Relativistic Heavy-Neighbor-Atom Effects on NMR Shifts: Concepts and Trends Across the Periodic Table [original]

Relativistic Heavy-Neighbor-Atom E ﬀ ects on NMR Shifts: Concepts
and Trends Across the Periodic Table
Jan V ı 
cha, Jan Novotny  , Stanislav Komorovsky, Michal Straka, * Martin Kaupp, * and Radek Marek *
Cite This: Chem. Rev. 2020, 120, 7065 − 7103
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ABSTRACT: Chemical shifts present crucial information about
an NMR spectrum. They show the in ﬂ uence of the chemical
environment on the nuclei being probed. Relativistic e ﬀ ects caused
by the presence of an atom of a heavy element in a compound can
appreciably, even drastically, alter the NMR shifts of the nearby
nuclei. A fundamental understanding of such relativistic e ﬀ ects on
NMR shifts is important in many branches of chemical and
physical science. This review provides a comprehensive overview
of the tools, concepts, and periodic trends pertaining to the
shielding e ﬀ ects by a neighboring heavy atom in diamagnetic
systems, with particular emphasis on the “ spin-orbit heavy-atom
e ﬀ ect on the light-atom ” NMR shift (SO-HALA e ﬀ ect). The
analyses and tools described in this review provide guidelines to
help NMR spectroscopists and computational chemists estimate
the ranges of the NMR shifts for an unknown compound, identify
intermediates in catalytic and other processes, analyze conforma-
tional aspects and intermolecular interactions, and predict trends in series of compounds throughout the Periodic Table. The present
review provides a current snapshot of this important sub ﬁ eld of NMR spectroscopy and a basis and framework for including future
ﬁ ndings in the ﬁ eld.
CONTENTS
1. Introduction and Historical Overview 7066
2. Theoretical Background for the Analysis of SO-
HALA Shifts 7068
2.1. Nuclear Magnetic Shielding and NMR Shift 7068
2.2. MO Analysis in the Nonrelativistic Domain 7069
2.3. Theory of Relativistic NMR Chemical Shifts 7069
2.3.1. Variational Inclusion of SO E ﬀ ects 7070
2.3.2. Perturbational Inclusion of SO E ﬀ ects 7071
2.3.3. Third-Order Perturbation Analysis 7072
2.4. HALA E ﬀ ects on 1 H NMR Shifts 7073
2.5. HALA E ﬀ ects on the NMR Shifts of 2p Atoms 7074
2.6. Connection between the SO-HALA Shift and
the Electron Density 7075
3. Chemical Concepts: the HA − LA Interaction 7075
3.1. Mechanism, Factors Involved, and Tools for
Interpretation 7075
3.1.1. Spin − Orbit/Fermi-Contact (SO/FC)
Mechanism of the SO-HALA Shift 7075
3.1.2. Visualization Tools: SOM-ISD and SO-
EDD 7078
3.1.3. Structural and Electronic In ﬂ uence of a
trans Ligand 7078
3.1.4. Long-Range SO E ﬀ ects 7079
3.1.5. “ Through-Space ” Supramolecular Ef-
fects 7080
3.2. Unifying the Chemical Concepts 7080
3.2.1. PT2 Analysis Based on Molecular Spinor
Pairs 7080
3.2.2. Transparent PT3 Analysis Based on
Nonrelativistic MOs 7081
4. Trends in SO-HALA Shifts Across the Periodic
Table 7083
4.1. HAs of the s-Block 7085
4.1.1. General Aspects 7085
4.1.2. Groups 1 and 2: Alkali and Alkaline-Earth
Metals 7085
4.2. HAs of the d-Block 7085
4.2.1. General Aspects 7085
Received: December 6, 2019
Published: June 23, 2020
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4.2.2. Early Transition Metals 7086
4.2.3. Group 8: Ruthenium and Osmium 7086
4.2.4. Group 9: Cobalt, Rhodium, and Iridium 7086
4.2.5. Groups 10 and 11: Platinum, Gold 7087
4.2.6. Group 12: Mercury 7089
4.3. HAs of the p-Block 7089
4.3.1. General Aspects 7089
4.3.2. Groups 13 and 14: Triels and Tetrels 7090
4.3.3. Group 15: Pnictogens 7090
4.3.4. Group 16: Chalcogens 7091
4.3.5. Group 17: Halogens 7091
4.3.6. Group 18: Aerogens 7092
4.4. HAs of the f-Block 7092
4.4.1. General Aspects 7092
4.4.2. Selected Examples and Applications 7092
4.5. Superheavy Elements (SHEs) as HAs 7094
4.6. Weak Interactions and “ Through-Space ” SO-
HALA E ﬀ ects 7094
4.7. Emerging Applications 7094
5. Summary and Prospects 7095
6. Technical and Computational Details 7095
6.1. Optimization of Structure 7095
6.2. Molecular orbital and bonding analysis 7095
6.2.1. Natural Bonding Orbital (NBO) Analysis 7095
6.2.2. Energy Decomposition Analysis-Natural
Orbitals for Chemical Valence (EDA-
NOCV) 7095
6.3. Calculation of SO-HALA Shifts 7095
6.4. Calculation and Visualization of the SO-and-
Magnetically Induced Spin Density (SOM-
ISD) and the Spin − Orbit Electron Deforma-
tion Density (SO-EDD) 7095
6.5. Analysis of SO-HALA Shifts 7095
6.5.1. Second-Order Perturbation Ana lysis
(PT2) 7095
6.5.2. Third-Order Perturbation Analysis (PT3) 7095
Author Information 7096
Corresponding Authors 7096
Authors 7096
Notes 7096
Biographies 7096
Acknowledgments 7096
Table of Abbreviations 7097
References 7097
1. INTRODUCTION AND HISTORICAL OVERVIEW
The NMR chemical shift, the resonant frequency of a particular
nucleus relative to a standard in an NMR spectrum, provides
crucial information about the molecular and electronic structure
of the given system. In order to optimize the information
obtainable from an NMR spectrum, a qualitative and
quantitative understanding of the shift values expected for a
given substance is fundamentally important. In this review we
will lay out the current level of understanding of the relativistic
e ﬀ ects of one or more heavy atoms (HAs) on the NMR shifts of
its neighboring atoms. The term “ heavy-atom e ﬀ ect ” , coined by
Pyykko  ,
1
initially referred to the nuclear shielding of the heavy
atom itself, i.e., the “ heavy-atom e ﬀ ect on the heavy-atom
shielding ” (HAHA e ﬀ ect).
2
The e ﬀ ect of a heavy atom on a
neighboring light atom was subsequently termed “ heavy-atom
e ﬀ ect on the light-atom shielding ” (HALA e ﬀ ect), even though
we now know that this e ﬀ ect also applies when the NMR nucleus
probed is itself a heavy atom. Keeping this fact in mind, we will
use the established HALA label throughout this review. Many
HALA e ﬀ ects are due to spin − orbit (SO) coupling, but there are
also scalar relativistic (SR, also known as spin-free relativistic,
SFR) HALA e ﬀ ects.
3
Because it is far more di ﬃ cult to derive
useful qualitative chemical concepts for the relativistic e ﬀ ects of
a heavy atom on its own shielding,
4
we will not cover the HAHA
e ﬀ ects in any detail here, but will mention the relevant literature
toward the end of this section. Moreover, while crucial for the
absolute shielding of the heavy atom, HAHA e ﬀ ects mostly
cancel out in the relative shifts that NMR spectroscopy typically
focuses on (with some exceptions, see refs 2 − 16 ). We also
restrict our coverage to formally diamagnetic systems.
As this revi ew will largel y focus on t he progres s in
understanding achieved since the last extensive reviews in
2004
3 , 13
(reviews includ ing the topic as part of a wider
discussion have appeared occasionally in the meantime
15 − 18
),
it seems appropriate to start with a brief overview of the state of
the ﬁ eld prior to 2004. Much of what is summarized in this
section is explained in much more detail in ref 3 .
The potential importance of the e ﬀ ects of special relativity in
describing the magnetic-resonance parameters, including NMR
chemical shifts, was recognized early on. The inner core − shells
of the atoms are often crucially involved and, in many instances,
the important role of the electronic spin introduces spin − orbit
coupling contributions. Because this applies to NMR shifts, the
e ﬀ ects of SO coupling will be the central focus of this work, but
we will also discuss the in ﬂ uence of SR e ﬀ ects on the NMR shifts
of neighboring atoms.
The ﬁ rst attempts by Nakagawa and co-workers to use
perturbation theory (PT) to extend the theory of nuclear
shielding in diamagnetic systems beyond the nonrelativistic
(NR) Ramsey equation (see also section 2 ) were published in an
organic chemistry journal
19
(and in the proceedings of a
domestic Japanese NMR conference).
20
Note also that these
ideas were preceded by the ﬁ rst theories of pseudocontact shifts
in paramagnetic systems,
21 − 25
which are also known to have an
SO origin. Nakagawa et al. laid out the third-order perturbation-
theory (PT3) framework that resulted from including SO
coupling as an additional perturbation in Ramsey ’ s theory and
pointed out the important analogy of the SO-induced HALA
e ﬀ ects with the Fermi-contact mechanism of indirect spin − spin
coupling constants, which was further exploited based on
quantitative computations almost 30 years later.
26 , 27
The ﬁ rst
explicit computations within such a third-order perturbation
scheme in the 19 70s and 1 980s rel ied on se miempi rical
approximate molecular-orbital (MO) theory.
28 − 32
The pertur-
bational treatment has been extended to accurate post-Hartree −
Fock wave functions as well as to Kohn − Sham DFT, and it has
been used for various applications and for detailed analyses.
3
We
will present the PT framework in section 2 and exploit it
extensively in section 3 .
The extension of the second-order perturbation theory (PT2)
Ramsey equations to a variational treatment of SO coupling in a
four-component framework (see section 2 ) dates from 1983,
when three independent research groups published pro-
posals.
33 − 35
The limited computational resources available at
the time forced initial implementations of such relativistic PT2
theories to use even more qualitative MO schemes (of the
extended-Hu  ckel type) than those used for PT3. Some crucial
understanding of the HALA e ﬀ ect on, e.g., the 1 H shielding in HI
in terms of the spin and spatial symmetry of four-component
spinors was extracted even at this early stage, many years before
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more quantitative ab initio or DFT four- or two-component
relativistic schemes were introduced to compute NMR
shieldings.
1
We note in passing that the ﬁ rst studies of the SR
neighbor-atom e ﬀ ects in the 1990s used e ﬀ ective-core potentials
(ECPs) on the heavy atom, typically at DFT levels, to e ﬃ ciently
incorporate the SR e ﬀ ects of the heavy atom on the overall
electronic structure and thus on the NMR shielding of the
neighboring atoms (e.g., ligand atoms in a heavy-metal
complex) .
36 − 39
While comp arison o f relat ivisti c and non-
relativistic ECP computations is still an e ﬃ cient way to study
the SR e ﬀ ects on the neighbor-atom NMR shieldings, in the
years since the 2004 review, relativistic all-electron methods of
one-, two-, or four-component types (see section 2 ) have
evolved to o ﬀ er easy access to similar information.
Methodological aspects will be described in detail in section 2 .
Here we summarize the qualitative understanding of HALA
e ﬀ ects on nuclear shieldings as of 2004. The analogy between
SO-HALA e ﬀ ects and the Fermi-contact (FC) mechanism of
indirect spin − spin coupling, implied qualitatively by Nakagawa
et al.,
20
was computationally tested in detail in a 1998 paper.
26
DFT calculations on a series of organoiodine complexes showed
closely parallel behavior of the SO-HALA contributions to 13 C
and 1 H NMR shieldings for di ﬀ erent positions in iodobenzene
and the corresponding I − C and I − H reduced spin − spin
coupling constants, an enhancement of both quantities with
carbon s-character in the C − I bond in iodoethane, iodoethene,
and iodoethyne, as well as the Karplus-type dependence of the
three-bond SO-HALA e ﬀ ects on the 1 H NMR shieldings on the
I − C − C − H dihedral angle in iodoethane. These ﬁ ndings clearly
established the validity and usefulness of Nakagawa ’ s analogy
and have been con ﬁ rmed by subsequent studies (see section 3 ).
The dominance of an FC-type mechanism for the SO-HALA
e ﬀ ects can best be understood and appreciated using the third-
order-perturbation framework (PT3) presented in section 2 ;
therefore, detailed explanations will be presented in section 3 .
The dependence of the FC-mechanism on the s-character of
the bonding at the spectator NMR atom
26
is shown by the fact
that (a) 1 H NMR shieldings are most sensitive to SO-HALA
e ﬀ ects because the hydrogen 1s-orbital predominates in
bonding, (b) 13 C SO-HALA e ﬀ ects increase, e.g., from sp 3 to
sp 2 to sp hybridization of carbon atoms, (c) for p-block elements
as LA, SO-HALA e ﬀ ects on the shifts tend to be largest for their
highest oxidation state, where the participation of the s-orbital in
bonding is maximized,
3 , 13 , 40
(d) SO-HALA e ﬀ ects thus tend to
be small for ﬂ uorine and oxygen, where p-orbitals dominate the
bonding,
41
(e) SO-HALA e ﬀ ects are also small for the early
transition-metal elements Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, and W
as LA in d 0 halide complexes because the metal − halogen bonds
in such systems are dominated by the metal d-orbitals,
3
but (f)
SO-HALA e ﬀ ects increase for the NMR shieldings of the metal
in late transition-metal halide complexes where the metal s-
character is again more signi ﬁ cant in the bonding.
3
These
considerations explain the predominance of the normal halogen
dependence (NHD) of shifts for p-block nuclei (in particular
those in their maximum oxidation state), the inverse halogen
dependence (IHD) for early d 0 systems (and the closely related
lanthanum
42
), and a return to NHD for the late transition
metals. NHD shows a decrease in the shifts (increased shielding)
from chloride to bromide to iodide substituent(s), whereas IHD
indicates the opposite behavior. NHD is most often, but not
always, caused by shielding SO-HALA e ﬀ ects, whereas IHD
tends to arise from the trend of the paramagnetic shielding
contributions when the SO-HALA e ﬀ ects are small. Many
examples of NHD induced by SO-HALA e ﬀ ects have been
studied (see section 4 ), and some mistakes in the interpretations
of NHD have been exposed, for example, for the 13 C NMR shifts
in trihalome thyl cations (CX 3 ;X=C l ,B r ,I ) .
43
Other
observations related to the s-character in bonding such as the
enhancement of hybridization defects
44 , 45
(caused by the size
mismatch between the valence s- and p-orbitals)
46
and thus of s-
character and SO-HALA e ﬀ ects, whereby more electronegative
substituents withdraw charge from the spectator NMR atom,
have been discussed in the 2004 review.
3
This last aspect
explains why the halomethanes CH 4 ‑ n X n (X = Br, I) exhibit a
“ nonlinear NHD ” with increasing n, whereas the corresponding
mixed complexes CY 4 ‑ n X n (e.g., Y = Br, X = I) show an essentially
linear decrease.
3
We note in passing that two computational
studies have shown that the dependence of SO-HALA e ﬀ ects on
the bond length in systems like HX (X = halogen) di ﬀ ers from
that of nonrelativistic shifts. As a consequence, SO-HALA e ﬀ ects
may alter the ro-vibrational corrections to these shifts and give
rise to unusual isotope e ﬀ ec ts a nd te mp er at ure d ep en d-
encies.
47 , 48
The importance of π -type lone electron pairs (LPs, or
nonbonding pairs, n ) for shielding SO-HALA e ﬀ ects (see
sections 2 and 3 ) had received substantial attention by 2004:
MO analyses showed that such π -LPs are important for shielding
SO-HALA e ﬀ ects. A comparative study of the 13 C shifts in the
series CF 3 IF n ( n = 0, 2, 4) demonstrated that the two iodine LPs
in CF 3 I dominate the 13 C SO-shifts. The latter are about halved
with one LP in CF 3 IF 2 and essentially vanish in the absence of π -
LPs in CF 3 IF 4 .
49
Similarly, chalcogenide substituents, which
tend to feature only one π -type LP, give rise to only about half of
the SO-HALA e ﬀ ects provided by halides.
3 , 50 − 52
Shielding SO-
HALA e ﬀ ects have also been identi ﬁ ed for the 13 C NMR shifts in
transition-metal carbonyl complexes and related systems.
39 , 53 , 54
In contrast, high-lying σ -bonding MOs in systems like XHgCH 3
or InCH 3 have been found to cause deshielding SO e ﬀ ects on
the 13 C NMR shifts.
38 , 39 , 55
The overall observations as of 2004
are summarized as follows: high-lying occupied nonbonding
orbitals with π -symmetry relative to the bond between the
NMR-active LA and the HA substituent(s) provide shielding
SO-HALA e ﬀ ects, while high-lying σ -bonding orbitals have
deshielding SO-HALA e ﬀ ects. Already in 1987, the 4-
component extended- Hu  ckel computation s on HI sho wed
that π -type iodine LPs make a shielding 1 H HALA contribution
while a σ -bond causes deshielding.
1
We will discuss in section 3
how a consistent interpretation within a third-order-perturba-
tion theory framework allows us to place these observations, and
particularly the sign of SO-HALA e ﬀ ects, into a more general
picture.
SR-HALA e ﬀ ects are easily evaluated computationally using
an ECP or all-electron approach, but their interpretation is not
as direct as that for the SO-HALA e ﬀ ects discussed above. They
are not governed by one dominant and distinct mechanism
56 , 57
like the SO/FC mechanism for SO-HALA e ﬀ ects that we will
describe in detail in section 2 . The SR e ﬀ ects also originate in the
core − shells of the HA(s), but it is their e ﬀ ect on the overall
electronic structure of the valence shell that matters for the
NMR shifts of neighboring atoms. This may happen via
relativistic changes in the shapes of MOs that involve both the
relativistically modi ﬁ ed AOs of the HA and those near the LA.
The paramagnetic shielding contributions may also be a ﬀ ected
by relativistically modi ﬁ ed energy denominators. These SR
e ﬀ ects tend to become important somewhat lower in the
Periodic Table than SO-HALA e ﬀ ects, typically in the sixth
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period and beyond. They also tend to obey shell-structure e ﬀ ects
known for SR e ﬀ ects on other properties, such as structures or
energetics. Thus, we expect to see a “ coinage-metal maximum ”
around group 11, related to the famous “ gold maximum ” .
58
Initial studies carried out in the 1990s and early 2000s included
the 17 O NMR shieldings of oxo complexes,
36 , 39
13 C and 17 O
NMR shieldings of carbonyl complexes,
39 , 53 , 54
as well as 13 C and
1 H NMR shieldings of organomercury complexes and mercury
hyd ri de s.
38 , 39
Bo th in di re ct S R e ﬀ ec t s du e to r ela ti vi st ic
structure changes and direct e ﬀ ects were analyzed. The
relativistic contraction of the bond length and the direct SR
e ﬀ ects at a given structure increased the 17 O NMR shielding in
oxo complexes. However, for organomercury complexes the 13 C
NMR shieldings were still increased by the relativistic bond-
length contraction but reduced by the direct SR e ﬀ ects at a given
structure (SO e ﬀ ects on structures tend to be smaller and have
not been studied as much
59
). These di ﬀ erences were attributed
to the di ﬀ erence in bonding character at the metal center, a
dominant 5d-expansion in the oxo complexes vs a dominant 6s
contraction in the mercury species. Further examples studied
included the 19 F NMR shifts of mixed uranium halide
complexes.
9 , 10 , 41
In this case the SO-HALA e ﬀ ects were rather
small because of an ine ﬃ cient SO/FC mechanism, whereas the
SR e ﬀ ects were sizable. We should also mention that SR- and
SO-HALA-e ﬀ ects are not additive. SR-induced changes in the
electronic structure (e.g., contraction/expansion of MOs,
modi ﬁ cation of the radial nodal structure, or changes in the
energy spectrum) may alter the SO e ﬀ ects.
60 − 62
Finally, we mention in passing the so-called HAHA e ﬀ ects.
These relate largely but not exclusively to the core − shells of the
relativistic HA.
2 , 3 , 5 − 16
Made up of both scalar and spin − orbit
contributions, they are generally included in full four-
component treatments but are less well described in some
quasi-relativistic treatments like the ZORA (zerot h-order
regular approximation) approach.
63
While analyses based on
Breit-Pauli (BP) perturbation theory yield a number of terms, it
often appears that a cross term between the Fermi-contact
operator and the kinetic spin-Zeeman term dominates the
HAHA e ﬀ ects.
14
Such contributions seem not to occur for spin-
rotation constants, and a historical approach frequently used to
extract the paramagnetic contributions to NMR shifts from spin-
rotation constants (see section 2 ) breaks down for heavier
atoms.
64 − 69
In the following, we will ﬁ rst summarize the theoretical and
computational background of relativistic e ﬀ ects on NMR shifts
( section 2 ) and then provide a general conceptual framework for
analyzing the SO-HALA e ﬀ ects ( section 3 ) before embarking on
a journey of the SO-HALA e ﬀ ects through the Periodic Table,
concentra ting o n wor k publis hed sinc e the 2004 rev iew
3
( section 4 ).
2. THEORETICAL BACKGROUND FOR THE ANALYSIS
OF SO-HALA SHIFTS
As a basis to better understand our analyses in the following
sections, we need to introduce some theoretical formalism based
on relativistic quantum chemistry and perturbation theory.
While we strive to keep matters simple, some key equations are
required. Relativistic e ﬀ ects in computat ional quantum
chemistry are de ﬁ ned as the di ﬀ erence between results obtained
by relativistic and nonrelativistic levels of theory.
1
As such, the
relativistic contribution to a particular molecular property is a
purely theor etical co ncept th at we can probe onl y by
computations. However, the study of relativistic e ﬀ ects is crucial
for our understanding of the molecular properties of systems
containing heavy atoms and for relating these properties to
electronic structure or chemical bonding. The inclusion of
relativistic e ﬀ ects in computations always entails a compromise
between accuracy and e ﬃ ciency. A more accurate description of
the relativistic e ﬀ ects usually require s a more demanding
computational methodology. Potentially relevant aspects of
quantum electrodynamics notwithstanding, relativistic four-
component (4c) theory based on the Dirac-Coulomb-Breit
Hamiltonian is considered
70 , 71
the gold standard for calculating
molecular properties, but because of its large computational
demands, more approximate quasi-relativistic two-component
(2c) theories have been developed. These theories include both
scalar and spin − orbit (SO) relativistic e ﬀ ects variationally, i.e.,
up to an in ﬁ nite order of Taylor expansion. Further down the
ladder are so-called one-component (1c) theories that neglect
spin − orbit e ﬀ ects, signi ﬁ cantly reducing the computational
demand. In areas where the spin − orbit e ﬀ ects are negligible,
such more expedient one-component treatments (SR or NR)
are t he obv io us cho ice . One- co mponen t ap pro ache s may
contain, to various degrees of accuracy, scalar-relativistic (SR)
e ﬀ ects or no relativistic e ﬀ ects at all. In the latter case, a
nonrelativistic (NR) theory featuring the well-known Schro  -
dinger Hamiltonian is obtained. Finally, at any level of
Hamiltonian (1c-NR, 1c-SR, 2c, or 4c), various corrections
can be added perturbatively, including, for example, SO e ﬀ ects
on top of 1c (NR or SR) computations.
From the computational point of view, all of the above
relativistic methods can be formulated at the all-electron or ECP
(e ﬀ ective core potential) level of theory, provided the core −
shells of the spectator NMR-active atom are covered, and the full
core nodal structure of its valence orbitals is thus treated
correctly. Of course, all-electron calculations are more computa-
tionally demanding because they take into consideration all of
the degrees of freedom for every electron variable. On the other
hand, ECP theory excludes core electron variables from the
computation. This approximation is based on the observation
that core electrons are not involved in chemical bonding and
therefore can be included in the calculation as an e ﬀ ective
potential. This technique leads to computationally e ﬃ cient
approaches, but because the number of electrons treated by the
ECP in ﬂ uences the ﬁ nal accuracy of the results, it requires
additional validation.
41
Moreover, any relativistic e ﬀ ects can be
included in the ECP as well. For example, it is popular to include
SR e ﬀ ects in the ECP but to treat the remaining valence
electrons at the NR level of theory. In this way the relativistic
e ﬀ ects are also included for the valence shell through the action
of the ECP. As relativistic all-electron approaches have become
more e ﬃ cient over the years, the need to use ECPs has
diminished to some extent.
2.1. Nuclear Magnetic Shielding and NMR Shift
With some exceptions (see below), an analysis of the e ﬀ ects of
the electronic structure on relative isotropic NMR chemical
shifts for diamagnetic species containing only light elements
typically r equires consideration of on ly the paramagn etic
contribution. In the following, we will rationalize this statement
in a nonrelativistic framework using a relation suggested by
Flyga re and Go odis man
72
and dis cuss why it can not be
transferred to relativistic theory.
The isotropic NMR chemical shift of the “ light atom ” of
interest (herein labeled LA) is de ﬁ ned as the di ﬀ erence between
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the isotropic nuclear shielding of a reference system and the
shielding of the system of interest
δ

σσ =− (LA) (LA) (LA)
ref (1)
The shielding constant and, thus, the chemical shift can be
broken down into diamagnetic and paramagnetic contributions
σσ σ =+ (LA) (LA) (LA)
dp (2)
δ

δδ =+ (LA) (LA) (LA)
dp (3)
Flygare and Goodisman
72
pointed out an important relation
for the diamagnetic shielding
σσ ≈+ C (LA) (LA) (LA)
dF A n u c (4)
where σ FA (LA) is the diamagnetic shielding of the free LA and
C nuc (LA) represents the nuclear contribution to the nuclear
spin-rotation constant (in ppm). When the results from eqs 1 − 4
are combined, σ FA (LA) cancels out, which gives us the following
simple expression:
δ

δ ≈− + CC (LA) (LA) (LA) (LA)
ref
nuc nuc p (5)
As C nuc depends only on the structure of the molecule, it is
clear that the electronic structure in ﬂ uences isotropic NMR
chemical shifts solely via the paramagnetic contribution.
Unfortunately, this result is not transferable to the calculation
of SO-HALA shifts. There is some indication that relation 4
(and by extension relation 5) is applicable in the relativistic
domain when the gauge origin is placed on the spectator atom
LA. However, in the presence of one or more HAs, this leads to
inaccurate results for any practical ﬁ nite basis set.
73
Fortunately,
the diamagnetic contribution to NMR shifts depends only
weakly on the electronic structure (usually by up to 2 ppm);
therefore, it is often su ﬃ cient to analyze only the paramagnetic
contribution to understand trends in NMR shifts. 1 H NMR
shifts are an exception, due to their small shielding range (i.e., 2
ppm is a large e ﬀ ect in this case) and the absence of a core − shell
in hydrogen. A small shift range may cause lithium shifts to be
borderline cases, where one should not a priori disregard
changes in the diamagnetic terms.
In the above discussion we assumed the use of common gauge
orig in ( CGO) met hodo lo gy f or cal cul ati on of the NM R
shielding. While the CGO is typically not used for practical
calculations, due to poor convergence of the basis set, the
analysis within a CGO picture is more transparent than, e.g.,
with the frequently used London orbitals (gauge-including
atomic orbitals, GIAOs).
74 , 75
This is mostly because GIAO
expressions are more complicated than those for CGO, while the
dependence of the GIAO and CGO diamagnetic contributions
on the electronic structure remains approximately the same.
Nev er th ele ss , GI AO- b ased ana ly se s can be fo u nd in t he
literature,
76 , 77
usually in cases where the higher precision of
the GIAO methodology is mandatory and in-depth MO analysis
is less crucial.
2.2. MO Analysis in the Nonrelativistic Domain
In view of the above description, we will focus further on the
analysis of the isotropic paramagnetic shielding contribution,
σ p (LA), or shift, δ p (LA). In the nonrelativistic (NR) domain
within Hartree − Fock or density functional theory, the σ p (LA)
contribution has a Ramsey-like form
78
(we will use Hartree
atomic units throughout section 2 )
∑∑∑
σ φφ φ φ
εε
= ⟨|
 |⟩ ⟨|  |⟩
−
== =
−
c
lr l
(LA) 2
3 uia
uu
NR, p
2
1
3
1
occ
1
vac
i
G
aa
LA
3 LA
i
ia (6)
where ε i ( ε a ) are occupied (vacant) one-electron energies, φ i
( φ a ) are occupied (vacant) molecular orbitals (MOs), r LA is the
electron position operator relative to the position of the LA
nucleus, r G is the electron position operator relative to the
position of the gauge origin, c is the speed of light, and
 =× lr
p

LA
LA ,  =× lr
p

G
G and p repre sent the electr on
momentum operator. In eq 6 , the angular-momentum operator,

l G , represents the interaction with an external magnetic ﬁ eld,
and the paramagnetic spin − orbit (PSO), r LA
− 3 l  u
LA , operator arises
from the interaction with the magnetic moment of the LA
nucleus. Note that PSO describes the (nuclear-)spin-
(electron-)orbit interaction and should not be confused with
the (electron-)spin-(electron -)orbit interaction discussed
throughout this work.
In the basis-set limit, the position of the gauge vector can be
chosen arbitrarily. However, for the MO analysis to be practical,
the size of the basis must be within reasonable limits. If there is
no HA in the system, the gauge vector is usually placed at the
position of the LA nucleus. This choice simpli ﬁ es the analysis
considerably, and at the same time, the size of the basis set can be
kept moderately large. In contrast, to keep basis requirements
reasonable for systems with one HA, the gauge vector must be
placed at the position of this HA, see eqs 8 , 18 , and 19 below.
79
Finally, if there is more than one HA in the system, it is not
possible to keep the basis-set requirements su ﬃ ciently low, and
methods employing GIAOs (or related approaches) must be
used. In the following, we will discuss systems containing just
one HA.
Equation 6 has been used for decades to successfully analyze
NMR shifts of “ non-relativistic cases ” , i.e., for systems without
HA(s). Molecular-orbital analysis based on eq 6 is performed in
three steps: ﬁ rst, identify the frontier MOs (other MOs are
insigni ﬁ cant because they bring in larger energy denominators),
then select MOs which have nonzero contributions from atomic
orbitals of the LA (the function r LA
− 3 enhances contributions from
these MOs), and ﬁ nally, ﬁ nd the most signi ﬁ cant occupied-
vacant MO ↔ MO * magnetic couplings using the angular-
momentum operator, l  u
LA . A textbook example of this procedure
is the n ↔ π * coupling in nitrogen heterocycles such as pyridine,
shown in Figure 1 .
2.3. Theory of Relativistic NMR Chemical Shifts
Given the central role of SO-HALA e ﬀ ects throughout this
review, we will concentrate the following discussion on
Figure 1. Schematic graphical representation of the Ramsey-type n ↔
π * coupling in pyridine, which dominates the isotropic 15 N NMR
shift.
80
Angular-momentum and paramagnetic spin − orbit operators are
shown in black and blue, respectively.
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relativistic SO e ﬀ ects. SR e ﬀ ects are brie ﬂ y discussed below.
Each subsection starts with a brief description of the underlying
methodology followed by a more detailed discussion focused on
the analysis of SO-HALA NMR shifts.
There are two concept ually di ﬀ erent approaches to
considering SO e ﬀ ects when calculating NMR shifts: two- or
four-component methods that describe SO e ﬀ ects variationally
and methods that include leading-order SO e ﬀ ects by using
perturbation theory.
81 − 85
2.3.1. Variational Inclusion of SO E ﬀ ects. Over the last
two decades, the development of new theoretical and numerical
techniques has allowed the practical application of relativistic
methods that include both SR and SO e ﬀ ects variationally.
Aided by growing computational power, these methods have
become increasingly popular because of their accuracy and
increasing e ﬃ ciency. The most rigorous methods capable of
treating relativistic e ﬀ ects on an equal footing across the Periodic
Table of Elements use the four-component Dirac-Coulomb(-
Breit) Hamiltonian.
86
These methods
73 , 87 − 89
were long
considered to be too computationally expensive, but it has
now been demonstrated that using modern computational
techniques, four-component methodology can be applied on a
regular basis to systems with more than 100 atoms,
90 − 92
or in an
extreme case with more than 300 atoms,
93
when combined with
Kohn − Sham density-functional theory (DFT). Two-compo-
nent quasi-relativistic methods
85
introduce various approxima-
tions at di ﬀ erent levels of theory to provide a more computa-
tionally e ﬃ cient framework. Therefore, in an ideal situation,
two-component methods are ﬁ rst validated against their more
accurate four-component counterparts on a particular class of
systems and are subsequently used in similar applications to
provide, hopefully, an improved compromise between e ﬃ ciency
and accuracy (example s of such comparison s range from
halogenated organic compounds
94
through transition met-
als
95 , 96
to actinide systems
97
). We can distinguish roughly
approximate two-component (A2C) and exact two-component
(X2C) methods. While in the former the transformation of the
one-electron Dirac equation is done only approximately, in the
latter it is done to machine accuracy. In both cases, the treatment
of the two-electron contributions in the transformation, in
particular for the SO e ﬀ ects, is still a matter of development and
is usually neglected in practical applications. While X2C-type
method s for nuc lear s hiel ding s have ve ry rec ently be gun
receiving attention,
98
most practical 2c computations of nuclear
shiel d ings ha ve been d one w ith the ze rot h- ord er regu la r
approximation (ZORA),
63
which is an A2C ansatz . Such SO-
ZORA computations and related approaches are currently being
applied, with reasonable approximations, to systems containing
hundreds of atoms, and they have become the most popular
methods for including SO e ﬀ ects variationally into computa-
tions of NMR shifts (note that the approximations involved in
the ZORA treatment give rise to substantial errors in calculated
absolute shieldings).
67
Detailed analysis of the NMR shifts within a 2c or 4c
framework is somewhat complicated, primarily by the complex
structures of both the magnetic moment operators and the
molecular spinors. Unlike nonrelativistic molecular orbitals,
molecular spinors consist of two (four) complex functions in the
case of 2c (4c) calculations. For simplicity, we will focus on 2c
theory. The isotropic paramagnetic NMR shielding can be
written formally as
∑∑∑
σ
φφ φ φ
εε
=
⟨|
 |⟩ ⟨|
 |⟩
−
μ
== =
9 mm
(LA)
2
3
(LA)
uia
i u aa
u i
ia
2c, p
1
3
1
occ
1
vac 2c B, 2c 2c 2c , 2c 2c
2c 2c
(7)
Here φ i
2c ( φ a
2c ) are 2c occupied (vacant) molecular spinors
(MSs), ε i
2c ( ε a
2c ) are 2c occupied (vacant) one-electron energies,

m B, 2
c

is a 2c magnetic-moment operator corresponding to the
external magnetic ﬁ eld,  μ
m (LA)
c ,2 is the 2c nuclear magnetic-
moment operator of the LA nucleus, and the symbol
9

speci ﬁ es
the real part of the following expression. To make the analysis of
eq 7 feasible and tractable but still meaningful in most cases, it is
best to use semilocal (nonhybrid) DFT potentials and to omit
DFT kernel. For a more detailed discussion of the necessary
approximations, see refs 99 and 100 .
Thanks to the time-reversal symmetry of the perturbation-free
Dirac-Coulo mb Hamilto nian, all MSs are at least doubly
degenerate. In the following, we will refer to two MSs with the
same one-electron energy as a molecular spinor pair (MSP), also
known as a Kramers pair.
101
Note that in addition to the
complex structures of MSs and magnetic moments, their form
di ﬀ ers signi ﬁ cantly depending on which 2c theory is used. A
pragmatic way to circumvent this problem in the analysis of eq 7
and provide a conceptual link to the PT3 approach based on the
NR wave functions described further below is to focus on only
the leading-order e ﬀ ects in a perturbation expansion, which are
the same for all 2c theories. These contributions can be
expressed as
100
Ä
Ç
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å É
Ö
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
∑∑∑
σ
σ
σσ π
φφ φ δ φ
εε
φφ φ δ φ
εε
≅+
⟨|
 |⟩ ⟨|
 |⟩
−
+ ⟨|
 |⟩ ⟨ |
 |⟩
−
== =
9 P c
l
l
1
1
(LA) (LA) 4
9 uia
i u aa u i
ia
i u aa u i
ia
2 c ,p N R ,p A B
2
1
3
1
occ
1
vac
SO HA nr nr LA nr
nr HA SO nr LA nr
(8)
where δ  LA = δ  ( r LA ) is the Dirac delta function centered on the
LA, 1 is the two-by-two identity matrix, σ is a vector composed of
Pauli matrices,
102
φ i
SO ( φ a
SO ) are occupied (vacant) linear-
response MSs perturbed by the SO interaction
∑ σ
φφ φ
φ
φφ
εε φ
≅+
=+ ⟨|  ·| ⟩
−
≠
≠±
− l
Z
c
r
4
pp p
p qp
qp
qp
pq q
2c nr SO
nr HA
2
1
all nr HA
3 HA nr
nr
(9)
Z N is the atomic number of nucleus N and φ i
nr , φ a
nr , and φ p
nr
denote occupied, vacant, and general (occupied or vacant) spin
orbitals, respectively. P AB is the permutation operator acting on
the operators in the brackets, leading to two additional terms in
brackets on the right-hand side (RHS) of eq 8 (AB+BA). The
linear SO correction to the one-electron energies is zero thanks
to the symmetry of the SO operator. Therefore, eqs 8 and 9 are
formulated only in terms of the nonrelativistic one-electron
energies, ε p (we refrain here from trying to start from an SR wave
function, as this would generate further ambiguities regarding
the nature of the perturbation operators). Note that each φ nr can
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be either an alpha or a beta spin orbital with the same one-
electron energy. In eq 9 we assume that the one-electron energy
level ε p is doubly degenerate and that the corresponding alpha
and beta spin orbitals ( φ p
nr and φ p ± 1
nr ) are excluded from the
summation. Furthermore, in the case of one heavy atom in the
system, we approximate the SO operator in eq 9 as
∑  ≅ 
=
−−
Zr l Z r l
N
N
Nu
N
u
1
nuc 3H A
HA
3 H
A

. We note in passing, that in the
sum-over-states theory the

l
1

HA
operator couples only states
with identical total spin, whereas both δ  LA σ u and σ
 ·
− l
r

HA
3 HA are
triplet operators and thus couple states with di ﬀ erent total spin.
Note also that after some tedious algebraic manipulations with
the terms in brackets on the RHS of eq 8 , we can recover the
third-order perturbation theory eqs 18 and 19 given below. All
necessary approximations to make the analysis at the 2c ( eqs 8
and 9 )o r1 c( eqs 18 and 19 ) level of theory feasible are given
below, and can be found in more detail in refs 99 and 100 .
To analyze the SO corrections to the isotropic paramagnetic
NMR shielding, the second term on the RHS of eq 8 , we use the
following guidelines:
• The energy denominat or in eq s 8 and 9 restricts
signi ﬁ cant contributions to frontier molecular spinors.
• Thanks to the double degeneracy of the MSPs, each
energy denominator in each of the four terms in brackets
of eq 8 corresponds to four possible combinations of spin
orbitals φ nr and the molecular response spinor φ SO .
Fortunately, in most cases, only two of these are non-
negligible, as illustrated in Figure 3 .
• Thanks to the shortsightedness of the function r HA
− 3 , we can
assume atomic-like SO splitting of the nonrelativistic spin
orbitals centered primarily on the HA when a single HA is
present. However, there are exceptions to this rule. In
general, these exceptions occur when the SOC mixes in
spin orbitals on neighboring atoms, see, e.g., the
description of the trans-ligand in ﬂ uence in section 3.2 .
• Because of the energy denominator in eq 9 and a usually
large HOMO − LUMO energy gap, the SO operator mixes
e ﬀ ectively either only occupied or only vacant molecular
spinors. In the case of small HOMO − LUMO gaps or
large SO-coupling integrals, there can be SO mixing
between occupied and vacant spin orbitals, as illustrated
by the TlH example in section 3 ( Figure 19 b).
• The ope ra tor δ  LA σ u couples onl y f unct ion s with s -
character on the LA. These functions are usually σ -
bonding and σ -antibonding orbitals. They can enter eq 8
as the stand-alone spin orbital φ nr or as part of the SO
correction to the MS, φ SO . Figure 3 below depicts the
former case.
To illustrate these points, in Figures 2 and 3 we deal with a
hypothetical system with one occupied σ -bonding MO, two
nonbonding MOs and one vacant σ -antibonding MO. Figure 2
represents eq 9 , i.e., the construction of the occupied MSP, φ i
2c
and φ i +1
2c , from NR orbitals. The NR spatial MOs depicted on the
left side of both ﬁ gures are used to construct alpha/beta spin
orbitals, φ nr ,i n eqs 8 and 9 . Each MS then is composed of alpha
and beta spin orbitals, φ i
nr and φ i +1
nr (left gray box in Figure 2 ),
and molecular response spinors, φ i
SO and φ i +1
SO (right gray box in
Figure 2 ), respectively. For localized SO mixing at the HA, the
SO correction φ SO generally contributes to the opposite spin
channel ( α / β ) of the total MS compared to the zeroth-order
correction φ nr . Figure 3 illustrates the ﬁ rst term in brackets on
the RHS of eq 8 , i.e., one part of the SO correction to the
isotropic paramagnetic nuclear shielding. Here only two out of
four possible combinations composed of doubly degenerate spin
orbitals φ nr and molecular response spinors φ SO give nonzero
contributions. Since the MOs that make up φ SO and φ nr are
orthonormal, the angular-momentum operator along the z -
direction in Figure 3 gives zero couplings. The vanishing
couplings then explain why the SO correction to the NMR
shielding makes a small contribution to the tensor component
parallel to the HA − LA bond.
99
2.3.2. Perturbational Inclusion of SO E ﬀ ects. As the
e ﬃ ciency of variational 2c and 4c methods has improved over
the years, a perturbational inclusion of SO coupling has become
less popular for practical calculations. A perturbational treat-
ment does not provide quantitative accuracy for systems with
atoms from the sixth period or below, and depending on the type
of perturbation theory, various numerical di ﬃ culties arise (e.g.,
variational instabilities and singular operators).
84
On the other
hand, PT treatment constitutes an excellent interpretational tool
when it is based on an NR wave function. In contrast to the
complex relativistic 2c or 4c MSs (see above), the use of
nonrelativistic (real) molecular orbitals results in a natural
language to discuss chemical phenomena. Therefore, in practice,
2c or 4c methods are used to obtain quantitative NMR shift data.
Then, provided a perturbational treatment of the SO e ﬀ ects
reproduces these results su ﬃ ciently well, one can use it to
analyze the data in the more convenient framework of NR (or
SR) MOs. In order to avoid any ambiguities in the nature of the
perturbation operators in di ﬀ erent SR frameworks, we will, as in
the 2c case above, not use SR wave functions as a starting point
in the following discussion. However, it is worth pointing out
that in principle the SR wave function can also be a valid starting
point for the perturbation analysis of SO e ﬀ ects only.
Within a single-determinantal Hartree − Fock or DFT treat-
ment of a closed-shell singlet system, the nonrelativistic Fock (or
Kohn − Sham) equations have the form
φε φ = F
p p
p

NR
(10)
where the index p denotes both occupied and vacant orbitals and
energies and φ p and ε p are the NR scalar MOs and one-electron
energies, respectively, as used also in eq 6 . In the nonrelativistic
theory for closed-shell systems, no operator mixes the alpha and
beta parts of a given spin orbital. Therefore, the alpha and beta
MOs come in pairs with the same spatial part φ p . The one-
electro n part o f the F ock o perator F NR is a well-kn own
nonrelativistic Schro  dinger operator h NR .T ot a k ei n t o
consideration the lowest-order ( ∼ c − 2 ) relativistic corrections
in the absence of the magnetic ﬁ elds, it is necessary to include
Figure 2. Graphical representation of the composition and creation of a
2c MSP from NR MOs. Every MS φ p
2c is composed of an NR spin orbital
φ p
nr and an SO correction φ p
SO . The SO correction to φ p
2c is constructed
from the NR spin orbitals φ q
nr by SO mixing within PT1 theory (see eq
9 ).
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three corrections to h NR . The so-called Pauli Hamiltonian then
has the form
82
=+ + + hh h h h
Pauli NR, 2c MV DW S
O

(11)
=− h c p
1

1
8
MV
2
4
(12)
∑
π δ = 
=
h c Z
1

2 N
N N
DW
2
1
nuc
(13)
∑ σ =  ·
=
− l h c Zr
1
4 N
N
N
N
SO
2
1
nuc
3
(14)
Here δ  N = δ  ( r N ) is the Dirac delta function centered on atom
N and r N is the electron position operator relative to the position
of nucleus N . The Pauli Hamiltonian ( eq 11 ) has a two-
component structure in contrast to the scalar form of eq 10 . This
di ﬀ erence is emphasized in a slightly di ﬀ erent de ﬁ nition of the
Schro  dinger operator, h NR,2c = h NR 1 . Two of the relativistic
operators, the mass velocity h MV and Darwin h DW terms, are
clearly SR operators. As they contribute via the identity matrix
on the diagonal of the Pauli Hamiltonian, they change only the
scalar part of the two-component MOs. In other words, SR
operators do not contain spin variables and are sometimes also
referred to as “ spin-free relativistic ” operators. Examples of the
well-known SR e ﬀ ects are the (direct) contraction of s-type core
orbitals and the (indirect) expansion of valence orbitals having
higher angular momentum.
103
On the other hand, the spin −
orbit operator h SO depends on the Pauli matrices (i.e., on spin
variables) and therefore mixes the alpha and beta parts of the
MOs and causes the SO splitting.
In the presence of magnetic ﬁ elds, there are additional
relativistic corrections to the Pauli Hamiltonian that we do not
list here for the sake of brevity.
11
Within perturbation theory, the
resulting isotropic paramagnetic NMR shielding can be written
formally as
σσ σ =+ Δ (LA) (LA) (LA)
pp p NR, REL, (15)
where σ NR, p is de ﬁ ned in eq 6 and Δ σ REL, p contains in total 19
relativistic contributions (up to order c − 4 )a sd e ﬁ ned in ref 11 .
Traditionally, the contributions containing h MV or h DW ( h SO ) are
termed SR (SO) relativistic corrections. The remainin g
contributions cannot be classi ﬁ ed easily as SR or SO, but they
can be characterized by their (in)dependence on the spin
operators.
56
Fortunately, the majority of the relativistic
contributions have negligible e ﬀ ects on the paramagnetic part
of the isotropic (relative) NMR chemical shift.
56 , 57
One reason
is that many of the SR contributions cancel out since they
depend only on the core electronic structure around the LA,
which is very similar for reference σ ref (LA) and spectator σ (LA)
isotropic NMR shielding (note the dependence of the Darwin
operator, eq 13 , on the Dirac delta function).
54 , 55
They can nevertheless be sizable, at least for HAs from the
sixth period downward (see section 4 for examples). In such
cases, both SR and SO e ﬀ ects must be analyzed.
36 − 39 , 104
For the
paramagnetic NMR shift we can write
δ

δδ =+ Δ (LA) (LA) (LA)
pp p NR, REL, (16)
If the system contains only one HA with directly bonded LA,
then there are ﬁ ve non-negligible relativistic contributions to
δ p (LA).
105
Two of them (arising from the mass-velocity and
Darwin terms) are SR contributions, which are discussed
separately (see Section 1 and below). Among the remaining
three terms, fortunately in most cases only one has a major
trend-de ﬁ ning e ﬀ ect on the isotropic shifts. This is the so-called
spin − orbi t/ Ferm i-c on tac t co ntri bu tion , σ SO /FC .
56 , 60 , 105
The
SO/FC term arises from the contact interaction (represented
by the Dirac delta function) between the nuclear and electron
spins, and therefore only s orbitals of the spectator LA make
nonvanishing contributions to this mechanism (the analogous
SO/SD term can also be sizable but does not control the main
trends
106
). For the chemical concepts arising from this
mechanism, see section 3 . As this term depends on the SO
contribution of the nearby HA, it typically will not cancel for
relative shifts. Therefore, we can express the dominant
relativistic contribution to the paramagnetic isotropic NMR
shift as
∑
δδ δ σ
σ
Δ≅ ≅ = −
=−
=
(LA) (LA) (LA) (LA)
1
3 (LA)
p
u
uu
REL, SO SO / FC SO / FC
1
3
SO / FC
(17)
The term δ SO (LA) is referred to as the SO-HALA shift and is
usually dominated by the SO/FC contribution, δ SO/FC (LA). In
sections 3 and 4 , the SO-HALA shift, δ SO (LA), is calculated as
the di ﬀ erence between the fully relativistic and scalar relativistic
values as described in section 6.3 . The quantitative analysis of
the approximate relation δ SO (LA) ≅ δ SO/FC (LA) is discussed, for
example, in ref 100 .
2.3. 3. Th ird- Or der P ertu rbat ion Anal ys is. Here we
discuss the dominant SO/FC contribution to the SO-HALA
shift in more detail, to provide the mathematical basis for the
MO analysis described in the following sections. In the third-
Figure 3. Graphical representation of the SO correction to isotropic paramagnetic NMR shielding at a LA in the framework of 2c relativistic theory. As
an example, we schematically visualize the ﬁ rst term in brackets on the RHS of eq 8 . The NR spin orbitals and the SO correction to the MS are coupled
using PT2 theory with 2c Fermi contact δ  LA σ u (in blue) and angular-moment l 
u
HA 1 (in black) operators.
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order perturbation theory (PT3) formalism, we can split σ uu
SO/FC
into two contributions (for example, see refs 106 and 107 ):
∑∑ ∑
σ π
φ δ φφ φφ φ
εε εε
≅−
⟨|  |⟩ ⟨|
 |⟩ ⟨|  |⟩
−−
===
−
Δ P Z
c
lr l
(LA) 2
3
() ()
uu
ij a
ai i
u jj u a
ia ja
SO / FC ABC HA
4
1
occ
1
occ
1
vac LA HA
HA
3 HA
(18)
∑∑∑
σ π
φδ φ φ φ φ φ
εε εε
≅
⟨|  |⟩ ⟨|
 |⟩ ⟨|  |⟩
−−
== =
−
∇ P Z
c
lr l
(LA) 2
3
() ( )
uu
ia b
ia a
u bb u i
ia i b
SO / FC ABC HA
4
1
occ
1
vac
1
vac LA HA
HA
3 HA
(19)
Here P ABC is the permutation operator acting on the operators
in brackets; thus, each of the two eqs 18 and 19 has three
contributions (ABC + CAB + BCA). Assuming there is only one
HA in the system, we have approximated the SO operator as
∑  ≅ 
=
−−
Zr l Z r l
N
N
Nu
N
u
1
nuc 3H A
HA
3 H
A

. We have made other approx-
imations, such as neglecting the two-electron SO interaction, as
well as ﬁ rst- and second-order kernels, and using a point-charge
model of the charge distribution (for more details, see refs 99
and 100 ), which in a semilocal DFT framework makes the
analysis of the HALA e ﬀ ect sim ultaneously f easible and
meaningful. Note that eqs 18 and 19 are mathematically
equivalent to the second term on the RHS of eq 8 .
Three simple observations from eqs 18 and 19 are
A The energy denominator restricts signi ﬁ cant contribu-
tions to frontier MOs. This restriction is even stronger for
energy di ﬀ erences >1 au than in the nonrelativistic case
( eq 6 ) due to the square-dependence of eqs 18 and 19 on
these energy di ﬀ erences. In addition, the energy
denominator is always positive, and thus it will not
in ﬂ uence the sign of eqs 18 or 19 .
B Thanks to the function r HA
− 3 , MOs φ i and φ a must contain
contributions from atomic orbitals centered at the HA.
C Similarly, thanks to the Dirac delta function δ  LA , MOs φ i
and φ j in eq 18 , and MOs φ a and φ b in eq 19 must include
a contribution from s-type atomic orbitals centered on the
LA.
These observations are used to analyze the NMR shifts of
hydrogen and 2p atoms in the following sections.
2.4. HALA E ﬀ ects on 1 H NMR Shifts
Considering that hydrogen as a LA uses almost exclusively its 1s
orbital in bonding, and taking the above three observations into
account, there are only four types of MOs making signi ﬁ cant
contributions to the summations in eqs 18 and 19 in the case of
1 H NMR shifts: the bonding σ HA − LA and the antibonding
σ HA − LA
* ,A O
HA , and AO HA
* . Here AO HA (AO HA
* ) denote occupied
(vacant) molecular orbitals containing appreciable contribu-
tions from atomic orbitals at the HA. An example of AO HA is a
nonbonding orbital at the HA ( n HA ), often denoted lone pair
(LP), or a bonding orbital with other ligand atoms (HA − L
bond). However, the latter type is typically lower in energy, and
the energy denominator thus diminishes its e ﬀ ect. In the
following, we abbreviate the sigma HA - LA orbitals as σ and σ * ,
when appropriate.
Equations 18 and 19 give rise to six di ﬀ erent terms, each of
which contains three integrals. Of these, one includes a delta
function, and two contain an angular-momentum operator. Two
examples of such terms are visualized in Figure 4 . The vertices of
the triangle represent the involved MOs, while the orientation of
the triangle corresponds to their energy (higher-lying vacant
orbitals at the top and lower-lying occupied orbitals at the
bottom). Each edge of the triangle coincides with an operator
that couples the MOs across the corresponding vertices. The
three operators involved in the SO/FC mechanism are the SO-
coupling, Fermi-contact, and angular-momentum operators.
The role of the SO-coupling term, which in analogy to the PSO
term in Ramsey ’ s equation has a rather local character, is to
provide a coupling between two HA-based p-, d-, or f-type
orbitals. In the following graphical representations this operator
( 
− l
r

HA
3 H
A

) is color-coded in orange to highlight its origin at the
HA. The Fermi-contact interaction term δ  LA , highlighted in blue,
is particularly sensitive to the electronic structure around the LA
(also shown in blue). It gives nonzero couplings only for σ -
bonding and σ -antibonding orbitals since only these orbitals
have a signi ﬁ cant s-type contribution at the LA. In contrast to the
Figure 4. Example of the nonlocal nature of the SO/FC contribution to the LA shielding, which originates at the HA and propagates to the LA via
either (a) the SO/FC Δ mechanism involving two occupied MOs: φ i ↔ φ j ↔ φ a (see eq 18 ) or (b) the SO/FC ∇ mechanism involving two vacant MOs:
φ i ↔ φ a ↔ φ b (see eq 19 ).
99
The interaction of an external magnetic ﬁ eld with the electronic orbital motion, given by an angular-momentum operator
l  HA, is color-coded in black. The perturbation caused by the nuclear magnetic moment of the LA via the FC interaction δ  LA is coded in blue, and the
SO coupling 
− l
r

HA
3 HA originating at the HA is coded in orange. Note that in contrast to the SO-HALA shielding shown here, the SO-HALA shift
discussed in sections 3 and 4 is of the opposite sign, see eq 17 .
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PSO term (see above), the FC term represents a direct
interaction of the magnetically induced electron-spin density
with the LA nuclear magnetic moment. The remaining coupling
term represents the applied external magnetic ﬁ eld through the
angular-momentum operator

l H
A

(analogous to the orbital
Zeeman term in Ramsey ’ s equation, eq 6 ). The mutual interplay
among all three perturbations is shown schematically on an MO
triangular diagram in Figure 4 . Note that, while the FC and SO
operators describe local interactions, their interplay results in the
nonlocal nature of the SO/FC contribution to the SO-HALA
shift. In the PT3 scheme, the SO/FC mechanism involves two
occupied MOs and one vacant MO for the SO/FC Δ term but
one occ upi ed MO a nd two va ca nt MOs fo r the SO /FC ∇
contribution, see Figure 4 and eqs 18 and 19 .
99
If the FC
operator couples two occupied MOs in SO/FC Δ or two vacant
MOs in SO/FC ∇ , then these MOs can be identical, see the ﬁ rst
triangle in each row in Figure 5 .
To explain trends of SO-HALA NMR shifts across the
Periodic Table
100
( sections 3 and 4 ), it is necessary to analyze
eqs 18 and 19 in detail. The signs of all six terms in eqs 18 and 19
(triangles in Figure 5 ) can be deduced by the following
reasoning. The ﬁ nal value of each of the six terms is invariant to
the change of the phase (sign) of the molecular orbitals.
Therefore, we can ﬁ x the sign of the HA AO contributing to the
σ HA − LA and σ HA − LA
* MOs to be positive. This choice has two
consequences:
1. The hydrogen 1s orbital contributes to σ HA − LA and σ HA − LA
*
with opposite sign, which results in the following signs of
the integrals involving the Dirac delta function: ⟨ σ | δ  LA | σ ⟩
>0 , ⟨ σ *| δ  LA | σ * ⟩ >0 , ⟨ σ | δ  LA | σ * ⟩ < 0, and ⟨ σ *| δ  LA | σ ⟩ <0 .
2. Second, if we assume that intra-atomic contributions
dominate interatomic ones, all four integrals, ⟨ X | l 
u
HA | σ ⟩ ,
⟨ X | l 
u
HA | σ * ⟩ , ⟨ X | r HA
− 3 l 
u
HA | σ ⟩ , and ⟨ X | r HA
− 3 l 
u
HA | σ * ⟩ , with X being
either AO HA or AO HA
* , have the same sign. This
assumption is less general, and we ’ ll discuss so me
counterexamples below.
The second assumption is well satis ﬁ ed for the SO integrals
( r HA
− 3 l 
u
HA ), as the function r HA
− 3 enhances AO contributions at the
HA while those from the other atoms are quenched.
Furthermore, it turns out that in practice the angular-
momentum integrals ( l 
u
HA ) follow the same trend, even without
the bene ﬁ t of such a short-sighted r HA
− 3 function. The second
consequence is also crucial in understanding the ﬁ nal signs in eqs
18 and 19 . It is cumbersome to determine the signs of all SO or
angular-momentum integrals, but since they always appear in
pairs, their signs are not needed, and their joint contribution is
always positive. Combining the above arguments, the ﬁ nal signs
of the di ﬀ erent contributions in eqs 18 and 19 can be easily
determined as illustrated in Figure 5 . See section 3 for chemical
interpretations.
While the determination of the signs of the di ﬀ erent triangles
described in Figure 5 works surprisingly well, there are still cases
where di ﬀ erent signs may occur:
• In the less common event that an orbital with σ -
antibonding character for the HA − LA bond is occupied,
i.e., when one sigma bonding orbital is replaced by an
occupied antibonding orbital, the corresponding triangle
will change sign. This can occur in cases where a given
MO has both bonding and antibonding characteristics in
nontrivial molecular systems.
• A similar situation may occur when a virtual MO has
partial σ -bonding character with respect to the HA − LA
bond.
• The simpler sign rules also come into question when the
interatomic contributions to magnetic couplings are non-
negligible (see point 2 above) as their sign is not governed
b ys i m p l er u l e sa si s t h ec a s ef o ri n t r a - a t o m i c
contributions. This can occur when the intra-atomic
contributions at HA vanish for reasons of symmetry (e.g.,
coupling of p- and d-atomic orbitals). One such case is the
trans-ligand in ﬂ uence described in section 3 .
2.5. HALA E ﬀ ects on the NMR Shifts of 2p Atoms
In the case of NMR shifts of 2p LAs, we must also consider
bonding and antibonding π orbitals ( π HA − LA , π HA − LA
* ). In this
paragraph, we will discuss only arguments that are speci ﬁ c to the
chemical shift of 2p elements, since the arguments presented
above for 1 H shifts are applicable in this case without any
changes. In most applications, we can assume that neither the
bonding π HA − LA nor the antibonding π HA − LA
* orbital contain s-
type AOs from the light atom LA. Then all π -type orbitals are
excluded from couplings with the delta function δ  LA , and the
only way π -type orbitals can in ﬂ uence the SO contribution to the
LA shielding, σ SO / FC (LA), is through AO HA and AO HA
* orbitals.
Note also that π -type orbitals are more likely to have non-
Figure 5. Schematic representation of the six MO coupling terms (triangles) active in the 1 H SO-HALA shielding, eq 18 (upper three triangles) and eq
19 (lower three triangles).
99
Four types of molecular orbitals are involved in the coupling, sigma HA − LA (anti)bonding MOs and occupied (vacant)
MOs with signi ﬁ cant AO HA (AO HA
* ) contribution (examples are nonbonding MOs, n HA , denoted alternatively as LP HA ). SO, FC, and angular-
momentum operators are shown in orange, blue, and black, respectively.
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negligible interatomic contributions as is demonstrated in the
discussion on trans-ligand in ﬂ uence in section 3 .
2.6. Connection between the SO-HALA Shift and the
Electron Density
In this section, we rationalize the connection of the relativistic
SO contribution to the LA NMR shift, and the SO-induced
electronic charge density. For closed-shell singlet systems, the
ﬁ rst nonzero response of the electronic charge density to SO
coupling is quadratic. In the following, we will refer to this
function as ρ 0
SO,SO or SO-EDD (spin − orbit-induced electron
deformation density).
99
The SO-EDD does not depend on the
external magnetic ﬁ eld. Its relation to the SO-HALA shift is thus
not immediately obvious.
It has been pointed out
108
that the ﬁ rst-order SO e ﬀ ects on
the NMR shielding tensor are generated exclusively by spin
currents
ρ

=− ∇×
j

spin
(20)
Here ρ represents the electron spin density. Focusing on the
Fermi-contact mechanism and the fact that σ SO/FC (LA) is
expressible as an integral of the spin-current j spin and the vector
potential generated by the magnetic moment of the LA nucleus,
one arrives at the relation
100
σ π ρρ
ρ
=[ +
+]
c
(LA) 8
9 (LA) (LA)
(LA)
x
B
y
B
z
B
SO / FC , SO , SO
,S O
xy
z (21)
where ρ B ,SO (LA) denotes the bilinear response of the electron-
spin density with respect to the magnetic ﬁ eld and the SO
operator evaluated at the position of the LA. We will use the
abbreviation SOM-ISD (spin − orbit-and-magnetically induced
spin density)
100
for the function ρ B ,SO .
In the case where only one HA is present in the system, a
simple proportionality connection of SOM-ISD
100
and SO-
EDD
99
can be derived from PT3 theory:
100
ρρρ
ρ
[+ +
]

∼
Z
c (LA) (LA) (LA)
(LA)
x
B
y
B
z
B
SO
HA ,S O ,S O ,S O
0
,S O
xyz
(22)
This expression follows from the assumption that angular-
momentum and SO integrals are proportional
φφ φ φ
⟨

|  |⟩ ∼ ⟨ |
 |
⟩

−
rl l
p u qp
u q
HA
3 HA HA
(23)
In practice, this assumption holds when eq 22 is evaluated at
the position of (or in close proximity to) the LA. In this case the
MOs involved in eq 23 arise from the set σ HA − LA , σ HA − LA
* , π HA − LA ,
π HA − LA
* ,A O
HA ,o rA O
HA
* . When eq 22 is evaluated closer to the
HA, other MOs, for which the proportionality of the integrals in
eq 23 breaks down become involved (see Figure 13 in section 3 ).
Collecting the results from eqs 17 , 21 , and 22 , we arrive at a
simple, approximate connection between the isotropic SO-
HALA shift, the SOM-ISD, and the SO-EDD at the position of
the LA:
100
δ π ρρ
ρ π ρ
Δ≅ − [ +
+] ∼ −
c
Z
(LA) 8
9 (LA) (LA)
(LA) 8
9 (LA)
p
x
B
y
B
z
B
REL, , SO , SO
,S O
HA 0
SO, SO
xy
z
(24)
As a result, the sign of Δ δ REL, p (LA) can be predicted just by
the changes in the SO-EDD.
99 , 100
This argument runs parallel to
typical (nonrelativistic) arguments in NMR studies that link
changes in the electron density near the NMR nucleus to its
shielding in di ﬀ erent chemical environments. It is well-known,
13
however, that such correlations may fail when other aspects,
such as energy denominators or angular momentum matrix
elements, become important. While for the present SO case the
link to the SO-EDD can likely explain the signs of the SO-HALA
contributions, we anticipate that it will not hold generally when
discussing more detailed trends.
3. CHEMICAL CONCEPTS: THE HA − LA INTERACTION
3.1. Mechanism, Factors Involved, and Tools for
Interpretation
As has been demonstrated in the previous sections, the NMR
chemical shift is intimately connected, albeit in a nontrivial way,
to the distribution of electron density in the molecular or
supramolecular system. Note that this does not mean just atomic
charges, as frequently argued and discussed in the literature, but
the full quantum electronic structure. In any case, chemists are
taught to work with, and think in terms of, “ chemical bonds ”
which somehow re ﬂ ect interactions between two or more atoms
in real space. A weakness of this broadly de ﬁ ned concept is that
there is no quantum-mechanical operator that links the concept
of a “ chemical bond ” to the quantum-mechanically de ﬁ ned
NMR shielding constant. Yet, there have been many attempts to
formulate links between the character of the HA − LA interaction
and the SO-HALA NMR shift at the qualitative or semi-
quantitative level. In the following we will discuss such links
from the perspective of a chemist.
3.1.1. Spin − Orbit/Fermi-Contact (SO/FC) Mechanism
of the SO-HALA Shift. a). “ The Fermi-contact step ” of the
SO/FC mechanism: the role of the s-character of the LA in the
HA − LA bond. As has been shown in section 2 , the SO-HALA
shift ( δ LA
SO in eq 17 ) is typically dominated by the Fermi-contact
mechanism, SO/FC. While the spin-dipole (SO/SD) term is
included in modern two- or four-component computations and
thus contributes to the quantitative SO-HALA data, it tends to
be less important
106
and is not considered in our qualitative
reasoning throughout this review.
The conceptual analysis of the SO/FC mechanism, from the
early work of Nomura et al.
19
over semiempirical implementa-
tions
30
to the ﬁ rst detailed treatment in DFT calculations,
26
has
been described in the historical pre-2004 overview in section 1 .
While the individual “ thought steps ” elaborated below are part of
one single SO/FC mechanism and thus act simultaneously, we
introduce and discuss them separately for explanatory purposes
and for the convenience of the reader.
Following the arguments presented in ref 26 , the close and
helpful analogy between the Fermi-contact mechanism of the
indirect nuclear spin − spin coupling constant ( J LA − X
FC ) and that of
the SO-HALA shift ( δ LA
SO/FC ) is illustrated in Figure 6 .
In both cases, the “ last step ” of the mechanism is linked to the
Fermi-contact interaction between the induced electron-spin
density surrounding the spectator NMR-active nucleus LA (not
necessarily a “ light atom ” , literally) and its nuclear magnetic
moment ( Figure 6 , left). In the FC mechanism of J LA − X , the spin
density around X is generated in the “ ﬁ rst step ” by the nuclear
magnetic moment of X ( Figure 6 , top right). In contrast, the spin
density around the HA in the “ ﬁ rst step ” of the SO/FC
mechanism of the SO-HALA NMR shift is induced by spin −
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orbit coupling in the presence of a magnetic ﬁ eld ( Figure 6 ,
bottom right). In an “ intermediate step ” , the induced electron-
spin density around the HA (or X) is propagated throughout the
system to reach the spectator LA. Here the spin density interacts
with the nuclear magnetic moment of the LA via the FC
interaction.
Clearly, a larger involvement of the valence s orbital of the LA
in the HA − LA bond, i.e., a high LA s-character of the hybrid
orbital employed by the LA for HA − LA bonding and e ﬃ cient
HA ↔ LA electron sharing, enables the spin-polarization
mechanism to induce a larger spin density near and at the LA
nucleus, causing a more signi ﬁ cant FC interaction and
enhancing δ LA
SO . The correlation between the LA s-character of
the HA − LA bond and δ LA
SO has been demonstrated widely and for
various classes of compounds, see section 1 . Textbook examples
of increasing s-character involve sp 3 ,s p
2 , and sp-hybridized
carbon atoms bonded to heavy halogen(s),
26
particularly iodine,
as shown in Figure 7 a.
Note that the e ﬀ ect of multiple-halogen substitution at the
carbon atom on the 13 C SO-HALA shift is nonlinear as shown in
Figure 7 b. As more electronegative halogens are attached to the
LA, increasing hybridization defects enhance the LA s-character
of the HA − LA bond and thus lead to nonadditive behavior.
3
A
similar dependence has been demonstrated for hydrides of
transition-metals
95
and actinides.
97
For various examples of
these e ﬀ ects, see Section 4 .
b). “ The spin-orbit step ” of the SO/FC me chanism:
Involvement of the HA orbitals in the HA − LA bond. In
addition to the LA character of the HA − LA bond discussed
above, one can look at the SO/FC mechanism from the opposite
direction, studying the involvement of the HA valence orbitals in
the HA − LA interaction. For example, the HA d -character of the
HA − LA bond in transition-metal complexes has been reported
to have a signi ﬁ cant e ﬀ ect on the electronic structure around the
HA, governing the “ initial spin-orbit step ” of the SO/FC
mechan ism.
109
This is demonstrat ed on a set of model
PyA u(I )X (P y = py ri dine ) co mpl ex es wi th va ri ous tra ns
substituents X which modulate the Au d -character of the HA −
LA bond in Figure 8 .
Two main factors in ﬂ uence the magnitude of the SO-based
magnetically induced spin density at the HA. They are linked
fundamentally to the electron g -factor in the EPR spectroscopy
of open-shell systems:
109 , 110
(i) The nuclear charge (atomic number) of the HA a ﬀ ects the
magnitude of the relativistic spin − orbit e ﬀ ects as shown
in eqs 14 , 18 , and 19 . The dependence of SO splitting on
the atomic number Z HA can be easily understood (albeit
not quantitatively) using semiclassical physical argu-
ments. In the electron frame, the HA nucleus revolves
Figure 6. Schematic representation of the analogy between the Fermi-
contact mechanism of the indirect spin − spin coupling constant ( J LA − X
FC )
and that of the SO-HALA shift ( δ LA
SO/FC ) based on the idea of ref 26 .
Note that for consistency with the SO-HALA mechanism, the NMR
spectator atom is labeled LA also in the case of indirect spin − spin
coupling.
Figure 7. E ﬀ ect of (a) carbon hybridization (sp 3 ,s p
2 , and sp) and of (b) the number of halogen atoms attached (1 − 3) on the magnitude of the 13 C SO-
HALA shift ( δ SO ). The data were calculated using methods speci ﬁ ed in section 6.1 (method 1) and section 6.3 . For analogous plots based on
perturbational SO treatments, see refs 3 and 26 .
Figure 8. Relationship between the HA d -character
109
of the HA − LA
bond in PyAu(I)X and δ LA
SO .
99
The data were obtained using methods
speci ﬁ ed in section 6.1 (method 1), section 6.2 , and section 6.3 .
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around the electron and generates an e ﬀ ective magnetic
ﬁ eld at the position of the electron. This magnetic ﬁ eld
depends li nearly on the nuclear charg e ( Z HA )a n d ,
coupled with the spin of the electron, gives the SO
interaction (an e ﬀ ective nuclear charge may be used when
also accounting for two-electron SO contributions).
(ii) The type of frontier MOs at the HA governs their SO
splitting. That is, SOC does not directly in ﬂ uence the s
orbitals, whereas valence p, d, and f orbitals are a ﬀ ected to
di ﬀ ere nt e xte nt s, d ep end ing a lso o n th ei r pr inci pa l
quantum number, for example, inner ( n − 1)d orbitals
vs outer np orbitals.
Considering (i), the nuclear charge is essentially a physical
property, not a chemical concept. However, di ﬀ erent HAs can
exhibit principally di ﬀ erent bonding with the LA, or with linking
atoms for e ﬀ ects across more than one bond, see section 3.1.4 .
Naturally, larger relativistic e ﬀ ects are expected going down the
Periodic Table (albeit energy gaps may counteract this trend for
transition metals, see the inverted V-shape dependence below),
whereas shell-structure e ﬀ ects cause the largest SO e ﬀ ects in
each period to be found toward the right side of the Periodic
Table, albeit exceptions do exist in the p-block, see section 4 .
Concerning (ii), the composition of the relevant valence MOs in
terms of contributing angular momentum at the HA is crucial.
For example, heavier elements of groups 2 (alkaline earth
metals) and 12 (zinc group) exhibit notable ( n − 1)d- and np-
character, respectively, in their chemical bonding. As a result,
they share many features with d(p) elements while formally
being considered s(d)-block HAs. For examples and e ﬀ ects on
δ LA
SO , see section 4 .
c). Link between the FC and SO steps: Covalency of the
HA − LA bond. The e ﬃ ciency with which the electron-spin
density induced by SO coupling at the HA in the presence of a
magnetic ﬁ eld is propagated to the spectator LA nucleus is
tightly connected to the degree of electron sharing between the
atomic basins of the HA and the LA, and thus to the covalency of
the HA − LA bond.
104 , 111 , 114
This aspect can be considered a
third piece of the puzzle in our understanding of the SO-HALA
mechanism, in addition to the individual characters of the HA
and the LA. Note, however, that the HA and LA character of the
HA − LA bonding, and the HA − LA covalency are mutually
interconnected. Naturally, the electron sharing for the HA − LA
interaction depends on the interatomic HA − LA distance as does
δ LA
SO ,
47
but this dependence is clearly nonlinear (see Figure 9 ).
The magnitude of electron sharing between the HA and the
LA can be described as follows:
(i) The covalency of the HA − LA bond is re ﬂ ected in the
relati ve pos itions of the fr ontie r-or bital energ ies, in
particular the gap between the occupied MOs dominating
the HA − LA interactions and the associated vacant MOs
of appropriate symmetry (see PT3 theory in section 2 ). A
clear correlation between the energy gap of the “ HALA
MOs ” and δ LA
SO has been demonstrated for a series of
CH 3 HgX complexes,
38
Figure 10 .
The role of the energy gap has also been used to explain the
rather striking observation of a counterintuitive “ inverted V-
shape ” dependence
95 , 113
of the SO-HALA e ﬀ ect down group 9,
an example of which is shown in Figure 11 . This is due to the
generally smaller ligand- ﬁ eld splitting in 3d systems caused by
the absence of a radial node in the 3d shell and the resulting
stretched-bond situation.
46
This reduces the (square) energy
denominators in the PT3 expressions ( eqs 18 and 19 in section
2 ) and renders the SO-HALA e ﬀ ects larger in 3d compared to
Figure 9. E ﬀ ect of the interatomic HA − LA distance on the magnitude
of δ SO ( 1 H) for AuH. This can be related to a similar e ﬀ ect of the
interatomic distance on the indirect nuclear spin − spin coupling.
112
The
data were obtained using methods speci ﬁ ed in section 6.1 (method 2)
and section 6.3 . The gray bar highlights the optimized equilibrium
distance.
Figure 10. Correlation of δ SO ( 13 C) in CH 3 HgX compounds with the
inverse of the energy di ﬀ erence between the occupied σ HA − LA and
vacant HA nonbonding ( n HA
* ) Kohn − Sham orbitals involved in the
dominant σ HA − LA ↔ n HA
* MO coupling. Data from ref 38 .
Figure 11. “ Inverted V-shape ” dependence of δ SO ( 1 H) for a set of group
9 model compounds. Data from ref 95 .
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the analogous 4d complexes. Indeed, the Rh complex exhibits a
less negative δ LA
SO than its Co analog ( Figure 11 ), as has also been
found for smaller three-bond 1 H NMR e ﬀ ects in a series of
polyamine complexes. While the energy denominators increase
further from Rh to Ir due to the SR expansion of the 5d orbitals,
the e ﬀ ect is more than compensated by the larger SO splitting of
the 5d element, leading, overall, to the inverted V-shape.
(ii) Second, the covalency of the HA − LA bond can be
quanti ﬁ ed by electron sharing between the atomic basins
of the HA and the LA as described, for example, by the
delocalization index (DI, HA ↔ LA)
114
of the Quantum
Theory of Atoms in Molecules (QTAIM). A larger
covalency is re ﬂ ected directly in a larger DI as has been
demonstrated for a series of transition-metal com-
plexes,
114 , 115
for which the DI correlates with the SO-
HALA shifts, Figure 12 a.
In a complementary fashion, the covalent character of
the HA − LA bond has been investigated by energy
decomposition analysis (EDA) and natural orbitals for
chemical valence (NOCV) analysis, with the results also
linked to δ LA
SO ,
99
Figure 12 b. Here, the larger covalency is
re ﬂ ected in both the more negative orbital stabilization
energy and the more negative energy of the ﬁ rst NOCV
bond channel, which re ﬂ ects the ability of the ligand
containing the LA (pyridine in Figure 12 ) to make a σ -
donation to the Au-X unit.
3.1.2. Visualization Tools: SOM-ISD and SO-EDD. The
propagation of the spin density induced at the HA by SO and the
magnetic ﬁ eld during the “ initial SO step ” through to the last
“ Fermi-contact step ” at the LA can be visualized by the real-
space distribution of the SO-and-magnetically induced spin
density (SOM-ISD, eq 24 in section 2 )
100
shown in Figure 13 a
for model Au I H and PhHg II H systems as examples. These plots
clearly demonstrate that the induced α or β spin density around
the spectator hydrogen LA is re ﬂ ected in a negative or positive
sign of δ SO ( 1 H), respectively.
It has also been shown previously that the SO-HALA shift can
be correlated with the SO-induced electron deformation density
(SO-EDD)
99
around the spectator NMR-active nucleus. This is
the change in the total ground-state charge density due to SO
coupling ( section 2 ) and is shown in Figure 13 b for the same two
systems. Despite the di ﬀ erent patterns in electronic structure
around the HA, the SO-induced charge density (SO-EDD)
surrounding the LA parallels the SOM-ISD patterns.
3.1.3. Structural and Electronic In ﬂ uence of a trans
Ligand. Particularly in the case of the HA being a transition
metal, the character of the HA − LA bond can be substantially
altered by the nature of the substituent or ligand bonded to the
HA in the trans-position relative to the LA (labeled as TL or X),
as had already been noted early on for d 10 LA-Hg-TL systems.
38
The structural and electronic trans-ligand in ﬂ uence (TLI, also
known as thermodynamic trans-e ﬀ ect or structural trans-e ﬀ ect)
on the SO-HALA shift can be modeled computationally by
changing the HA − TL bond length as has been demonstrated
ﬁ rst for an octahedral Ir(III) complex.
118
While both scalar-
relativistic and spin − orbit contributions to the NMR shifts are
a ﬀ ected by the changes in the HA-TL distance, the modulation
of the SO-HALA contribution is by far the more signi ﬁ cant (see
also section 4 ). The e ﬀ ect of the Au-TL distance on the 1 H SO-
HALA shift of the trans hydride in HAu I TL is shown in Figure
14 . Note the reversal of the sign of the SO-HALA shift near the
equilibrium distance of about 1.95 Å, highlighted by a gray
vertical bar. The structural and electronic TLI is also re ﬂ ected in
Figure 12. Relationship between the 15 N SO-HALA shifts ( δ LA
SO ) in PyAuX compounds and (a) the delocalization index (HA ↔ LA) for the HA − LA
bond
114
and (b) the orbital contribution to the EDA
116
bond energy (red) and ligand donation (orange) as quanti ﬁ ed by the ﬁ rst NOCV-channel
117
of
the EDA/NOCV analysis (see section 6.2 ).
99
Data from ref 99 .
Figure 13. (a) Visualization of the SOM-ISD in Au I H (left) and
PhHg II H (right). The local spatial predominance of α or β induced spin
density is shown in blue and red, respectively. The calculated 1 H SO-
HALA shifts ( δ LA
SO ) are also shown for comparison. (b) Visualization of
the SO-EDD in Au I H and HHg II Ph. The accumulation and depletion of
electron density induced by SO coupling is shown in blue and red,
respectively. The data were obtained using methods speci ﬁ ed in section
6.1 (method 1), section 6.3 , and section 6.4 .
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an inverted spin density around the LA as shown by the SOM-
ISD for the short (left) and long (right) HA-TL distance.
The TLI on the SO-HALA shift has been elaborated upon
extensively in a series of papers on iridium, platinum, gold, and
mercury complexes.
99 , 109 , 114 , 115 , 119 , 120
The electronic TLI has
been rationalized by the simultaneous involvement of a central
HA orbital with both the TL and the LA. In other words, the two
trans-arranged ligands share a single HA atomic orbital (AO) in
their bonding. As a result, the polarization of HA-centered AOs
by one of the substituents (e.g., the TL) induces a signi ﬁ cant
repolarization of the same orbital on the side of the second
substituent (e.g., the LA).
99
The two substituents thus mutually
in ﬂ uence the character of their bonding with the HA center,
where the increasing polarity of the bond to the TL (smaller HA
AO character in the HA − TL bond for a weaker trans-ligand)
results in a more covalent bonding with the LA. The e ﬀ ect of the
nature of the TL on the SO-HALA shift is shown in Figure 15 .
We note that matters are rather di ﬀ erent for actinide systems,
where the intrinsically ungerade character of the 5f-AOs, as
opposed to the gerade character of d-type AOs in transition-
metal complexes, is known to contribute to an inverse TLI for
structural and electronic properties.
121 − 124
Such an inverse TLI
has been found to a ﬀ ect 13 C SO-HALA shifts in U VI complexes
in a relatively complicated manner.
97
As the valence p-orbitals
are also intrinsically ungerade in nature, substituents in di ﬀ erent
positions likely a ﬀ ect SO-HALA shifts di ﬀ erently in heavy p-
block compounds, although this has yet to be investigated in
detail.
3.1.4. Long-Range SO E ﬀ ects. As shown in Figure 6 above,
there is a clear parallel between the FC mechanism of SO-HALA
shifts and the FC contribution to indirect nuclear spin − spin
couplings (see also Introduction). Thus, the SOM-induced spin
density around the HA (see section 3.1.1 ) for the SO-HALA
e ﬀ ect and the spin density around the X atom induced by the
nuclear magnetic moment in case of J -coupling are propagated
similarly by spin-polarization mechanisms, governed by
exchange interactions, to generate a spin density at the position
of the spectator LA. The schematic representation of the
propagation of spin polarization from the HA to the LA via one,
two, and three chemical bonds as governed by the Pauli principle
and Hund ’ s rule is shown in Figure 16 a. The SOM-ISD
distributions in iodobenzene
26
and in PhHgH highlight the local
predominance of α (blue) or β (red) induced spin density and
its alternation throughout the conjugated system, resulting in
alternation of the sign of δ LA
SO for di ﬀ erent atoms, see Figure 16 b.
We note in passing that a similar analogy applies to the pathways
of spin-density propagation in the context of hyper ﬁ ne
couplings.
21 , 125 , 126
This has led to the de ﬁ nition of new 3D
quantities termed “ coupling electron deformation den-
sity ”
127 , 128
and “ hyper ﬁ ne structure deformation density ” .
129
Figure 14. E ﬀ ect of the HA-TL distance on the SO contributions
118
to
the 1 H NMR shift in HAuNH 3 . Note the change in the sign of the SO-
HALA contribution around the equilibrium distance (indicated by a
gray vertical bar). The SOM-ISD calculated for the short and the long
HA-TL distances is shown on the left and the right, respectively. Note
the change in color (opposite induced spin density) around the
hydrogen atom. The data were calculated using methods speci ﬁ ed in
sections 6.3 and 6.4 .
Figure 15. In ﬂ uence of the trans-ligand (X) on the equilibrium HA − LA
distance, r , and the SO contribution to the 15 N NMR shift ( δ LA
SO ) for a
pyridine ligand.
99
The data were obtained using methods speci ﬁ ed in
section 6.1 (method 1) and section 6.3 .
Figure 16. (a) Schematic representation of the propagation of spin
polarization from the HA to the LA via one, two, and three bonds using
the Dirac vector model, which is also commonly used to rationalize the
alternation of the sign of indirect spin − spin coupling constants. (b)
DFT-calculated distribution of SOM-ISD in PhI
26
(left) and
PhHg II H
130
(right). The spatial predominance of α and β SOM-ISD
is shown in blue and red, respectively. The data were obtained using
methods speci ﬁ ed in section 6.1 (method 1), section 6.3 , and section
6.4 .
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Further examples of long-range SO-HALA e ﬀ ects are given in
Section 4 .
In analogy to the Fermi-contact mechanism of indirect
nuclear spin − spin coupling, the Fermi-contact mechanism of
SO-HALA shifts is governed by the electronic structure on the
interaction path between the HA and the spectator LA. Thus,
the Karplus-like torsion dependence of three-bond δ LA
SO e ﬀ ects
has also been rationalized by its analogy to nuclear spin − spin
coupling pa thways.
26
An example of the conformational
dependence of δ SO ( 1 H) for a “ three-bond ” interaction in Te 
CH − CH 3 is shown in Figure 17 . More examples are provided in
section 4 .
3.1.5. “ Through-Space ” Supramolecular E ﬀ ects. The
propagation of the spin density induced during the above-
mentioned SO step ( section 3.1.1 ) toward the LA can also occur
in va ri ous s upr a mol ecu l ar in ter a ct ion s su ch a s hyd ro gen ,
halogen, or chalcogen bonding. Note that this requires an
uninterrupted pathway of su ﬃ ciently large electron density
between the HA and the LA that can be polarized. In principle,
this can be observed for various supramolecular bonding,
nonbonding, and antibonding situations. However, the most
e ﬃ cient currently known examples of this e ﬀ ect are those
exhibiting supramolecular covalency (orbital interaction and
HA ↔ LA electron sharing),
131
which can also make a signi ﬁ cant
contribution to the total binding energy. As an example of this
phenomenon, we plot (i) δ LA
SO , (ii) the change in the NMR shift,
calculated at the SR level, induced by the formation of a
supramolecular bond ( Δ δ LA
SR/SB ), and (iii) the SOM-ISD map in
Figure 18 for a) the hydrogen-bonded (HB) ion-pair complex
NH 4 + ··· I − and b) the halogen-bonded (XB) complex NH 3 ··· IF.
We note that in the ﬁ rst case, the SO-HALA e ﬀ ects reduce the
overall (positive) complexation shift somewhat, whereas in the
second case, the (negative) complexation shift is enhanced by
the SO contributions.
3.2. Unifying the Chemical Concepts
3.2.1. PT2 Analysis Based on Molecular Spinor Pairs. A
PT2 analysis of SO-HALA e ﬀ ects based on molecular spinors
calculated at the two-component (or four-component) level is
analogous to the standard NR or SR analysis of paramagnetic
contributions to the NMR shift within the Ramsey formalism,
except that in the presence of SO coupling the perturbation
operators for the nuclear magnetic moment include the FC and
SD hyper ﬁ ne contributions that underlie the SO-HALA e ﬀ ect
(see section 2 ). The analysis for diamagnetic (closed-shell)
systems is typically done using molecular spinor pairs (MSPs),
also known as Kramers pairs. This allows an MSP ↔ MSP *
coupling analysis while accounting for SO e ﬀ ects and provides
di ﬀ erences from the analogous SR (or NR) values for the
relevant MO ↔ MO * couplings.
A comparison between the MOs (NR or SR level) and MSPs
(2c or 4c level) can be used to estimate the role of mixing MOs
of di ﬀ erent symmetry by SO coupling for the magnetic response
of the system. This mixing in turn can be obtained from the
nonrelativistic MOs using ﬁ rst-order perturbation theory (PT1),
as demonstrated in Figure 2 . As an example, we show the SO-
induced mixing of occupied σ -bonding and nonbonding ( n HA )
MOs at the HA for AuH in Figure 19 a. The resulting MSP
(HOMO − 3) of mixed σ and π character dominates ( Σ δ SO =
− 16 ppm) the enhancement of the paramagnetic 1 H NMR
shielding by SO coupling (re ﬂ ected in the δ tot
SO = − 13 ppm
relative to the SR data). The e ﬃ cient and most signi ﬁ cant
magnetic coupling of the HOMO − 3 with the LUMO ( δ SO =
− 14 ppm) is shown schematically in Figure 19 a. In contrast to
AuH, the SO-induced mixing of vacant ( n HA
* ) MOs at the HA in
TlH results in an extremely e ﬃ cient deshielding HOMO ↔
LUM O ma gn eti c cou pli ng ( δ SO =+ 1 0 7p p m ) ,
132
which
dominates the total SO-HALA shift ( δ tot
SO = +122 ppm), Figure
19 b.
The advantage of the PT2 analysis is that SO e ﬀ ects are
included variationally and thus more accurately and completely.
A disadvantage is that the MSPs are complex quantities (see
Figure 2 ) that are di ﬃ cult to visualize and interpret completely,
see section 2 . One needs to transform the data in order to
visualize them. For example, within the two-component ZORA
implementation in the ADF code,
133 , 134
only one real function is
typically displayed ( Figure 19 ) instead of the required two
complex functions. This has been used previously to interpret
the roles played by the electronic structure around the metal
atom
114
and the TLI
99 , 119
on the SO-HALA shifts in various
transition-metal complexes. To relate the MSPs to the NR MOs
Figure 17. Karplus-like dependence of three-bond δ SO ( 1 H) in Te 
CH-CH 3 on the rotation of the central C − C bond. The data were
obtained using methods speci ﬁ ed in section 6.1 (method 1) and section
6.3 .
Figure 18. (a) δ SO ( 1 H), Δ δ SR/HB ( 1 H), and a plot of the SOM-ISD for
the hydrogen-bonded complex between NH 4 + and I − (for the H atom
involved in the hydrogen bond). (b) δ SO ( 15 N), Δ δ SR/XB ( 15 N), and a
plot of the SOM-ISD for the halogen-bonded complex between NH 3
and IF. The predominance of α and β induced spin density is shown in
blue and red, respectively. The data were obtained using methods
speci ﬁ ed in section 6.1 (method 1), Section 6.3 , and section 6.4 .
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and thus obtain further insight, one would have to apply PT1 to
the MSPs ( Figure 2 and section 2 ) and thus to sacri ﬁ ce the
higher accuracy of the two- or four-component treatment.
Moreover, the mechanism and sign of the SO-HALA shift are
not fully transparent in Figure 19 . To understand the sign and
mechanism of the SO-HALA e ﬀ ect in more detail, we must
resort to the PT3-based analysis of δ LA
SO , which is elaborated in
the following section.
3.2.2. Transparent PT3 Analysis Bas ed on Non-
relativistic MOs. The PT3 scheme starts from nonrelativistic
(NR) or scalar-relativistic (SR) MOs and adds SO coupling as
another perturbation ( section 2 ). It provides a particularly
transparent and ﬁ ne-grained interpretation in terms of standard
nonrela tivist ic MOs , d ecoding their role i n the m agnet ic
response of the system.
As described in detail in section 2 , the three perturbations
involved in the SO/FC mechanism are the SO-coupling, Fermi-
contact, and angular-momentum operators. The mutual inter-
play of all three perturbations has been discussed thoroughly in
section 2 and is shown schematically in Figure 20 , for examples
of the triangular diagrams
99
that dominate the shielding and
deshielding SO/FC contributions to δ LA
SO in AuH and TlH,
respectively. For the convenience of the reader, the interaction
of an external magnetic ﬁ eld with the electronic orbital motion,
given by the angular-momentum operator l

, is color-coded in
black. The perturbation caused by the nuclear magnetic moment
of the spectator atom LA via the Fermi-contact (FC) interaction
is coded in blue, and the SO coupling originating at the HA is
coded in orange.
In the PT3 scheme, the SO/FC mechanism involving two
occupied MOs and one vacant MO for the SO/FC Δ term
(within the reasonable approximations discussed in section 2 )
results in a shielding SO contribution ( δ SO <0 , Figure 20 a)
unless both occupied MOs are of σ (HA − LA)-bonding
character, which changes the overall sign ( δ SO >0 , Figure
20 b).
99
The two examples shown in Figure 20 are directly
related to the PT2 analysis of AuH and TlH shown in Figure 19 .
Given the PT3 examples in Figure 20 , we now proceed to
analyze the sign of the SO-HALA contribution to the NMR shift,
as it relates to the electronic structure of the HA, which is
crucially important to understanding the periodic trends
discussed in section 4 .
100
As discussed in section 2 , the leading
Figure 19. SO-induced (a) mixing of the occupied σ -bonding (HOMO and HOMO − 5) and nonbonding ( n HA , HOMO − 3, HOMO − 4) SR MOs for
AuH, and an example of the MSP ↔ MSP * coupling of the resulting HOMO − 3 MSP with a LUMO MSP having signi ﬁ cant LA s-character, and (b)
mixing of two vacant n HA
* MOs (LUMOs) and the occupied σ -bonding HOMO for TlH, and an example of the MSP * ↔ MSP coupling of the resulting
LUMO MSP with an HOMO MSP having signi ﬁ cant LA s-character. The contributions to δ LA
SO are shown for the individual MSPs. The data were
obtained using methods speci ﬁ ed in section 6.1 (method 1) and section 6.5 (PT2).
Figure 20. Schematic graphical representation of the SO/FC Δ mechanism involving two occupied MOs (see also eq 18 and Figure 4 ).
99
(a) The
shielding SO contribution that dominates the negative δ SO ( 1 H) in AuH, and (b) the deshielding SO contribution that dominates the positive δ SO ( 1 H)
in TlH. The occupied MO in TlH can be viewed as a superposition of two MOs, antibonding s (Tl)- s (H) and bonding p (Tl)- s (H). Because s functions
of the Tl atom have no e ﬀ ect on magnetic and SO coupling integrals, this occupied MO is considered as a σ -bonding orbital in the PT3 analysis. The
data were obtained using the method speci ﬁ ed in section 6.5 .
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MO ↔ MO couplings must arise from the frontier MOs, i.e.,
high-lying occupied and low-lying vacant MOs. Furthermore,
the electronic structure around the HA can be analyzed in terms
of MOs with signi ﬁ cant coe ﬃ cients from AOs on the HA
(AO HA ). These AO HA s are represented by nonbonding orbitals
at the HA ( n HA , or lone electron pair - LP), π HA − LA and δ HA − LA
bond types, and σ HA − L with another ligand (L). For the sake of
simplicity, let us consider the simplest HA − H example with only
a σ (HA − H) bond and with nonbonding orbitals at the HA (e.g.,
AuH). Four types of frontier orbitals then play roles in the SO-
HALA mechanism: nonbonding orbitals at the HA, including
both occupied ( n HA ) and vacant ( n HA
* ) MOs, and occupied
HA − LA bonding ( σ HA − LA ) and vacant antibonding ( σ HA − LA
* )
MOs. We denote them for simplicity as n , n * , σ , and σ * .
The two limiting cases of electronic con ﬁ gurations at the HA
are shown in Figure 21 . A representative electronic con ﬁ g-
uration with an available occupied nonbonding orbital n at the
HA ( n HA ) but without n HA
* is shown in Figure 21 a, whereas one
without n HA but with an available vacant nonbonding orbital n HA
*
is shown in Figure 21 b. The presence of ﬁ lled nonbonding MOs
in the valence shell of the HA ( Figure 21 a) causes the active
shielding contributions ( δ LA
SO < 0) described as n HA ↔ σ HA − LA
( * ) in
the SO/FC mechanism (examples are transition-metal com-
plexes with d 2 − d 8 con ﬁ gurations and halogen compounds,
section 4 ). Note that σ ( * ) in the text stands for σ and σ * , which
are both present in all of the triangles due to the role of the FC
operator. In the absence of occupied nonbonding MOs, only
deshielding contributions ( δ LA
SO > 0) labeled σ HA − LA
( * ) ↔ n HA
* in
Figure 21 b are nonzero, as found, for example, for p 0 ,d
0 , and f 0
con ﬁ gurations at the HA in the molecule (not necessarily the
atomic ground state of the HA).
100
In general, both the SO deshielding and shielding
contributions may be active simultaneously, with the total δ LA
SO
value governed by the energy separations between the relevant
frontier MOs and the coupling MO matrix elements.
99
Excellent
examples representing a delicate balance between two
contributions of opposite sign are d 10 systems (e.g., Au(I)
complexes),
99 , 119
where the properties of the ligand trans to the
Figure 21. Schematic representation of two limiting electronic con ﬁ gurations at the HA with (a) occupied ( n , left) and (b) vacant ( n * , right) HA-
based nonbonding orbitals.
100
The sign of δ LA
SO/FC is governed by the availability of occupied/vacant HA-based orbitals of suitable energy as shown by
the MO coupling triangles for the SO/FC Δ (bottom) and SO/FC ∇ (top) mechanisms.
99
Triangles shown in gray are not accessible for a given
electronic con ﬁ guration. Note that MF stands for the interaction of an external magnetic ﬁ eld ( B 0 ) with the electronic orbital motion given by the
angular-momentum operator l
 (see section 2 ), and σ ( * ) in the text stands for σ and σ * , which are both present in all of the triangles due to the role of the
FC operator (shown in blue).
Figure 22. (a) Structural and electronic trans-ligand in ﬂ uence of a weak X w like F or a strong X s like CH 3 on the relative energy of the σ HA − LA MO
entering the coupling triangles and a ﬀ ecting the SO-HALA shift as discussed above.
99 , 119
(b) The PT3 coupling triangle governing δ LA
SO ( 1 H) > 0 in
HAuNH 3 for a short Au − N distance (strong X), which highlights the role of 2p AOs of the trans nitrogen atom (shown in green), cf. Figure 14 .M F
stands for the interaction of an external magnetic ﬁ eld ( B 0 ) with the electronic orbital motion, given by the angular-momentum operator l
 (see section
2 ). The data were obtained using the method speci ﬁ ed in section 6.5 .
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LA can modulate or even invert the overall sign of δ LA
SO , see Figure
15 . For weak trans-ligands (X w ) in a linear LA-HA-X complex of
gold(I), the HA − LA bond is relatively strong; and the
corresponding MO of σ HA − LA character is energetically stabilized
and thus plays a marginal role in the SO-HALA mechanism. In
this case, the SO-HALA shift is governed by many small
shielding SO contributions (negative δ LA
SO ) originating in n HA ↔
σ HA − LA
( * ) type (HA-based 5d ↔ 5d * ) couplings. A strong trans-
ligand (X s , covalent HA − X bond) destabilizes the HA − LA
bond by a structural and electronic trans-ligand in ﬂ uence (TLI),
and the energy of the corresponding σ HA − LA orbital is increased,
Figure 22 a. It therefore contributes signi ﬁ cantly, and it typically
governs the SO deshielding (positive δ LA
SO ).
The PT3 mechanism of this SO-HALA deshielding
particularly highlights the role of the trans ligand as shown for
HAuNH 3 in Figure 22 b. The SO (and angular-momentum)
operator couples the occupied and vacant MOs with the
following AO contributions: 2p N
σ ↔ 2p N
π ,2 p
N
σ ↔ 6p Au
π , and 5d Au
σ
↔ 2p N
π , all involving the trans-nitrogen atom and contributing to
the SO deshielding. The SO-and-magnetically induced electron-
spin density interacts with the magnetic moment of the LA
nucleus via the FC operator, acting on occupied MOs with s-
character at the hydrogen (LA) atom. Induction and
propagation of spin density thus involve AOs of the trans
atom X, particularly for strong TLs ( Figure 15 ) and short HA-
TL distances ( Figure 14 ).
The trans-ligand in ﬂ uence shown in Figure 22 b is a good
example of an arrangement where one of the assumptions we
used to derive the rules for the sign of the SO-HALA e ﬀ ect
( Figure 5 ) does not hold. Speci ﬁ cally, the MO ↔ MO couplings
between the occupied and vacant HA AOs vanish, and the SO
( 
− l
r

HA
3 H
A

) and MF ( l

) coupling is dominated by the interatomic
contributions. Here, the FC coupling (the occupied MOs are
identical) and the product of SO and MF coupling (equal signs)
are both positive. The ﬁ nal SO-HALA deshielding (positive δ LA
SO )
is then given by the negative sign in eq 18 .
The theoretical foundations of the SO-HALA shift
summarized in section 2 and demonstrated in this section on
selected examples in terms of chemical concepts allow for a more
complete understanding of previously observed trends across
the Periodic Table of the Elements (PTE)
100
than had hitherto
been possible. In section 4 we will focus on these general trends
and selected examples from the time period 2004 − 2019 to
illustrate the usefulness of these concepts and to reveal the broad
importance of such e ﬀ ects throughout the Periodic Table.
4. TRENDS IN SO-HALA SHIFTS ACROSS THE
PERIODIC TABLE
As shown in sections 2 and 3 , SO-HALA NMR e ﬀ ects are
connected to the electronic structure of the neighboring heavy
atom (HA) and its ligand sphere.
99 , 100 , 119
With few exceptions,
trends i n SO-HALA e ﬀ ects have bee n inv estiga ted usi ng
relatively small groups of HA compounds. Recent e ﬀ orts to
understand the phenomenon on a wider scale encompass three
large studies
99 , 100 , 119
that have revealed in detail how the sign of
the SO-HALA shift relates to the electronic structure at the HA
and how the position of the HA within the PTE, its oxidation
state, and its bonding environment (e.g., the TLI in transition-
metal complexes), in ﬂ uence the sign and magnitude of δ SO . The
link between the electron con ﬁ guration of the HA and the SO-
HALA shift was generalized when the triangular PT3 concept
99
was applied systematically to compounds of the sixth-row
100
to
determine whether or not the SO-HALA e ﬀ ect shows periodic
trends. A large set of 1 H NMR shifts for compounds containing
HAs of the sixth period, ranging from Cs I to At I , has been
analyzed, with periodic trends in the sign of δ SO ( 1 H) revealed
and rationalized.
100
Most importantly, it has been shown that
the sign of δ SO (LA) correlates systematically with the presence
of occupied/vacant nonbonding orbitals centered at the HA
(this is valid also for heavier atoms such as 13 C and 15 N), as will
be discussed in detail in this section. Note that throughout
section 4 we will use the notation LA for light atoms and also for
heavier spectator NMR atoms (e.g., Se or Te) in the vicinity of
an even heavier HA. Based on refs 51 , 91 , 95 , 97 , 99 , 100 , 111 ,
113 , 115 , 119 , and 132 − 138 , the following general rules can be
derived for periodic trends in the SO-HALA shifts:
1. A formally empty valence shell at the HA (p 0 ,d
0 ,o rf
0
con ﬁ guration) gives rise to SO-HALA deshielding of
LA(s) bonded directly. This is the case particularly for
HAs with 5d 0 and a positive oxidation state Cs I − Os VIII , for
6p 0 Hg II − Bi III (the 6s 2 shell is present but unimportant in
this context), and for 6d 0 5f 0 Ac III − U VI .
51 , 100 , 132 , 137 , 138
For
the corresponding lighter analogs, the e ﬀ ect exists in
diminished form ( Figure 23 ). (Note that the electronic
state relates to the HA bonded in a molecule not its
atomic ground state.) In such cases, the SO-HALA
deshielding is dominated by the σ HA − LA
( * ) ↔ n HA
* coupling
mechanism ( Figures 5 and 21 b), which requires the
presence of a low-lying vacant orbital(s) at HA.
100 , 132
2. Partially ﬁ lled, energetically high-lying valence subshells
(d 2 − d 8 ,p
2 − p 4 ) at the HA induce SO-HALA shielding at
the LA, particularly in 5d 2n (Hf II − Au III )o r6 p
2n (Bi I − At I )
compounds and, in diminished form, for their corre-
sponding lighter analogs ( Figure 23 ).
51 , 95 , 100
The SO-
HALA shielding is dominated by the n HA ↔ σ HA − LA
( * )
coupling mechanism ( Figures 5 and 21 a) and requires
the presence of one or more nonbonding ( n HA , or lone
electron pair, LP HA ) orbitals at the HA. This e ﬀ ect is to a
certain extent additive when more than one suitable
nonbonding MO is present, for example, along the 5d 2
through 5d 4 to 5d 6 and 6p 2 to 6p 4 series of hydride
complexes.
51 , 95 , 100 , 132
A considerably smaller shielding
e ﬀ ect is observed when the HA AOs are not LPs but are
involved in bonding interactions to ligands other than the
LA or the trans ligand (see rule 4), for example, in σ HA − L
bonds.
100
3. The oxidation state of the HA is related directly to the
occupat ion of n HA and thus determines δ SO (LA),
according to rules 1 and 2 above.
99 , 100 , 132
For instance,
an LA in a 5d 2 complex of W IV experiences SO-HALA
shielding, i.e., a negati ve δ SO (LA), whil e SO-HALA
deshielding, i.e., positive δ SO (LA), would be expected in
a5 d
0 W VI complex.
100
In most cases, the above-
mentioned mechanisms act simultaneously, an d one
usually dominates the other. Sometimes, for example in
Bi III (6p 0 ) and Au I (5d 10 ) systems, these two mechanisms
are so similar in magnitude that the sign of the SO NMR
shift can be dictated by other factors, or the SO-HALA
e ﬀ ect can be vanishingly small (see rule 4).
4 . Additional factors, such as the trans-ligand in ﬂ u-
ence,
99 , 109 , 115 , 118 , 119
(TLI, section 3.1.3 ) and a recently
described cis-ligand in ﬂ uence
120
further modulate the
δ SO .T h eT L Ii n ﬂ uences the relative energi es and
com posi ti ons of fr on tier σ ( * ) HA − LA orb it als , the reby
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a ﬀ ecting the SO-HALA mechanism. Strong trans ligands
may even invert the sign of δ SO , particularly in d 8 and d 10
complexes, see below.
99 , 115 , 119
The e ﬀ ects of cis and/or
other than trans ligands are typically smaller than the TLI
but should be considered, e.g., for Bi III (6p 0 ) and Au I
(5d 10 ) complexes (see below).
5. The vertical trends for HAs of the s- and p-blocks follow
the expected dependence of relativistic e ﬀ ects on Z HA (the
atomic number of the HA nucleus).
56
The trend is
analogous to the well-known normal halogen dependence
(NHD, i.e., the SO shielding increases going down the
given group for the HA, section 1 ) observed for groups
14 − 18, whereas the opposite behavior (growing SO
deshielding going down the group) is observed for
compounds with HAs of s-block
100 , 139
and group 13.
51
The latter trend is analogous to i nverse halogen
dependence (IHD, section 1 )
42
but its origin is di ﬀ erent
(the SO-HALA e ﬀ ect instead of scalar paramagnetic
shield ing). For HAs w ith par tially ﬁ lled d-shells,
particularly for octahedral d 6 and square-planar d 8
complexes, the SO-HALA e ﬀ ects generally follow an
inverted V-shape trend,
113 , 138 , 140
resulting from competi-
tion between the growing SO splittings and the likewise
growing energy denominators down the PTE (explained
in section 3 and Figure 11 ).
For convenience, we have collected the available occurrences
of typical SO-HALA (de)shielding from the studies reviewed
here and presented them schematically throughout the Periodic
Table ( Figure 23 ). Red and blue are used to show SO-HALA
deshielding and shielding e ﬀ ects, respectively, in compounds of
a given HA in its typical (closed shell) oxidation state(s). The
color-coded description of electronic con ﬁ gurations located
beneath the seventh period has been added to further emphasize
the elements which can, at least hypothetically, be found in an
ns 0 ( n − 1)d 0 or np 0 con ﬁ guration. In that case, vacant valence p-
or d-orbitals lead to deshielding SO-HALA e ﬀ ects, while the
more electron-rich con ﬁ gurations at the HA feature partially
occupied orbitals and thus make shielding SO contributions. For
instance, deshielding SO-HALA e ﬀ ects are expected in W VI
complexes with a 6s 0 5d 0 con ﬁ guration, whereas SO-HALA
shielding is expected in W IV (5d 2 ) and W II (5d 4 ) complexes,
increasingly so for the latter. The same applies for the majority of
early transition-metal complexes. Similarly, transition-m etal
con ﬁ gurations for which the SO-HALA e ﬀ ects are in ﬂ uenced
considerably by the availability of n p orbitals
99 , 114 , 119
are
denoted ( n − 1)d m ,n p
0 . Here d m corresponds to an even-
numbered d-con ﬁ guration, i.e., d 6 ,d
8 ,o rd
10 . The in ﬂ uence of a
TLI (see sections 2 and 3 ), which can switch the sign of the SO-
HALA e ﬀ ect for some HAs, is emphasized by both red and blue
stripes for the given element (see, e.g., Pt and Au). The red and
blue stripes in group 14 and Bi emphasize the possible large
deshielding SO-HALA e ﬀ ects in HA II when the HA is a group 14
element and in Bi III complexes (compared to shielding for higher
oxidation states, HA IV and Bi V ).
91 , 132
Exceptions to these
elaborated general trends may occur in speci ﬁ c cases (e.g., exotic
oxidation states, involvement of the atom in a cluster, strong
TLI).
More detailed descriptions of the known SO-HALA trends in
s-, d-, p-, and f-block compounds are given in sections 4.1 − 4.5
below, together with an overview of the individual works dealing
with calculating and predicting SO and (where available) SR
contributions to LA NMR shifts in heavy-element compounds.
As outlined in section 1 ,S Re ﬀ ects have not been investigated
extensively, and only a few general concepts have been derived
for them, although some basic observations are outlined in the
text. Schematic illustrations of the known ranges of SO-HALA
shifts for complexes of a given element (group), collected from
the works covered in this comprehensive review, are presented at
the beginning of each subsection. These are accompanied by
Figure 23. General trends in SO-HALA NMR shifts across the Periodic Table. Red shows SO-HALA deshielding ( δ SO > 0) and blue SO-HALA
shielding ( δ SO < 0) for a given HA in its typical oxidation state(s). The relative magnitude of the SO-HALA e ﬀ ect is represented by lighter (less) and
darker (more) shading. HAs for which SO-HALA e ﬀ ects have not yet been evaluated or for which closed-shell complexes are unknown, are indicated
by a white background. The color-coded con ﬁ gurations presented beneath the 7th period and beneath actinides indicate which of the elements in the
groups above can occur in ( n − 2)f 0 /(n − 1)d 0 /np 0 con ﬁ gurations (n is principal quantum number), which may lead to deshielding SO-HALA e ﬀ ects
caused by vacant valence p-, d-, or f-orbitals, and which of the more electron-rich situations tend to be more shielded (d m /p m con ﬁ gurations, index m
represents even-numbered electron con ﬁ guration).
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examples of applications of SO-HALA e ﬀ ects in the elucidation
of molecular and electronic structure and chemical reactivity.
4.1. HAs of the s-Block
4.1.1. General Aspects. SO-HALA e ﬀ ects of HAs of the s-
block have scarcely been investigated, likely because the ns
valence orbitals themselves do not contribute to SO e ﬀ ects.
However, low-lying ( n -1)d orbitals of the heavy alkaline earth
elements (Ca, Sr, Ba, and Ra) participate in their chemical
bonding, making them “ honorary d-elements ” .
141
As a result, the
low-lying ( n − 1)d HA orbitals contribute to the SO-HALA
mechanism in a way similar to that for genuine d 0 systems (as
well as for the lanthanides), particularly to the σ HA − LA
( * ) ↔ n HA
*
deshielding ( Figure 23 ).
100
An example of such deshielding
orbital magnetic coupling (OMC) involving contributions from
5d and 5d * orbitals in BaH 2 is illustrated in Figure 24 .
100
Because of the d 0 character, weakly deshielding SO-HALA
e ﬀ ects are found in heavier, formally ns 0 complexes. The range of
curre ntly kno wn δ SO ( 1 H) values
100
for hy drid es of thes e
elements is shown in Figure 25 .
We note in passing that alkalide anions from Na − through Cs −
with the electronic con ﬁ guration ns 2 at the HA have been
observed experimentally, including NMR measurements,
142
but
no data regarding SO-HALA e ﬀ ects or LA NMR shifts have
been reported. We speculate that the SO-HALA e ﬀ ects in such
compounds are in any case small, because of the strongly ionic
nature of alkalide bonding.
4.1.2. Groups 1 and 2: Alkali and Alk aline- Earth
Metals. Computational analysis of 1 H NMR shifts in gas-
phase s-block hydrides
100
has shown that in group 2 hydrides the
SO-HALA e ﬀ ect increases by about 4 ppm from CaH 2 to
BaH 2 ,
100
see Figure 25 . This correlates well with experimental
1 H NMR shifts in the solid-state hydrides, which increase by 4.2
ppm, from 4.5 ppm (CaH 2 ) to 8.7 ppm (BaH 2 ).
143
While earlier
attempts to explain this trend using classical arguments, such as
the electronegativity of the alkaline earth metal, were not
convincing,
143
the SO-HALA e ﬀ ect provides a natural
explanation for the observations.
Relativistic deshielding of 1 H NMR resonances can also be
found in relatively complex materials. For instance, the hydride
signals in hydride-doped strontium mayenite have been found
experimentally at δ tot up to +6 ppm, a region that is usually
associated with the 1 H NMR resonances of OH groups.
144
These results could not be reproduced using scalar relativistic
calculations, although the inclusion of scalar-relativistic
corrections slightly improved the agreement with experiment.
144
Clearly, the inclusion of SO-HALA e ﬀ ects in the calculations is
necessary to reproduce the 1 H NMR spectra of hydride-doped
materials of the heavier s-block metals.
Relativistic e ﬀ ects were also calculated for 19 F NMR shifts in
solid-state Li, Na, Ca, Rb, Sr, Cs, and Ba ﬂ uorides.
139
The
reported δ SO ( 19 F) values were between − 3 ppm (CaF 2 ) and +2
ppm (CsF). The relatively small SO-contributions are likely due
to the high polarity and particularly to the low s-character of the
ﬂ uorine in the M − F bond. That δ SO ( 19 F) = − 3 ppm for CaF 2
may re ﬂ ect the numerical uncertainties of these calculations
(di ﬀ erences between SO-ZORA and SR-ZORA data) is
suggested in the original work.
139
4.2. HAs of the d-Block
4.2.1. General Aspects. Due to the high sensitivity of 1 H
shifts to SO-HALA e ﬀ ects (e ﬃ cient SO/FC mechanism, see
section 3 ), and the ensuing spectacular 1 H shifts, relativistic
contributions to the 1 H NMR shifts in transition-metal hydride
complexes have been at the center of various computational
analyses of SO-HALA e ﬀ ects.
95 , 100 , 115 , 120 , 137 , 145
Indeed, many
general SO-HALA trends in d-block systems have been derived
primarily from hydride NMR shifts. While not as spectacular in
magnitude, due to the larger “ standard ” NMR spectral ranges,
similar trends are also found for other LAs, such as carbon and
nitrogen, provided an e ﬃ cient SO/FC mechanism is oper-
ative.
96 , 99 , 100 , 109 , 114 , 119 , 130 , 146 , 147
Calculated ranges of the SO-
HALA shift for 1 H, 13 C, and 15 N nuclei in transition-metal
complexes are summarized in Figure 26 . The data were collected
from the works covered in section 4.2 .
Figure 24. Example of the 5d-5d * σ HA − LA
( * ) ↔ n HA
* deshielding orbital
magnetic coupling in 5d 0 6s 0 Ba II H 2 , adapted from ref 100 . Copyright
2018 American Chemical Society.
Figure 25. Range of computed 1 H SO-HALA shifts in s-block hydride
complexes (LA bonded directly to the HA).
100
Figure 26. Ranges of the computed 1 H, 13 C, and 15 N SO-HALA shifts
for individual d-block metal HAs (LA bonded directly to the HA). Note
that lighter shades are used for 3d and 4d to highlight the trend in
δ SO (LA) for compounds of 5d elements.
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In early transition-metal complexes with d 0 con ﬁ guration, the
deshielding σ HA − LA
( * ) ↔ n HA
* orbital coupling dominates the SO-
HALA shifts. Hence, positive SO-HALA contributions are
found in Figure 26 for early transition-metal complexes featuring
d 0 con ﬁ gurations (mainly from groups 3 − 8). As the d-shell
becomes populated in d 2 ,d
4 d 6 , and d 8 systems, the shielding n HA
↔ σ HA − LA
( * ) coupling mechanism becomes dominant.
100
This
trend, together with additional shielding due to the Bucking-
ham-Stephens e ﬀ ect
148 , 149
(shielding at the LA, caused by o ﬀ -
center paramagnetic ring currents from the incomplete valence
shell of the HA and found particularly in hydride complexes), is
responsible for the predominantly low-frequency 1 H NMR
signals in d 2 to d 8 d-metal hydrides ( Figure 26 ). A local
extremum in δ SO ( 1 H) for d-block hydrides is observed for 5d 6
Ir III complexes with a computed δ SO of − 30 ppm (and even more
negative overall shifts).
95 , 100
We note that the lowest total 1 H
NMR shift ever experimentally observed was − 59.1 ppm in a 5d 6
Ir III porphyrin hydride complex.
150
Vacant HA 6p orbitals start to play an important role for SO-
HALA e ﬀ ects in 5d 8 complexes, see Pt and Au in Figure 26 .
114
These orbitals mix into the frontier MOs and thus participate in
the deshielding σ HA − LA
( * ) ↔ n HA
* mechanism that reduces the
overall SO-HALA shielding. The 6p-6p * -based σ HA − LA
( * ) ↔ n HA
*
mechanism then typically dominates the SO-HALA e ﬀ ects in d 10
compounds and beyond (particularly in group 12, see Hg in
Figure 26 ) causing largely positive δ SO (LA) values.
99 , 130 , 151
For
borderline cases, such as 5d 10 Au I complexes, both mechanisms
play similar roles and can be strongly modulated by the TLI
( section 3.1.3 ).
99 , 109 , 115 , 118 , 119
Hence the broad range of SO-
HALA shifts in Pt and Au complexes shown in Figure 26 .
Except for the d 0 and d 10 con ﬁ gurations, the vertical trends in
the SO-HALA shift in the d-block tend to be characterized by an
inverted V-shape dependence ( Figure 11 ),
95 , 113
although this
can be a ﬀ ected by TLI when a strong trans ligand is present.
99 , 119
The inverted V-shape re ﬂ ects greater (shielding) SO-HALA
contributions for the 3d and 5d compared to the 4d transition-
metal complexes, due to the increase in ligand- ﬁ eld splitting with
respect to 3d analogs and thus the relevant energy denominators
for the 4d case ( section 3.1.1 ). A detailed explanation of the V-
trend can be found in the discussion of group 9 complexes
below.
Before we dis cuss i ndivi dual ex ampl es for d-b lock HA
systems, it should be noted that a number of the computational
stud ies o f SO -HAL A sh ifts in tran sit ion -met al comp lexe s
mentioned in the following discussion used a higher admixture
of exact exchange in the DFT functional (typically 40%), to
obtain better agreement with the experimental NMR shifts at the
two-component SO-ZORA level, e.g., refs 104 , 109 , 114 , 118 ,
137 , 147 , and 152 However, the need for a higher exact-
exchange admixture was later identi ﬁ ed as an artifact caused by
the absence of an exchange-correlation kernel in the SO-ZORA
implementation used.
96 , 97 , 153
Such a high exac t-excha nge
admix ture s sho ul d be avo ide d when usi ng co mput ati onal
codes that treat the kernel properly.
134 , 154
4.2.2. Early Transition Metals. The SO-HALA e ﬀ ects in
early transition-metal complexes have not been investigated as
much as those of the late transition metals but a short overview
of relativistic NMR e ﬀ ects in d 0 complexes of early transition
metals was contained in a recently published larger review.
138
Available dat a con ﬁ rm that the SO-HALA e ﬀ ects for d 0
complexes are uniformly deshielding, in contrast to the
increasingly shielding contribut ions in d 2 and d 4 sys-
tems.
100 , 137 , 146 , 155
An analysis of the relativistic 13 C NMR shifts in early
transition-metal complexes was used to elucidate the link
between the NMR shifts and the electronic structure in d 0
Schrock alkylidene methathesis catalysts based on Mo, Ta, W,
and Re.
156
The high-frequency 13 C signals in these complexes
were attributed to the large paramagnetic contributions caused
by the small energy separations between the σ M − C and π M − C
*
orbitals. In addition, the orientations of some of the components
of the 13 C shielding tensor were found to be in ﬂ uenced by the
angle between the metal atom and the  C − H fragment. These
factors are very similar to those responsible for the modulation
of SO-H ALA e ﬀ ect s, par ticul arly t he in ﬂ uenc e of ene rgy
denominators. Also, the relevant orbital couplings resemble
those of the deshielding σ HA − LA
( * ) ↔ n HA
* mechanism, see Figure
21 b.
4.2.3. Group 8: Ruthenium and Osmium. The hydride
1 H NMR shift in 4d 6 Ru complexes with N -heterocyclic
carbenes increases from − 41 ppm to +5 ppm with increasing
TLI (trans ligand: none, F − ,C l
− , MeCN, N 2 ,H
2 ,H
− ,P
4 ,S O
2 ,
CO, and O 2 ).
157
A striking outlying value ( δ ( 1 H) = +5 ppm) has
been experimentally observed for an Ru complex with an η
2
-O 2
ligand in the trans position. Detailed analysis revealed strong π -
type interactions between the O 2 2p and the Ru 5d xy orbitals
(HOMO) along with substantial destabilization of the Ru 5d yz
orbital. The Ru 5d yz orbital, occupied in other complexes,
becomes the LUMO in the O 2 complex. The original LUMO,
which is in fact the σ Ru − H
* MO, thus becomes less available for the
shielding n Ru ↔ σ Ru − H
( * ) magnetic coupling, which diminishes the
SO shielding mec hanism (the SO contri bution remains
negative). Simultaneously, paramagnetic deshielding couplings
between the Ru 5d xz HOMO and the Ru 5d yz LUMO further
increase the total NMR shift at the hydrogen nuclei in question.
The SO-HALA shielding contribution to the 29 Si NMR
signals of − SiX 3 groups (X = H, C, and Cl) in 4d 6 Ru silyl
complexes
158
was found to increase with the electronegativity of
substituent X, with the maximum SO-HALA value of − 35 ppm
for X = Cl.
158
It has been suggested that this could be due to a
smaller energy gap between the occupied and vacant MOs.
However, an alternative explanation would be that the
electronegative substituents increase the positive charge at Si
and thus also the hybridization defects that increase the 3s-
character of Si in the Si − M bond,
46
thereby enhancing the SO/
FC mechanism of the SO-HALA e ﬀ ect ( section 3.1.1 ).
26
A recent detailed study of the e ﬀ ect of TLI on the mechanism
of 1 H SO-HALA shifts in a series of transition-metal complexes
also included d 6 osmium(II) compounds.
119
This study is
discussed more thoroughly with groups 10 and 11 below.
4.2.4. Group 9: Cobalt, Rhodium, and Iridium. An
experimental solid-state NMR study of 1 H shift tensors in 4d 6
Rh III and 5d 6 Ir III hydride complexes
159
revealed that the unusual
low- ﬁ eld isotropic shifts are accompanied by very large chemical
shift an isotropies (CSAs), particularl y for the Ir systems ,
con ﬁ rming predictions based on large computed SO-HALA
anisotropies.
95
T h e s ea r i s ef r o mt h et e n s o rc o m p o n e n t s
perpendicular to the M − H bond, increasing from Rh to Ir.
159
Four-component relativistic computations of the 1 H NMR
shifts of iridium hydride, shifted to a lower frequency by SO-
HALA e ﬀ ects, were used to elucidate the mechanism of the
hy dro s il yl ati on o f ca rbo ny l c om po und s us i ng B ro ok ha rt ’ s
iridi um pincer com plex.
160
A detailed compar ison of the
calculated 1 H NMR shifts of nine di ﬀ erent hydride complexes
with experimental 1 H NMR data revealed a fast equilibrium
between a dihydride-silane Ir III complex and a trihydride silyl Ir V
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complex formed by oxidative addition of the silane.
160
Similar
computational predictions of relativistic 1 H NMR shifts for
hydride complexes and intermediates have been reported for
Si − H activation across an Ru − S bond.
161
These results could
not be veri ﬁ ed experimentally because of rapid decomposition
reactions.
Two-component relativistic calculations of 1 H and 13 C NMR
chemical shifts in d 8 Ir I and Rh I complexes
145
interacting with
metha ne wer e per form ed t o pro vid e an i nsi ght into the
mechanisms of activation of the C − H bond. δ SO has been
found to dominate the overall 1 H and 13 C NMR shifts, causing
considerable shielding (overall shifts exceeding − 40 ppm) of
both types of nuclei. Only proper treatment of SO-HALA e ﬀ ects
and consideration of long-range-corrected DFT functionals
enabled identi ﬁ cation of the correct structures of methane σ -
complex intermediates, assignment of their signals in the NMR
spectra, and explanations for the underlying reaction mecha-
nism.
145
Solid-state NMR and relativistic DFT calculations of a hexa-
iridium cluster
162
provided explanations for unusually shielded
13 C NMR resonances of bridging carbonyl ligands resonating at
+191 ppm. Analyses showed that this is caused by a δ SO ( 13 C) for
the bridging carbonyls ( − 27 ppm) much larger than that for the
terminal ligands ( − 2 ppm). This is likely due to the bonding of
the bridging carbon atom to two iridium centers.
163 , 164
A higher
covalency of the bridging Ir − CO bonds due to stronger back-
bonding may also contribute via a more e ﬃ cient SO/FC
mechanism ( section 3.1.1 ).
109 , 114
Systematic studies of three-bond SO-HALA e ﬀ ects in group 9
chelates have been carried out
113 , 140
to understand experimental
1 H NMR titration results in a series of d 6 Co, Rh, and Ir
polyamine complexes. Upon deprotonation of the aqua, alcohol,
or polyalcohol (carbohydride) coligands in these systems, one
would have expected better charge donation by the anionic
coligand (OH) generated upon deprotonation to enhance
shielding of all CH 2 protons within the polyamine ligand.
However, for certain methylene positions trans to the
deprotonation site, substantial deshielding was observed. This
can even lead to a nonmonotonous pH-dependence of the 1 H
shift titration curves for multiple deprotonation sites.
Computations at NR levels could not reproduce the
observations, while inclusion of a PT3 treatment of SO-HALA
e ﬀ ects provided the correct trends and insights.
113 , 140
Interruption of SO/FC pathways from the metal center to the
methylene proton site by the larger TLI of the deprotonated
coligand combined with the antiperiplanar M − N − C − H
arrangement in the protonated system enabling maximal SO/
FC contributions (rationalized by a Karplus-type dependence,
sectio n 3.1.4 ) explain ed the di ﬀ eren tial SO -HALA e ﬀ ec ts
causing the observed anomalous pH dependencies. Notably,
smaller SO-HALA e ﬀ ects were found for the 4d 6 Rh complexes
compared to both the 3d 6 Co and 5d 6 Ir analogs, indicating for
the ﬁ rst time the inverted V-shape dependence ( section 3.1.1 )o f
the SO-HALA e ﬀ ects down the group, due to the increased
ligand- ﬁ eld splittings (energy denominators) partly counter-
acting larger SO coupling (cf. eq 18 in section 2 ).
98
While the
larger energy gap renders the SO-HALA e ﬀ ects smaller for the
Rh compared to the Co system, the still much larger SO splitting
explains the largest SO-HALA shifts for the Ir complex. This
rati ona liza ti on ha s been co n ﬁ rm ed su bseq uen tl y in oth er
studies, e.g., for d 8 metallasilatrane complexes,
111
and for d 6 −
d 10 transition metal hydrides.
95
A modulation of δ SO ( 15 N) of about 25 ppm caused by
changing the metal trans-ligand distance was computed in an
octahedral 5d 6 Ir III complex.
118
Notably, SR and NR
contributions both changed by only 6 ppm in total. This
initiated the ﬁ rst systematic study of the mechanisms of TLI in
the SO-HALA e ﬀ ects. It was performed on a set of model d 6 Ir III
complexes featuring di ﬀ erent oxygen- and sulfur-based trans
ligands,
109
and found a linear correlation of δ SO (LA) with the Ir
5d-character of the HA − LA bond ( section 3.1.1 ). Essentially,
covalently bound trans substituents tend to diminish the metal
d-character in the HA − LA bond because they compete for the
available metal d-orbitals. In contrast, substituents bound more
ionically leave a larger fraction of the HA d-orbitals available for
HA − LA bonding, thus increasing the metal d-character in the
HA − LA bond. The analysis also provided a link between the
SO/FC part of the SO-HALA shielding and the g- and A-tensors
of EPR spectroscopy, which are known to be modulated by the
metal d-character. The role of the metal d-character was later
embedded into a more general relationship between covalency
and the magnitude of the SO-HALA e ﬀ ects ( section 3.1.1 ).
114
This concept has been developed further by several studies of
the TLI (see below).
99 , 115 , 119 , 120
4.2.5. Groups 10 and 11: Platinum, Gold. As discussed
above, SO-HALA e ﬀ ects in platinum and, especially, gold
complexes are strongly in ﬂ uenced by contributions from the
frontier 6p-orbitals participating in the SO/FC mechanism and
by TLI. Overall, the LAs are more shielded in Pt complexes than
in Au systems, partly due to the predominance of d 8 Pt II in the
examples studied (a d 10 con ﬁ guration would require Pt 0
complexes, which tend to be very reactive). However, even for
d 10 Au I , a weak TLI can lead to shielding SO-HALA e ﬀ ects (see
Figure 26 ).
Interest in computational studies of SO-HALA shifts in Pt and
Au complexes peaked in 2011 with three independent works.
One study used relativistic DFT methods to investigate SR and
SO-HALA e ﬀ ects on the ligand atoms of 2-phenylpyridine
ligands in related square-planar Pt and Au complexes.
104
Large
SR e ﬀ ects on 13 C and 15 N NMR shifts ( δ SR )u pt o − 28 ppm were
calculated for the Au system, likely re ﬂ ecting the well-known
“ gold maximum ” of SR e ﬀ ects.
58
Relativistic contributions to the
13 C and 15 N NMR shifts in the Pt system were, in contrast,
dominated by the SO-HALA e ﬀ ects. The much smaller 15 N SO-
HALA shift in the Au complex (near 0 ppm) compared to the Pt
system ( − 18 ppm) was rationalized by the rather ionic (less
covalent) character of the Au − N bonds (due in part to
di ﬀ e r e n c e si nt h et r a n sl i g a n d ,a sw a ss h o w n l a t e r
114
).
Alternating signs of the long-range SO-HALA e ﬀ ects correlating
with indirect spin − spin couplings have also been noted (cf.
Figure 16 in section 3.1.4 ).
The unusually shielded 29 Si resonances in certain metal-
lasilatrane complexes
165
were interpreted in terms of SO-HALA
shielding e ﬀ ects in ﬂ uenced by TLI when the trans-ligand was
changed from Cl to I. A role for the Si − M bond covalency in the
propagation of the SO-HALA was hinted at. The calculated
values of δ SO ( 29 Si) = − 20, − 17, and − 33 ppm for Ni, Pd, and Pt
complexes, respectively, follow the inverted V-shape depend-
ence
165
(cf. Figure 11 in section 3.1.1 ).
A third study in 2011 focused on the well-known high- ﬁ eld
hydride 1 H NMR shifts in heavy transition-metal hydride
complexes.
95
In the 1960s, the shielded hydride resonances had
been attributed to o ﬀ -center paramagnetic ring-current e ﬀ ects in
the Buckingham-Stephens model.
148 , 149
That model was
con ﬁ rmed by SR-DFT calculations in 1996.
77
The 2011 work
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showed that for the heaviest d 6 and d 8 systems, including Pt and
Ir complexes, up to two-thirds of these spectacular high- ﬁ eld
shifts can be due to SO-HALA e ﬀ ects. This was contrasted to the
deshielding SO-HALA e ﬀ ects in d 10 group 11 and 12 hydride
complexes. This work also clearly con ﬁ rmed the inverted V-
shape dependence of SO-HALA shifts for the d 6 and d 8 cases.
95
Pt and Au complexes were also central to subsequent studies
on the importance of the covalency of the HA − LA bond for SO-
HALA e ﬀ ects and TLI.
114 , 115
Reference 114 correlated 13 C and
15 N SO-HALA shifts with the covalence of the HA − LA bond
expressed by the delocalization index of QTAIM (DI; cf. Figure
12 ). This work further established the role of 6p-type HA
orbitals (and the e ﬀ ect of decreasing 5d character in bonding) in
SO-HALA deshielding contributions in Au complexes, an e ﬀ ect
that becomes even more pronounced in group 12 systems. In the
second study,
115
the 1 H shifts in a series of square-planar 5d 8 Pt
hydride complexes trans-H-PtL 2 X (L = PMe 3 , X = trans ligand)
were studied systematically as a function of the strength of the
trans ligand. It was found that, while paramagnetic shieldings
also contributed to the e xperimentally well-known larger
shielding with weaker trans ligands, the SO-HALA contributions
dominated the trend. While greater shielding also correlates with
the Pt − H distances, closer analysis has established that both
trends are consequences of the stronger trans ligands competing
for the metal 5d orbitals with the hydride bond and thereby
weakening it.
115
Two independent computational studies from our groups in
2017 used complementary analytical tools to shed more light on
the TLI for SO-HALA e ﬀ ects.
99 , 119
While ref 119 relied on two-
component spinors from ZORA computations and applied PT2
analyses, ref 99 in addition to PT2 introduced PT3 analysis of
δ SO based on NR MOs to understand the electronic-structure
origins of the SO-HALA trends ( section 2.3.3 ). The systems
studied in these two works overlapped to some extent, allowing a
comparison of the insights gained from PT2 analysis based on
MSPs (molecular spinor pairs, section 2 ) and PT3 analysis based
on NR MOs. The generality of the TLI for SO-HALA e ﬀ ects for
1 H, 13 C, and 15 N shifts all the way from 5d 6 Os II and Pt IV
complexes via 5d 8 Pt II and Au III systems to 5d 10 Au I and Hg II
compounds with a broad range of trans ligands was
demonstrated in these works. For 5d 8 Pt II and d 10 Au I , the
TLI may even change the sign of the SO-HALA contribution
(and even the sign of the total 1 H hydride shifts) from shielding
for weak trans ligands to deshielding for strong trans ligands.
119
Interestingly, a comparison of linear d 10 Cu, Ag, and Au
complexes showed, that the V-shape dependence of SO-HALA
shifts down the group for d 6 and d 8 systems extends to the d 10
case for weak trans ligands but changes to a monotonous
increase for strong trans ligands,
119
due to enhanced SO-HALA
deshielding contributions. Modulation of the sign of the SO-
HALA contributions by the interplay of the metal center and the
trans ligand has also been correlated with SO e ﬀ ects on the
electron density (SO-EDD,
99
see sections 2 and 3 ).
A recent study of the interplay between cis- and trans-ligand
e ﬀ ects on SO-HALA shifts in a series of square-planar d 8 Au III
hydride complexes explained the broad span of experimentally
observed 1 H NMR signals from − 8.5 ppm to +7 ppm.
120
The cis
and trans ligands in ﬂ uence the δ SO ( 1 H) in opposite way. That is,
stronger ligands in the cis position stabilize the higher-lying
σ Au − H bonding MOs an d hence we aken the SO-HALA
deshielding mechanism. The largest deshielding due to the cis
ligand (up to +5 ppm) was therefore found for neutral weak σ -
donors (S-, N-, and O-based ligands and CO). As in previous
studies, the 1 H NMR shifts were found to correlate with the Au −
H distance and the ionicity of the bond, which in turn correlated
with trends in reactivity.
Combined DFT and experimental solid-state NMR inves-
tigations were performed for a diphosphacyclobutadiene
cobaltate sandwich complex of Au I .
166
Inclusion of relativistic
e ﬀ ects (both SR and SO-HALA e ﬀ ects) in the DFT
computations was crucial for the correct assignment of the 31 P
NMR resonances. For instance, the nonrelativistic 31 P NMR
chemical shift of the Au I -bound phosphorus atoms was − 64
ppm (using an SR-ECP at gold), while the relativistic two-
component SO-ZORA value was +23 ppm, indicating
substantially deshielding SO-HALA e ﬀ ects. The need to revisit
the 31 P NMR spectra assignments of similar compounds using
relativistic DFT calculations was pointed out by the authors.
A sizable δ SO ( 14 N), − 45 ppm, was calculated at the N α atom of
an azide ligand directly bound to Pt.
167
Signi ﬁ cant SO-HALA
shielding was also predicted for N β ( − 21 ppm) and N γ ( − 7
ppm) as a result of the high electron delocalization within the
azide ligand. However, in spite of the large δ SO ( 14 N), the N α and
N β nuclei have experimental NMR chemical shifts similar to
those in free N 3
− . Analysis showed that the SO-HALA shielding
contributions at the N α and N β nuclei partly compensate the
paramagnetic deshielding contributions be cause electron
density is withdrawn upon formation of the Pt − N α bond,
including trans ligand e ﬀ ects. The authors suggested the term
hidden heavy-atom ef fect when strong SO-HALA e ﬀ ects are
present but are disguised in the spectrum by other contributions.
Probably the most dramatic example of a TLI for SO-HALA
shifts can be found in a 2005 computational study,
168
where the
unusually low δ ( 13 C), 135.2 ppm, of a carbon atom trapped
inside a phosphine-ligand-stabilized Au 6 cluster was inves-
tigated, see Figure 27 . Signals of interstitial carbon atoms in
“ non-d 10 ” transition-metal clusters tend to resonate at rather
high frequencies, above +400 ppm.
168
Computations on the
unsubstituted parent CAu 6 2+ cluster gave δ SO ( 13 C) = − 240 ppm
( Figure 27 a), which would lead to the even more exotic δ tot ( 13 C)
of − 288 ppm. Inclusion of the phosphine ligands into the model
resulted in δ SO ( 13 C) = +28 ppm (see Figure 27 b), which means
that more than 300 ppm of the total chemical shift at the
interstitial carbon results from the combined TLI of the six
phosphine ligands! Inversion of the sign of δ SO ( 13 C) due to the
presence of strong trans ligands in Au complexes is now well-
known,
99 , 114 , 119
but in 2005 it prompted Le Guennic et al. to
carefully re-examine their computational methodology. The
large SO-HALA e ﬀ ects were con ﬁ rmed, with a “ ﬁ nal ” calculated
δ tot ( 13 C) of +129 ppm, in excellent agreement with experiment.
Figure 27. Structures of (a) the central CAu 6 2+ cluster and (b) the
entire cluster stabilized by phosphine ligands. Data from ref 168 .
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While the origin of the large shielding of the interstitial carbon
atom was not explained, the observed modulation of δ SO is
remarkable.
4.2.6. Group 12: Mercury. While we discuss the group 12
elements in the d-block section, we note that they are often
viewed as “ post transition-metal ” or “ representative ” p elements,
as their ( n − 1)d-orbital participation in bonding tends to be
minor. Indeed, the SO-HALA e ﬀ ects for Hg II complexes are
dominated by the n p orbitals, leading to considerable SO-HALA
deshielding, Figure 26 , which is due to the 6p ↔ 6p * -based
σ HA − LA
( * ) ↔ n HA
* deshielding mechanism ( Figure 20 b).
26 , 99 , 100 , 114
Building on the pre-2004 work discussed in section 1 ,
38
more
recent (experimental and computational) studies of relativistic
e ﬀ ects (both SR and SO) on the 13 C NMR shielding constants in
organomercury compounds have shown deshielding SO
contributions up to +52 ppm (f or sp- hybri dized c arbon
atoms).
130
Notably, SR deshielding e ﬀ ects were observed at
the LA in contrast to the shielding δ SR in Pt and Au complexes
discussed above.
104
Alternating signs of Hg-derived long-range
SR e ﬀ ects on neighboring atoms were noted,
130
with δ SR = +28
ppm, − 10 ppm, and +2 ppm for C α ,C
β , and C γ , respectively.
While the alternation of the sign of δ SR resembles the oscillations
of δ SO that have been linked to FC-base d J -coupling
mechanisms,
26
the SR e ﬀ ects likely operate via inductive e ﬀ ects,
polarizing the intervening bonds. Notably, a δ SR ( 13 C α ) of only
+1 ppm has been calculated at the sp-hybridized acetylide α -
carbon in CH 3 HgCCH, compared to the much larger δ SR ( 13 C α )
of +20 ppm for the sp 3 hybridized methyl group. This is the exact
opposite of δ SO (cf. Figure 7 ), which is enhanced by the
increasing LA s-character in the HA − LA bond. A TLI on the 13 C
NMR shifts in LAHg II X complexes was noted earlier
38
and has
been analyzed in detail in the PhHg II X series
99
(see discussion of
groups 10 and 11 above).
4.3. HAs of the p-Block
4.3.1. General Aspects. The periodic trends of the SO-
HALA e ﬀ ect in the p-block are governed predominantly by the
occupation or vacancy of n p n HA / n HA
* orbitals, which directly
relates to the oxidation state of the HA.
100
Calculated SO-HALA
NMR shift ranges in p-block element complexes, based on
references discussed in this section, are summarized in Figure
28 .
The presence of the low-lying vacant n p n HA
* orbitals together
with a small HOMO − LUMO energy separation maximizes the
SO-HALA deshielding mechanism ( Figure 20 b) in group 13
and 14 compounds with the HA in oxidation state I and II,
respectively. Maximum deshielding (up to several hundreds of
ppm for 13 C LAs) has been computed for Tl I and Pb II
compounds.
91 , 132
Increased deshielding was also calculated for
In I complexes.
169
On the other hand, involvement of the HA n s
orbitals in bonding (incomplete hybridization of s and p orbitals
of rather di ﬀ erent orbital sizes) disrupts the e ﬃ cient orbital
overlaps in the magnetic couplings and increases the HOMO −
LUMO separation, which results in SO-HALA e ﬀ ects in Tl III
and Pb IV compounds that may be an order of magnitude smaller
than for their Tl I /Pb II counterparts.
Vertical trends in the p-block using typical oxidation states are
compared in Figure 29 for the δ SO ( 13 C) of the sp-hybridized C α
atoms of acetylide substituents bound to di ﬀ erent p-block
HAs.
51
Note that nonbonding MOs are absent from the group
13 and 14 examples. In the form er, the σ HA − LA
( * ) ↔ n HA
*
deshielding mechanism ( Figure 21 b) dominates and leads to
an inverse dependence, while the group 14 HA IV compounds
lack low-lying n HA
* orbitals and thus exhibit small shielding
contributions which originate in the σ HA − L ↔ σ HA − LA
*
contributions (see rule 2 in the beginning of this section).
We recall that the absence of larger shielding SO-HALA
contributions due to ine ﬃ cient SO/FC pathways may lead to a
trend that is inverse to halogen dependence for certain LAs with
small s-character in the HA − LA bond, such as early d-block
elements in their halide complexes. In that case the IHD is
mostly of paramagnetic origin.
42
However, for HAs of group 13,
the inverse trend (analogous to IHD) down the group is due to
increasingly deshielding SO-HALA e ﬀ ects ( Figure 29 ). This has
been termed a triel dependence .
51
In groups 15 − 17, the SO-HALA shielding increases with the
group number and with decreasing oxidation state of the HA,
i.e., with an increasing number of occupied LP HA (s). Group 15
compounds actually represent a borderline where the SO-HALA
mechanisms of deshielding and shielding largely compensate
each other, often resulting in a small overall (in absolute value)
δ SO .
51 , 100
Maximum SO-HALA shielding at the LA is observed
in group 17 complexes in low oxidation states, i.e., halogens in
oxidation state + I/-I with two occupied π -type LPs at the
HA.
49 , 51 , 100
Regarding group 18, LA SO-HALA shifts were
studied only for Xe compounds and are generally smaller than in
group 17 compounds, except the outlier δ SO ( 1 H) in HXeF,
which explains the extended 1 H NMR range in Figure 28 .
Figure 28. Computed ranges of the 1 H and 13 C SO-HALA shifts for
individual groups in the p-block (LA bonded directly to the HA).
Figure 29. Comparison of trends in δ SO ( 13 C α ) for acetylide substituents
bound to heavy p-block elements (with 0, 0, 1, 2, and 3 n HA for groups
13, 14, 15, 16, and 17, respectively). Data from ref 51 .
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4.3.2. Groups 13 and 14: Triels and Tetrels. The SO-
HALA-induced 13 C chemical shifts in group 13 HA III and group
14 HA IV complexes have been thoroughly investigated using
model et hane -, et hene -, and et hyne- subs tit uted sy stem s.
51
Distinct SO-HALA deshielding was noted down the group 13
HA III com pl exes , rang ing from δ SO ( 13 C) = +0 .9 pp m in
H 2 Al III (CCH) to δ SO ( 13 C) = +57 ppm in H 2 Tl lII (CCH).
Comparatively smaller shielding δ SO ( 13 C) was calculated in
group 14 H 3 M IV (CCH) complexes (M = Si IV to Pb IV ) ranging
from − 0.3 ppm to − 6 ppm in H 3 Pb IV (CCH).
51
This is not
surprising, as the av ailabili ty of low-lying HA-centered
unoccupied MOs, important for the SO-HALA deshielding
mechanism, is diminished as we move from tricoordinated
group 13 to tetra-coordinated group 14 systems, and the SO-
HALA shielding e ﬀ ects originating in σ HA − L ↔ σ HA − LA
*
mechanism prevail. Notably, the overall reported relativistic
contributions to the 13 C NMR resonances in group 14 HA IV
systems are deshielding (up to +11 pm) as a result of dominant
deshielding SR e ﬀ ects ( δ SR up to +17 ppm for the C α of an alkyl
Pb IV complex), which are only partially compensated by SO-
HALA shielding contributions ( δ SO = − 6 ppm in the same
complex).
51
Similar results had been reported previously for
group 14 hydrides (HA IV H 4 , HA = C, Si, Ge, Sn, and Pb) using
complete active space self-consistent ﬁ eld (CASSCF) calcu-
lations,
170
where small 1 H SO-HALA shieldings and over-
compensating deshielding δ SR contributions were reported. A
deeper PT2 analysis of those model Pb IV systems, performed for
the purpose of this review, reveals a number of small SO-HALA
shielding and deshielding MO couplings balanced by several
factors, mostly related to the substituents present.
A delicate dependence of both SR and SO-HALA e ﬀ ects on
electronic structure has also been pointed out in a fully
relativistic DFT study of δ ( 13 C) in group 14 (M IV ) alkynyl
compounds.
92
For instance, a deshielding δ SR up to +16 ppm
and a shielding δ SO up to − 13.5 ppm were calculated for C α in
tetra -alkyn yl co mplex es of Pb IV , in goo d agre emen t wit h
previous ﬁ ndings.
51 , 170
However, when three out of the four
alkynyls were replaced by methyl groups, the signs of both the
SR and the SO contributions on the remaining alkynyl C α
switched, to δ SR = − 7 ppm and δ SO = +4 ppm. No explanation for
this observation was given in that work, but we speculate that it
may be related to a larger covalency of the Pb − CH 3 than the
Pb − alkynyl bonds, in ﬂ uencing the energy and composition of
the Pb − L bon di ng o rbi ta ls (p a rt icu la rly orb i tal s wi th π -
character relative to the Pb − LA bond) and thus also a ﬀ ecting
the corresponding orbital magnetic couplings. In summary, the
SO-HALA e ﬀ ects in the M IV L 4 tetrel complexes of group 14 are
relatively small, and their sign depends heavily on the
substituents present.
Moving to the lower oxidation states in groups 13 and 14, the
recently predicted extremely deshielding δ SO (LA) in Tl I and Pb II
compounds
91 , 132
can be up to +200 ppm for 13 C, to +1000 ppm
for 29 Si, and to +80 ppm for 1 H NMR chemical shifts. The origin
of this extraordinarily large SO-HALA contribution relates to
the geometrical and electronic structu res of Tl I and Pb II
compounds. Large deshielding σ HA − LA
( * ) ↔ n HA
* orbital magnetic
couplings arise from the interplay of occupied and vacant 6p-
based MOs, which are rotated by 90 degrees relative to each
other ( Figure 20 b). This results in nearly perfect magnetically
induced orbital overlap, further complemented by rather small
energy gap(s) between σ HA − LA and n HA
* MOs.
132
Extreme deshielding SO-HALA e ﬀ ects readily explain why
certain NMR signals were absent in the experimental spectra of
some previously studied Tl I and Pb II compounds; such signals lie
far outside the typical NMR spectral ranges,
132
see Figure 28 .
Hence, only a few months after the computational prediction,
91
a record 1 H NMR shift of more than 35 ppm in a dinuclear Pb II
hydride was experimentally con ﬁ rmed.
171
Since then, a growing
number of new experimental studies have been motivated by the
discovery of extreme SO-HALA e ﬀ ects in Pb II compounds, and
old attempts to prepare the Pb II hydrides and investigate their
chemistry have been revived.
171 − 173
An analogous but an order of a magnitude smaller deshielding
δ SO has been calculated for some Sn II compounds.
91 , 132
It should
be noted, however, that in certain circumstances, such as the
involvement of the HA in clusters, relatively large shielding SO-
HALA e ﬀ ects can also be found for Sn II species. In earlier
computations on Sn − P cages,
174
sizable SO-HALA shielding
contributions to the 31 P shifts were calculated ( − 80 ppm at the
phosphorus atoms adjacent to three Sn II centers). This is a
rather extreme example of the in ﬂ uence of structural aspects on
the SO-HALA e ﬀ ect.
The magnitude of δ SO ( 1 H) in both Sn II and Pb II hydrides has
been reported to depend on the coordination number of the
central atom.
91
This was rationalized by the variable
involvement of Sn and Pb n p * orbitals in ligand binding,
making them less accessible for the SO-HALA deshielding
mechanism in case of higher coordination numbers of the HA.
SO-HALA shifts in Sn II and Pb II compounds thus sensitively
probe the coordination sphere of the central metal atom. A
practical application of the dependence of the SO-HALA e ﬀ ect
on electronic structure
175
showed that a sizable decrease of
δ SO ( 13 C α ) in cyclic plumbanes from ∼ 80 to ∼ 20 ppm can be
used to probe the formation of adducts with Lewis bases. This
decrease results from the involvement of the Pb 6p * orbitals in
adduct formation, which increases the energy denominators in
the SO/FC formula ( section 2.3.3 ).
175
Four-component relativistic DFT calculations of the NMR
chemical shifts in a dimeric plumbylene,
176
[Fe{( η 5-C 5 H 4 )-
NSiMe 3 } 2 Pb:], which can activate aromatic C − H bonds under
normal conditions, revealed a large di ﬀ erence between the
δ SO ( 15 N) values of the two nitrogen atoms bound directly to the
lead center (76 ppm vs 9 ppm). This di ﬀ erence was rationalized
based on the covalency of the individual Pb − N bonds, which
was in turn used to explain the unexpected shielding of one of
the 207 Pb resonances.
4.3.3. Group 15: Pnictogens. The SO-HALA e ﬀ ects in
compounds with an HA of group 15 are mostly shielding, and
the role of the oxidation state of the HA seems less pronounced
(so far it is reported to be notable only for bismuth). For
instance, the calculated
51
SO-HALA shielding at 13 C α alkynyl
atoms ranges from δ SO = − 1 ppm in H 2 P III CCH to δ SO = − 31
ppm in H 2 Bi III CCH. A weak 1 H SO-HALA shielding of − 0.2
ppm for hydrogen atoms bound to phosphorus has been
noted.
65
The rovibrational, thermal, and relativistic corrections
to the 19 F shielding constants in group 15 tri ﬂ uorides (NF 3 ,P F
3 ,
and AsF 3 ) have been investigated using a combination of
nonrelativistic coupled-cluster with single, double, and triple
excitatio ns (CCSDT ) and relativ istic DFT cal culations .
177
Weakly shielding δ SO ( 19 F) up to − 4 ppm was observed. A
similarly large SO-HALA 13 C shielding down to δ SO = − 4 ppm
was calculated at a carbon atom bound to As in a number of
naturally occurring organo-arsenic compounds, including a
variety of isomers of Arsenicin A, for which an adamantane-like
structure has been proposed.
178
Inclusion of SO-HALA e ﬀ ects
in the calculations was found necessary in order to distinguish
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between the individual isomers of Arsenicin A, enabling the
con ﬁ rmation of the adamantane-like structure.
A rather small SO-HALA shielding δ SO ( 13 C) = − 4 ppm was
predicted at 13 C ipso in triphenylbismuth.
179
The C ipso of phenyl
(Ph) substituents was later used to study the dependence of the
SO-HALA e ﬀ ect on the oxidation state of Bi in Bi III PhCl 2 and
Bi V PhCl 4 .
132
Small SO-HALA shielding at C ipso was predicted in
PhBi V Cl 4 ( δ SO = − 2 ppm), probably resulting from a low
availability of low-lying LP * MOs for the SO-HALA deshielding
(compare to Pb IV compounds above) and stabilization of the
HA-L MOs, which diminishes the SO-HALA shielding
mechanism.
132
A notably larger SO-HALA deshielding ( δ SO =
+50 ppm) was calculated for PhBi III Cl 2 . The latter value is nearly
an order of magnitude larger than the one predicted for the same
ligand C ipso (P h) in t riphe nylbi smuth (s ee abov e).
179
The
reasons for this e ﬀ ect are explained at the bottom of next
paragraph.
Another example of the importance of the ligand sphere for
SO-HALA shifts can be found for Bi III hydrides. The simple
hydride HBi III (CH 3 ) 2 ( Figure 30 ) has an 1 H NMR shift of 2.1
ppm,
180
with a small δ SO ( 1 H) of 0.7 ppm.
100
Replacing the
methyl groups with a pair of large terphenyl substituents
181
results in a total 1 H shift of 19.4 ppm and in δ SO ( 1 H) = 14.4 ppm
(presented in this work, calculated at the same level as used in ref
91 ). The delicate dependence of δ SO on the nature of the
substituents can be rationalized by a competition between the
shielding and desh ielding SO-HAL A mechanisms i n Bi III
compounds, which has been explained for a Bi III hydride using
PT3 theory,
100
see Figure 30 . Essentially, di ﬀ erences in the
covalency of the Bi − L bonds promote one mechanism at the
expense of the other. More electron-donating substituents (i.e.,
the terphenyl group in these hydrides) stabilize those Bi-L
bonding orbitals that contribute most to the SO shielding
mechanism. This diminishes the negative, shielding σ HA − L ↔
σ HA − LA
* SO-HALA contributions, while the positive (deshield-
ing) SO-HALA contributions remain largely una ﬀ ected. Hence,
considerably larger SO deshielding may be observed at the LA
when more charge-withdrawing substituents are present in the
molecule. This applies also to the Pb IV compounds, see above.
4.3.4. Group 16: Chalcogens. The SO-HALA e ﬀ ects in
heavy chalcogen compounds have been well investigated, at least
partly because the favorable spin 1/2 of 77 Se and 125 Te nuclei
give ris e to exper imenta l mult inucle ar NMR stu dies t hat
frequently also include the lighter nuclei. Relativistic e ﬀ ects on
the NMR chemical shifts in heavy chalcogen compounds have
been studied mostly by relativistic DFT methods.
51 , 164 , 182 − 185
Generally, the SO-HALA shielding increases uniformly with Z HA
down group 16 and also increases with the number of occupied
LPs at the HA, i.e, for a lower HA oxidation state. Maximal SO-
HALA shieldings (>50 ppm) are reached for the heavier
chalcogen compounds (Se and Te) in oxidation state − II, when
a double bond to the LA is present (as in seleno- and
telluroketones).
164 , 182 , 185
This bonding situation clearly favors
the propagation of the SO-HALA e ﬀ ects from the HA to the LA
in p-blo ck spec ies and additionall y invol ves small energy
denominators. Even a relatively “ light ” HA like sulfur can
induce an overall δ SO shielding of − 8 ppm at the adjacent 13 C
nuclei of CS 2 via the C  S double bonds.
164 , 185
In addition to
the two factors mentioned above, a large carbon 2s character in
the C − S bonds (formal sp 2 hybri dization) also aids this
shielding. A thorough study of the SO-HALA e ﬀ ects in heavy
chalcogen complexes at the Coupled-Cluster with Single and
Double and Perturbative Triple excitations CCSD(T) and Breit-
Pauli Perturbation Theory (BPPT) levels of theory was
performed to evaluate secondary isotope e ﬀ ects on nuclear
shielding in CSe 2 .
185 , 186
Large SO-HALA e ﬀ ects propagating over several bonds in
experimentally known s elen o- and telluro-ketones were
calculated using relativistic DFT methods.
187
The long-range
SO-HALA e ﬀ ects on the 13 C shifts were found to be − 23, +18,
and − 9 ppm for C α ,C
β , and C γ , respectively. This alternation of
the sign re ﬂ ects the analogy of the SO/FC mechanism with that
of J-coupling
26
( section 3.1 ). Moreover, δ SO (C γ ) ful ﬁ lled the
Karplus double-cosine dependence on the M-C α -C β -C γ dihedral
angle, also known for J-coupling.
26 , 140
In a follow-up study,
188
the authors found similar relations (sign alternation and Karplus
dependence), for H β and H γ nuclei in analogous systems. These
results con ﬁ rm earlier ﬁ ndings
113 , 140
that, when su ﬃ ciently
large, δ SO can be used to determine dihedral angles and
isomerism.
4.3.5. Group 17: Halogens. The ﬁ rst mentioned
19 , 20
and
most intensively studied SO-HALA e ﬀ ect is the normal halogen
dependence (NHD), which refers to the increasing SO-HALA
shielding at the LA with increasing Z HA of an adjacent halide
substituent for both organic and inorganic compounds. While
we saw analogous dependencies throughout groups 14 − 17 (see
Figure 29 in section 4.3.1 ), the SO-HALA e ﬀ ects and thus the
NHD-type behavior tends to be most pronounced for HAs from
group 17. Given that the SO-HALA e ﬀ ect depends on the
number of halogen LPs,
49
NHD has been studied largely for
halide substituents, and the number of studies is so large that we
do not attempt a comprehensive coverage of the literature in this
case. For example, since 2004, computational investigations of
NHD for 1 H and 13 C NMR shifts have targeted hydrogen
189
and
methyl
135 , 189
halides, halogen-substituted acetonitrile,
190
cyclo-
hexane,
191
pyran,
191
purines,
105
and carbazoles.
192
Studies of
NHD on the NMR chemical shifts and shielding tensors of 11 B,
19 F, and 29 Si nuclei at various levels of theory
193 − 195
led to
essentially the same trends and conclusions as for 1 H and 13 C
nuclei. Higher halogen oxidation states have been mentioned in
Section 1 .
49
Figure 30. Schematic PT3 visualization of the δ SO ( 1 H) deshielding
(top) and shielding (bottom) orbital magnetic couplings computed for
HBi(CH 3 ) 2 .A d a p t e df r o mr e f 100 . Copyright 20 18 American
Chemical Society.
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One of the computationally most advanced and detailed
studies focused on the 13 C and 1 H shift tensors in the methyl
halides using combined ab initio and DFT analyses with
perturbation theory (PT), complemented by vibrational
corrections and thermal averaging.
135
The SO origin of the
NHD was con ﬁ rmed (the SO/FC term dominates the PT
contributions), although SR e ﬀ ects of opposite sign (deshield-
ing) were found important in order to obtain quantitative
agreement with experiment. Note that the SR e ﬀ ects do not
follow the SO-HALA trends but tend to be generally deshielding
for HAs of the p-block. Relativistic DFT and BPPT computa-
tional approaches were also applied to the propagation of SO-
HALA e ﬀ ects in heteroaromatic 6-halopyridines.
105
The authors
concluded that relativity, solvent e ﬀ ects, and the choice of
reference compound are important for achieving quantitative
agreement between theor y and experiment. Long-range
relativistic e ﬀ ects were also studied, and the previously discussed
sign alternation of the SO-HALA contributions
47
was also seen
in these heterocyclic systems.
Interestingly, the magnitude of the SO-HALA e ﬀ ect induced
by heavy halide substituents depends on the conformation of
stereoisomers. The propagation of the SO-HALA e ﬀ ect was
diminished in halogenated hydrocarbons
191
when stro ng
hyperconjugative interactions involving σ * C − X orbitals (X =
halogen) were present. The authors calculated various con-
formers of halocyclohexanes and 2-halo-tetrahydropyran and
analyzed the SO-HALA e ﬀ ects on the 13 C shifts. The SO
contribution in the axial conformer of iodo-cyclohexane with
stronger hyperconjugative interactions was found to be 7 ppm
lower than in the equatorial conformer ( δ SO = 15.8 ppm vs 22.8
ppm), con ﬁ rming the experimentally observed di ﬀ erence of 8
ppm. Hyperconjugation was shown to stabilize the occupied
LP HA orbitals and to thereby increase the relevant energy
denominat ors. The changes in the SO-H ALA e ﬀ ect thus
determine the observed di ﬀ erences in shift and may be used
to distinguish individual conformers. The authors pointed out,
however, that this may apply only to structurally similar groups
of molecules (conformers, i somers) where other factors
in ﬂ uencing the SO-HALA e ﬀ ect (e.g., hybridization at the
LA) are comparable. Subsequent work on the in ﬂ uence of
stereoelectronic interactions on the SO and paramagnetic
contributions to the 13 C NMR shielding tensors in dihaloe-
thenes
196
came to similar conclusions: the hyperconjugative
interactions involving the halogen LPs and σ * C − X antibonding
orbitals (X = halogen), together with steric interactions between
the halogen atoms, were found to be a major cause of the
di ﬀ erence in δ ( 13 C) between the cis vs trans isomers. These two
studies nicely demonstrate the possible use of SO-HALA shifts
to investigate isomerism and hyperconjugation in compounds
with heavy halogen substituents.
4.3.6. Group 18: Aerogens. For obvious reasons, SO-
HALA e ﬀ ects originating from noble-gas (Ng) heavy atoms have
been studied less than those for group 17 HAs. A combination of
ab initio and relativistic DFT levels for noble-gas hydride cations
(NgH + ;N g=N e − Xe)
136
found the total relativistic e ﬀ ect on the
1 H shifts (the SO-HALA was not investigated separately) to
range from ca. 0 ppm in NeH + to ca. − 18.1 ppm in XeH + .A
perturbational treatment of relativistic e ﬀ ects typically gave
smaller SO-HALA e ﬀ ects for a series of HXeY
197 , 198
compounds
than for HI,
189
probably due to the lower covalency of the Rg − H
bonds. The HXeF system with an extreme value of δ SO ( 1 H) =
− 35 ppm, probably caused by the TLI of ﬂ uorine, was an
exception.
197 , 198
While no structure-related applications of SO-HALA e ﬀ ects
have yet been reported for Ng compounds, the known
correlations between δ SO and the covalency of the HA − LA
bond
114
can be useful in determining the character of Ng − LA
bonds and the oxidation state of the Ng by using a combination
of relativistic calculations and NMR studies, as suggested in refs
197 and 198 .
4.4. HAs of the f-Block
4.4.1. General Aspects. Investigation of the SO-HALA
trends in lanthanide and actinide complexes has been limited to
the closed-shell 4f 0 (La III ,C e
IV ), 4f 14 (Lu III ), and 5f 0 (Th IV ,P a
V ,
and U VI ) electronic con ﬁ gurations, as others typically feature
paramagnetic metal centers and thus di ﬀ erent physics of nuclear
shielding. In these closed-shell con ﬁ gurations, the f-elements
have formally empty, low-lying valence ( n -1)d and/or ( n -2)f
shells. Hence , the deshielding σ HA − LA
( * ) ↔ n HA
* coupling
mechanism dominates, similar to the p 0 and d 0 cases ( Figure
23 ).
97 , 100 , 137 , 199
A relatively small deshielding δ SO ( 1 H) ≈ +5
ppm, has been calculated for La III and Lu III hydrides.
100
This is
comparable to 5d 0 transition-metal hydrides, which may reach
δ SO ( 1 H) values up to ca. + 10 ppm (e.g., Hf IV ,T a
V ,W
VI , and
Re VII , Figure 26 ). A larger deshielding e ﬀ ect of δ SO ( 1 H) ≈ +20
ppm was predicted for a Ce IV hydride, see below. The ranges of
the SO-HALA 1 H and 13 C shifts in 4f and 5f complexes that have
been predicted are summarized in Figure 31 .
The availability of low-lying 5f-orbitals and the involvement of
the 5f-shell in chemical bonding is decisive for the 5f 0 systems.
Moderat ely deshielding SO-HAL A e ﬀ ects (5 − 12 ppm),
comparable to 5d 0 systems, are found for Th IV complexes
137 , 200
because 6d and 7s orbitals dominate Th IV bonding with only
minor involvement of the relatively di ﬀ use 5f orbitals.
137 , 200
A
protact inium m odel hyd ride comp lex [H PaF 5 ] − ha s been
predicted to have δ SO ( 1 H) > +50 ppm, which indicates that
5f-orbitals which are more available can strongly enhance the
deshielding SO-HALA e ﬀ ects.
12
This e ﬀ ect is further enhanced
for U VI hydride complexes with δ SO ( 1 H) predicted to be well
over +200 ppm in model hydrides and δ SO ( 13 C) of nearly +400
ppm for carbon atoms directly bound to the U VI centers in
organometallic complexes (see below).
95 , 97 , 201
The oxidation state again plays a potentially important role in
the size and sign of the SO-HALA e ﬀ ects, as it does for the d- and
p-block elements. In contrast to the large deshielding δ SO (LA) in
5f 0 6d 0 U VI complexes, a strongly shielding δ SO ( 1 H) of − 37 ppm
has been predicted for a hypothetical closed-shell 5f 2 6d 0 U IV
complex.
100
While this 5f 2 model system is unlikely to re ﬂ ect any
real U IV ground-state complex, the computations con ﬁ rm that
the general mechanism outlined above (SO shielding due to LPs
at the HA)
100
can also be applied to 5f actinide systems. The δ SO
of this hypothetical closed-shell U IV complex is not included in
Figure 31 because no closed-shell 5f 2 complexes are known to
exist.
4.4.2. Selected Examples and Applications. SO-HALA
e ﬀ ects in lanthanide complexes have, until now, been
Figure 31. Computed ranges of the 1 H and 13 C SO-HALA shifts in 4f
and 5f complexes (LA bonded directly to the HA).
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investigated only very sparsely, probably due to a limited
number of known closed-shell species.
100 , 199
Very small SO-
HALA e ﬀ ects (<1 ppm) were computed for the nitrogen atoms
of a phenanth rolin e ligand bou nd direct ly to lantha num,
probably due to the highly ionic La − N bonding.
199
On the
other hand, a large 1 H SO-HALA deshielding ( δ SO = +20 pm,
δ tot ≈ +31 ppm) has been predicted for a Ce IV hydride
100
and
rationalized by the involvement of formally vacant Ce 4f orbitals
in the Ce − H bonding and thus also in the σ HA − LA
( * ) ↔ n HA
*
deshielding mechanism, in analogy to U VI , see Figure 32 .
However, neither this prediction for Ce IV nor the U VI case
described below has been con ﬁ rmed experimentally.
The NMR shifts of closed-shell actinide complexes have been
studied exte nsive ly ove r t he yea rs. Ear ly rel ativi sti c DFT
computations of the 19 F shifts in U VI ﬂ uoride complexes gave
modest SO-HALA shifts up to about 30 ppm,
41 , 202
because the
low ﬂ uorine 2s-character in the U − F bond disfavors the FC
mechanism. Much of the early work focused on the challenges of
computing the NMR shifts for actinide complexes, e.g., choosing
the sizes of the ECP cores and the exact-exchange admixtures to
use in the DFT functionals.
97 , 137 , 201
The calculation of 19 F NMR
shifts in actinide systems has been reviewed critically.
203
A two-component ZORA study of the 1 H shifts in U VI hydride
complexes
137
was aimed explicitly at investigating potentially
large SO-HALA shifts. The results indeed suggested giant
deshielding of more than +100 ppm in some of the model
systems studied. As no experimental NMR data for such species
are yet available (the only known U VI hydride complex, H 2 UO 2 ,
has been studied only by matrix-isolation IR spectroscopy),
204
selecting a suitable functional to make valid predictions required
that other reference data be available. These were the 13 C NMR
shifts of some U VI -bound carbon atoms in organometallic
complexes. Except for the system discussed in the next
paragraph, comparison with the available experimental 13 C
data suggested the use of a hybrid functional with 40% exact-
exchange admixture. Notably, this was due to the absence of an
exchange-correlation kern el in the 2-component ZORA
implementation used at the time.
96 , 97 , 153
A systematic
reinvestigation of the 1 H and 13 C NMR shifts in U VI complexes
including the exchange-correlation kernel (in comparison with
4-component results)
97
suggested that functionals with a lower
exact-exchange admixture (20 − 25%) are more appropriate.
Nevertheless, the giant SO-HALA shifts were con ﬁ rmed.
The one 13 C case for which the predictions of ref 137
disagreed starkly with the available experimental data was an
unusual U VI -hexaalkyl complex. Here the uranium-bound
methylene 13 C NMR shift had been assigned originally to a
signal at +34 ppm, which was likely a contaminant in the in situ
spectrum,
205
while the computations suggested values closer to
400 ppm because of the extremely large SO-HALA shifts. Such
shifts are completely outside the usual expectations for an alkyl
carbon atom, and consequently the original measurements had
been restricted to a range between − 50 and +130 ppm. A new
measurement with a larger spectral range gave a value of 434
ppm,
201
in line with the computational predictions, and
con ﬁ rming the existence of giant SO-HALA shifts of more
than +300 ppm. The predictive quality of the computational
methods suggested a blind test of computed and measured 13 C
NMR shifts for a newly found tetraalkyluranylate complex. The
computed shift of +250 ppm agreed excellently with the
independently measured value of +243 ppm, again with sizable
SO-HALA contributions of about +177 ppm.
201
These results were still obtained with 40% exact-exchange
admixture and without a kernel in the perturbation treatment.
The above-mentioned reinvestigation of ref 97 with a kernel and
other functionals con ﬁ rmed overall the large SO-HALA shifts
also for these cases while providing some corrections to the
quantitative data. That study also gave further predictions, in
particular an extension of the range of U VI -bound 13 C NMR
shifts up to +550 ppm ( δ SO = +390 ppm) for organometallic
model complexes, and it suggested the positions of missing
signals in experimentally characterized complexes.
97
Systematic
analyses correlated the SO-HALA shifts with the covalency of
the U − C bond and the position of the LA in the complex. For
example, atoms opposite to strong π -donors (e.g., in uranyl
complexes) were found to have smaller SO-HALA shifts than
nuclei positioned trans to less π -donating ligands, e.g., in the
above-mentioned hexaalkyl complex.
97
Covalency was also studied in rare actinide-chalcogen Th IV
and U VI complexes by 77 Se and 125 Te NMR.
206
Curiously,
selenium and tellurium constitute the “ LA ” in these systems, and
their chemical shifts include SO-HALA e ﬀ ects arising from the
actinide center. The δ SO ( 77 Se) and δ SO ( 125 Te) were found to
increase linearly with the covalency of the actinide − chalcogen
bond (R 2 = 0.92 − 0.99) expressed as the delocalization index. In
one of the U VI complexes studied, a 77 Se NMR shift of 4900 ppm
was measured, which extends the known chemical shift range for
this nucleus in diamagnetic systems from the previous record of
3300 ppm. Even larger 77 Se shifts, up to 8000 ppm, have been
predicted computationally.
206
Uranyl 17 O NMR chemical shifts have been investigated
rece ntly
207
by c omb in ed ex p eri me nta l an d co mpu t ati on al
approaches. The ab initio calculations revealed that the 17 O
resonances were very sensitive to the local environment, a fact
which could be used to provide better understanding of the
structure and crystal packing of such compounds in the solid
sta te. A t horo ugh ev alu at ion o f the DF T ap proa che s to
calculating various properties of uranium complexes, including
computation of the relativistic NMR shift, has been reported
recently.
208
Figure 32. Example of the orbital magnetic coupling in a Ce IV hydride
complex (top) and schematic representation of the AOs contributing to
the SO deshielding (bottom). Adapted from ref 100 . Copyright 2018
American Chemical Society.
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4.5. Superheavy Elements (SHEs) as HAs
Given the short half-lives of SHEs, the chances for experimental
NMR studies of their compounds are slim at best. Therefore,
computational studies have also been almost nonexistent. So far,
only one set of computations (done at four- and two-component
levels) is available for a number of postactinide complexes with
6d 6 ,6 d
8 , and 6d 10 HA con ﬁ gurations.
209
Interestingly, the SO-
HALA shifts in the 6d 6 complexes are slightly smaller or only
marginally larger (in absolute value) than in their 5d 6 analogs.
This is due to greater ligand- ﬁ eld splitting (caused by the
improved bonding interactions of the relativistically expanded
6d-orbitals) counteracting the larger SO matrix elements.
113
The SO-shifts in 6d 8 complexes are somewhat larger than in
their 5d 8 analogs. Notably, the ligand- ﬁ eld e ﬀ ects do not apply to
the d 10 con ﬁ guration, and in consequence the deshielding SO-
HALA s hift s ar e not abl y enh ance d in t he 6d 10 hyd rides
compared to their 5d 10 analogs, e.g. + 23.7 ppm for
[HRg(NHC)] vs +3.3 ppm for [HAu(NHC)] (NHC is the
simplest possible imidazolium-based N-heterocyclic carbene-
model ligand, Rg is roentgenium) or +33.1 ppm for [HCnPh]
(Cn is copernicium) compared to +9.0 ppm for [HHgPh].
209
All data given are predictions based on two-component SO-
ZORA computations.
134
4.6. Weak Interactions and “ Through-Space ” SO-HALA
E ﬀ ects
The above-mentioned studies concentrated mostly on SO-
HALA e ﬀ ects at an LA chemically bound to the HA (or involved
in a covalent bonding network including the HA). However, in
the same way as the FC mechanism of J -coupling operates via
the polarization of the spin density even in the absence of
“ chemical bonds ” ( “ through-space ” J -coupling),
210
one can
expect this to hold also for the SO-HALA chemical shifts, see
section 3.1.5 .
The possibility of “ through-space ” propagation of SO-HALA
e ﬀ ects in pyridinium halide ionic liquids has been investigated
computationally.
211
Using vario us comput ationa l models,
δ SO ( 13 C) up to +3 ppm was predicted at the pyridinium carbon
atoms when pyridinium was involved in an ion pair with I 3 − , but
this e ﬀ ect vanished with the Cl 3
− analog. The authors concluded
that it may be challenging to ﬁ nd experimental evidence to
support these computational predictions, owing to the short
lifetime of the investigated ion pairs in a liquid environment.
Notably, in contrast to typical shielding SO-HALA e ﬀ ects for
bonded halogen substituents, the through-space SO-HALA
contribution from X 3 − to the pyridinium carbon nuclei was
found to be deshielding. This would correspond to an IHD, and
the authors argued that the spin polarization is transmitted via
the π -system.
Spe ctro sc opi c and com put atio na l evi den ce of a n int ra-
molecular Au I ··· H − N hydrogen bond in a gold carbene complex
was reported recently.
212
While the SO-HALA e ﬀ ect can
propagate only via an Au ··· H hydrogen bond, δ SO values of − 6
ppm and − 0.1 ppm were predicted for the 15 N and 1 H nuclei,
respectively, involved in the Au I ··· H + − N hydrogen bond. The
SO-EDD
99
calculated for this system is shown in Figure 33 . The
low stability of the molecule in solution prohibited detailed
experimental NMR investigations.
During the revision of this review, solid-state 1 H NMR and
DFT study of a series of 4-(dimethylamino)pyridinium with Cl,
Br, and I counteranions has been reported.
213
Discrepancy
between the experimental and theoretical NMR shifts was
attributed to an absence of the SO e ﬀ ects in the calculations of
hydrogen-bonded ion pairs at the SR DFT level. However,
agreement between the theoretical and experimental NMR data
was signi ﬁ cantly improved at the SO level that demonstrated
propagation of the SO e ﬀ ects through hydrogen bonding which
resulted in “ through-space ” SO-HALA shifts.
The involvement of the HA in weak interactions may also
indirectly in ﬂ uence the δ SO at a bonded LA. As heavier halogens,
particularly iodine, induce considerable (shielding) δ SO at an
adjacent LA, their involvement in halogen bonding modulates
the size of δ SO in analogy to the TLI reported for transition-metal
complexes (see above), thus providing a nother tool to
investigate halogen bonding.
214
Already in 2004,
215
following
still earlier work,
216 , 217
the e ﬀ ect of halogen bonding on the 13 C
NMR chemical shifts of C α in haloarenes in di ﬀ erent solvents
was studied. The authors noted that nonrelativistic deshielding
of 6 − 8 ppm at C α caused by the formation of a halogen bond was
compensated by an induced SO-HALA shielding of similar size.
One might thus mistakenly assume the absence of halogen
bonding based on the total observed 13 C α NMR chemical shift.
Recently, a multinuclear solid-state NMR study investigated
halogen bonding in various cocrystals.
218
The carbon atoms
attached to a heavy halogen ( 13 C α ) were found to be deshielded
by +6 ppm when iodine derivatives were involved in halogen
bonding. Two-component relativistic SO-ZORA calculations
showed that modulation of both the paramagnetic and SO-
HALA contributions to the 13 C α chemical shifts was the main
cause of the observed deshielding. The relativistic 13 C NMR
chemical shift was proposed as a possible indicator of halogen
bonding, although the exact mechanism was not investigated.
The calculated deshielding correlated with increasing C − I and
C − Br bond lengths, as con ﬁ rmed by other studies.
219
We
speculate that the main cause of the observed deshielding is a
decrease of the p-character/covalency of the C − X bond upon
formation of the halogen bond,
114 , 132
which in turn weakens the
SO/FC propagation of the SO-HALA e ﬀ ect.
114
4.7. Emerging Applications
In Section 4 , we have explained and summarized how the SO-
HALA shifts relate to the electronic structure of the heavy atom
across the Periodic Table, that is, the periodic trends in SO-
HALA e ﬀ ects. We hope the examples provided show that
applications of SO-HALA shifts are not limited to merely aiding
the assignments of experimental spectra or to improving the
agreement between experimental and computational data. As
demonstrated in sections 3 and 4 , the SO-HALA shifts
sensitively re ﬂ ect − and thus can be used to study − various
chemical phenomena, such as the structural trans and cis
in ﬂ uences, the covalency of the HA − LA bond, molecular
Figure 33. SO-EDD
99
for an intramolecular Au I ··· H − N hydrogen bond
in a gold carbene complex.
212
The SO-induced increase and decrease in
the electron density, which correspond to the sign of SO-HALA shift,
are shown in blue and red, respectively.
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conformation and isomerism, and weak supramolecular
interactions. The SO-HALA shifts also provide insights and
links to chemical reactivity and may guide experimental e ﬀ orts
by revealing new and exotic ranges of NMR chemical shifts. A
thorough understanding of the SO-HALA shift thus represents a
powerful tool for probing the structure and chemical properties
of heavy-element compounds.
5. SUMMARY AND PROSPECTS
Relativistic e ﬀ ects can in ﬂ uence the neighbor-atom NMR shifts
in heavy-element compounds appreciably, with substantial and
often analytically characteristic consequences for the NMR
spectra. Relativistic neighbor-atom e ﬀ ects on NMR shifts are
typically dominated by spin − orbit and magnetically induced
spin polarization arising at the heavy atom and propagating to
the spectator nucleus via a Fermi-contact-type interaction. This
phenomenon, generally known as the spin − orbit heavy-atom
e ﬀ ect on the light-atom NMR shift (SO-HALA shift), has been
known for some time but is still underappreciated in the NMR
community. Signi ﬁ cant progress in our understanding of these
e ﬀ ects across the Periodic Table since the last general review in
2004 and the recent availability of powerful MO-by-MO analysis
and interpretative tools rooted in third-order perturbation
theory called for a new and comprehensive exposition and
critical overview of the theoretical background of SO-HALA
shifts, of the conceptual tools available, and of the trends
observed throughout the Periodic Table of the Elements.
Throughout this review we have compared both second-order
perturbation theory based on relativistic molecular spinor pairs
and third-order perturbation theory based on nonrelativistic (or
scalar relativistic) MOs. The latter o ﬀ ers a more ﬁ ne-grained
analysis of SO-HALA shifts, in particular regarding the question
of the sign of the e ﬀ ect − shielding vs deshielding. Provided a
perturbational treatment of the SO e ﬀ ects does not cause too
large errors, the PT3 treatment gives the clearer picture. It has
been central to clarifying many of the concepts laid out in this
overview. The terms obtained in the PT3 framework for the SO/
FC mechanism of SO-HALA shifts can be connected directly to
the bonding and electronic structure of the given system and can
be displayed in a pictorial representation based on triangles
linking the MOs and their energetics, as well as the perturbation
oper ato rs. F urt her gr aph ic al ins igh ts ca n be p rovi ded by
comparisons to the spin − orbit and magnetically induced spin
density (SOM-ISD) and to changes in charge density induced
by spi n − orbit co uplin g (spin − or bit e lec t ro n de fo rm at io n
density, SO-EDD). Together, these analyses and graphical
tools allow for chemically intuitive interpretations of the SO
e ﬀ ects on the neighbor-atom NMR shifts. This understanding is
expected to greatly assist experimental chemists and NMR
spectroscopists in estimating the ranges of the NMR shifts for
unknown compounds, in the identi ﬁ cation of intermediates in
catalysis or other processes, in analyzing conformational aspects
or intermolecular interactions, and in the prediction of trends in
series of compounds throughout the Periodic Table.
Given the substantial progress made since the 2004 review
referred to in section 1 and the fast-growing literature with ever
more examples of relativistic neighbor-atom e ﬀ ects on NMR
shifts and their utilization in many ﬁ elds of chemistry, it seems
safe to expect further insights and a further widening of this topic
over the coming years.
6. TECHNICAL AND COMPUTATIONAL DETAILS
To have a consistent data set on which to base our discussion,
and to illustrate the concepts in Section 3 , new data were
calculated using the methods given below.
6.1. Optimization of Structure
Method 1: The structures were optimized using the Turbomole
program (releases 6.3 or 7.3).
220
The PBE0 functional,
221
scalar-
relativistic e ﬀ ective core potentials (ECPs) for the HA, and def2-
TZVPP basis sets were used.
Method 2 : The structures were optimized using the ORCA 4.1
program.
222
The scalar-relativistic all-electron DKH approach,
the PBE0 functional, and def2-TZVPP basis sets were employed.
6.2. Molecular orbital and bonding analysis
6.2.1. Natural Bonding Orbital (NBO) Analysis.
Contributions of the AOs (e.g., d orbitals) to the HA − LA
bond were calcul ated using the N BO 3.1
223
module as
implemented in Gaussian16.
224
The PBE0 functional, an ECP
for the HA, and def2-TZVPP basis sets were used.
6.2.2. Energy Decomposition Analysis-Natural Orbi-
tals for Chemical Valence (EDA-NOCV).
225
Charge and
energy analysis of orbital contributions to the binding energy
was performed in the ADF 2014 program.
226
The computational
setup was based on the ZORA/PBE0/TZP level of theory.
6.3. Calculation of SO-HALA Shifts
The SO-HALA data calculated for this work were obtained as
the di ﬀ erence between fou r-component relat ivistic DFT
calculations of NMR shifts
73
and calculations neglecting spin −
orbit coupling (scalar-relativistic values - the contribution of the
SO operator was down-scaled to 0),
110
as implemented in the
ReSpect program.
93 , 154
The Dirac-Kohn − Sham (DKS) ap-
proach was used with the PBE0
221 , 227
functional, Dyall-vtz basis
sets
228 − 230
for the HA, and pcS-2 basis sets
231 , 232
for the other
atoms (DKS/PBE0/Dyall-vtz/pcS-2 level).
6.4. Calculation and Visualization of the
SO-and-Magnetically Induced Spin Density (SOM-ISD) and
the Spin − Orbit Electron Deformation Density (SO-EDD)
The SOM-ISD maps
100
were calculated as implemented in the
ReSpect program.
93 , 154
The SO-EDD maps
99
were obtained as
di ﬀ erences between the 3D electron density maps calculated at
four-component and scalar-relativistic levels (to obtain scalar-
relativistic values, the contribution of the SO operator was
down-scaled to 0).
110
6.5. Analysis of SO-HALA Shifts
6.5.1. Second-Order Perturbation Analysis (PT2). The
SO-HALA shifts were obtained as the di ﬀ erence between 2c-
ZORA and 1c-ZORA values calculated in the ADF program.
The analysis of MO ↔ MO and MSP ↔ MSP couplings was
used as implemented in the ADF program.
134
The composition
of 2c MSPs in terms of 1c-ZORA MOs was determined by
employing standard fragment analysis.
134
The graphical display
of MSPs used only one real function of the complex, transformed
spinor of a given Kramers pair.
6.5.2. Third-Order Perturbation Analysis (PT3). The
SO-HALA shifts were calculated and analyzed using the PT3
approach as implemented
99
in the ReSpect program.
93 , 154
This
implementation follows the third-order response theory of ref
107 for the static (perturbation-free) case. The applied analysis
considers only the one-electron contributions to the SO/FC
NMR shielding and neglects all HF and DFT kernels.
Assuming the presence of a single heavy atom in the system
Chemical Reviews pubs.acs.org/CR Review
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Chem. Rev. 2020, 120, 7065 − 7103
7095

allows us to approximate the SO operator as
∑  ≅ 
=
−−
Zr l Z r l
N
N
Nu
N
u
1
nuc 3H A
HA
3 H
A

, see section 2.3.3 . The unper-
turbed wave function has been calculated at the PBE/Dyall-vtz/
pcS-2 level.
AUTHOR INFORMATION
Corresponding Authors
Michal Straka − Institute of Organic Chemistry and Biochemistry,
Czech Academy of Sciences, CZ-16610 Prague, Czechia;
orcid.org/0000-0002-7857-4990 ; Email: straka@
uochb.cas.cz
Martin Kaupp − Institute of Chemistry, Technische Universita  t
Berlin, D-10623 Berlin, Germany; orcid.org/0000-0003-
1582-2819 ; Email: [email protected]
Radek Marek − CEITEC - Central European Institute of
Technology and Department of Chemistry, Faculty of Science,
Masaryk University, CZ-62500 Brno, Czechia; orcid.org/
0000-0002-3668-3523 ; Email: [email protected]
Authors
Jan V ı

cha − Centre of Polymer Systems, Tomas Bata University in
Zl ı 
n, CZ-76001 Zl ı 
n, Czechia; orcid.org/0000-0003-3698-
8236
Jan Novotny  − CEITEC - Central European Institute of
Technology, Masaryk University, CZ-62500 Brno, Czechia;
orcid.org/0000-0002-1203-9549
Stanislav Komorovsky − Institute of Inorganic Chemistry, Slovak
Academy of Sciences, SK-84536 Bratislava, Slovakia;
orcid.org/0000-0002-5317-7200
Complete contact information is available at:
https://pubs.acs.org/10.1021/acs.chemrev.9b00785
Notes
The authors declare no competing ﬁ nancial interest.
Biographies
Jan V ı 
cha studied inorganic chemistry at the Masaryk University in
Brno and received his Ph.D. in biochemistry under the supervision of
Radek Marek in 2014. After a postdoctoral stay at the Central European
Institute of Technology, Brno, he moved to Tomas Bata University in
Zl ı 
n, where he obtained a position as a Senior Researcher in 2017. His
research interests focus on the chemistry of heavy-element complexes
with anticancer activity and on better understanding the relationship
between the electronic structure and relativistic e ﬀ ects in NMR
spectroscopy.
Jan Novotny  studied biomolecular chemistry at the Masaryk University
in Brno. He obtained his Ph.D. with Radek Marek in 2015. As a postdoc
in the group of Radek Marek, he received the Petr Sedmera Award in
NMR spectroscopy in 2018. He is currently teaching NMR spectros-
copy, biophysical chemistry, and computational quantum chemistry at
the Masaryk University. His main research interests include NMR
spectroscopy, paramagnetic and relativistic e ﬀ ects in NMR, structure
and spectroscopic properties of transition-metal complexes, supra-
molecules, and noncanonical forms of nucleic acids.
Stanislav Komorovsky studied theoretical physics at the Comenius
University in Bratislava. He received his Ph.D. in chemical physics at the
Comenius University with Vladim ı 
r Malkin in 2009. After postdocs in
Toulouse with Trond Saue and in Tromsø with Kenneth Ruud, he was
awarded a Marie Curie Fellowship at the Slovak Academy of Sciences.
His research focuses on the development of relativistic methods for the
calculation of spectroscopic properties with stress on the magnetic
properties of open-shell systems. He is one of the main authors of the
ReSpect program ( respectprogram.org ), a computer simulatio n
package dedicated to predicting and understanding molecular proper-
ties obtained by ﬁ rst-principle relativistic DFT calculations.
Michal Straka received his M.Sc. in physical chemistry at the Comenius
University in Bratislava in 1996 and his Ph.D. with Pekka Pyykko  at the
University of Helsinki in 2001. After a postdoc with Martin Kaupp at
the University of Wu  rzburg and a Marie Curie Fellowship with Juha
Vaara at the University of Helsinki, he moved to Prague, where he
became a senior researcher at the Czech Academy of Sciences. He
enjoys solving theoretical problems on the border of inorganic, physical,
and organic chemistry. His research interests include theoretical
spectroscopy, chemical bonding, heavy-atom compounds, relativistic
e ﬀ ects, and endohedral fullerenes.
Martin Kaupp studied chemistry at universities in Stuttgart and
Cincinnati. He obtained his Ph.D. at Universita  t Erlangen-Nu  rnberg
with Paul v. R. Schleyer. After postdocs in Stuttgart with Hans-Georg
von Schnering and in Montreal with Dennis Salahub, he completed his
Habilitation in Stuttgart before starting his ﬁ rst professorship at
Universita  tW u
 rzburg in 1999, teaching inorganic and theoretical
chemistry. Since 2010, he has led the quantum chemistry unit at
Technische Universita  t Berlin, and since 2017 he has been the deputy
director of the institute of chemistry at TU. His awards include the
Heinz-Ma ier-L eibnitz-Pr eis 1994, the DFG Heisenb erg and FCI
Docent Scholarships in 1998, as well as the Dirac medal of WATOC
(2001). The research interests of M.K. cover a wide range of topics in
computational and theoretical chemistry, including method develop-
ment in mag netic- res onan ce p aramet er co mput ati ons, rel at ivis tic
quantum chemistry and new density functionals, and various
applications in catalysis, materials research, fundamental questions of
chemical bonding, and di ﬀ erent kinds of spectroscopies.
Radek Marek is a Professor of Chemistry at the Masaryk University in
Brno. During his chemistry studies, he spent several months with Roger
Dommisse at the University of Antwerp and obtained his Ph.D. with
Milan Pota  c  ek in 1995. After a postdoc in NMR spectroscopy with
Vladim ı 
r Sklena  r  he completed his Habilitation in 2004 before starting
his professorship at the Masaryk University in 2011, teaching NMR
spectroscopy and structural and computational chemistry. He received
the Prix de Chimie (Jean-Marie Lehn Prize) for the best PhD thesis in
chemistry (1996), the Alfred Bader Prize (Sigma-Aldrich Comp.) for
the top young researcher in bioorganic chemistry (2002), and the Petr
Sedmera Award in NMR spectroscopy (2018). The main research areas
of R.M. ’ s group at the Central European Institute of Technology
(CEITEC) and Faculty of Science, MU are NMR spectroscopy,
paramagnetic and relativistic e ﬀ ects in NMR, fundamental aspects of
chemical bonding, DFT applied to bonding and molecular properties,
supramolecular structure and interactions, host − guest chemistry, and
anticancer metallodrugs.
ACKNOWLEDGMENTS
This work has received support from the Czech Science
Foundation (Grant No. 18-05421S); the Ministry of Education,
Youth, and Sports of the Czech Republic (Grant Nos. LQ1601,
LO1504 , and LTA USA1914 8); t he Grant Agency of the
Masaryk University (Grant No. MUNI/E/1335/2019); and
the Slovak Grant Agencies VEGA and APVV (Contract Nos. 2/
0116/17 and APVV-15-0726). Work in Berlin has bene ﬁ tted
from funding by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany ’ s Excellence
Strategy, EXC 2008/1-390540038, and by the DFG collabo-
rative research center CRC1349 ( “ Fluoro-speci ﬁ c interactions ” ,
Chemical Reviews pubs.acs.org/CR Review
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Chem. Rev. 2020, 120, 7065 − 7103
7096

Project 387284271). Computational resources were provided
by the CESNET (Grant No. LM2015042), the CERIT Scienti ﬁ c
Cloud (Grant No. LM2015085), and the IT4Innovations
National Supercomputing Center (Grant No. LM2015070).
TABLE OF ABBREVIATIONS
1c one-component
2c two-component
4c four-component
A2C approximate two-component
AO atomic orbital/occupied atomic orbital
AO * vacant atomic orbital
BP Breit-Pauli
BPPT Breit-Pauli perturbation theory
CASSCF complete active space self-consistent ﬁ eld
CCSD(T) c o u p l e d - c l u s t e r w i t h s i n g l e a n d d o u b l e a n d
perturbative triple excitations
CCSDT co up led -c lus ter w it h sing l e, do uble , an d tri p le
excitations
CGO common gauge origin
DFT density-functional theory
DHK Douglas − Kroll − Hess
DI delocalization index
ECP e ﬀ ective core potential
EDA energy decomposition analysis
EPR electron paramagnetic resonance
FC Fermi-contact
GIAO gauge-including atomic orbitals
HA heavy atom
HAHA heavy-atom on the heavy-atom e ﬀ ect
HALA heavy-atom on the light-atom e ﬀ ect
HB hydrogen bond
HF Hartree − Fock
HOMO highest occupied molecular orbital
IHD inverse halogen dependence
KS Kohn − Sham
LA light atom
LP lone electron pair
LUMO lowest unoccupied molecular orbital
Me methyl
MF magnetic ﬁ eld (external, B 0 )
MO molecular orbital/occupied molecular orbital
MO * vacant molecular orbital
MS molecular spinor
MSP molecular spinor pair
NBO natural bonding orbital
Ng noble gas
NHC N -heterocyclic carbene
NHD normal halogen dependence
NMR nuclear magnetic resonance
NOCV natural orbitals for chemical valence
NR non-relativistic
Ph phenyl
PSO paramagnetic spin − orbit
PT1 ﬁ rst-order perturbation theory
PT2 second-order perturbation-theory
PT3 third-order perturbation-theory
PTE Periodic Table of Elements
Py pyridine
QTAIM quantum theory of atoms in molecules
RHS right-hand-side
SD spin-dipolar
SFR spin-free relativity
SHE super-heavy elements
SO spin − orbit
SO/FC spin − orbit/Fermi-contact term
SO/FC ∇ spin − orbit/Fermi-contact term involving one MO
and two MO * s
SO/FC Δ spin − orbit/Fermi-contact term involving two MOs
and one MO *
SOC spin − orbit coupling
SO-EDD spin − orbit induced electron deformation density
SO-HALA spin − orbit heavy-atom e ﬀ ect on the light-atom
(NMR shift)
SOM-ISD spin − orbit-and-magnetically induced spin density
SR scalar relativity/scalar relativistic
TL trans-ligand
TLI trans-ligand in ﬂ uence
X2C exact two-component
XB halogen bond
Z atomic number
ZORA zeroth-order regular approximation
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