.99 6.02 6.16 6.29 6.31
83
Table 3.11: Vertical excitation energies E (in eV), dominant excitation character (Exc),
PR
NTO
, and their weights (
2
) in BIA and BOL; B3LYP/cc-pVTZ.
State Exc PR
NTO

2
E
BIA
S
1
n

1.02 0.99 2.71
S
2
n
0

1.13 0.94 4.50
S
9

2.08 0.59, 0.36 7.76
T
1
n

1.03 0.99 2.15
T
2
n
0

1.24 0.89 3.90
T
3

1.13 0.94 5.00
BOL
S
4

1.07 0.97 6.70
S
9
n

1.06 0.97 8.70
T
1

1.01 1 3.97
T
8
n

1.03 0.99 8.28
Table 3.12: Vertical excitation energies (E, in eV), and dominant excitation character (Exc)
of psoralen molecules; B3LYP/cc-pVDZ.
PsoralenOO PsoralenOS PsoralenSO
State Exc E Exc E Exc E
S
1

H
!

L
3.95
H
!

L
3.68
H
!

L
3.77
S
2
n
H2
!

L
4.44 n
H2
!

L
3.86
H1
!

L
4.29
S
3

H1
!

L
4.62
H1
!

L
4.37 n
H2
!

L
4.40
T
1

H
!

L
3.01
H
!

L
2.88
H1
!

L
2.99
T
2

H1
!

L
3.22
H1
!

L
3.11
H
!

L
3.09
T
3

H
!

L+1
3.82 n
H2
!

L
3.52
H
!

L+1
3.57
T
4
n
H2
!

L
4.10
H
!

L+1
3.52 n
H2
!

L
4.06
84
Table 3.13: NTO descriptors of excited states in psoralen molecules; B3LYP/cc-pVDZ.
PsoralenOO PsoralenOS PsoralenSO
State PR
NTO

2
PR
NTO

2
PR
NTO

2
S
1
1.19 0.92 1.17 0.92 1.15 0.93
S
2
1.00 1.00 1.00 1.00 1.55 0.78
S
3
1.42 0.83 1.75 0.71 1.00 1.00
T
1
1.04 0.98 1.04 0.98 1.15 0.93
T
2
1.16 0.93 1.27 0.88 1.06 0.97
T
3
1.31 0.87 1.01 1.00 1.17 0.92
T
4
1.01 1.00 1.39 0.83 1.01 1.00
85
3.7 Appendix B: One electron SOCCs
Table 3.14: One-electron SOCCs in formaldehyde and acetone computed with TD-DFT/TDA
(B3LYP, PBE0,!PBEh,!B97X-D, and!B97M-V) and EOM-EE-CCSD using cc-pVTZ.
Transition B3LYP PBE0 !PBEh !B97 !B97 EOM-EE-
X-D M-V CCSD
Formaldehyde
GS/
3
n

100.08 98.87 98.94 99.61 98.05 97.98
GS/
3

0 0 0 0 0 0
1
n

/
3
n

0 0 0 0 0 0
1
n

/
3

70.22 68.89 69.05 69.42 68.2 81.55
Acetone
GS/
3
n

94.85 93.90 94.31 94.95 93.85 94.73
GS/
3

0.38 0.37 0.36 0.37 0.45 0.36
1
n

/
3
n

0.06 0.06 0.06 0.06 0.07 0.09
1
n

/
3

68.58 67.77 68.04 68.21 67.04 79.16
86
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93
Chapter 4: Origin of Magnetic Anisotropy
in Nickelocene Molecular Magnet and
Resilience of its Magnetic Behavior
4.1 Introduction
Molecular magnets have potential applications as building blocks of spin-based memory devices.
The individual molecules can be deposited on a surface or self-assembled into 3D architectures,
giving rise to scalable magnetic materials.
1–3
Using molecular magnetic units affords high chemi-
cal tunability. To be a good magnet, molecule should possess magnetic anisotropy—orientational
dependence of the ability to magnetize. Microscopically, magnetic anisotropy originates from a
large spin-orbit coupling (SOC) that gives rise to zero-field splitting (ZFS) of the magnetic sub-
levels.
4
Such magnetic anisotropy yields slow magnetic relaxation, providing energy levels that are
well-defined by their spinS and spin projectionM
S
quantum numbers and well-separated in en-
ergy. Assuming that coherence times are sufficiently long,
5
in order to realize a molecule-based
quantum device, its states must then be easy to address by inducing transitions with light or mi-
crowave fields.
6, 7
Addressability—the ability of controlling such energy levels and generating
superposition states—permits initialization, manipulation, and read-out of the individual molecu-
lar magnetic units of a quantum algorithm. To develop individually addressable molecular mag-
nets, one should deposit molecules on a surface and then investigate their electronic structure and
magnetic behavior with a probing technique.
1, 2
94
Spin-flip transitions between the magnetic sublevels of the system can be induced by mi-
crowaves within an electron paramagnetic resonance (EPR) setting,
8
or via inelastic electron
tunneling, i.e., within the junction of a scanning tunneling microscope (STM),
9–11
revealing
also the energy spacing between the magnetic sublevels (i.e., spin-orbit splitting or magnetic
anisotropy). STM affords atomic-scale spatial resolution, allowing one to address individual
molecules, whereas EPR experiments require larger ensembles of magnetic units (e.g., solutions or
molecular crystals). Additionally, since the electronic structure of the molecule and, consequently,
its response to an applied magnetic field may change significantly by the environment due to, for
example, charge transfer or polarization, it is important to verify that spin states and related spin
dynamics (coherence or magnetic relaxation times) are retained upon adsorption on a surface.
12
When grafted onto a surface or adsorbed at the apex of a scanning probe tip, molecular mag-
nets can also serve as molecular spin probes capable of detecting states and properties of nearby
nano-magnets through their magnetic interactions.
13
These interactions lead to a shift of the spin
sensor’s energy levels and to a perturbation of its static and dynamic magnetic properties, which,
if measured, enables sensing of the nearby magnetic species. For example, when functionalizing
a metallic STM tip by such spin-probe molecules, the electrons tunneling through the molecular
magnet-terminated tip can induce spin-flip transitions in magnetic molecules or arrays of molecules
deposited on a support, thus addressing their spin-orbit levels, magnetic anisotropy, exchange in-
teractions, and spin-phonon couplings.
Recently, the nickelocene molecular magnet (NiCp
2
, Cp = cyclopentadienyl) adsorbed on metal
surfaces has been investigated by STM experiments.
14–18
Due to the robustness of NiCp
2
’s mag-
netic anisotropy upon adsorption and the ability to address its energy levels by STM, NiCp
2
has
also been used to functionalize the STM’s metallic tip, probing double spin-flip excitations as
well as exchange interactions and coupled spin-vibration transitions in the NiCp
2
dimer (i.e., one
NiCp
2
is anchored to the tip and positioned above another NiCp
2
deposited on a surface).
15–18
NiCp
2
shows robust magnetic anisotropy after adsorption on a substrate and is minimally affected
by its local environment, while being strongly sensitive to the nearby magnetic molecules. Thus,
95
to maximize the sensitivity of the spin sensor towards the desired magnetic quantity, one should
be able to mitigate the undesired interactions with the surrounding.
19, 20
Despite the rise of STM
techniques and potential applications of NiCp
2
as a spin-sensing probe, microscopic understand-
ing of the origin of NiCp
2
magnetic anisotropy and the resilience of its magnetic behavior is still
incomplete. Furthermore, magnetic behavior of NiCp
2
adsorption complexes has not been yet the-
oretically characterized beyond DFT (density functional theory) or DFT+U (Hubbard correction to
DFT). Thus, affordable yet reliable computational methods capable of investigating complex and
extended model systems, such as molecular magnets on a surface or their self-assemblies, are still
to be developed.
3
Detailed understanding of electronic structure and magnetic behavior of molecular magnets is
key to interpreting experiments and designing novel magnetic materials by structural modification
of their building units. In the context of molecular magnets, one target quantity is the spin-reversal
barrier (i.e., magnetic anisotropy), which determines the extent to which the system can be mag-
netized by an applied magnetic field (i.e., susceptibility) and influences the rate of magnetization
switching (magnetic relaxation time), also enabling a better control over the spin states.
Ab initio description of molecular magnets’ spin states is challenging because of closely lying
electronic states and their multi-configurational wave functions, which are not amenable to stan-
dard single-reference treatments, such as Kohn-Sham DFT, perturbation theory (PT), and coupled-
cluster (CC) methods. It is desirable to have a reliable and affordable method capable of tackling
both dynamical and non-dynamical correlation in such molecular and extended systems. One
strategy to treat molecular magnets entails using multi-reference (MR) methods or a broken sym-
metry (BS)-DFT approach, extracting magnetic anisotropy from phenomenological spin Hamilto-
nians.
21
Recently, we reported an alternative approach based on the equation-of-motion coupled-
cluster (EOM-CC) framework, implemented in the ezMagnet module
22
of the ezSpectra suite.
23
ezMagnet enables extracting magnetic anisotropy (from the spin-orbit splitting) and computing
macroscopic magnetic properties (magnetization and susceptibility) starting from EOM-CC cal-
culations of magnetic states and relevant properties (SOCs and angular momentum operators).
96
Calculated macroscopic quantities can then be directly compared with experiment, bypassing the
spin-Hamiltonian formalism.
The EOM-CC family of methods
24–26
extends the hierarchy of black-box single-reference
methods to strongly correlated systems. Among EOM-CC variants, the spin-flip (SF) method
27–29
provides access to multi-configurational states of polyradicals, making EOM-SF-CC suitable for
applications to molecular magnets.
22, 30–33
Calculations of spin-related properties also requires
SOCs and Zeeman interactions (i.e., the interaction of the electrons with an applied magnetic field).
ezMagnet treats these relativistic interactions using a two-step state-interaction scheme.
34–36
First,
zero-order states are obtained from non-relativistic EOM-CC calculations. Second, in the basis
of these zero-order non-relativistic EOM-CC states, a perturbed Hamiltonian is formed by includ-
ing SOC and Zeeman terms. Using the EOM-CC wave functions, SOCs are computed as matrix
elements of the Briet-Pauli spin-orbit Hamiltonian,
37, 38
as recently implemented in the Q-Chem
software
39, 40
by Krylov and co-workers.
41–43
Then, by applying Boltzmann statistic, ezMagnet
computes macroscopic properties from the resulting partition function, following the protocol de-
veloped by Neese and co-workers.
44
Ideally, the theory should be able to provide not only state energies and experimental observ-
ables, but also tools for interpreting the computed properties, thus furnishing insights. In many-
body correlated theories (such as EOM-CC), one can derive a molecular orbital interpretation by
employing reduced quantities, i.e., one-particle transition density matrix, and corresponding nat-
ural transition orbitals (NTOs).
42, 45–51
In this way, the computed properties can be described in
terms of a simple orbital picture, which is rigorously defined for many-body wave-functions and
is not sensitive to the basis set or the correlation treatment, such that the comparisons between
different methods can be carried out.
43, 49, 51, 52
Within this orbital picture, hole-particle NTO pairs
describe the transitions between spin-orbit coupled states,
42
so that trends in magnetic anisotropy
and barrier for spin-inversion can be rationalized based on the shape of the NTO pairs contribut-
ing to the SOC,
22
and strength of exchange couplings between metal centers can be related to the
frontier natural orbitals’ character and occupation.
53
97
In this contribution, we characterize magnetic properties of the NiCp
2
molecular magnet us-
ing this methodology, recently implemented in ezMagnet and the Q-Chem software,
40
in order to
investigate whether NiCp
2
’s electronic structure and magnetic behavior are sensitive to its coordi-
nation environment or not. To this end, we first consider isolated NiCp
2
using EOM-SF-CCSD. We
then modify the local ligand field and consider six differently ring-substituted NiCp
2
compounds
and the adsorption complex of a single NiCp
2
molecule on a model surface. Our set of modified
nickelocene compounds includes both electron withdrawing and donating substituents as well as
two bent structures, which are used as precursors of nickelocene chains. As a model surface, we
consider magnesium oxide and its most commonly exposed (001) plane. Owing its well-defined
ionic character and simple cubic structure, MgO is suitable for testing our approach using simpli-
fied cluster models of NiCp
2
on MgO(001). Insulating thin MgO layers are often used to mediate
the interaction between the magnetic species and the metallic substrate, improving the magnetic
bistability of the nano-magnet.
54, 55
While EOM-CC methods (as well as other many-body ap-
proaches) scale steeply with the system size, which precludes brute-force applications to large
molecular magnets, the combination of the spin-flip approach with TD-DFT (SF-TD-DFT)
56, 57
circumvents this problem, allowing us to investigate spin states of sufficiently large clusters of the
NiCp
2
/MgO(001) complex. Benchmark studies of Cu(II)- and Fe(III)-based molecular magnets
have shown good agreement between SF-TD-DFT and EOM-SF-CCSD calculations and/or ex-
periment.
32, 53
The benchmarks consistently show
32, 53, 57
that SF-TD-DFT performs the best when
deployed with the non-collinear kernel
57, 58
and hybrid and range-separated hybrid functionals,
such as PBE0 and LRC-!PBEh. Another benchmark study of a set of binuclear transition-metal
complexes have reported that the SF-TD-DFT approach provided more accurate magnetic coupling
constants (i.e., spin-state energy gaps) than BS-DFT, with the additional advantage of avoiding the
ambiguities associated with the use of non-unique spin projectors.
59
We note that the present methodology (implemented in ezMagnet) for computing molecular
magnetic properties is general and can be combined with any electronic structure method that can
furnish SOCs and transition angular momentum. Because calculations of SOCs
41–43
and transition
98
angular momentum are formulated using reduced density matrices, their extension to a broader
class of methods is straightforward. Recent formulations of SOCs in combination with computa-
tionally more efficient than EOM-SF-CC yet reliable approaches, such as RASCI
43
and SF-TD-
DFT,
60
extend the scope of applicability of the algorithms implemented in ezMagnet to larger
magnetic systems. Taking advantage of this generality, here we investigate magnetic behavior of
NiCp
2
on MgO(001) using SF-TD-DFT spin states. This is the first application of SF-based ap-
proaches to describe magnetic molecules on surfaces, illustrating the utility of this methodology.
For the NiCp
2
molecule, we analyze the SOCs between the spin-orbit interacting states using NTOs
and explain the origin of magnetic anisotropy. We then consider spin states’ ordering and their or-
bital character as the key descriptors of the magnetic behavior of NiCp
2
in a modified surrounding.
Preservation of such descriptors is the prerequisite for robust magnetic anisotropy against changes
of the local environment, interpretation corroborated by our magnetic property calculations on the
ring-substituted NiCp
2
compounds and on the NiCp
2
/MgO(001) adsorption model.
The chapter is organized as follows. Section 4.2 presents the theoretical background. We begin
with a brief introduction of the SF approach and of the molecular orbital picture introduced to
interpret the computed properties, followed by a recap of our formalism for computing SOCs,
Zeeman interactions, and magnetic properties. Section 4.3 provides the computational details and
describes model systems. Section 4.4 presents the results. Our concluding remarks are given in
Section 4.5.
4.2 Theoretical background
We describe open-shell molecular magnets using the EOM-SF-CC method.
27–29
In SF approaches,
a single-determinant high-spin state is used as a reference from which all multi-configurational
lower-spin states can be obtained by spin-flipping excitations. Within the EOM-SF-CC formalism,
target-state wave functions have the following form:
j i
S;S1
M
S
=S1
=R
M
S
=1
e
T
j
0
i
S
M
S
=S
; (4.1)
99
wherej
0
i is the high-spin reference Slater determinant,e
T
j
0
i is the CC wave function, andR
is an excitation operator that flips the spin of an electron. EOM-SF provides a balanced treatment
of relevant spin states (i.e., M
S
= S 1 states), does not require choosing an active space, and
describes dynamical and non-dynamical correlation in a single computational step. EOM-SF has
been successfully used to treat ground and excited states of various molecular magnets.
22, 30–33
We
describe singlet and triplet states of NiCp
2
and of the six ring-substituted NiCp
2
derivatives by
EOM-SF-CCSD starting from a high-spind
8
triplet reference. To describe the NiCp
2
/MgO(001)
model complex, we use SF within the TD-DFT ansatz.
56, 57
We note that in SF calculations, only
M
S
= 0 states are used to compute energy gaps and state and transition properties, and that the
high-spin state only serves as the reference.
27, 29, 32, 61
Magnetic properties arise from spin-orbit and magnetic field (Zeeman) interactions. We de-
scribe these effects by a two-step state-interaction procedure:
22
first, EOM-CC states are computed
and then these states are used to evaluate matrix elements of the spin-orbit (H
SO
) and Zeeman (H
Z
)
operators. The latter describes the interaction of the electrons with the external magnetic fieldH
H
Z
=
B
H (L +g
e
S); (4.2)
whereg
e
= 2:0023 is the free sping-factor,
B
is the Bohr magneton, andS andL are the total
spin and orbital angular momentum operators, respectively. Spin-orbit interactions are treated by
using the Breit-Pauli Hamiltonian.
37, 38
Spin-orbit and field-perturbed states are then obtained by
diagonalization of the zero-order Hamiltonian (H
0
) augmented byH
SO
andH
Z
:
H =H
0
+H
SO
+H
Z
: (4.3)
In Q-Chem,
40
calculation of SOCs and matrix elements of the Zeeman operator is currently im-
plemented for EOM-SF-CC, RASCI, and SF-TD-DFT,
41, 43, 60
and can be easily extended to any
100
method that can provide reduced density matrices. Note that by virtue of the Wigner-Eckart theo-
rem, only theM
S
= 0 transition density matrix needs to be explicitly computed in the SF calcula-
tions.
To gain insight into the nature of electronic states and transitions, we analyze magnetic behav-
ior of NiCp
2
using transition density matrix between spin-orbit interacting states and the resulting
NTOs. In previous studies, such analysis has been carried out for diradical and iron-based molec-
ular magnets, whose SOCs are determined by one NTO contribution only. Here, we extend this
treatment to the cases in which two leading NTO pairs contribute to the transition property.
The key quantity is the one-electron transition density matrix:

FI
pq
=h
F
ja
y
p
a
q
j
I
i; (4.4)
wherea
y
p
anda
q
are the creation and annihilation operators associated with the
p
and
q
molecular
orbital basis. A singular-value decomposition
62, 63
of
FI
pq
yields the most compact description of
the one-electron excitation in terms of a unique set of orbitals — hole and particle NTOs — which
are independent of the method/basis set choice thus allowing a meaningful comparison of different
levels of theory. In the basis of these NTOs, expectation values of one-electron operators (
^
A) can
be computed as:
h
F
j
^
Aj
I
i =
X
k;l

k

l
h
p
k
j
^
Aj
h
l
i; (4.5)
where are singular values associated to the NTO pairs. Except for the trivial case of one dom-
inant NTO pair, observables can be affected from the contribution of cross terms, which may
lead to very different properties for transitions between states characterized by equivalent NTO
pairs.
50, 64, 65
Here, we show that nickelocene’s states with identical NTOs can have very differ-
ent SOC based on the different form of their two-configurational wave function (symmetric or
antisymmetric combination), see discussion in Section 4.4 and in Appendix E.
101
SOC is responsible for the zero-field splitting of the 2S + 1 degenerate components of the spin
S ground state, giving rise to spin-inversion barrier (U =jDjS
2
) and magnetic anisotropy (quan-
tified by the spin-Hamiltonian parameterD). Temperature- and field-dependent magnetization and
susceptibility are obtained from first- and second-order derivatives, respectively, of the resulting
partition function (Z =
P
n
e
En(H)=RT
) with respect to the field,
44
M(T;H) =NkT
@lnZ(T;H)
@H
(4.6)
and
(T;H) =NkT
@
2
lnZ(T;H)
@H
2
; (4.7)
which allows direct comparison with experiment without relying on the spin-Hamiltonian formal-
ism. In Eqs. (4.6) and (4.7), H is the magnetic field vector, while E
n
(H) are the SOC- and
field-perturbed energies of the magnetic sublevels contributing to the partition functionZ(T;H).
This computational approach is implemented in the ezMagnet software and was shown to yield
accurate results for a set of small- and medium-sized iron-based SMMs.
22
This approach is gen-
eral and can be combined with any other ab initio method, which provides angular momentum and
spin-orbit matrix elements, e.g., SF-TD-DFT and RASCI for more extended systems. Calculation
of matrix elements of the spin operator entering Eq. (4.2) can be easily derived from spinS and
spin projectionM
S
of the states involved.
22
4.3 Computational details and model systems
To investigate the origin of magnetic anisotropy in the nickelocene (NiCp
2
) molecular magnet,
we first consider an isolated NiCp
2
molecule. In NiCp
2
, the Ni(II) transition metal ion is sand-
wiched between two cyclopentadienyl (Cp or C
5
H

5
) rings. NiCp
2
has a triplet ground state with
a d
8
electronic configuration. The molecule possesses a C
5
rotational axis collinear with the z
direction. Here, we consider the centrosymmetric staggered molecular structure (ligand field of
D
5d
symmetry) with mean Ni–C distance of 2.185
˚
A, taken from single-crystal X-ray diffraction
102
studies.
66
Magnetic susceptibility measurements revealed a deviation from the Curie-Weiss law
below 70 K.
67, 68
This departure from the Curie isotropic behavior of a paramagnetic molecule has
to be ascribed to the large SOC and, thus, ZFS of the magnetic sublevels (withM
S
=1; 0) of the
triplet ground state. Fig. 4.1 shows NiCp
2
staggered configuration, electronic states (both triplet
and singlet), and spin-orbit splitting of the triplet ground state as computed from EOM-SF-CCSD
calculations.
Figure 4.1: a) Structures of the nickelocene with staggered rings (coordinates are from Ref.
66). The bond of the metal with the Cp centroid is shown with blue dash lines. Color code: Ni
— purple, C — gray, and H — white. b) Electronic configurations of the high-spinM
S
= 1
triplet reference and the low-spin target states: M
S
= 0 triplet (j1i) and singlets (j2i,j3i,
andj4i). State energies and character are obtained from EOM-SF-CCSD calculations. The
two closed-shell configurations in statesj3i andj4i have equal weights. Based on natural
orbital analysis, the two unpaired electrons reside on thed
yz
andd
xz
orbitals. c) Four lowest
electronic states and spin-orbit splitting of the S = 1 ground state. Spin-orbit splitting is
computed via a two-step state-interaction scheme. Magnetic anisotropyD is obtained from
the energy splitting between the magnetic sublevels of the triplet ground state.
We also examine ligand-field effects on NiCp
2
electronic structure (e.g., spin state ordering and
energy gaps) and magnetic behavior (e.g., magnitude of SOCs) and select a series of six differently
ring-substituted NiCp
2
derivatives. Below we refer to the individual molecules by the numbers
shown in Fig. 4.2. In complex 1, one C–H group for each C
5
H

5
ring is replaced by a phosphorus
atom, following the strategy adopted in the design of bis-cyclopentadienyl dysprosium molecular
magnets to enhance spin-reversal energy barrier and operating temperature.
69, 70
We then examine
three systems in which one hydrogen atom for each Cp framework is substituted by an isovalent
functional group: complex 2 includes two methyl groups, complex 3 has two cyano groups, and
103
complex 6 includes two aromatic ring substituents. Preparation of complex 2, i.e., Ni(C
5
H
4
R)
2
with R = CH
3
, is reported in Refs. 71 and 72. Complex 4 is selected as precursor of NiCp
2
-based
chains to be used as memory devices.
73
Complex 5 is an example of bridged NiCp
2
in which the
two Cp units are linked by naphthalene.
74
Structures of complex 1, 2, 3, and 6 were optimized with
!B97X-D/cc-pVDZ for the high-spin triplet state, whereas for complex 4 and 5, experimental bent
structures were considered, as reported in Ref. 73 and Ref. 74, respectively.
Figure 4.2: Structures of six ring-substituted nickelocene derivatives. In complex 1 two C–H
groups are substituted with two P atoms. In complex 2, 3, and 6, two H atoms are substituted
with methyl, cyano, and aromatic groups, respectively. Complexes 4 and 5 are bent structures
taken from Ref. 73 and Ref. 74, respectively. The bond of the metal with the Cp centroid is
shown with blue dash lines. Color code: Ni — purple, P — orange, N — blue, C — gray, and
H — white.
We also investigate electronic structure and magnetic behavior of NiCp
2
adsorbed on a support
using MgO(001) as a model surface. MgO is chosen due to its insulating character and low phonon
density, thereby suppressing magnetic relaxation via negligible spin-phonon coupling.
75
To obtain
a reliable finite-cluster model of NiCp
2
on the MgO(001) surface, we used an embedded cluster
approach, which is often employed in computational catalysis to describe isolated point defects
or isolated adsorbed molecules on ionic surfaces.
76
This structural model is well suited for the
description of individual molecular magnets on a surface, whereas using periodic boundary condi-
tions would require large supercell to minimize the artificial molecule-molecule interactions with
the periodic images. Further details can be found in Appendix F.
First, we performed a DFT structure optimization of the NiCp
2
/Mg
49
O
49
model cluster em-
bedded in a sufficiently large array of point charges resembling the ionic MgO surface. Fig. 4.3
104
shows our embedded cluster model. We increase the number of point charges to converge adsorp-
tion energy and equilibrium distance between the NiCp
2
and MgO(001) surface (see Table 4.8).
Second, we considered a smaller cut-out (i.e., NiCp
2
/Mg
25
O
25
) of the optimized NiCp
2
/Mg
49
O
49
model cluster and investigate spin states and magnetic properties of NiCp
2
on MgO(001) using
the SF-TD-DFT ansatz. For DFT structure optimizations, we employed the PBE0
77, 78
functional,
whereas to compute spin states, we followed recommendations in Refs. 32, 53 for transition-metal
compounds and used SF-PBE0 and SF-LRC-!PBEh
79
within the non-collinear formulation of SF-
TD-DFT.
57, 58
The size and shape of the NiCp
2
/Mg
25
O
25
quantum mechanical region was chosen
to minimize the computational cost while providing a structural model comparable with models
already available in the literature for adsorbates on MgO(001).
80, 81
STM images of NiCp
2
on
Cu(001)
14–16
and Ag(110)
17, 18
show that NiCp
2
is bonded perpendicularly to the surface through a
Cp ring, whereas the other Cp ring is exposed to the vacuum. For these reasons, we take inspira-
tion from previous studies on metal substrates
14–18
and select the NiCp
2
perpendicularly adsorbed
on MgO(001) as guess structure. Additionally, we investigate two possible adsorption sites, i.e.,
Ni(II) ion on top of oxygen and magnesium.
Figure 4.3: a) The embedded cluster setup used for structure optimization: the all-electron
QM region (NiCp
2
/Mg
49
O
49
) is treated with PBE0/6-31G

, while the outermost region con-
tains point charges. The QM region is shown as lifted for clarity. Here, NiCp
2
is on-top of
Mg
2+
adsorption site. b) Top and side views of the embedded NiCp
2
/Mg
49
O
49
PBE0 region.
c) Top and side views of a smaller cut-out (NiCp
2
/Mg
25
O
25
) used for the SF-TD-DFT calcula-
tions. The bond of the metal with the Cp centroid is shown with blue dash lines. Color code:
Ni — purple, Mg — green, O — red, C — gray, and H — white.
105
For NiCp
2
and its six derivatives, triplet ground states and singlet excited states are computed
using EOM-SF-CCSD, whereas for the model complex of NiCp
2
on MgO(001) we employed SF-
TD-DFT. All SF calculations are performed starting from a high-spin triplet state, as shown in Fig.
4.1. This reference state is represented by a single determinant (high-spinM
S
= 1) and is well
described by the CC expansion of the wave function or by Kohn-Sham DFT. From this reference,
single SF excitations generate the triplet state linear combination of two M
S
= 0 determinants
with equal weights and three singlet states of open-shell character. Given that NiCp
2
compounds
have a tripletS = 1 ground state, the energy barriers for spin inversionU =jDjS
2
is equal to
the magnetic anisotropyD. Following previous studies,
14–18
we obtainD from the ZFS between
the three components (M
S
= 0;1) of the lowest triplet state (S = 1, denoted asj1i in Fig. 4.1).
The split states (andD) are obtained by diagonalizing the spin-orbit perturbed Hamiltonian of Eq.
(4.3).
For isolated NiCp
2
and complexes 1-4, we used Dunning’s cc-pVTZ basis set.
82–84
The SOC
calculations of complexes 5 and 6 and of NiCp
2
/Mg
25
O
25
adsorption complex were performed
using Dunning’s cc-pVDZ basis set.
82–84
As noted before,
22
using cc-pVTZ rather than cc-pVDZ
has a negligible effect on the state energy, magnetic anisotropy, and computed magnetization and
susceptibility data (deviations are less than 1 cm
1
forD and within 1% for the average magne-
tization and susceptibility, see Tables 4.4 and 4.5 and Fig. 4.14. For isolated NiCp
2
, SF-PBE0
reproduces well state ordering, magnetic anisotropy, and magnetic properties as compared with
EOM-SF-CCSD (Section 4.7), but the relative energies are quite different (e.g., triplet-singlet en-
ergy gap is about 9800 cm
1
for EOM-SF-CCSD and 6500 cm
1
for SF-PBE0). Importantly, the
calculated SOCs of isolated NiCp
2
are not sensitive to the basis set (cc-pVDZ versus cc-pVTZ),
neither to the ab initio method employed (EOM-SF-CCSD or SF-PBE0), see Table 4.11.
Open-shell reference states were treated using unrestricted Hartree-Fock (HF). For NiCp
2
, spin
contamination of the reference and EOM-SF-CCSD states is small: the correspondinghS
2
i values
106
are 2.00 forS = 1 and between 0.06 and 0.13 for the singlet states (see Table 4.2). For the six ring-
substituted NiCp
2
compounds, spin contamination is less than 0.23, whereas, for the NiCp
2
/MgO
model complex, deviations are less than 0.08, see Table 4.3 and 4.9.
To speed up the EOM-SF-CCSD calculations, we applied: (i) Frozen-core approximation; (ii)
Cholesky decomposition
85
with a threshold of 10
3
for two-electron integral calculations, (iii)
open-shell frozen natural orbital (OSFNO)
86
truncation of the virtual space with the total popula-
tion threshold of 99%, and (iv) single precision execution.
87
All electronic structure calculations were performed using the Q-Chem software.
39, 40
4.4 Results and discussion
For isolated NiCp
2
, we computed four spin states: a triplet ground state and three singlet excited
states (see Fig. 4.1). Because of the near-degeneracy of the two frontier orbitals (here,d
yz
andd
xz
orbitals), electronic structure of nickolocene follows a diradical pattern (see, for example, Refs. 88
and 52)—the manifold of the low-lying states are derived by distributing two electrons in the two
near-degenerate frontier orbitals. The high-spin triplet state withM
S
= +1 is single-determinantal,
whereas wave functions ofM
S
= 0 triplet and singlet states are two-configurational (require linear
combination of two Slater determinants). The effective number of unpaired electrons,
89
n
n;nl
, of
each state is exactly 2, indicative of a strong diradical and open-shell character (see Table 4.2 ).
Fig. 4.9 shows frontier natural orbitals for the four states. These are nearly perfect d
xz
and d
yz
orbitals, indicating that unpaired electrons are mainly localized on the Ni(II) metal center, with
almost no involvement of the Cp rings. For each transition between the spin-orbit coupled states,
two NTO pairs contribute to the overall SOC (the number of non-zero singular values is about 2).
Fig. 4.4 shows the two leading NTO pairs between states 1 and 2. Transition between states 1
and 2 involve an orbital torque fromd
yz
tod
xz
(first NTO pair) and vice versa (second NTO pair).
Similar picture describes the transition between the triplet ground state and higher singlet excited
states (see Figs. 4.10 and 4.11).
107
Figure 4.4: Hole and particle NTO pairs of the spinless density matrix, giving rise to SOC
within the states 1 and 2 of nickelocene (EOM-SF-CCSD/cc-pVTZ). Singular values are
1.19 and 1.15, respectively. Red, green, and blue axes indicatex,y, andz coordinates axes,
respectively.
The SOC between the triplet ground state and the singlet excited states lifts the degeneracy of
the NiCp
2
triplet ground state, which results in a low-spin (S = 1, M
S
= 0) ground state and a
high-spin (S = 1,M
S
=1) doubly degenerate excited state (see Fig. 4.1). The energy barrier
between the ground state and the first excited state is about 15 cm
1
. Due to the axial symmetry
of NiCp
2
and to the spin S = 1 character of its ground state, this energy barrier corresponds to
the axial magnetic anisotropy (D). The computed value ofD = 15 cm
1
agrees reasonably well
with the experimentally derivedD parameter, which ranges between 25.6
67
and 33.6
68
cm
1
, and
with the NEVPT2/def2-TZVP value of 40 cm
1
.
90
Our calculations confirm that D is positive,
which implies that magnetic sublevel ordering is inverted with respect to molecules with negative
D values, such as single-molecule magnets. The latter have the states with highest spin projection
(M
S
=S) that are degenerate and lowest in energy, while for NiCp
2
the doubly degenerate level
with spin projectionM
S
=1 is the first excited state.
Fig. 4.5 shows the computed magnetizationM for a field oriented parallel and perpendicular to
theC
5
rotational axis; powder magnetization (M
av
) is obtained by numerical averaging over a large
108
set of field orientations. The magnetization curves rise with the field strength, but do not saturate
at strong magnetic fields. Fig. 4.5 shows the temperature dependence of the computes susceptibil-
ity main values (i.e.,
Xm
,
Ym
,
Zm
) obtained by diagonalization of the susceptibility 2nd-rank
tensor.
Zm
T and
Xm
T go to zero whenT approaches zero;
Zm
T continuously decreases, while

Xm
T passes through a maximum. We note that these field- and temperature-dependencies of
the magnetization and susceptibility are fingerprints of a spin-triplet molecule with positive axial
zero-field splittingD.
91
The measured temperature dependence of the inverse of the susceptibility
(black and green curves in Fig. 4.6) shows that beyond 70 K results follow the Curie-Weiss law,
i.e., =C=(T ), whereC is the Curie constant and is the Weiss constant.
67, 68
Below 70 K,
however, a deviation from this law is observed.
67, 68
In agreement with experiments, the computed
T value at 298 K is close to the expected value of 1.0 cm
3
K/mol for a spin-onlyS = 1 system
that follows the Curie law (see Section 4.8), and predict a deviation from linearity (Curie law) at
low temperatures (red curve in Fig. 4.6).
Figure 4.5: Calculated field-dependent magnetizations (on the left) of NiCp
2
at low temper-
ature (T = 2 K). Magnetization is in Bohr magneton (
b
) units. Calculated
Zm
T ,
Xm
T
(
Ym
T ), and
av
T (on the right) of NiCp
2
in the temperature range from 5 to 300 K and
under an applied field of 1 T. “av” stands for isotropic powder averaging.
To explain the nature of this magnetic behavior, we combine the NTO-picture of Fig. 4.4
(as well as of Figs. 4.10 and 4.11) and the transition-density matrix based analysis of calculated
orbital angular-momentum matrix elements and SOCs of Table 4.7 with their explicit calculation
109
Figure 4.6: Calculated temperature dependence of the inverse susceptibility (1/
av
) of NiCp
2
in the temperature range from 5 to 250 K (on the left) and from 5 to 80 K (on the right), and
under an applied field of 1 T. Calculated curves including three and four electronic states are
in blue and red, respectively. Experimental susceptibility data, i.e. black and green curves,
are taken from Ref. 67 and Ref. 68, respectively. Experimental magnetization data are not
available.
using wave functions of triplet and singlet states for a two-electron in two-orbital system of Section
4.10. Table 4.7 reportshL
i
i withi =x;y;z and SOC matrix elements. Orbital angular momentum
between state 1 and states 2, 3, and 4 is zero. However, while the calculated SOC is small between
states 1 and 2 and states 1 and 3, SOC is large between states 1 and 4, which is the origin of the
anisotropic magnetic behavior of NiCp
2
and deviation from the Curie law at low temperatures.
We note that for the transition between states 1 and 4, it is the average valueh
^
H
SO
L
i
i of the
^
H
SO
Lz
component of the spin-orbit operator to be large, whileh
^
H
SO
Lx
i andh
^
H
SO
Ly
i values are small. Indeed,
if the spin-orbit interaction between state 1 and the higher in energy singlet excited state (state 4) is
not included, the calculations fail to reproduce the experimental trend at low temperatures (see blue
curve in Fig. 4.6). This can be explained by looking at the eigenvalue analysis of the transition
density matrix. Each NTO pair contributes with the same weight to the overall SOC. However,
while for states 1 and 2 and for states 1 and 3, the sign of the two leading NTO contributions is
opposite, leading to the SOC cancellation, for states 1 and 4, the sign of the two SOC contributions
is the same, giving rise to large SOC. El-Sayed’s rules
92
explain why the cross terms in Eq. (4.5),
i.e.,hd
xz
j
^
H
SO
Lz
jd
xz
i andhd
yz
j
^
H
SO
Lz
jd
yz
i, are zero because of no orbital torque in between. This can
110
be rationalized by considering the proportionality between the spin-orbit
^
H
SO
Lz
and orbital angular
momentumL
z
operators, and that, in the basis of real harmonics (d orbitals in the case ofL = 2),
average valueshL
z
i, thush
^
H
SO
Lz
i, are zero between two identicald orbitals. In addition, by knowing
triplet and singlet wave functions of NiCp
2
(i.e., two electrons localized in two orbitals
A
and
B
),
one can explicitly compute transition orbital momentum and SOC between theS = 1 ground state
and the threeS = 0 excited states (see Section 4.10). The computed orbital angular momentum
between states 1 and states 2, 3, and 4 is zero because of their different spin (triplet versus singlet),
consistent with the results in Table 4.7 obtained from ab initio calculations. In contrast, we obtain
a zeroh1jH
SO
j2i andh1jH
SO
j3i SOC, while SOC between states 1 and 4 is large, as predicted by
the full calculation of the SOC matrix elements (see Table 4.7). This result can be ascribed to the
different nature of the wave functions of the three singlet states:
j2i (
A
(1)
B
(2) +
B
(1)
A
(2))((1)(2)(1)(2)) (4.8)
j3i (
A
(1)
A
(2)
B
(1)
B
(2))((1)(2)(1)(2)) (4.9)
j4i (
A
(1)
A
(2) +
B
(1)
B
(2))((1)(2)(1)(2)): (4.10)
Wave functionsj2i,j3i, andj4i share the same spin part. Wave functionsj3i andj4i have simi-
lar spatial component except for a sign: in statej4i the two configurations appear with the same
sign, while in statej3i they appear with the opposite sign (the weights of these configurations are
fixed by symmetry, as in other diradicals with symmetry-degenerate frontier orbitals, for example,
trimethylenemethane
88
). Spatial component of wave functionj2i is different. Due to the common
spin component of these wave functions, for all spin-orbit couplings, onlyhmjL
z
S
z
jni term of the
spin-orbit operator (i.e.,H
SO
L
+
S

+L

S
+
+L
z
S
z
) can survive due to a non-zerohS
z
i con-
tribution. However, due to the same sign in the spatial part of wave functionj4i, onlyh1jL
z
S
z
j4i
and thush1jH
SO
j4i is large because of a non-zerohL
z
i term, whilehL
z
i terms between states 1
and 2 and states 1 and 3 vanish, and so does the SOC.
111
We also consider six differently substituted NiCp
2
molecules (Fig. 4.2), with the goal to inves-
tigate weather changes in the local ligand fields (i.e., electron withdrawing/donating substituents
or bent structures) influence the triplet-singlet energy gap and so the SOC affecting their magnetic
behavior. Equilibrium structure of nickelocene does not change significantly upon substitution of
hydrogen atoms with methyl, cyano, and aryl groups (Ni–Cp ring height is about 1.82
˚
A). After
integration of P, the P–C distance increases from the average value of 1.38 (for C–C bond) to 1.78
˚
A, but the sandwich structure is retained. Therefore, nickelocene structure is resistant against ring
substitution. Moreover, our calculation show that all substituted NiCp
2
molecules preserve the
same spin state ordering (Table 4.3) and orbital character (Figs. 4.12 and 4.13). Additionally, also
the spin-orbit splitting (quantified by the energy barrierU or by the parameterD in Table 4.3) and
the anisotropic magnetic behavior (Fig. 4.7) do not change. This confirms that magnetic anisotropy
of NiCp
2
is not affected by changes of the ligand field.
If one desires to use NiCp
2
molecular magnets to trigger spin-flip transitions in magnetic
molecules, atoms, or arrays of magnetic unit on a support, it is important to verify whether NiCp
2
magnetic behavior is affected by the interactions with the support. If not, any change in the spin
excitation spectrum of nickelocene would reflect electronic structure and magnetic properties of
the magnetic molecule in the proximity, and NiCp
2
could be used a spin sensor. To verify this
assumption, we consider NiCp
2
adsorbed on a model surface (i.e., MgO(001)) and investigate its
chemical and electronic structure. From the DFT structure optimizations of NiCp
2
/MgO(001), we
identified the O-top as the most favorable adsorption site (see Table 4.8). NiCp
2
does not deform
upon adsorption. The staggered configuration of NiCp
2
is retained and NiCp
2
is found to adsorb
perpendicularly to the surface through one Cp ring, similarly to NiCp
2
on metal substrates.
14–18
The Ni–Cp ring height decreases from 1.82 to 1.78
˚
A upon adsorption on MgO(001), which corre-
sponds to a small charge transfer from the molecule to the surface. The adsorption energy is small
(about -7 kcal/mol), indicative of physisorption. Fig. 4.18 shows the spin density for the isolated
NiCp
2
and for the NiCp
2
/MgO model cluster. For NiCp
2
, spin density is localized on the frontier
d
xz
and d
yz
orbitals of the Ni atom. Upon adsorption, spin density is preserved; the molecule
112
Figure 4.7: Calculated field-dependent magnetizations (top) of (i) NiCp
2
, (ii) six ring-
substituted NiCp
2
compounds, and (iii) NiCp
2
/Mg
25
O
25
adsorption complex (T = 2 K). Mag-
netization is in Bohr magneton (
b
) units. Calculated
Zm
T ,
Xm
T (
Ym
T ), and
av
T (bottom)
in the temperature range from 5 to 300 K and under an applied field of 1 T. “av” stands for
isotropic powder averaging. Properties are obtained by EOM-SF-CCSD/cc-pVTZ calcula-
tions, with the exception of complexes 5 and 6 for which we employed EOM-SF-CCSD/cc-
pVDZ and of NiCp
2
/Mg
25
O
25
for which we used SF-PBE0/cc-pVDZ.
retains S=1 and there is no significant spin polarization from the surface. Table 4.8 compares
electronic states of isolated NiCp
2
with its electronic structure on the MgO(001) surface. Whereas
electronic states of the isolated molecule are computed with EOM-SF-CCSD, EOM-SF-MP2 and
SF-TD-DFT, the electronic states of the adsorption complex are accessible by SF-TD-DFT only.
Table 4.9 shows the full comparison. We observe that the relative energies depend significantly on
the electronic structure method, however, the spin-state ordering and orbital character of the states
for NiCp
2
on MgO(001) are unchanged. Moreover, similarly to the isolated molecule, Fig. 4.8
113
shows that the two leading NTO pairs between triplet and singlet states of NiCp
2
on MgO involve
an orbital rotation betweend
xz
andd
yz
orbitals. This analysis indicates that magnetic anisotropy
and magnetic properties of NiCp
2
on MgO(001) would also be retained, which is confirmed by
our magnetic properties calculations (see magnetization and susceptibility plots of Fig. 4.7 and
magnetic anisotropy of Table 4.10). Our calculations also reveal that SOC and related spin-orbit
splitting (i.e.,D) of the molecule are not sensitive to the adsorption on the model MgO(001) sur-
face (see Table 4.10 and 4.11). These results for NiCp
2
/MgO(001) are consistent with previous
STM studies investigating the adsorption of NiCp
2
on a metal surface.
15–18
When placed above
a Cu(001)
15
(or Ag(110))
17
surface, a magnetic anisotropy energy D of 26 cm
1
(or 31 cm
1
)
was determined by STM experiments, which is consistent with experimentalD values for NiCp
2
powder samples (25.6
67
and 33.6
68
cm
1
), and with our results for the isolated NiCp
2
and the
NiCp
2
/MgO(001) adsorption complex.
Our results indicate that electronic properties of NiCp
2
are extremely robust and are not per-
turbed by the variations of the ligand field (by modifying substituents) and by its local environment
(by adsorption on a metal
15–18
or insulating surface as investigated in this work), which is key prop-
erty for molecular spin sensors capable of targeting magnetic interactions in the proximity rather
than modification of its surrounding.
Table 4.1: Energies (in cm
1
) of the four lowest states of the isolated NiCp
2
and of NiCp
2
on top of MgO(001), computed using EOM-SF-CCSD/cc-pVTZ and SF-TD-DFT/cc-pVTZ
with PBE0. “NiCp
2
/(Mg)O” and “NiCp
2
/Mg(O)” stand for the adsorption complexes of
NiCp
2
/MgO(001) with the Ni atom on-top of Mg
2+
and O
2
adsorption sites, respectively
(cluster model: NiCp
2
/Mg
25
O
25
). Differences between NiCp
2
and NiCp
2
/MgO with SF-PBE0
are reported in parenthesis.
NiCp
2
NiCp
2
/(Mg)O NiCp
2
/Mg(O)
EOM-SF-CCSD SF-PBE0 SF-PBE0 SF-PBE0
j1i 0.0 0.0 0.0 0.0
j2i 9805.5 6552.1 6438.1 (144.0) 6434.4 (117.4)
j3i 9894.2 6585.6 6450.7 (134.9) 6464.8 (120.7)
j4i 15144.0 12240.5 12052.9 (187.6) 12041.1 (199.4)
114
Figure 4.8: Hole and particle NTO pairs of the spinless density matrix between states 1 and
2 of the NiCp
2
/Mg
25
O
25
adsorption complex (SF-PBE0/cc-pVTZ). Ni atom is on-top of O
2
.
Singular values are 0.5 and 0.5, respectively. Red, green, and blue axes indicatex,y, andz
coordinates axes, respectively.
4.5 Conclusions
We investigated electronic structure and magnetic behavior of the NiCp
2
molecular magnet and of
its derivatives (i.e., six ring-substituted NiCp
2
and the NiCp
2
/MgO adsorption model) using EOM-
SF in combination with the ezMagnet software. Our NO and NTO analyses of electronic states
and excitations, and corresponding transition properties (e.g., SOCs) are directly extracted from
the many-body correlated EOM-SF-CCSD wave functions, providing both pictorial representation
of the spin-orbit coupled states and quantitative analysis of their contributions to nickelocene’s
magnetic anisotropy.
For the isolated NiCp
2
, calculated magnetic anisotropy and susceptibility curves agree well
with experiment, reproducing the deviation from the Curie law in the low temperature regime.
We demonstrated that such anisotropic magnetic behavior originates from the SOC between the
triplet ground state (state 1) and the third singlet state (state 4), which is large, while coupling
with lower singlet states (states 2 and 3) is small and does not affect NiCp
2
’s magnetic behavior.
Although NiCp
2
singlet excited states are characterized by two nearly equivalent NTO transitions
115
(same nature and same weight), the NTO contributions to the SOC are canceling out between
states 1 and 2 (or 3), while for the highest in energy singlet state, the NTO contributions have
same sign and sum up, leading to a substantial SOC. This analysis of the SOCs is then combined
with explicit calculation of SOCs using wave functions of triplet and singlet states for a two-
electron in two-orbital system. By doing so, we found that SOC between states 1 and 4 survives
because of the nature of the singlet wave function of state 4, i.e., symmetric combination of two
Slater determinants, while it vanishes between states 1 and 2 (or 3), whose wave functions are
characterized by an antisymmetric combination of two leading configurations.
Qualitatively, the magnitude ofD in SMMs can often be interpreted by using a simple molec-
ular orbital picture, as has been done in earlier studies (e.g., see Refs. 93 and 94). However, as
pointed out before,
93
this approach is valid only for orbitally nondegenerate ground states. The
working expressions in terms of molecular orbitals and their energies have been derived for cases
where all relevant states can be expressed by a single Slater determinant (which does not admit
spin-flipped states) and state energy differences can be approximated by orbital energy differences
(which does not admit contributions from degenerate states).
93, 94
Therefore, this model is not suit-
able for nickelocene, whose magnetic anisotropy originates from the interaction between the three
components of the triplet state and two-determinantal singlet excited states.
Upon modification of NiCp
2
ligand field and adsorption of NiCp
2
on the MgO(001) surface
model, we observed that NiCp
2
’s geometry, its state ordering, and orbital character of the spin
states are unchanged. State ordering and character can be used as key descriptors. On the basis
of such descriptors, one would expect that a robust electronic structure would be equivalent to
a robust magnetic behavior. Observation that is then confirmed by explicit calculation of SOC,
magnetic anisotropy, and magnetic properties. Therefore, our calculations on NiCp
2
in different
local environments confirm resilience of NiCp
2
magnetic behavior and support using NiCp
2
as a
spin sensor. These results advance our understanding of magnetic behavior of molecular magnets
on surface models that can be used as spin sensors.
116
This work on the NiCp
2
adsorption model is the first study using EOM-SF approaches for
tackling such complex molecule/surface adducts. This has been possible by combining the SOC
implementation of SF-TD-DFT with the generality of the ezMagnet software, which can be inter-
faced with any method providing orbital angular momentum and SOCs. In addition, our approach
has the advantage that it does not rely on spin-Hamiltonian formalism, selection of active space,
or spin projections associated to the BS-DFT approach. Compared with periodic DFT+U, our
SF-TD-DFT protocol does not introduce additional system-dependent parameters apart from the
ones already defining the exchange-correlation functional. Moreover, our approach does not ap-
ply periodic boundary conditions neither uses plane waves. Rather, it obtains reliable structures
of NiCp
2
on the ionic MgO(001) plane from a quantum mechanical cluster–point charges com-
bined approach. The latter has the advantage to be able to investigate individual magnetic species
when deposited on a surface without the need of large supercells. However, it has the limitation
to be applicable to ionic surfaces only. However, chemical structures of nano-objects with metal
substrates might be described well by DFT-based approaches. In this regard, the computational
protocol implemented in the ezMagnet software has the potential to be combined (in addition to
EOM-CC and SF-TD-DFT) with density embedding formalism
95, 96
(e.g., EOM-SF-CC-in-DFT or
SF-TD-DFT-in-DFT) allowing us to investigate electronic structure of magnetic species on metal
surfaces.
We hope that our study will motivate further applications of SF-TD-DFT combined with vari-
ous embedding techniques to study even larger magnetic systems, and will inspire the development
of robust methodologies suitable for tackling complex electronic structures in extended structural
models.
117
4.6 Appendix A: Wave function analysis
Table 4.2: Wave function properties of the target states of NiCp
2
; EOM-SF-CCSD/cc-pVTZ.
EOM-SF-CCSD/cc-pVTZ E, cm
1
n
u
n
u;nl
hS
2
i D
j1i 0.0 1.96 2.01 2.00
j2i 9805.5 2.03 2.00 0.07
j3i 9894.2 2.05 2.01 0.07
j4i 15144.0 2.12 2.00 0.13 15.4
Effective numbers of unpaired electrons (n
u
andn
u;nl
) are computed using Head-Gordon’s
formulas.
89
ExperimentalD parameter is in the range of 25.6 (Ref. 67) - 33.6 (Ref. 68) cm
1
.
CalculatedD parameter is obtained from the spin-orbit splitting usingU =jDjS
2
forS = 1.
Figure 4.9: Singly occupied natural orbitals (SONOs) of triplet (state 1) and singlet states
(states 2, 3, and 4); EOM-SF-CCSD/cc-pVTZ.n = n

+n

and n = n

n

(in paren-
thesis) are provided. n

and n

values are obtained from the occupancies of the and
natural orbitals, respectively. Red, green, and blue axes indicatex,y, andz coordinates axes,
respectively.
118
Figure 4.10: Hole and particle NTO pairs of the spinless density matrix, contributing to the
overall SOC between the states 1 and 3 of NiCp
2
(EOM-SF-CCSD/cc-pVTZ). Singular values
are 1.19 (above) and 1.16 (below). Red, green, and blue axes indicatex,y, andz coordinates
axes, respectively.
119
Figure 4.11: Hole and particle NTO pairs of the spinless density matrix, contributing to the
overall SOC between the states 1 and 4 of NiCp
2
(EOM-SF-CCSD/cc-pVTZ). Singular values
are 1.17 (above) and 1.11 (below). Red, green, and blue axes indicatex,y, andz coordinates
axes, respectively.
120
Figure 4.12: Hole and particle NTO pairs of the spinless density matrix, contributing to the
overall SOC between the states 1 and 2 of complexes 1, 2, and 3 (EOM-SF-CCSD/cc-pVTZ).
Color code: Ni — purple, P — orange, N — blue, C — gray, and H — white. Red, green, and
blue axes indicatex,y, andz coordinates axes, respectively.
121
Figure 4.13: Hole and particle NTO pairs of the spinless density matrix, contributing to the
overall SOC between the states 1 and 2 of complexes 4, 5, and 6 (EOM-SF-CCSD/cc-pVTZ,
with the exception of complexes 5 and 6 for which cc-pVDZ basis set has been used). Color
code: Ni — purple, C — gray, and H — white.
122
Table 4.3: Wave function properties of the target states of the six ring-substituted NiCp
2
compounds; EOM-SF-CCSD/cc-pVTZ (unless specified otherwise).
2P (1) E, cm
1
n
u
n
u;nl
hS
2
i D
j1i 0.0 1.96 2.02 2.00
j2i 9038.9 2.10 2.01 0.12
j3i 9403.3 1.27 1.32 0.11
j4i 14295.9 1.48 1.43 0.23 14.6
2CH
3
(2) E n
u
n
u;nl
hS
2
i D
j1i 0.0 1.96 2.01 2.00
j2i 9702.1 2.00 1.99 0.08
j3i 9774.9 2.06 2.01 0.08
j4i 14834.5 2.10 2.00 0.17 15.1
2CN (3) E n
u
n
u;nl
hS
2
i D
j1i 0.0 1.96 2.01 2.01
j2i 9676.4 2.08 2.01 0.10
j3i 9895.2 1.60 1.74 0.09
j4i 14973.5 1.72 1.75 0.20 14.8
Bent1 (4) E n
u
n
u;nl
hS
2
i D
j1i 0.0 1.96 2.01 2.00
j2i 9285.0 1.41 1.52 0.07
j3i 9520.6 2.05 2.01 0.07
j4i 14776.9 1.51 1.54 0.13 15.6
Bent2 (5)
a
E n
u
n
u;nl
hS
2
i D
j1i 0.0 1.96 2.01 2.00
j2i 8990.4 1.56 1.69 0.07
j3i 9177.4 2.06 2.01 0.07
j4i 14200.5 1.66 1.71 0.13 15.8
Aryl (6)
a
E n
u
n
u;nl
hS
2
i D
j1i 0.0 1.96 2.01 2.00
j2i 9246.1 2.07 2.01 0.09
j3i 9356.4 1.98 1.99 0.08
j4i 14182.9 2.08 1.99 0.16 15.5
a
EOM-SF-CCSD/cc-pVDZ.
Effective numbers of unpaired electrons (n
u
andn
u;nl
) are computed using Head-Gordon’s
formulas.
89
CalculatedD parameter is obtained from the spin-orbit splitting usingU =jDjS
2
forS = 1.
123
4.7 Appendix B: Basis-set effects
Table 4.4: Wave function properties of the target states of NiCp
2
obtained by EOM-SF-CCSD
and SF-PBE0 calculations using the cc-pVDZ and cc-pVTZ basis sets.
EOM-SF-CCSD/cc-pVDZ E, cm
1
n
u
n
u;nl
hS
2
i
j1i 0.0 1.96 2.01 2.00
j2i 9636.9 2.03 2.00 0.07
j3i 9683.5 2.04 2.01 0.06
j4i 14884.3 2.12 2.01 0.13
EOM-SF-CCSD/cc-pVTZ E, cm
1
n
u
n
u;nl
hS
2
i
j1i 0.0 1.96 2.01 2.00
j2i 9805.5 2.03 2.00 0.07
j3i 9894.2 2.05 2.01 0.07
j4i 15144.0 2.12 2.00 0.13
SF-PBE0/cc-pVDZ E, cm
1
n
u
n
u;nl
hS
2
i
j1i 0.0 2.02 2.00 2.03
j2i 6624.4 1.95 1.99 0.04
j3i 6675.6 2.00 2.00 0.03
j4i 12394.7 1.96 1.99 0.08
SF-PBE0/cc-pVTZ E, cm
1
n
u
n
u;nl
hS
2
i
j1i 0.0 2.02 2.00 2.03
j2i 6552.1 1.94 1.99 0.04
j3i 6585.6 2.01 2.00 0.04
j4i 12240.5 1.96 1.99 0.08
Effective numbers of unpaired electrons (n
u
andn
u;nl
) are computed using Head-Gordon’s
formulas.
89
124
Table 4.5: CalculatedD parameter of NiCp
2
as extracted from the spin-orbit splitting of the
triplet ground state (S = 1,U =jDjS
2
= D) using both the EOM-SF-CCSD and SF-PBE0
methods and the cc-pVDZ and cc-pVTZ basis sets.
Method D, cm
1
EOM-SF-CCSD/cc-pVDZ 15.6
EOM-SF-CCSD/cc-pVTZ 15.4
SF-PBE0/cc-pVDZ 15.8
SF-PBE0/cc-pVTZ 15.8
Exp. 25.6
a
-33.6
b
a
Ref. 67.
b
Ref. 68.
Figure 4.14: Calculated magnetization (left) and susceptibility (right) plots of NiCp
2
obtained
with the cc-pVDZ and cc-pVTZ basis sets. Both EOM-SF-CCSD and SF-PBE0 methods are
used.
125
4.8 Appendix C: Molar susceptibility within the Curie law
Molecules that have a ground states with no first-order angular momentum and large separation in
energy from the first excited state (such that coupling with the ground state is neglected) follow the
Curie Law:
=
Ng
2

2
3KT
S(S + 1); (4.11)
whereN is the Avogadro number,g is the g-factor, is the Bohr magneton, andk is the Boltzmann
constant. The susceptibility varies asC=T , where the constantC depends only on the multiplicity
of the ground state. The two essential constants in magnetismk and are then given by:
Boltzmann constantk = 1:380658 10
16
erg K
1
Bohr magneton = 9:274015 10
21
erg T
1
so that the constantN
2
=3K appearing in the Curie law is equal to 0.125048612 cm
3
mol
1
(very
close to 1/8). Values of (4/g
2
)T for different spin values are calculated from the Curie law and
are reported in Table 4.6.
Table 4.6: Values of (4/g
2
)T as a function of the spin.
S (4/g
2
)T
[cm
3
Kmol
1
]
1/2 0.375
1 1.000
3/2 1.876
2 3.000
5/2 4.377
3 6.002
7/2 7.878
126
4.9 Appendix D: Analysis of orbital angular momentum and
spin–orbit matrix elements
Table 4.7: Spin-orbit mean-field reduced matrix elements of NiCp
2
. Only A(lowest energy)
! B(higher energy) transition is shown. values are the singular values. The sum of the
contribution from the two leading NTO pairs recovers with accuracy the reduced spin-orbit
matrix elements (full EOM-SF-CCSD values).
a
1! 2 h
p
1
jH
SO
j
h
1
i x
1
h
p
2
jH
SO
j
h
2
i x
2
full EOM-SF-CCSD
h2jL
z
j1i 0:00i
hSjjH
SO
L

jjS
0
i 17:12 12:71i 6:27 + 1:64i 11:20 11:24i
hSjjH
SO
L
0
jjS
0
i 432:89i 414:11i 20:41
hSjjH
SO
L
+
jjS
0
i 17:17 + 12:73i 6:23 1:58i 11:20 + 11:24i
1! 3 h
p
1
jH
SO
j
h
1
i x
1
h
p
2
jH
SO
j
h
2
i x
2
full EOM-SF-CCSD
h3jL
z
j1i 0:00i
hSjjH
SO
L

jjS
0
i 4:19 17:44i 16:27 + 7:42i 12:63 10:23i
hSjjH
SO
L
0
jjS
0
i 428:44i 419:81i 9:32i
hSjjH
SO
L
+
jjS
0
i 4:13 + 17:52i 16:34 7:41i 12:63 + 10:23i
1! 4 h
p
1
jH
SO
j
h
1
i x
1
h
p
2
jH
SO
j
h
2
i x
2
full EOM-SF-CCSD
h4jL
z
j1i 0:00i
hSjjH
SO
L

jjS
0
i 10:14 2:64i 16:68 18:64i 27:91 22:25i
hSjjH
SO
L
0
jjS
0
i 396:85i 376:08i 840:43i
hSjjH
SO
L
+
jjS
0
i 10:14 + 2:58i 16:73 + 18:73i 27:91 + 22:25i
a
EOM-SF-CCSD/cc-pVTZ
127
4.10 Appendix E: Calculation of orbital angular momentum
and spin–orbit coupling
To explain magnetic anisotropy in NiCp
2
, we compute orbital angular-momentum and spin–orbit
coupling between the tripletS = 1 ground state and the three singletS = 0 excited states (see Fig.
4.15).
Figure 4.15: Electronic configurations of the high-spin M
S
= 1 triplet reference and the
low-spin target states:M
S
= 0 triplet (i.e.,j1i) and singlets (i.e.,j2i,j3i, andj4i).
Associated wave functions are:
j1i =
1
p
2
(
A
(1)
B
(2)
B
(1)
A
(2))((1)(2) +(1)(2)) (4.12)
j2i =
1
p
2
(
A
(1)
B
(2) +
B
(1)
A
(2))((1)(2)(1)(2)) (4.13)
j3i =
1
p
2
(
A
(1)
A
(2)
B
(1)
B
(2))((1)(2)(1)(2)) (4.14)
j4i =
1
p
2
(
A
(1)
A
(2) +
B
(1)
B
(2))((1)(2)(1)(2)); (4.15)
where
A
and
B
denote orbitals localized on the metal center, such as nickeld
xz
andd
yz
orbitals.
Let us start with consideringhnjLjmi between these electronic states. To do so, we recall that
theL orbital angular-momentum operator is a spin-free operator and that corresponding integrals
(L
p;q
= L
pq

) vanish for opposite spin (6= ), see also Ref. 97 for a general description of
spin-free operators. Therefore, transition orbital angular momentumhnjLjmi is zero between the
triplet and singlet states of nickelocene.
Now, let us computehnjH
SO
jmi between state 1 and states 2, 3, and 4. The spin-orbitH
SO
operator is a mixed, spin and space, operator, meaningH
SO
affects both the spatial and spin parts
128
of the wave function. To describe spin-orbit effects, we here consider the effective spin-orbit
interaction operator:
98
H
SO
=
N
el
X
i
L(i)S(i); (4.16)
whereN
el
is the number of unpaired electrons (N
el
= 2 for NiCp
2
),L(i) andS(i) are the orbital
and spin angular-momentum operators for electroni, and the parameter is related to the spin-orbit
coupled constant through =

2S
(negative for a shell more than half full and vice versa).
98
An
alternative form of the effective spin-orbit operator is obtained by expressing Eq. (4.16) in terms
of the shift operators:
H
SO
=
N
el
X
i

1
2
L
+
(i)S

(i) +
1
2
L

(i)S
+
(i) +L
z
(i)S
z
(i)

; (4.17)
where the orbital-shift operators are given by
L

(i) = L
x
(i)iL
y
(i) (4.18)
S

(i) = S
x
(i)iS
y
(i) (4.19)
Let us first compareh1jH
SO
j3i toh1jH
SO
j4i. States 3 and 4 wave functions have same spin
part, while the spatial parts are different by the sign of the two electronic configurations. Spin-orbit
coupling between states 1 and 3 is:
h1jH
SO
j3i =
1
p
2

1
p
2
h(
A
(1)
B
(2)
B
(1)
A
(2))((1)(2)
+(1)(2))jH
SO
j(
A
(1)
A
(2)

B
(1)
B
(2))((1)(2)(1)(2))i
(4.20)
First, we insert Eq. (4.17) into Eq. (4.20) and compute the first termh1jL
+
(1)S

(1)j3i for
electron 1.
129
h1jL
+
(1)S

(1)j3i =
1
2
h(
A
(1)
B
(2)
B
(1)
A
(2))((1)(2)
+(1)(2))jL
+
(1)S

(1)j
(
A
(1)
A
(2)
B
(1)
B
(2))((1)(2)(1)(2))i
=
1
2
h(
A
(1)
B
(2)
B
(1)
A
(2))jL
+
(1)j
(
A
(1)
A
(2)
B
(1)
B
(2))i
h((1)(2) +(1)(2))jS

(1)j((1)(2)(1)(2))i
(4.21)
To do so, we first focus on the spin part of Eq. (4.21), i.e., h((1)(2) +
(1)(2))jS

(1)j((1)(2) (1)(2))i. To compute this term, one has to consider that (i)
S

= andS

= 0 and that (ii) and are orthonormal functions.
h((1)(2) +(1)(2))jS

(1)j((1)(2)(1)(2))i =
h(1)(2)jS

(1)j(1)(2)ih(1)(2)jS

(1)j(1)(2)i
+h(1)(2)jS

(1)j(1)(2)ih(1)(2)jS

(1)j(1)(2)i
=h(1)(2)j(1)(2)i +h(1)(2)j(1)(2)i = 0
(4.22)
Therefore, alsoh1jL
+
(1)S

(1)j3i will be zero. Similarly, for electron 2, we have that also
h1jL
+
(2)S

(2)j3i is zero. Secondly, let us compute the second termh1jL

(1)S
+
(1)j3i for electron
1:
130
h1jL

(1)S
+
(1)j3i =
1
2
h(
A
(1)
B
(2)
B
(1)
A
(2))((1)(2)
+(1)(2))jL

(1)S
+
(1)j
(
A
(1)
A
(2)
B
(1)
B
(2))((1)(2)(1)(2))i
=
1
2
h(
A
(1)
B
(2)
B
(1)
A
(2))jL

(1)j
(
A
(1)
A
(2)
B
(1)
B
(2))i
h((1)(2) +(1)(2))jS
+
(1)j((1)(2)(1)(2))i
(4.23)
Again, we first focus on the spin part of Eq. (4.23), and consider that (i)S
+
= andS
+
= 0
and that (ii) and are orthonormal functions.
h((1)(2) +(1)(2))jS
+
(1)j((1)(2)(1)(2))i
=h(1)(2)jS
+
(1)j(1)(2)ih(1)(2)jS
+
(1)j(1)(2)i
+h(1)(2)jS
+
(1)j(1)(2)ih(1)(2)jS
+
(1)j(1)(2)i
=h(1)(2)j(1)(2)ih(1)(2)j(1)(2)i = 0
(4.24)
Therefore, alsoh1jL

(1)S
+
(1)j3i is zero. Similarly, for electron 2, alsoh1jL

(2)S
+
(2)j3i is
zero. Finally, we shall compute the last term involvingh1jL
z
(1)S
z
(1)j3i.
h1jL
z
(1)S
z
(1)j3i =
1
2
h(
A
(1)
B
(2)
B
(1)
A
(2))((1)(2) +(1)(2))j
L
z
(1)S
z
(1)j(
A
(1)
A
(2)
B
(1)
B
(2))
((1)(2)(1)(2))i
=
1
2
h(
A
(1)
B
(2)
B
(1)
A
(2))j
L
z
(1)j(
A
(1)
A
(2)
B
(1)
B
(2))i
h((1)(2) +(1)(2))jS
z
(1)j((1)(2)(1)(2))i
(4.25)
131
We look at the spin part and consider that (i)S
z
=
1
2
andS
z
=
1
2
and that (ii) and
are orthonormal functions.
h((1)(2) +(1)(2))jS
z
(1)j((1)(2)(1)(2))i =
h(1)(2)jS
z
(1)j(1)(2)ih(1)(2)jS
z
(1)j(1)(2)i+
h(1)(2)jS
z
(1)j(1)(2)ih(1)(2)jS
z
(1)j(1)(2)i
=
1
2
1 +
1
2
0 +
1
2
0 +
1
2
1 = 1
(4.26)
For electron 2:hS
z
(2)i =
1
2
1
1
2
0
1
2
0
1
2
1 =1. We observe that this result satisfies
the selection rules for SOCs between singlet and triplet states:
h (S = 0;M
S
= 0)jH
SO
j
0
(S = 1;M
S
= 0)i =hH
SO
z
i (4.27)
Since only thehS
z
i term is non-zero for both electrons, we compute the orbital part for the L
z
operator. To do so, we consider that (i)
A
isd
xz
and that (ii)
B
isd
yz
. For electron 1, we have:
h
A
(1)
B
(2)jL
z
(1)j
A
(1)
A
(2)ih
A
(1)
B
(2)jL
z
(1)j
B
(1)
B
(2)i
h
B
(1)
A
(2)jL
z
(1)j
A
(1)
A
(2)i +h
B
(1)
A
(2)jL
z
(1)j
B
(1)
B
(2)i
=hd
xz
jL
z
jd
xz
ihd
yz
jd
xz
ihd
xz
jL
z
jd
yz
ihd
yz
jd
yz
ihd
yz
jL
z
jd
xz
ihd
xz
jd
xz
i
+hd
yz
jL
z
jd
yz
ihd
xz
jd
yz
i =(i) 1 (i) 1 = 0 (4.28)
For electron 2, we have:
h
A
(1)
B
(2)jL
z
(2)j
A
(1)
A
(2)ih
A
(1)
B
(2)jL
z
(2)j
B
(1)
B
(2)i
h
B
(1)
A
(2)jL
z
(2)j
A
(1)
A
(2)i +h
B
(1)
A
(2)jL
z
(2)j
B
(1)
B
(2)i
=hd
yz
jL
z
jd
xz
ihd
xz
jd
xz
ihd
yz
jL
z
jd
yz
ihd
xz
jd
yz
ihd
xz
jL
z
jd
xz
ihd
yz
jd
xz
i
+hd
xz
jL
z
jd
yz
ihd
yz
jd
yz
i = +ii = 0 (4.29)
132
Therefore, we conclude that the spin-orbit coupling between state 1 and 3 is zero. Following
similar protocol and selection rules of Eq. (4.27), we compute the spin-orbit coupling between
states 1 and 4, which should be large based on the full calculation (see SOC between states 1 and
4 in Table 4.7).
h1jH
SO
j4i =
1
2
h(
A
(1)
B
(2)
B
(1)
A
(2))((1)(2) +(1)(2))jH
SO
j(
A
(1)
A
(2)
+
B
(1)
B
(2))((1)(2)(1)(2))i
=h1jL
z
(1)S
z
(1) +L
z
(2)S
z
(2)j4i
(4.30)
Since the spin part ofj3> andj4> wave functions does not change, thehS
z
i between states 1
and 4 is already known from Eq. 4.26. Therefore, we only need to compute thehL
z
i. To do so, we
consider that (i)
A
isd
xz
and (ii)
B
isd
yz
. For electron 1, we have:
h
A
(1)
B
(2)jL
z
(1)j
A
(1)
A
(2)i +h
A
(1)
B
(2)jL
z
(1)j
B
(1)
B
(2)i
h
B
(1)
A
(2)jL
z
(1)j
A
(1)
A
(2)ih
B
(1)
A
(2)jL
z
(1)j
B
(1)
B
(2)i
=hd
xz
jL
z
jd
xz
ihd
yz
jd
xz
i +hd
xz
jL
z
jd
yz
ihd
yz
jd
yz
ihd
yz
jL
z
jd
xz
ihd
xz
jd
xz
i
hd
yz
jL
z
jd
yz
ihd
xz
jd
yz
i =ii =2i (4.31)
Therefore,h1jL
z
(1)S
z
(1)j4i =
1
2
2ihS
z
i =
1
2
2i 1 =i For electron 2, we have:
h
A
(1)
B
(2)jL
z
(2)j
A
(1)
A
(2)i +h
A
(1)
B
(2)jL
z
(2)j
B
(1)
B
(2)i
h
B
(1)
A
(2)jL
z
(2)j
A
(1)
A
(2)ih
B
(1)
A
(2)jL
z
(2)j
B
(1)
B
(2)i
=hd
yz
jL
z
jd
xz
ihd
xz
jd
xz
i +hd
yz
jL
z
jd
yz
ihd
xz
jd
yz
ihd
xz
jL
z
jd
xz
ihd
yz
jd
xz
i
hd
xz
jL
z
jd
yz
ihd
yz
jd
yz
i =i (i) = 2i (4.32)
Therefore,h1jL
z
(2)S
z
(2)j4i =
1
2
2ihS
z
i =
1
2
2i(1) =i. Finally, we need to sum the contribu-
tion from electron 1 to the one of electron 2, which gives:h1jL
z
(1)S
z
(1) +L
z
(2)S
z
(2)j4i =2i,
leading to a non-zero SOC between states 1 and 4. Such spin-orbit coupling serves to split the
133
triplet ground state into 2S + 1 magnetic sublevels (M
S
= 0;1). For the free Ni
2+
ion, the
spin-orbit coupling parameter is 630 cm
1
.
98
Taking such parameter as approximated value for
the nickelocene molecular magnet as well leads to a of315 cm
1
and spin-orbit coupling of
h1jL
z
(1)S
z
(1) +L
z
(2)S
z
(2)j4i = 630i.
For states 1 and 2, by using similar arguments,h((1)(2) + (1)(2))jS

j((1)(2)
(1)(2)i matrix elements are zero, thus bothh1jL

S
+
j2i andh1jL
+
S

j2i terms ofh1jH
SO
j2i
vanish. ForS
z
,hS
z
(1)i = 1 andhS
z
(2)i =1, buthL
z
(1)i andhL
z
(2)i are zero, thush1jL
z
S
z
j2i
term of the spin orbit coupling is also zero.
Summarizing, we compute a zeroh1jH
SO
j2i andh1jH
SO
j3i spin-obit coupling, while spin-
orbit coupling between states 1 and 4 is large. This confirms the results obtained from the full ab
initio calculations.
134
4.11 Appendix F: Nickelocene on the MgO surface
The MgO(001) surface model employed in this work is finite and is constructed by periodic rep-
etition of a 4x4 supercell of MgO(001) (as shown in Fig. 4.16). This 4x4 supercell is taken from
periodic PBE+D calculations and consists in a four-layer MgO(001) slab.
80, 81
The lattice parame-
tera of this 4x4 supercell is 4d, whered is the PBE+D bulk optimized Mg–O distance of 211.93
pm. The NiCp
2
/MgO(001) adsorption complex under study is then obtained by adsorbing nick-
elocene perpendicularly to the MgO(001) surface model (see Fig. 4.16). Structure optimizations
are performed by selecting a quantum mechanical (QM) region, i.e., NiCp
2
/Mg
25
O
25
in Fig. 4.16,
which is treated with PBE0/6-31G

and is embedded in an array of point charges. During structure
optimizations, one layer of atoms at the border of the the QM region is kept frozen and external
charges are blured with Gaussian functions.
Figure 4.16: From the 4x4 supercell of MgO(001) (on the left) to the NiCp
2
/MgO(001) ad-
sorption complex (on the right) consisting in a QM cluster (NiCp
2
/Mg
25
O
25
) embedded in
point charges. The QM region is lifted for clarity. Here, NiCp
2
is on-top of Mg
2+
adsorption
site. The number of point charges is 2206, corresponding to a 6x6 supercell of the original
MgO(001) slab model. The bond of the metal with the Cp centroid is shown with blue dash
lines. Color code: Ni — purple, Mg — green, O — red, C — gray, and H — white.
135
Figure 4.17: Adsorption complex distances (R(Ni–Mg)) of NiCp
2
on the MgO(001) surface.
Here, NiCp
2
is on-top of Mg
2+
adsorption site. Model system: QM cluster (NiCp
2
/Mg
25
O
25
).
Blue dash line indicatesR(Ni–Mg) distance. Color code: Ni — purple, Mg — green, O —
red, C — gray, and H — white.
Table 4.8: Adsorption energies (E) and adsorption complex distances (R(Ni–Mg/O)) of
NiCp
2
on the MgO(001) surface. Model system: QM cluster (NiCp
2
/Mg
49
O
49
) embedded
into an array of point charges. Method: PBE0/6-31G

. “NiCp
2
/(Mg)O” and “NiCp
2
/Mg(O)”
stand for the adsorption complexes of NiCp
2
/MgO(001) with the Ni atom on-top of Mg
2+
and
O
2
adsorption sites, respectively.
R(Ni–Mg/O), pm E, kJ/mol
a
NiCp
2
/(Mg)O 548.0 27:0
NiCp
2
/Mg(O) 513.1 32:0
NiCp
2
/Mg(O)
b
518.8 32:3
a
Adsorption energies are computed considering the relaxed structures of the surface MgO(001)
(model: Mg
49
O
49
cluster embedded into an array of point charges) and of the NiCp
2
in the gas
phase, i.e., E =E(NiCp
2
/MgO)E(MgO)E(NiCp
2
).
b
The number of point charges is
increased from 2206 to 9118.
136
Figure 4.18: Calculated spin density of a) isolated NiCp
2
, b) adsorbed NiCp
2
on (Mg)O, and
c) adsorbed NiCp
2
on Mg(O). Color code: Ni — purple, Mg — green, O — red, C — gray,
and H — white.
137
Table 4.9: Energies (in cm
1
) of the target states of NiCp
2
in different environments: iso-
lated and on MgO(001). State energies are computed for the NiCp
2
/Mg
25
O
25
cluster using
cc-pVTZ basis set and EOM-SF-CCSD, EOM-SF-MP2, and SF-TDDFT with both PBE0
and LRC-!PBEh methods. “NiCp
2
/(Mg)O” and “NiCp
2
/Mg(O)” stand for the adsorption
complexes of NiCp
2
/MgO(001) with the Ni atom on-top of Mg
2+
and O
2
adsorption sites,
respectively.S
2
values are in parenthesis.
NiCp
2
EOM-SF-CCSD EOM-SF-MP2 SF-PBE0 SF-LRC-!PBEh
j1i 0.0 (2.00) 0.0 (1.98) 0.0 (2.03) 0.0 (2.03)
j2i 9805.5 (0.07) 11454.7 (0.13) 6552.1 (0.04) 6505.3 (0.04)
j3i 9894.2 (0.07) 11510.4 (0.13) 6585.6 (0.04) 6553.6 (0.04)
j4i 15144.0 (0.13) 17558.0 (0.26) 12240.5 (0.08) 12242.5 (0.07)
NiCp
2
/(Mg)O SF-PBE0 SF-LRC-!PBEh
j1i 0.0 (2.03) 0.0 (2.03)
j2i 6438.1 (0.04) 6380.4 (0.04)
j3i 6450.7 (0.04) 6409.1 (0.04)
j4i 12052.9 (0.08) 12055.2 (0.07)
NiCp
2
/Mg(O) SF-PBE0 SF-LRC-!PBEh
j1i 0.0 (2.03) 0.0 (2.03)
j2i 6434.7 (0.04) 6373.2 (0.04)
j3i 6464.9 (0.04) 6422.4 (0.04)
j4i 12041.1 (0.08) 11993.7 (0.07)
138
Table 4.10: Wave function properties of the target states of the isolated NiCp
2
and of
NiCp
2
/Mg
25
O
25
adsorption complex obtained by EOM-SF-CCSD and SF-PBE0 calculations
using cc-pVDZ basis set. The NiCp
2
/Mg(O) adsorption model is considered.
NiCp
2
– EOM-SF-CCSD/cc-pVDZ E, cm
1
n
u
n
u;nl
hS
2
i D
j1i 0.0 1.96 2.01 2.00
j2i 9636.9 2.03 2.00 0.07
j3i 9683.5 2.04 2.01 0.06
j4i 14884.3 2.12 2.01 0.13 15.6
NiCp
2
– SF-PBE0/cc-pVDZ E, cm
1
n
u
n
u;nl
hS
2
i D
j1i 0.0 2.02 2.00 2.03
j2i 6624.4 1.95 1.99 0.04
j3i 6675.6 2.00 2.00 0.03
j4i 12394.7 1.96 1.99 0.08 15.8
NiCp
2
/Mg(O) – SF-PBE0/cc-pVDZ E, cm
1
n
u
n
u;nl
hS
2
i D
j1i 0.0 2.02 2.00 2.03
j2i 6515.0 2.02 2.00 0.04
j3i 6528.4 2.00 2.00 0.04
j4i 12124.9 2.02 2.00 0.08 15.8
Effective numbers of unpaired electrons (n
u
andn
u;nl
) are computed using Head-Gordon’s
formulas.
89
ExperimentalD parameter is in the range of 25.6 (Ref. 67) - 33.6 (Ref. 68) cm
1
.
CalculatedD parameter is obtained from the spin-orbit splitting usingU =jDjS
2
forS = 1.
139
Table 4.11: Spin-orbit mean-field reduced matrix elements of the isolated NiCp
2
(full
EOM-SF-CCSD and full SF-PBE0 values with cc-pVDZ and cc-pVTZ basis sets) and of
NiCp
2
/Mg
25
O
25
adsorption complex (SF-PBE0/cc-pVDZ values only). The NiCp
2
/Mg(O) ad-
sorption model is considered.
NiCp
2
– EOM-SF-CCSD/cc-pVDZ 1! 2 1! 3 1! 4
hSjjH
SO
L

jjS
0
i 10:69 + 17:28i 19:02 10:20i 27:47 + 21:92i
hSjjH
SO
L
0
jjS
0
i 21:47i 4:56i 838:27i
hSjjH
SO
L
+
jjS
0
i 10:69 17:28i 19:02 + 10:20i 27:47 21:92i
NiCp
2
– EOM-SF-CCSD/cc-pVTZ 1! 2 1! 3 1! 4
hSjjH
SO
L

jjS
0
i 11:20 11:24i 12:63 10:23i 27:91 22:25i
hSjjH
SO
L
0
jjS
0
i 20:41i 9:32i 840:43i
hSjjH
SO
L
+
jjS
0
i 11:20 + 11:24i 12:63 + 10:23i 27:91 + 22:25i
NiCp
2
– SF-PBE0/cc-pVDZ 1! 2 1! 3 1! 4
hSjjH
SO
L

jjS
0
i 0:37 + 0:97i 0:78 0:23i 24:69 21:77i
hSjjH
SO
L
0
jjS
0
i 15:36i 5:07i 765:87i
hSjjH
SO
L
+
jjS
0
i 0:37 0:96i 0:78 + 0:23i 24:69 + 21:77i
NiCp
2
– SF-PBE0/cc-pVTZ 1! 2 1! 3 1! 4
hSjjH
SO
L

jjS
0
i 0:92 + 0:62i 1:03 0:99i 24:21 + 21:84i
hSjjH
SO
L
0
jjS
0
i 16:42i 3:51i 761:04i
hSjjH
SO
L
+
jjS
0
i 0:92 0:62i 1:03 + 0:99i 24:21 21:84
NiCp
2
/Mg(O) – SF-PBE0/cc-pVDZ 1! 2 1! 3 1! 4
hSjjH
SO
L

jjS
0
i 0:09 0:25i 0:07 + 0:40i 11:99 + 4:48i
hSjjH
SO
L
0
jjS
0
i 0:09i 6:10i 758:29i
hSjjH
SO
L
+
jjS
0
i 0:09 + 0:25i 0:07 0:40i 11:99 4:48i
140
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Chapter 5: Future work
In the preceding chapters, I presented the implementation of spin–orbit couplings using TD-DFT
and SF-TDDFT, and the applications of SF-TDDFT to extract exchange coupling constants in
Fe(III) binuclear and tetranuclear complexes, and to characterize the magnetic properties of nick-
elocene molecular magnet. In this chapter, I propose future work in this direction.
5.1 Implementation of mixed-reference non-collinear SF-
TDDFT
It is well-known that the standard spin-flip solutions are not pure spin eigenfunctions.
1–3
Several
methods have been developed to tackle the spin-contamination problem in SF-TDDFT.
2, 4
For ex-
ample, tensor equation-of-motion approaches can produce correct spin eigenstates by applying ten-
sor operators to a tensor reference.
2
Although for all examples in this work the spin-contamination
of spin-flip states was small, it is desirable to completely eliminate spin-contamination by using
mixed-reference (MR) SF-TDDFT (MR-SF-TDDFT), which was recently introduced by Lee et
al.
4–6
In the original SF method, a high-spin M
s
= +1 triplet state is taken as a reference upon
which the SF excitation operator acts to yield the target states.
7, 8
However, as illustrated in Fig.
5.1, some of the configurations miss their “spin complements” (shown in red) and this leads to
spin-contaminated solutions. The spin-contamination error can be mitigated by using the MR re-
duced density matrix (RDM) which combines the RDMs ofM
s
= +1 andM
s
= -1 triplet-ground
states with equal weights. MR-SF-TDDFT method (illustrated in Fig. 5.1) can be implemented
with non-collinear formulation of SF-TDDFT in theQ-Chem electronic structure package.
9, 10
149
Figure 5.1: MR-SF-TDDFT employs two references shown in the upper panel by black and
red arrows. In the lower panel, electronic configurations generated by mixed reference spin-
flip operator are shown in blue, black, and red arrows. The blue ones are generated by both
references, while the black and red ones are generated fromM
s
= +1 andM
s
= -1 references
respectively. Gray arrows correspond to configurations that cannot even be obtained in MR-
SF-TDDFT. Reproduced with permission from Ref. 6; Copyright 2022 American Chemical
Society.
5.2 Magnetic properties of a spin-frustrated trinuclear copper
complex
We can utilize all theoretical methods presented in this dissertation to study the magnetic properties
of a spin-frustrated trinuclear copper complex shown in Fig. 5.2. With a T
2
coherence time of 591
ns (in pyridine-d
5
solution), this complex is a promising candidate as a building block for molecular
spintronics.
11
We can compute exchange interactions using SF-TDDFT as explained in Chapter
2 and magnetic anisotropy and magnetic susceptibility with theezMagnet module (employing the
SOC code) using SF-TDDFT, as explained in Chapter 4. The experimental results for this complex
150
already exist and can validate the theoretical results.
11
This would be a comprehensive study of the
microscopic exchange interactions and macroscopic magnetic properties of a multinuclear qubit
system.
Figure 5.2: Molecular structure for the trinuclear copper complex. Hydrogen atoms are
omitted for clarity. Reproduced from Ref. 11 with permission from the Royal Society of
Chemistry; Copyright 2018 Royal Society of Chemistry.
151
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153

Abstract (if available)

Abstract Shut yourself up in a room, and have with you an unbiased 25-cent coin. Keep tossing the coin. Suppose, on one day, you get tails as the first outcome and heads every other time you toss the coin. Amazed by the asymmetry of this configuration, you follow this experiment the next day to have heads as the first outcome and tails every other time. These different configurations on day 1 and 2 are merely different microstates available to the system, and can be obtained from an all-heads configuration with a single flip and multiple flips. It turns out that the low-energy spectra of a system of interacting spins can be described by just a single spin-flip from a high-spin configuration and is equivalent to a configuration produced by multiple spin-flips in terms of its physical properties. The work presented here elucidates electronic structure of single-molecule magnets (SMMs) within a single spin-flip approach.

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Creator Kotaru, Saikiran (author)

Core Title Electronic structure of strongly correlated systems: method development and applications to molecular magnets

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry

Degree Conferral Date 2023-08

Publication Date 06/20/2023

Defense Date 06/16/2023

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag electronic structure theory,exchange interactions,OAI-PMH Harvest,single-molecule magnets,spin-flip,spin-orbit coupling,strongly correlated systems

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Language English

Contributor Electronically uploaded by the author (provenance)

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