Relativistic effects are usually taken into account in heavy-element-containing species, bringing to the scientific community stimulating cases of study. Scalar and spin–orbit effects are required to properly evaluate both the geometrical and electronic structures of such species, where, generally, scalar corrections are included. In order to take into account the spin–orbit term resulting from the interaction between the spatial and spin coordinates, double-valued point groups of symmetry are required, leading to total angular momenta (j) functions and atomic or molecular spinors, instead of pure orbital-angular momenta (l) and atomic or molecular orbitals. Here, we reviewed the role of spin–orbit coupling in bare and ligand-protected metallic clusters, from early to current works, leading to a more comprehensive relativistic quantum chemistry framework. As a result, the electronic structure is modified, leading to a variation in the calculated molecular properties, which usually improves the agreement between theory and experiment, allowing furthering rationalize of experimental results unexpected from a classical inorganic chemistry point of view. This review summarizes part of the modern application of spin–orbit coupling in heavy-elements cluster chemistry, where further treatment on an equal footing basis along with the periodic table is encouraged in order to incorporate such term in the general use vocabulary of both experimental and theoretical chemist and material scientist.
REFERENCES
1.
H.
Kragh
, “
The fine structure of hydrogen and the gross structure of the physics community, 1916–26
,” Hist. Stud. Phys. Sci
15
(2
), 67
–125
(1985
).2.
L. Y.
Xie
,
J. G.
Wang
, and
R. K.
Janev
, “
Relativistic effects in the photoionization of hydrogen-like ions with screened Coulomb interaction
,” Phys. Plasmas
21
(6
), 063304
(2014
).3.
P.
Pyykko
, “
Relativistic effects in structural chemistry
,” Chem. Rev.
88
(3
), 563
–594
(1988
).4.
P.
Pyykkö
, “
Strong closed-shell interactions in inorganic chemistry
,” Chem. Rev.
97
(3
), 597
–636
(1997
).5.
P.
Schwerdtfeger
, “
Relativistic effects in properties of gold
,” Heteroat. Chem.
13
(6
), 578
–584
(2002
).6.
H.
Schmidbaur
, “
The fascinating implications of new results in gold chemistry
,” Gold Bull.
23
(1
), 11
–21
(1990
).7.
F.
Scherbaum
,
A.
Grohmann
,
B.
Huber
,
C.
Krüger
, and
H.
Schmidbaur
, “
‘Aurophilicity’ as a consequence of relativistic effects: The Hexakis(triphenylphosphaneaurio)methane dication [(Ph3PAu)6C]2⊕
,” Angew. Chem., Int. Ed. Engl.
27
(11
), 1544
–1546
(1988
).8.
H.
Schmidbaur
and
A.
Schier
, “
Aurophilic interactions as a subject of current research: An up-date
,” Chem. Soc. Rev.
41
(1
), 370
–412
(2012
).9.
M.
Reiher
and
A.
Wolf
, Relativistic Quantum Chemistry
(
Wiley-VCH Verlag GmbH & Co. KGaA
,
Weinheim, Germany
, 2009
).10.
M.
Iliaš
,
V.
Kellö
, and
M.
Urban
, “
Relativistic effects in atomic and molecular properties
,” Acta Phys. Slovaca
60
(3
), 259
(2010
).11.
P.
Schwerdtfeger
, Relativistic Electronic Structure Theory
(
Elsevier Science
,
New York
, 2004
).12.
J.
Autschbach
, “
Perspective: Relativistic effects
,” J. Chem. Phys.
136
(15
), 150902
(2012
).13.
P.
Pyykkö
, “
The RTAM electronic bibliography, version 17.0, on relativistic theory of atoms and molecules
,” J. Comput. Chem.
34
(31
), 2667
–2667
(2013
).14.
P.
Pyykkö
, “
Relativistic quantum chemistry
,” Adv. Quantum Chem.
11
, 353
–409
(1978
).15.
P.
Pyykko
and
J. P.
Desclaux
, “
Relativity and the periodic system of elements
,” Acc. Chem. Res.
12
(8
), 276
–281
(1979
).16.
G.
Malli
and
J.
Oreg
, “
Relativistic self‐consistent‐field (RSCF) theory for closed‐shell molecules
,” J. Chem. Phys.
63
(2
), 830
–841
(1975
).17.
I. P.
Grant
, “
Relativistic self-consistent fields
,” Proc. R. Soc. A.
262
, 555
–576
(1961
).18.
I. P.
Grant
, “
Relativistic self-consistent fields
,” Proc. Phys. Soc.
86
(3
), 523
–527
(1965
).19.
Relativistic Quantum Theory of Atoms and Molecules
, Springer Series on Atomic, Optical, and Plasma Physics Vol.
40
, edited by
I. P.
Grant
(
Springer
,
New York, NY
, 2007
).20.
K.
Faegri
, Jr. and
K. G.
Dyall
, Introduction to Relativistic Quantum Chemistry
(
Oxford University Press
, 2007
).21.
Y. J.
Franzke
and
F.
Weigend
, “
NMR shielding tensors and chemical shifts in scalar-relativistic local exact two-component theory
,” J. Chem. Theory Comput.
15
(2
), 1028
–1043
(2019
).22.
A. J. S.
Valentine
and
X.
Li
, “
Intersystem crossings in late-row elements: A perspective
,” J. Phys. Chem. Lett.
13
(13
), 3039
–3046
(2022
).23.
S.
Moncho
and
J.
Autschbach
, “
Molecular orbital analysis of the inverse halogen dependence of nuclear magnetic shielding in LaX3, X = F, Cl, Br, I
,” Magn. Reson. Chem.
48
(S1
), S76
–S85
(2010
).24.
P. R.
Batista
,
L. C.
Ducati
, and
J.
Autschbach
, “
Solvent effect on the 195Pt NMR properties in pyridonate-bridged PtIII dinuclear complex derivatives investigated by ab initio molecular dynamics and localized orbital analysis
,” Phys. Chem. Chem. Phys.
23
(22
), 12864
–12880
(2021
).25.
M.
Straka
,
P.
Lantto
, and
J.
Vaara
, “
Toward calculations of the 129Xe chemical shift in Xe@C60 at experimental conditions: Relativity, correlation, and dynamics
,” J. Phys. Chem. A
112
(12
), 2658
–2668
(2008
).26.
J.
Autschbach
and
E.
Zurek
, “
Relativistic density-functional computations of the chemical shift of 129Xe in Xe@C60
,” J. Phys. Chem. A
107
(24
), 4967
–4972
(2003
).27.
S.
Standara
,
P.
Kulhánek
,
R.
Marek
, and
M.
Straka
, “
129Xe NMR chemical shift in Xe@C60 calculated at experimental conditions: Essential role of the relativity, dynamics, and explicit solvent
,” J. Comput. Chem.
34
(22
), 1890
–1898
(2013
).28.
J. M.
Kasper
,
X.
Li
,
S. A.
Kozimor
,
E. R.
Batista
, and
P.
Yang
, “
Relativistic effects in modeling the ligand K-edge X-ray absorption near-edge structure of uranium complexes
,” J. Chem. Theory Comput.
18
(4
), 2171
–2179
(2022
).29.
H. D.
Ludowieg
,
M.
Srebro‐Hooper
,
J.
Crassous
, and
J.
Autschbach
, “
Optical activity of spin‐forbidden electronic transitions in metal complexes from time‐dependent density functional theory with spin‐orbit coupling
,” ChemistryOpen
11
(5
), e202200020
(2022
).30.
A. H.
Greif
,
P.
Hrobárik
,
V.
Hrobáriková
,
A. V.
Arbuznikov
,
J.
Autschbach
, and
M.
Kaupp
, “
A relativistic quantum-chemical analysis of the trans influence on 1H NMR hydride shifts in square-planar platinum(II) complexes
,” Inorg. Chem.
54
(15
), 7199
–7208
(2015
).31.
L. A.
Truflandier
,
K.
Sutter
, and
J.
Autschbach
, “
Solvent effects and dynamic averaging of 195Pt NMR shielding in cisplatin derivatives
,” Inorg. Chem.
50
(5
), 1723
–1732
(2011
).32.
R.
Feng
,
X.
Yu
, and
J.
Autschbach
, “
Spin–orbit natural transition orbitals and spin-forbidden transitions
,” J. Chem. Theory Comput
17
(12
), 7531
–7544
(2021
).33.
A. C.
Castro
,
D.
Balcells
,
M.
Repisky
,
T.
Helgaker
, and
M.
Cascella
, “
First-principles calculation of 1H NMR chemical shifts of complex metal polyhydrides: The essential inclusion of relativity and dynamics
,” Inorg. Chem.
59
(23
), 17509
–17518
(2020
).34.
A. C.
Castro
,
H.
Fliegl
,
M.
Cascella
,
T.
Helgaker
,
M.
Repisky
,
S.
Komorovsky
,
M. Á.
Medrano
,
A. G.
Quiroga
, and
M.
Swart
, “
Four-component relativistic 31P NMR calculations for trans-platinum (II) complexes: Importance of the solvent and dynamics in spectral simulations
,” Dalt. Trans.
48
(23
), 8076
–8083
(2019
).35.
F.
Aquino
,
N.
Govind
, and
J.
Autschbach
, “
Electric field gradients calculated from two-component hybrid density functional theory including spin−orbit coupling
,” J. Chem. Theory Comput.
6
(9
), 2669
–2686
(2010
).36.
S.
Sun
,
T. F.
Stetina
,
T.
Zhang
,
H.
Hu
,
E. F.
Valeev
,
Q.
Sun
, and
X.
Li
, “
Efficient four-component Dirac–Coulomb–Gaunt Hartree–Fock in the Pauli spinor representation
,” J. Chem. Theory Comput.
17
(6
), 3388
–3402
(2021
).37.
S.
Sun
,
J.
Ehrman
,
Q.
Sun
, and
X.
Li
, “
Efficient evaluation of the Breit operator in the Pauli spinor basis
,” J. Chem. Phys.
157
(6
), 064112
(2022
).38.
F.
Neese
, “
Software update: The ORCA program system—Version 5.0
,” Wires Comput. Mol. Sci.
12
(5
), e1606
(2022
).39.
M.
Repisky
,
S.
Komorovsky
,
M.
Kadek
,
L.
Konecny
,
U.
Ekström
,
E.
Malkin
,
M.
Kaupp
,
K.
Ruud
,
O. L.
Malkina
, and
V. G.
Malkin
, “
ReSpect: Relativistic spectroscopy DFT program package
,” J. Chem. Phys.
152
(18
), 184101
(2020
).40.
D.
Peng
,
N.
Middendorf
,
F.
Weigend
, and
M.
Reiher
, “
An efficient implementation of two-component relativistic exact-decoupling methods for large molecules
,” J. Chem. Phys.
138
(18
), 184105
(2013
).41.
A.
Grofe
,
J.
Gao
, and
X.
Li
, “
Exact-two-component block-localized wave function: A simple scheme for the automatic computation of relativistic ΔSCF
,” J. Chem. Phys.
155
(1
), 014103
(2021
).42.
H.
Hu
,
A. J.
Jenkins
,
H.
Liu
,
J. M.
Kasper
,
M. J.
Frisch
, and
X.
Li
, “
Relativistic two-component multireference configuration interaction method with tunable correlation space
,” J. Chem. Theory Comput.
16
(5
), 2975
–2984
(2020
).43.
G.
Te Velde
,
F. M.
Bickelhaupt
,
E. J.
Baerends
,
C.
Fonseca Guerra
,
S. J. A.
van Gisbergen
,
J. G.
Snijders
,
T.
Ziegler
,
G. T. E.
Velde
,
C. F.
Guerra
, and
S. J. A.
Gisbergen
, “
Chemistry with ADF
,” J. Comput. Chem.
22
(9
), 931
–967
(2001
).44.
C. E.
Hoyer
,
H.
Hu
,
L.
Lu
,
S.
Knecht
, and
X.
Li
, “
Relativistic Kramers-unrestricted exact-two-component density matrix renormalization group
,” J. Phys. Chem. A
126
(30
), 5011
–5020
(2022
).45.
Q.
Zhou
and
B.
Suo
, “
New implementation of spin‐orbit coupling calculation on multi‐configuration electron correlation theory
,” Int. J. Quantum Chem.
121
(20
), e26772
(2021
).46.
L.
Lu
,
H.
Hu
,
A. J.
Jenkins
, and
X.
Li
, “
Exact-two-component relativistic multireference second-order perturbation theory
,” J. Chem. Theory Comput.
18
(5
), 2983
–2992
(2022
).47.
T.
Shiozaki
and
W.
Mizukami
, “
Relativistic internally contracted multireference electron correlation methods
,” J. Chem. Theory Comput.
11
(10
), 4733
–4739
(2015
).48.
P.
Sharma
,
A. J.
Jenkins
,
G.
Scalmani
,
M. J.
Frisch
,
D. G.
Truhlar
,
L.
Gagliardi
, and
X.
Li
, “
Exact-two-component multiconfiguration pair-density functional theory
,” J. Chem. Theory Comput.
18
(5
), 2947
–2954
(2022
).49.
Y. J.
Franzke
,
N.
Middendorf
, and
F.
Weigend
, “
Efficient implementation of one- and two-component analytical energy gradients in exact two-component theory
,” J. Chem. Phys.
148
(10
), 104110
(2018
).50.
D.
Peng
and
M.
Reiher
, “
Local relativistic exact decoupling
,” J. Chem. Phys.
136
(24
), 244108
(2012
).51.
T.
Saue
,
R.
Bast
,
A. S. P.
Gomes
,
H. J. A.
Jensen
,
L.
Visscher
,
I. A.
Aucar
,
R.
Di Remigio
,
K. G.
Dyall
,
E.
Eliav
,
E.
Fasshauer
,
T.
Fleig
,
L.
Halbert
,
E. D.
Hedegård
,
B.
Helmich-Paris
,
M.
Iliaš
,
C. R.
Jacob
,
S.
Knecht
,
J. K.
Laerdahl
,
M. L.
Vidal
,
M. K.
Nayak
,
M.
Olejniczak
,
J. M. H.
Olsen
,
M.
Pernpointner
,
B.
Senjean
,
A.
Shee
,
A.
Sunaga
, and
J. N. P.
van Stralen
, “
The DIRAC code for relativistic molecular calculations
,” J. Chem. Phys.
152
(20
), 204104
(2020
).52.
G.
Jha
and
T.
Heine
, “
Spin–orbit coupling corrections for the GFN-xTB method
,” J. Chem. Phys.
158
(4
), 044120
(2023
).53.
R.
Jin
, “
Atomically precise metal nanoclusters: Stable sizes and optical properties
,” Nanoscale
7
(5
), 1549
–1565
(2015
).54.
M.
Zhu
,
C. M.
Aikens
,
F. J.
Hollander
,
G. C.
Schatz
, and
R.
Jin
, “
Correlating the crystal structure of a thiol-protected Au25 cluster and optical properties
,” J. Am. Chem. Soc.
130
(18
), 5883
–5885
(2008
).55.
I.
Chakraborty
and
T.
Pradeep
, “
Atomically precise clusters of noble metals: Emerging link between atoms and nanoparticles
,” Chem. Rev.
117
(12
), 8208
–8271
(2017
).56.
Y.
Levi-Kalisman
,
P. D.
Jadzinsky
,
N.
Kalisman
,
H.
Tsunoyama
,
T.
Tsukuda
,
D. A.
Bushnell
, and
R. D.
Kornberg
, “
Synthesis and characterization of Au102(p-MBA)44 nanoparticles
,” J. Am. Chem. Soc.
133
(9
), 2976
–2982
(2011
).57.
T.
Tsukuda
,
H.
Tsunoyama
, and
H.
Sakurai
, “
Aerobic oxidations catalyzed by colloidal nanogold
,” Chem. Asian J.
6
(3
), 736
–748
(2011
).58.
Y.
Zhu
,
H.
Qian
, and
R.
Jin
, “
Catalysis opportunities of atomically precise gold nanoclusters
,” J. Mater. Chem.
21
(19
), 6793
–6799
(2011
).59.
K.
Kwak
,
S. S.
Kumar
, and
D.
Lee
, “
Selective determination of dopamine using quantum-sized gold nanoparticles protected with charge selective ligands
,” Nanoscale
4
(14
), 4240
–4246
(2012
).60.
N.
Sakai
and
T.
Tatsuma
, “
Photovoltaic properties of glutathione-protected gold clusters adsorbed on TiO2 electrodes
,” Adv. Mater.
22
(29
), 3185
–3188
(2010
).61.
Z.
Wu
,
M.
Wang
,
J.
Yang
,
X.
Zheng
,
W.
Cai
,
G.
Meng
,
H.
Qian
,
H.
Wang
, and
R.
Jin
, “
Well-defined nanoclusters as fluorescent nanosensors: A case study on Au25(SG)18
,” Small
8
(13
), 2028
–2035
(2012
).62.
R. W.
Murray
, “
Nanoelectrochemistry: Metal nanoparticles, nanoelectrodes, and nanopores
,” Chem. Rev.
108
(7
), 2688
–2720
(2008
).63.
J. M.
Galloway
,
J. P.
Bramble
,
A. E.
Rawlings
,
G.
Burnell
,
S. D.
Evans
, and
S. S.
Staniland
, “
Biotemplated magnetic nanoparticle arrays
,” Small
8
(2
), 204
–208
(2012
).64.
Y.-C.
Shiang
,
C.-C.
Huang
,
W.-Y.
Chen
,
P.-C.
Chen
, and
H.-T.
Chang
, “
Fluorescent gold and silver nanoclusters for the analysis of biopolymers and cell imaging
,” J. Mater. Chem.
22
(26
), 12972
(2012
).65.
J.
Sun
and
Y.
Jin
, “
Fluorescent Au nanoclusters: Recent progress and sensing applications
,” J. Mater. Chem. C
2
(38
), 8000
–8011
(2014
).66.
Z.
Wu
and
R.
Jin
, “
On the ligand's role in the fluorescence of gold nanoclusters
,” Nano Lett.
10
(7
), 2568
–2573
(2010
).67.
L.
Li
,
H.
Liu
,
Y.
Shen
,
J.
Zhang
, and
J.-J.
Zhu
, “
Electrogenerated chemiluminescence of Au nanoclusters for the detection of dopamine
,” Anal. Chem.
83
(3
), 661
–665
(2011
).68.
H.
Qian
,
Y.
Zhu
, and
R.
Jin
, “
Size-focusing synthesis, optical and electrochemical properties of monodisperse Au38(SC2H4Ph)24 nanoclusters
,” ACS Nano
3
(11
), 3795
–3803
(2009
).69.
R. L.
Whetten
,
J. T.
Khoury
,
M. M.
Alvarez
,
S.
Murthy
,
I.
Vezmar
,
Z. L.
Wang
,
P. W.
Stephens
,
C. L.
Cleveland
,
W. D.
Luedtke
, and
U.
Landman
, “
Nanocrystal gold molecules
,” Adv. Mater.
8
(5
), 428
–433
(1996
).70.
L. A.
Oro
,
P.
Braunstein
, and
P. R.
Raithby
, Metal Clusters in Chemistry
(
Wiley-VCH, Weinheim
,
Germany
, 1999
), Vol.
3
.71.
M.-C.
Daniel
and
D.
Astruc
, “
Gold nanoparticles: Assembly, supramolecular chemistry, quantum-size-related properties, and applications toward biology, catalysis, and nanotechnology
,” Chem. Rev.
104
(1
), 293
–346
(2004
).72.
M. W.
Heaven
,
A.
Dass
,
P. S.
White
,
K. M.
Holt
, and
R. W.
Murray
, “
Crystal structure of the gold nanoparticle [N(C8H17)4][Au25(SCH2CH2Ph)18]
,” J. Am. Chem. Soc.
130
(12
), 3754
–3755
(2008
).73.
M.
Walter
,
J.
Akola
,
O.
Lopez-Acevedo
,
P. D.
Jadzinsky
,
G.
Calero
,
C. J.
Ackerson
,
R. L.
Whetten
,
H.
Grönbeck
, and
H.
Häkkinen
, “
A unified view of ligand-protected gold clusters as superatom complexes
,” Proc. Natl. Acad. Sci. U. S. A.
105
(27
), 9157
–9162
(2008
).74.
P.
Jena
, “
Beyond the periodic table of elements: The role of superatoms
,” J. Phys. Chem. Lett.
4
(9
), 1432
–1442
(2013
).75.
S.
Knoppe
,
I.
Dolamic
, and
T.
Bürgi
, “
Racemization of a chiral nanoparticle evidences the flexibility of the gold-thiolate interface
,” J. Am. Chem. Soc.
134
(31
), 13114
–13120
(2012
).76.
H.
Qian
,
M.
Zhu
,
Z.
Wu
, and
R.
Jin
, “
Quantum sized gold nanoclusters with atomic precision
,” Acc. Chem. Res.
45
(9
), 1470
–1479
(2012
).77.
A. W.
Castleman
and
S. N.
Khanna
, “
Clusters, superatoms, and building blocks of new materials
,” J. Phys. Chem. C
113
(7
), 2664
–2675
(2009
).78.
D. M. P.
Mingos
, “
Molecular-orbital calculations on cluster compounds of gold
,” J. Chem. Soc. Dalt. Trans.
1976
(13
), 1163
–1169
.79.
D. M. P.
Mingos
,
J.
Lewis
, and
M. L. H.
Green
, “
Some theoretical and structural aspects of gold cluster chemistry
,” Philos. Trans. R. Soc., A
308
(1501
), 75
–83
(1982
).80.
R. H.
Adnan
,
J. M. L.
Madridejos
,
A. S.
Alotabi
,
G. F.
Metha
, and
G. G.
Andersson
, “
A review of state of the art in phosphine ligated gold clusters and application in catalysis
,” Adv. Sci.
9
(15
), 2105692
(2022
).81.
W.
Weltner
and
R. J.
Van Zee
, “
Transition metal molecules
,” Annu. Rev. Phys. Chem.
35
(1
), 291
–327
(1984
).82.
J. A.
Howard
and
B.
Mile
, “
Study of the genesis, structure, and reactions of small metal clusters using a rotating cryostat
,” Acc. Chem. Res.
20
(5
), 173
–179
(1987
).83.
S.
Ogawa
and
S.
Ino
, “
Formation of multiply-twinned particles on alkali halide crystals by vacuum evaporation and their structures
,” J. Cryst. Growth
13–14
, 48
–56
(1972
).84.
S.
Iijima
and
T.
Ichihashi
, “
Structural instability of ultrafine particles of metals
,” Phys. Rev. Lett.
56
(6
), 616
–619
(1986
).85.
H.
Hofmeister
, “
Forty years study of fivefold twinned structures in small particles and thin films
,” Cryst. Res. Technol.
33
(1
), 3
–25
(1998
).86.
H.
Yang
,
Y.
Wang
,
X.
Chen
,
X.
Zhao
,
L.
Gu
,
H.
Huang
,
J.
Yan
,
C.
Xu
,
G.
Li
,
J.
Wu
,
A. J.
Edwards
,
B.
Dittrich
,
Z.
Tang
,
D.
Wang
,
L.
Lehtovaara
,
H.
Häkkinen
, and
N.
Zheng
, “
Plasmonic twinned silver nanoparticles with molecular precision
,” Nat. Commun.
7
(1
), 12809
(2016
).87.
Y.-M.
Su
,
Z.
Wang
,
S.
Schein
,
C.-H.
Tung
, and
D.
Sun
, “
A Keplerian Ag90 nest of Platonic and Archimedean polyhedra in different symmetry groups
,” Nat. Commun.
11
(1
), 3316
(2020
).88.
Y.-M.
Su
,
B.-Q.
Ji
,
Z.
Wang
,
S.-S.
Zhang
,
L.
Feng
,
Z.-Y.
Gao
,
Y.-W.
Li
,
C.-H.
Tung
,
D.
Sun
, and
L.-S.
Zheng
, “
Anionic passivation layer-assisted trapping of an icosahedral Ag13 kernel in a truncated tetrahedral Ag89 nanocluster
,” Sci. China Chem.
64
(9
), 1482
–1486
(2021
).89.
B. A.
Collings
,
K.
Athanassenas
,
D.
Lacombe
,
D. M.
Rayner
, and
P. A.
Hackett
, “
Optical absorption spectra of Au7, Au9, Au11, and Au13, and their cations: Gold clusters with 6, 7, 8, 9, 10, 11, 12, and 13 s‐electrons
,” J. Chem. Phys
101
(5
), 3506
–3513
(1994
).90.
K. J.
Taylor
,
C. L.
Pettiette‐Hall
,
O.
Cheshnovsky
, and
R. E.
Smalley
, “
Ultraviolet photoelectron spectra of coinage metal clusters
,” J. Chem. Phys.
96
(4
), 3319
–3329
(1992
).91.
H.
Roulet
,
J.-M.
Mariot
,
G.
Dufour
, and
C. F.
Hague
, “
Size dependence of the valence bands in gold clusters
,” J. Phys.
10
(5
), 1025
–1030
(1980
).92.
S.-T.
Lee
,
G.
Apai
,
M. G.
Mason
,
R.
Benbow
, and
Z.
Hurych
, “
Evolution of band structure in gold clusters as studied by photoemission
,” Phys. Rev. B
23
(2
), 505
–508
(1981
).93.
P.
Sudraud
,
C.
Colliex
, and
J.
van de Walle
, “
Energy distribution of EHD emitted gold ions
,” J. Phys. Lett.
40
(9
), 207
–211
(1979
).94.
D. A.
Shirley
, “
High-resolution x-ray photoemission spectrum of the valence bands of gold
,” Phys. Rev. B
5
(12
), 4709
–4714
(1972
).95.
R.
Arratia‐Perez
and
D. A.
Case
, “
Relativistic effects on molecular hyperfine interactions: Application to XeF and CsO
,” J. Chem. Phys.
79
(10
), 4939
–4949
(1983
).96.
R.
Arratia-Pérez
and
L.
Hernández-Acevedo
, “
Relativistic electronic structure of an icosahedral Au12Pd cluster
,” Chem. Phys. Lett.
303
(5–6
), 641
–648
(1999
).97.
R.
Arratia-Perez
,
A. F.
Ramos
, and
G. L.
Malli
, “
Calculated electronic structure of Au13 clusters
,” Phys. Rev. B
39
(5
), 3005
–3009
(1989
).98.
K. G.
Dyall
and
E.
van Lenthe
, “
Relativistic regular approximations revisited: An infinite-order relativistic approximation
,” J. Chem. Phys.
111
(4
), 1366
–1372
(1999
).99.
E.
van Lenthe
,
E.-J.
Baerends
, and
J. G.
Snijders
, “
Relativistic total energy using regular approximations
,” J. Chem. Phys.
101
(11
), 9783
–9792
(1994
).100.
W.
Liu
and
D.
Peng
, “
Exact two-component Hamiltonians revisited
,” J. Chem. Phys.
131
(3
), 031104
(2009
).101.
D.
Peng
and
M.
Reiher
, “
Exact decoupling of the relativistic Fock operator
,” Theor. Chem. Acc.
131
(1
), 1081
(2012
).102.
S.
Varga
,
B.
Fricke
,
H.
Nakamatsu
,
T.
Mukoyama
,
J.
Anton
,
D.
Geschke
,
A.
Heitmann
,
E.
Engel
, and
T.
Baştuǧ
, “
Four-component relativistic density functional calculations of heavy diatomic molecules
,” J. Chem. Phys.
112
(8
), 3499
–3506
(2000
).103.
T.
Saue
, “
Relativistic Hamiltonians for chemistry: A primer
,” ChemPhysChem
12
(17
), 3077
–3094
(2011
).104.
K.
Balasubramanian
, “
Relativistic double group spinor representations of nonrigid molecules
,” J. Chem. Phys.
120
(12
), 5524
–5535
(2004
).105.
F.
Cotton
, Chemical Applications of Group Theory
(
Wiley-Interscience
,
Michigan
, 1990
).106.
H.
Bethe
, “
Termaufspaltung in Kristallen
,” Ann. Phys.
395
(2
), 133
–208
(1929
).107.
A.
Le Paillier-Malécot
and
L.
Couture
, “
Properties of the single and double D4d groups and their isomorphisms with D8 and C8v groups
,” J. Phys.
42
(11
), 1545
–1552
(1981
).108.
R. L.
Flurry
, Jr., Applications of Symmetry and Group Theory
(
Prentice-Hall Inc
.,
New Jersey
, 1980
).109.
G. F.
Koster
,
J. O.
Dimmock
,
R. G.
Wheeler
, and
H.
Statz
, The Properties of the Thirty-Two Point Groups
(
The MIT Press
, 1963
).110.
H. T.
Toivonen
and
P.
Pyykkö
, “
Relativistic molecular orbitals for the double group D3h
,” Int. J. Quantum Chem.
11
(4
), 695
–700
(1977
).111.
L.
Visscher
, “
On the construction of double group molecular symmetry functions
,” Chem. Phys. Lett.
253
(1–2
), 20
–26
(1996
).112.
S. L.
Altmann
, “
Improper double groups and the bases for the representations of O(3)
,” J. Phys. A.
20
(14
), 4587
–4593
(1987
).113.
Y.
Onodera
and
M.
Okazaki
, “
Tables of basis functions for double point groups
,” J. Phys. Soc. Jpn.
21
(11
), 2400
–2408
(1966
).114.
G. K.
Woodgate
, Elementary Atomic Structure
(
Oxford University Press
,
New York
, 1999
).115.
D.
Jiang
,
M.
Kühn
,
Q.
Tang
, and
F.
Weigend
, “
Superatomic orbitals under spin-orbit coupling
,” J. Phys. Chem. Lett.
5
(19
), 3286
–3289
(2014
).116.
C.
Liao
,
M.
Zhu
,
D.
Jiang
, and
X.
Li
, “
Manifestation of the interplay between spin–orbit and Jahn–Teller effects in Au25 superatom UV-Vis fingerprint spectra
,” Chem. Sci.
14
, 4666
–4671
(2023
).117.
S.
Koseki
,
N.
Matsunaga
,
T.
Asada
,
M. W.
Schmidt
, and
M. S.
Gordon
, “
Spin–orbit coupling constants in atoms and ions of transition elements: Comparison of effective core potentials, model core potentials, and all-electron methods
,” J. Phys. Chem. A
123
(12
), 2325
–2339
(2019
).118.
S. R.
Sahoo
and
S.-C.
Ke
, “
Spin-orbit coupling effects in Au 4f core-level electronic structures in supported low-dimensional gold nanoparticles
,” Nanomaterials
11
(2
), 554
(2021
).119.
H.
Ramanantoanina
,
W.
Urland
,
F.
Cimpoesu
, and
C.
Daul
, “
Ligand field density functional theory calculation of the 4f2 → 4f15d1 transitions in the quantum cutter Cs2KYF6:Pr3+
,” Phys. Chem. Chem. Phys.
15
(33
), 13902
(2013
).120.
A. A.
Rusakov
,
E.
Rykova
,
G. E.
Scuseria
, and
A.
Zaitsevskii
, “
Importance of spin-orbit effects on the isomerism profile of Au3: An ab initio study
,” J. Chem. Phys.
127
(16
), 164322
(2007
).121.
R.
Arratia-Pérez
,
L.
Hernández-Acevedo
, and
J. S.
Gómez-Jeria
, “
Calculated paramagnetic properties of matrix isolated Au3 cluster
,” Chem. Phys. Lett.
236
(1–2
), 37
–42
(1995
).122.
G. A.
Bishea
and
M. D.
Morse
, “
Resonant two‐photon ionization spectroscopy of jet‐cooled Au3
,” J. Chem. Phys.
95
(12
), 8779
–8792
(1991
).123.
A.
Baldes
,
R.
Gulde
, and
F.
Weigend
, “
Jahn–Teller distortion versus spin–orbit splitting: Symmetry of small heavy-metal atom clusters
,” J. Clust. Sci.
22
(3
), 355
–363
(2011
).124.
K.
Balasubramanian
and
D.
Majumdar
, “
Spectroscopic properties of lead trimer (Pb3 and Pb3+): Potential energy surfaces, spin–orbit and Jahn–Teller effects
,” J. Chem. Phys.
115
(19
), 8795
–8809
(2001
).125.
R.
Craciun
,
D.
Picone
,
R. T.
Long
,
S.
Li
,
D. A.
Dixon
,
K. A.
Peterson
, and
K. O.
Christe
, “
Third row transition metal hexafluorides, extraordinary oxidizers, and lewis acids: Electron affinities, fluoride affinities, and heats of formation of WF6, ReF6, OsF6, IrF6, PtF6, and AuF6
,” Inorg. Chem.
49
(3
), 1056
–1070
(2010
).126.
T.
Drews
,
J.
Supeł
,
A.
Hagenbach
, and
K.
Seppelt
, “
Solid state molecular structures of transition metal hexafluorides
,” Inorg. Chem.
45
(9
), 3782
–3788
(2006
).127.
L.
Alvarez-Thon
,
J.
David
,
R.
Arratia-Pérez
, and
K.
Seppelt
, “
Ground state of octahedral platinum hexafluoride
,” Phys. Rev. A
77
(3
), 034502
(2008
).128.
C. Y.
Yang
,
R.
Arratia-Perez
, and
J. P.
Lopez
, “
Electronic structure of tungsten hexacarbonyl
,” Chem. Phys. Lett.
107
(2
), 112
–116
(1984
).129.
R.
Arratia-Perez
and
D. A.
Case
, “
Electronic structure of octachloroditungstate(II)
,” Inorg. Chem.
23
(20
), 3271
–3273
(1984
).130.
W. W.
Lukens
,
M.
Speldrich
,
P.
Yang
,
T. J.
Duignan
,
J.
Autschbach
, and
P.
Kögerler
, “
The roles of 4f- and 5f-orbitals in bonding: A magnetochemical, crystal field, density functional theory, and multi-reference wavefunction study
,” Dalt. Trans.
45
(28
), 11508
–11521
(2016
).131.
N.
Moreno
,
F.
Ferraro
,
E.
Flórez
,
C. Z.
Hadad
, and
A.
Restrepo
, “
Spin-orbit coupling effects in AumPtn clusters (m + n = 4)
,” J. Phys. Chem. A
120
(10
), 1698
–1705
(2016
).132.
P.
Błoński
,
S.
Dennler
, and
J.
Hafner
, “
Strong spin–orbit effects in small Pt clusters: Geometric structure, magnetic isomers and anisotropy
,” J. Chem. Phys.
134
(3
), 034107
(2011
).133.
H. K.
Yuan
,
H.
Chen
,
A. L.
Kuang
, and
B.
Wu
, “
Spin–orbit effect and magnetic anisotropy in Pt clusters
,” J. Magn. Magn. Mater.
331
, 7
–16
(2013
).134.
M. N.
Huda
,
M. K.
Niranjan
,
B. R.
Sahu
, and
L.
Kleinman
, “
Effect of spin-orbit coupling on small platinum nanoclusters
,” Phys. Rev. A
73
(5
), 053201
(2006
).135.
A.
Castro
,
M. A. L.
Marques
,
A. H.
Romero
,
M. J. T.
Oliveira
, and
A.
Rubio
, “
The role of dimensionality on the quenching of spin-orbit effects in the optics of gold nanostructures
,” J. Chem. Phys.
129
(14
), 144110
(2008
).136.
A.
Shayeghi
,
L. F.
Pašteka
,
D. A.
Götz
,
P.
Schwerdtfeger
, and
R.
Schäfer
, “
Spin–orbit effects in optical spectra of gold–silver trimers
,” Phys. Chem. Chem. Phys.
20
(14
), 9108
–9114
(2018
).137.
L.
Xiao
and
L.
Wang
, “
From planar to three-dimensional structural transition in gold clusters and the spin–orbit coupling effect
,” Chem. Phys. Lett.
392
, 452
–455
(2004
).138.
M. A.
Flores
and
E.
Menéndez-Proupin
, “
Spin-orbit coupling effects in gold clusters: The case of Au13
,” J. Phys.
720
, 012034
(2016
).139.
M. H.
Khodabandeh
,
N.
Asadi-Aghbolaghi
, and
Z.
Jamshidi
, “
Influence of scalar-relativistic and spin–orbit terms on the plasmonic properties of pure and silver-doped gold chains
,” J. Phys. Chem. C
123
(14
), 9331
–9342
(2019
).140.
Y.-R.
Liu
,
T.
Huang
,
Y.-B.
Gai
,
Y.
Zhang
,
Y.-J.
Feng
, and
W.
Huang
, “
Three-dimensional assignment of the structures of atomic clusters: An example of Au8M (M=Si, Ge, Sn) anion clusters
,” Sci. Rep.
5
(1
), 17738
(2016
).141.
B.
Anak
,
M.
Bencharif
, and
F.
Rabilloud
, “
Time-dependent density functional study of UV-visible absorption spectra of small noble metal clusters (Cun, Agn, Aun, n = 2–9, 20)
,” RSC Adv.
4
(25
), 13001
(2014
).142.
J.
Li
,
X.
Li
,
H.-J.
Zhai
, and
L.-S.
Wang
, “
Au20: A tetrahedral cluster
,” Science
299
(5608
), 864
–867
(2003
).143.
X.
Yang
,
J.
Zhou
,
H.
Weng
, and
J.
Dong
, “
Spin-orbit interaction in au structures of various dimensionalities
,” Appl. Phys. Lett.
92
(2
), 023115
(2008
).144.
Q.
Zeng
,
X.
Chu
,
M.
Yang
, and
D.-Y.
Wu
, “
Spin–orbit coupling effect on Au–C60 interaction: A density functional theory study
,” Chem. Phys.
395
, 82
–86
(2012
).145.
A.
Muñoz-Castro
,
D.
Paez-Hernandez
, and
R.
Arratia-Perez
, “
Spin-orbit effect into isomerization barrier of small gold clusters. Oh ↔ D2h Fluxionality of the Au62+ cluster investigated by relativistic methods
,” Chem. Phys. Lett.
683
, 404
–407
(2017
).146.
M.
Iliaš
and
T.
Saue
, “
An infinite-order two-component relativistic hamiltonian by a simple one-step transformation
,” J. Chem. Phys.
126
(6
), 064102
(2007
).147.
J.
Autschbach
,
B. A.
Hess
,
M. P.
Johansson
,
J.
Neugebauer
,
M.
Patzschke
,
P.
Pyykkö
,
M.
Reiher
, and
D.
Sundholm
, “
Properties of WAu12
,” Phys. Chem. Chem. Phys.
6
(1
), 11
–22
(2004
).148.
P.
Pyykkö
and
N.
Runeberg
, “
Icosahedral WAu12: A predicted closed-shell species, stabilized by aurophilic attraction and relativity and in accord with the 18-electron rule
,” Angew. Chem., Int. Ed.
41
(12
), 2174
–2176
(2002
).149.
X.
Li
,
B.
Kiran
,
J.
Li
,
H.-J.
Zhai
, and
L.-S.
Wang
, “
Experimental observation and confirmation of icosahedral W@Au12 and Mo@Au12 molecules
,” Angew. Chem., Int. Ed. Engl.
41
(24
), 4786
–4789
(2002
).150.
H.
Häkkinen
, “
Electronic structure: Shell structure and the superatom concept
,” in Protected Metal Clusters: From Fundamentals to Applications
, edited by
H.
Hakkinen
and
T.
Tsukuda
(
Elsevier Science
, 2015
), pp 189
–222
.151.
P.
Jena
, Superatoms: Principles, Synthesis and Applications
, edited by
Q.
Sun
(
Wiley
, 2022
).152.
A.
Muñoz-Castro
and
R.
Arratia-Perez
, “
Spin-orbit effects on a gold-based superatom: A relativistic jellium model
,” Phys. Chem. Chem. Phys.
14
(4
), 1408
–1411
(2012
).153.
G.-J.
Cao
,
W. H. E.
Schwarz
, and
J.
Li
, “
An 18-electron system containing a superheavy element: Theoretical studies of Sg@Au12
,” Inorg. Chem.
54
(7
), 3695
–3701
(2015
).154.
P.
Pyykkö
,
C.
Clavaguéra
, and
J.-P.
Dognon
, “
The 32-Electron principle
,” in Computational Methods in Lanthanide and Actinide Chemistry
, edited by
M.
Dolg
(
John Wiley & Sons Ltd
:
Chichester, UK
, 2015
), pp 401
–424
.155.
J.-P.
Dognon
,
C.
Clavaguéra
, and
P.
Pyykkö
, “
Towards a 32-electron principle: Pu@Pb12 and related systems
,” Angew. Chem., Int. Ed.
46
(9
), 1427
–1430
(2007
).156.
J.-P.
Dognon
,
C.
Clavaguéra
, and
P.
Pyykkö
, “
Chemical properties of the predicted 32-electron systems Pu@Sn12 and Pu@Pb12
,” C. R. Chim.
13
(6–7
), 884
–888
(2010
).157.
L.-F.
Cui
,
X.
Huang
,
L.-M.
Wang
,
J.
Li
, and
L.-S.
Wang
, “
Pb122–: Plumbaspherene
,” J. Phys. Chem. A
110
(34
), 10169
–10172
(2006
).158.
A.
Muñoz-Castro
,
D.
Mac-Leod Carey
, and
R.
Arratia-Pérez
, “
Inside a superatom: The M7q (M=Cu, Ag, Q=1+, 0, 1–) case
,” ChemPhysChem
11
(3
), 646
–650
(2010
).159.
T.
Li
,
Z.
Feng
,
C.
Jing
,
F.
Hong
,
S.
Cao
, and
J.
Zhang
, “
Importance of spin–orbit coupling in M@Pb12 clusters (M=3d and 4d atoms)
,” Chem. Phys. Lett.
543
, 106
–110
(2012
).160.
L.
Visscher
, “
Four-component relativistic electronic structure theory
,” in Encyclopedia of Computational Chemistry
(
John Wiley & Sons, Ltd
,
Chichester, UK
, 2004
).161.
T.
Tsukuda
and
H.
Häkkinen
, Protected Metal Clusters: From Fundamentals to Applications
(
Elsevier
, 2015
).162.
X.
Kang
,
H.
Chong
, and
M.
Zhu
, “
Au25(SR)18: The captain of the great nanocluster ship
,” Nanoscale
10
(23
), 10758
–10834
(2018
).163.
M. S.
Devadas
,
S.
Bairu
,
H.
Qian
,
E.
Sinn
,
R.
Jin
, and
G.
Ramakrishna
, “
Temperature-dependent optical absorption properties of monolayer-protected Au25 and Au38 clusters
,” J. Phys. Chem. Lett.
2
(21
), 2752
–2758
(2011
).164.
O.
Toikkanen
,
V.
Ruiz
,
G.
Rönnholm
,
N.
Kalkkinen
,
P.
Liljeroth
, and
B. M.
Quinn
, “
Synthesis and stability of monolayer-protected Au38 clusters
,” J. Am. Chem. Soc.
130
(33
), 11049
–11055
(2008
).165.
L.
Cheng
,
C.
Ren
,
X.
Zhang
, and
J.
Yang
, “
New insight into the electronic shell of Au38(SR)24: A superatomic molecule
,” Nanoscale
5
(4
), 1475
(2013
).166.
L.
Liu
,
P.
Li
,
L.-F.
Yuan
,
L.
Cheng
, and
J.
Yang
, “
From isosuperatoms to isosupermolecules: New concepts in cluster science
,” Nanoscale
8
(25
), 12787
–12792
(2016
).167.
A.
Muñoz-Castro
, “
Single, double, and triple intercluster bonds: Analyses of M2Au36(SR)24 (M = Au, Pd, Pt) as 14-, 12- and 10-ve superatomic molecules
,” Chem. Commun.
55
(51
), 7307
–7310
(2019
).168.
G. L.
Malli
, “
Thirty years of relativistic self-consistent field theory for molecules: Relativistic and electron correlation effects for atomic and molecular systems of transactinide superheavy elements up to ekaplutonium E126 with g-atomic spinors in the ground state
,” Theor. Chem. Acc.
118
(3
), 473
–482
(2007
).169.
A.
Muñoz-Castro
, “
A superatomic molecule under the spin-orbit coupling: Insights from the electronic properties in the thiolate-protected Au38(SR)24 cluster
,” Int. J. Quantum Chem.
118
(6
), e25508
(2018
).170.
R.
Arratia-Pérez
,
L.
Hernández-Acevedo
, and
L.
Alvarez-Thon
, “
Calculated paramagnetic hyperfine structure of pentagonal bipyramid Ag7 cluster
,” J. Chem. Phys.
108
(14
), 5795
–5798
(1998
).171.
R. J.
Van Zee
and
W.
Weltner
, “
Cu7 cluster: Pentagonal bipyramid
,” J. Chem. Phys.
92
(11
), 6976
–6977
(1990
).172.
R.
Arratia‐Perez
and
G. L.
Malli
, “
Dirac scattered‐wave calculations for Ag2+3, Auq+3, and Auq+4 (q = 1, 2) clusters
,” J. Chem. Phys.
84
(10
), 5891
–5897
(1986
).173.
R.
Arratia-Perez
and
G. L.
Malli
, “
Bonding, optical, and magnetic properties of paramagnetic Ag41+ and Ag43+ clusters
,” J. Magn. Reson.
73
(1
), 134
–142
(1987
).174.
C. E.
Forbes
and
M. C. R.
Symons
, “
An electron spin resonance study of dimeric, trimeric, and tetrameric paramagnetic silver atom clusters
,” Mol. Phys.
27
(2
), 467
–471
(1974
).175.
R. S.
Eachus
and
M. C. R.
Symons
, “
Unstable intermediates. Part LXX. An electron spin resonance study of silver radicals trapped in frozen aqueous sulphuric acid
,” J. Chem. Soc.
1970
, 1329
.176.
A. V.
Arbuznikov
,
J.
Vaara
, and
M.
Kaupp
, “
Relativistic spin-orbit effects on hyperfine coupling tensors by density-functional theory
,” J. Chem. Phys.
120
(5
), 2127
–2139
(2004
).177.
J. A.
Howard
,
R.
Sutcliffe
, and
B.
Mile
, “
Cryochemical studies. 6. Electron spin resonance spectrum of silver cluster (Ag5)
,” J. Phys. Chem.
87
(13
), 2268
–2271
(1983
).178.
J. A.
Howard
,
R.
Sutcliffe
, and
B.
Mile
, “
The geometric and electronic structures of small metal clusters of group 1B metals
,” Surf. Sci.
156
, 214
–227
(1985
).179.
A.
Van Der Pol
,
E. J.
Reijersen
,
E.
De Boer
,
T.
Wasowicz
, and
J.
Michalik
, “
The electron paramagnetic resonance spectrum of Ag2+3
,” Mol. Phys.
75
(1
), 37
–42
(1992
).180.
L.
Alvarado-Soto
,
R.
Ramírez-Tagle
, and
R.
Arratia-Pérez
, “
Spin–orbit effects on the aromaticity of the Re3Cl9 and Re3Br9 clusters
,” Chem. Phys. Lett.
467
(1–3
), 94
–96
(2008
).181.
L.
Alvarado-Soto
,
R.
Ramírez-Tagle
, and
R.
Arratia-Pérez
, “
Spin−orbit effects on the aromaticity of the Re3 X92− (X = Cl, Br) cluster ions
,” J. Phys. Chem. A
113
(9
), 1671
–1673
(2009
).182.
A.
Vásquez-Espinal
,
R.
Pino-Rios
,
L.
Alvarez-Thon
,
W. A.
Rabanal-León
,
J. J.
Torres-Vega
,
R.
Arratia-Perez
, and
W.
Tiznado
, “
New insights into Re3(μ-Cl)3 Cl6 aromaticity. Evidence of σ- and π-diatropicity
,” J. Phys. Chem. Lett.
6
(21
), 4326
–4330
(2015
).183.
W. A.
Rabanal‐León
,
A.
Vásquez‐Espinal
,
O.
Yañez
,
R.
Pino‐Rios
,
R.
Arratia‐Pérez
,
L.
Alvarez‐Thon
,
J. J.
Torres‐Vega
, and
W.
Tiznado
, “
Aromaticity of [M3(μ‐X)3X6]0/2– (M = Re and Tc, X = Cl, Br, I) clusters confirmed by ring current analysis and induced magnetic field
,” Eur. J. Inorg. Chem.
2018
(28
), 3312
–3319
.184.
L.
Alvarez-Thon
and
W.
Caimanque-Aguilar
, “
Spin-orbit effects on magnetically induced current densities in the M42− (M=B, Al, Ga, In, Tl) clusters
,” Chem. Phys. Lett.
671
, 118
–123
(2017
).185.
L.
Alvarez-Thon
and
N.
Inostroza-Pino
, “
Spin-orbit effects on magnetically induced current densities in the M5− (M = N, P, As, Sb, Bi, Mc) clusters
,” J. Comput. Chem.
39
(14
), 862
–868
(2018
).186.
X.
Bai
,
J.
Lv
,
F.-Q.
Zhang
,
J.-F.
Jia
, and
H.-S.
Wu
, “
Spin-orbit coupling effect on structural and magnetic properties of Con Rh13−n (n = 0–13) clusters
,” J. Magn. Magn. Mater.
451
, 360
–367
(2018
).187.
B.
Le Guennic
and
J.
Autschbach
, “
[Pt@Pb12]2− − A challenging system for relativistic density functional theory calculations of 195Pt and 207Pb NMR parameters
,” Can. J. Chem.
89
(7
), 814
–821
(2011
).188.
E. N.
Esenturk
,
J.
Fettinger
, and
B.
Eichhorn
, “
The Pb122− and Pb102− Zintl ions and the M@Pb122− and M@Pb102− cluster series where M = Ni, Pd, Pt
,” J. Am. Chem. Soc.
128
(28
), 9178
–9186
(2006
).189.
A.
Li
,
Y.
Wang
,
D. O.
Downing
,
F.
Chen
,
P.
Zavalij
,
A.
Muñoz‐Castro
, and
B. W.
Eichhorn
, “
Endohedral plumbaspherenes of the group 9 metals: Synthesis, structure and properties of the [M@Pb12]3− (M=Co, Rh, Ir) ions
,” Chem. - Eur. J.
26
(26
), 5824
–5833
(2020
).190.
A. C.
Castro
,
E.
Osorio
,
J. O. C.
Jiménez-Halla
,
E.
Matito
,
W.
Tiznado
, and
G.
Merino
, “
Scalar and spin−orbit relativistic corrections to the NICS and the induced magnetic field: The case of the E122− spherenes (E = Ge, Sn, Pb)
,” J. Chem. Theory Comput.
6
(9
), 2701
–2705
(2010
).191.
D.
Smith
, “
The ‘anomalous’ ionization potential of bismuth
,” J. Chem. Educ.
52
(9
), 576
(1975
).192.
J. A.
Chamizo
, “
The ‘anomalous’ electron affinity of lead: An example of isoelectronic behavior
,” J. Chem. Educ.
61
(10
), 874
(1984
).193.
W.-K.
Li
and
S. M.
Blinder
, “
Introducing relativity into quantum chemistry
,” J. Chem. Educ.
88
(1
), 71
–73
(2011
).194.
H. G.
Raubenheimer
and
H.
Schmidbaur
, “
The late start and amazing upswing in gold chemistry
,” J. Chem. Educ.
91
(12
), 2024
–2036
(2014
).195.
D. W.
Smith
, “
Effects of exchange energy and spin-orbit coupling on bond energies
,” J. Chem. Educ.
81
(6
), 886
(2004
).196.
J.
Even
,
L.
Pedesseau
,
J.-M.
Jancu
, and
C.
Katan
, “
Importance of spin–orbit coupling in hybrid organic/inorganic perovskites for photovoltaic applications
,” J. Phys. Chem. Lett.
4
(17
), 2999
–3005
(2013
).197.
W. P.
Carbery
,
A.
Verma
, and
D. B.
Turner
, “
Spin–orbit coupling drives femtosecond nonadiabatic dynamics in a transition metal compound
,” J. Phys. Chem. Lett.
8
(6
), 1315
–1322
(2017
).198.
S.
Manna
and
S.
Mishra
, “
The role of spin–orbit coupling in the double-ionization photoelectron spectra of XCN2+ (X = Cl, Br, and I)
,” J. Phys. Chem. A
120
(9
), 1554
–1561
(2016
).199.
E.
Ronca
,
F.
De Angelis
, and
S.
Fantacci
, “
Time-dependent density functional theory modeling of spin–orbit coupling in ruthenium and osmium solar cell sensitizers
,” J. Phys. Chem. C
118
(30
), 17067
–17078
(2014
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
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