Relativistic Effects on NMR Chemical Shifts

552

P. Schwerdtfeger (Editor) Relativistic Electronic Structure Theory, Part 2: Applications Theoretical and Computational Chemistry, Vol. 14 9 2004 Elsevier B.V. All rights reserved.

Chapter 9

Relativistic Effects on N M R C h e m i c a l Shifts M. Kaupp Institut ftir Anorganische Chemie, Universit~it Wiirzburg, A m Hubland, D - 9 7 0 7 4 Wiirzburg, G e r m a n y

ABSTRACT Our current understanding of relativistic effects on NMR chemical shifts is analyzed. After briefly reviewing four-, two-, and one-component quantum chemical formalisms to relativistically compute chemical shifts, a main focus of the article is placed on understanding spin-orbit effects that dominate the important ,,heavy-atom effects on shieldings of light atoms" (HALA). Many computational examples are provided, and an analogy between ,,spinorbit chemical shifts" and the FC mechanism of indirect spin-spin coupling is emphasized. This leads to a natural understanding of many periodic trends in chemical shifts, such as ,,normal" or ,,inverse" halogen dependence, or of the large sensitivity of hydrogen shifts to spin-orbit effects. Both shielding and deshielding spin-orbit effects may be understood, and long-range spin-orbit contributions follow the same pathways as spin-spin coupling constants. Scalar relativistic effects are important for both chemical shifts of nuclei neighboring heavy atoms, and for the heavy nuclei themselves. The dominant relativistic contributions to ,,heavyatom effects on the shielding of the heavy atom itself' (HAHA) appear to arise from an interesting cross term between kinematic corrections to the nuclear spin Zeeman term and the Fermi-contact operator. Most of the HAHA effects tend to cancel in relative chemical shifts and are thus less easily observed experimentally than the HALA effects.

553 1. INTRODUCTION

Few physico-chemical properties of atoms, molecules or solids are as intrinsically connected to relativistic quantum mechanics as the parameters of magnetic resonance spectroscopy. This has to do with two facts: a) Often the spatial regions close to the nuclei are involved, b) Often electronic spin is a basic variable. This brings automatically spin-orbit coupling into play, sometimes induced by the magnetic fields. The quest for a deeper understanding and interpretation of magnetic resonance spectra thus leads naturally into relativistic theory. As a historically interesting aspect, it may be noted that Pekka Pyykk6 got initially drawn into relativistic quanttma mechanics when trying to understand NMR spectra. He subsequently shared a large part of the responsibility in making chemists aware of the importance of relativistic effects [1]. Today, the role relativistic effects play for NMR and EPR parameters has been appreciated to very different extents for different properties and by different communities of experimentalists and theoreticians. For example, it has been known early on in the EPR community that the electronic g-tensors of EPR spectroscopy are basically dominated by spin-orbit coupling and are thus intrinsically relativistic [2]. On the other hand, in spite of much early work on relativistic theories of NMR chemical shifts, and much associated recent computational developments and applications [3,4,5,6,7], most users of NMR spectroscopy still seem largely unaware of the important role of relativistic effects. This holds in particular for the role of spin-orbit effects, in what is often simply called 'heavy-atom effects" on NMR chemical shifts. This can be seen easily when inspecting most NMR textbooks and much of the research literature. In this article, we will concentrate on NMR chemical shifts, for which the importance of spin-flee (scalar) relativistic (SFR) and spin-orbit (SO) contributions needs to be better appreciated. Reviews of relativistic calculations of spin-spin coupling constants are available [6,7]. Articles on contemporary quantum chemical calculations of electronic g-tensors have been published elsewhere [5,8]. Other EPR parameters like zero-field splittings and hyperfme coupling constants are also strongly affected by relativity and are covered,

554 together with NMR parameters, in several reviews in an upcoming book [9]. Nuclear quadrupole coupling constants may also feature large relativistic contributions and are covered comprehensively in another chapter in this book [ 10], and elsewhere (P. Schwerdtfeger, M. Pempointner, W. Nazarewicz in [9]). The treatment in this article will not be comprehensive regarding the theoretical formalism or the computational methods. We will try to provide enough theory to enable the reader to appreciate the role various relativistic contributions to NMR parameters have on spectroscopic properties. A major goal of this paper is to emphasize cases in which a relativistic discussion is mandatory for an understanding of the observed chemical shifts. Regarding reviews of nonrelativistic ab initio methodology of NMR parameters, the reader is referred to [11], whereas DFT approaches have been reviewed by several authors, including various aspects of relativity [3,4,5,6,7]. Various chapters on relativistic formalism will appear in [9]. A series of annual reviews of physical and theoretical aspects of chemical shieldings also accounts for relativistic calculations [12]. The setup of this article is as follows: In section 2, the theoretical background is reviewed, section 3 discusses the extensive field of SO effects on shieldings of 'light" neighbor atoms, section 4 SFR effects on neighbor atoms, and section 5 relativistic effects on heavy-atom shieldings. Concluding remarks are provided in section 6.

2. T H E O R E T I C A L B A C K G R O U N D 2.1 Nonrelativistic Theory

Nuclear shielding reflects changes to the nuclear Zeeman splitting by an external magnetic field and is phenomenologically defined by eq. 1, where the effective magnetic field B ~ff at the position of the nucleus in question is modified relative to the external field B ~ by the shielding constant (tensor) c. n eff

=

B~

- ~)

(1)

From a perturbation theory point of view, the cartesian tensor components of c are derived as:

555

crN.,v

. . .

d2E d/~N.u~v,U,V ,

x,y,z

(2)

where E is the total energy of the system, and PN.u and Bv are cartesian components of the magnetic moment of nucleus N and of the external magnetic field, respectively. Chemical shifts, measured experimentally, are defined with respect to the nuclear shielding of a reference compound, ~ref. = (l~ref'G')/(1-~ref) ~'

~ref'G

(for O'ref<
(3)

Isotropic nuclear shieldings are defined as: cr~o = 1/3 Tr(CtN)

(4)

and an analogous equation holds for chemical shifts. The suitable nonrelativistic starting point for understanding NMR chemical shifts is Ramsey's equation (5) [13]. o = o d+ o p

(5)

Ramsey based his theory on the Pauli Hamiltonian, treated eq. 2 by RayleighSchrSdinger second-order perturbation theory, and neglected SO coupling and any other relativistic corrections. A component of the diamagnetic shielding tensor o a is given as (in SI-based atomic units, used also throughout this paper)

(r'~176

o-~.~a=_~c2 1 (W0]~,

(6)

where r~N is the position vector of the electron relative to the nucleus, and r~o is the position vector of the electron relative to the gauge origin. A u, v cartesian component of the paramagnetic shielding tensor oP reads

crN'u=v 2c2 nr (VnlL~176176 E0-1En p

1 ~

....

(7)

In this second-order expression, W0 and W, are the many-electron wavefunctions of ground and n th excited singlet states (we will deal here exclusively with formally closed-shell molecules), respectively, and E0 and lEa provide the corresponding total energies. The two matrix elements in the numerators of eq. 7

556 reflect interactions with a) the external magnetic field ('larbital Zeeman term" OZ),

and

with

b)

the

magnetic

moment

of the nucleus

in question

(']3aramagnetic spin-orbit term" -PSO). Except for proton shieldings, it is usually assumed that changes in t~p dominate relative chemical shifts 8 for different chemical environment of nucleus N (however, see below for spin-orbit effects). Equations 6 and 7 have been formulated for a common gauge origin at the nucleus of interest and need to be modified for methods with distributed gauge origins such as GIAO ('~gauge-including-atomic orbitals" [14]) or IGLO ('2qdividual-gauge-for-localized orbitals" [15]). This means also that erd and erp are not individually gauge invariant, and care has to be taken in their separate interpretation. Historically, the first attempts to extend Ramsey's equation to take into account relativistic effects were by perturbation theoretical inclusion of SO contributions [16]. This leads to a third-order perturbation treatment that lends itself very nicely to a detailed interpretation of SO effects on chemical shifts. We will come back to the third-order perturbation theoretical a n s a t z in section 2.4. Before, we will concentrate on approaches, in which SO coupling is included in the zeroth-order Hamiltonian, in a relativistic f o u r - o r

two-

component framework. This is probably the preferred framework from a purely theoretical point of view, and it is most appropriate when relativistic effects are large, i.e. for systems containing very heavy atoms.

2.2 Relativistic Four-Component Methodology At a four-component level, several groups derived independently and almost simultaneously relativistic variants of Ramsey's equation within the framework of the one-electron Dirac equation [ 17,18,19]. The perturbing Hamiltonian, H'N, reads c

,4

"

(8)

It is linear in the magnetic vector potential .g,, which incorporates the vector potential for both the external magnetic field and the nuclear magnetic dipole; is the vector containing the Dirac matrices. Inserting this into eq. 2, and using a

557 Dirac Hamiltonian and 4-spinor wavefunctions, one arrives at an expression that is similar to that for t~p (eq. 7) in the nonrelativistic treatment but that incorporates all relevant relativistic corrections (e.g. those related to hyperfme interactions, see below). A relativistic analogue to r d does not arise in this framework. It may be shown that instead, part of the '~aramagnetic" term involves summation over positronic states, which then accounts for the 'ttiamagnetic" parts [ 20]. An explicit summation over positronic states can be avoided by introducing the external magnetic field by finite perturbation theory, as in [21 ]. Pyper [18] proposed a Gordon decomposition that recovers the terms of the nonrelativistic theory. In a careful analysis, Aucar et al. showed more recently that the 'ttiamagnetic" oontributions are due to a '~'edressin g" of the electronic ground state by the magnetic fields [22]. In a series of approximations, the expectation value of nonrelativistic theory can be reobtained (see also ref. [23] for a recent alternative procedure). Moreover, within relativistic random-phase approximation (RPA) the summation over positronic states can be avoided by introducing a 'ttiamagnetic approximation"[ 22] (see [24] for a discussion of the limitations of this approach). Most recently, Kutzelnigg has shown in detail [25] that the diamagnetic and paramagnetic contributions to magnetic susceptibilities (and thus analogously to nuclear shielding) arise naturally also in a 4-component framework, when an appropriate unitary transformation of the Dirac operator in a magnetic field is used. Kutzelnigg recommended a combination of this unitary transformation with the Gordon decomposition of the induced current density and discouraged the interpretation or computation of diamagnetic contributions in terms of negativeenergy states [25]. The first numerical calculations within a four-component framework were done at the semi-empirical relativistic extended-Hiickel (REX) level [26]. This provided the first qualitative insights into SO contributions to the proton shieldings in the hydrogen halides and related cases (cf. section 3). Attempts to obtain more quantitative results include the finite-perturbation Dirac-Fock implementation of Ishikawa, Nakatsuji and coworkers [21 ], and the more recent Dirac-Fock implementation of Visscher et al. [27]. The calculations of ref. [21] used basis sets too small to provide useful numerical results (later calculations employed improved basis sets [28]). Both methods were applied mainly to

558 atoms and to the hydrogen halide molecules, with common gauge origin at the halogen. In ref. [27], reasonable agreement with SO-corrected nonrelativistic RHF results was found. The method of Visscher et al. uses relativistic RPA (random phase approximation, equivalent to a coupled-perturbed Hartree-Fock scheme) and employed the abovementioned 'ttiamagnetic approximation". Another four-component implementation due to Quiney et al. is based on an approximate sum-over-states formalism and has up to now been tested for the water molecule only [29]. The four-component approaches are important as benchmark methods, but computationally they tend to become exceedingly expensive for larger molecules. Only systems with one heavy atom and a few hydrogen atoms have been treated as yet at the Dirac-Fock level [21,27,30]. Consequently, no four-component ab initio calculations of magnetic resonance parameters with explicit inclusion of electron correlation have as yet been reported.

2.3 Relativistic Two-Component Hamiltonians Visscher pointed out correctly [27] that, as one tries to reduce the complexity of the zeroth-order wavefunction and thus the computational effort of its calculation, the formalism of the magnetic resonance parameters tends to become more complicated (see also [31]). The first step on this way are twocomponent approaches, in which the positronic degrees of freedom in the wavefunction have been eliminated (transformed into operator form), but SO coupling is still included in the ground-state Hamiltonian. Then second-order perturbation theory may be used to calculate nuclear shieldings but to nevertheless include SO effects. As has been pointed out early on in the context of hyperfine coupling [32,33,34], this should be done properly using only nonsingular SO and hyperfme operators, which have undergone the same transformation to a two-component picture as the wavefunction (the '~ictt~e change effect'). Examples, in which Breit-Pauli SO operators and nonrelativistic hyperfine operators have been used with two-component Douglas-Kroll-Hess transformed waveffmctions [35] are not internally consistent and have been criticized [36]. In another implementation by Schreckenbach and Ziegler [37], the complete Breit-Pauli (BP) Hamiltonian has been used, initially in a scalar relativistic approximation, later augmented by SO coupling in a two-component

559 approach [38]. Variational collapse has been avoided here by using relativistic frozen-core shells. In [39],

'hctive" SFR contributions (cf. discussion in 2.4)

were also implemented. Probably the most complete two-component treatment currently

available

is within the

framework

of the

zero-order

regular

approximation (ZORA) [40] as implemented at the DFT level in the ADF program [41]. The method does not require frozen cores, and it incorporates both SFR and SO effects in a two-component framework. Interestingly, a scalar relativistic DFT-ZORA implementation (i.e. as yet without SO coupling) of chemical shifts has recently been reported even within the framework of an augmented plane-wave basis for extended solids [42]. See [43] for a recent Hartree-Fock ZORA implementation. Still effectively two-component in nature is the SO-UHF approach of Nakatsuji et al. [44], in which both external magnetic field and SO coupling are included in the zeroth-order UHF wavefunction (which consists then of complex molecular orbitals) within a finite perturbation theory (FPT) ansatz. Apart from the lack of electron correlation and the use of common gauge origins (and of frequently too small basis sets), a possible criticism [27a] of the approach employed i__n_itiallyis the use of the strongly singular BP SO operators as a fmite perturbation. The approach was later extended [35] to include SFR effects at the level of the Douglas-Kroll-Hess (DKH [45]) Hamiltonian. The DKH Hamiltonian involves somewhat more complicated operators than ZORA if implemented consistently (and ZORA integrals are more suitable for numerical techniques). As pointed out above, the initial implementations used typically SO and magnetic operators that were not properly transformed to the DKH framework and were thus inconsistent. Recently, in two steps most of these deficiencies were removed: a 'l'nagnetic interaction term" correction was added [46,47] (which means that the magnetic peturbation operators were transformed to the DKH picture), at least the one-electron part of the SO Hamiltonian was transformed to the DKH level (the two-electron SO terms appear to be still at BP level) [28], and a fmite Gaussian nuclear model was added. Moreover, a GIAO treatment was finally added to account for the gauge-origin problem [48]. Methods based on DKH are currently lagging somewhat behind ZORA implementations in terms of completeness, in particular as no full treatment at correlated levels is available so far. Note that core-orbital energies within a

560 DKH framework should be a closer approximation to the 4-component picture than within a ZORA framework [49]. In the long run, the DKH approach may be expected to go a somewhat longer way, in particular when higher-order relativistic contributions are considered variationally. 'Picture change effects" [50], i.e. the difference between using nonrelativistic and properly transformed relativistic operators, were found to be important in DKH-based calculations, both for the valence- and core-orbital contributions to heavy-atom shifts [28,48]. To date, the majority of calculations of NMR chemical shit, s of heavy nuclei have been performed at some level of the UHF-DKH treatment, at the 'tluasirelativistic" DFT level (BP Hamiltonian, frozen cores), or at the ZORADFT level. Theoretical considerations leads one to the conclusion that the ZORA approach should be superior to the frozen-core Pauli treatment. It appears that the main disadvantage of the latter is the fact that increased basis sets led eventually to a deterioration of results, due to variational collapse [39]. Thus, with modest basis sets the results of the two approaches were not dramatically different, but within the ZORA framework the agreement with experiment could be enhanced significantly by enlargening the basis sets. In particular, the larger basis sets enhanced the treatment of the core wiggles of the valence orbitals [39], which is very important to describe properly both the interaction to the nuclear magnetic moment and the spin-orbit effects (both terms weight strongly the parts of the valence orbitals close to the nucleus). Cases with significant discrepancies between frozen-core-Pauli and ZORA treatment are provided by a recent study of uranium shiits [51,52]. Due to the absence of experimental data, the study could not establish beyond doubt which of the methods performed better. For both methods, gauge-including atomic orbital (GIAO [14]) methods have been used to deal with the problem of the gauge origin of the magnetic vector potential. We will come back to some of the interpretations at these levels further below.

2.4 Perturbational Treatment of Relativistic Effects

The next step on the way towards the nonrelativistic limit is to treat SO coupling as a perturbation, based on nonrelativistic or scalar relativistic onecomponent wavefunctions (SFR effects may be included for example by ECP

561 approaches or perturbationally, see below). It is remarkable that already in the late 1960's Nakagawa et al. [ 16] worked out the basic treatment of the thirdorder SO effects that dominate the relativistic contributions to neighbor-atom shieldings (cf. below). They used the theory to qualitatively explain heavy-atom effects on carbon and hydrogen shifts in halogen-substituted aromatic compounds. The first implementations into computer programs were done within semi-empirical MO schemes [53,54,55] using a number of sometimes serious approximations of both matrix elements and energy denominators (interestingly, solid-state physicists later rediscovered the SO effects [56] and used them in a semi-empirical framework to explain the 2~ chemical shifts in PbTe semiconductor [57]). However, already Cheremisin and Schastnev in their INDO-level studies [55] used a semi-empirical variant of the GIAO approach, included both FC and SD terms, and obtained reasonable semi-quantitative results. Implementation can either be done purely analytically, based on quadratic response formalisms, or some terms may be included by finite perturbation theory. Before we mention some of the more recent implementations, the basics of the formalism should be outlined. More detailed accounts of the (Breit-)Pauli-based perttu'bational approach to relativistic effects on NMR parameters are provided elsewhere [58] (see also [23,59,60]). This includes SO and SFR effects, but also spin-dependent 'laon-SO" oontributions, as discussed further below. Let us start with the field-free SO effects. Perturbation by SO coupling mixes some triplet character into the formally closed-shell ground-state wavefunction. Therefore, electronic spin has to be dealt with as a fi~her degree of freedom. This leads to hyperfine interactions between electronic and nuclear spins, in a BP framework expressed as Fermi-contact (FC) and spin-dipolar (SD) terms (in other quasirelativistic frameworks, the hyperfme terms may be contained in a single operator, see e.g. [34,40,39]). Thus, in addition to the firstorder c d and second-order ~P at the nonrelativistic level (eqs. 5-7), third-order contributions c s~ to nuclear shielding (8) arise, that couple the one- and twoelectron SO operators (9) and (10) to the FC and SD Hamiltonians (11) and (12), respectively. Throughout this article, we will follow the notation introduced in [58,61,62], where these spin-orbit shielding contributions were denoted t~s~

562 aSO-I N,,uv

= trFC(1)-I ,,,.FC(2)-I .,,.SD(1)-I trSD(2).-I V N, uv + O N, uv + U N.uv + V N, uv

HSO(1) ~.

..-I~NZNLiNu

(8)

(9)

~

riN

HSO,,,

(10) \j~,

r~

..

=-Tr(r~)

(11)

HsD _ (r,Nr~8,v - 3r,u,r~v.) Nl)uv -5 riN

(12)

This leads to third-order perturbation expressions like [62]

_

o'

o';~'-" - 0P,.,0B0.,

[

> .Eo

(E 0-'Eo)(E0-' Eo )

for any combination of H s~ or H s~ with H K = H vc or H sD (for example, o.~co)-~ couples Hso(I) and HFC to contribute to the shielding of nucleus N; the appropriate cartesian components u, v of the perturbation operators have been assumed and are left out for simplicity). As the latter two operators and H s~ involve electronic spin, at least one of the two intermediate-state wavefunctions involved is now a triplet state. In the particular permutation shown, n is a singlet excited state, and m is a triplet excited state. The other permutations indicated but not shown explicitly in eq.

13 feature different orders of operator

components and wavefunctions. Among the terms in eq. 8, the or,,,., - ~ " ) - ' term is generally the by far most important SO contribution to the shielding of light nuclei in the vicinity of heavy atoms (heavy-atom effect on the light atom, '~IALA" [ 26]), to some extent diminished by the rr ~(')-' term. The o'~('>-'and ~ N )uv ~ N ,uv O'~,~:'-' terms are typically of lesser importance for light-atom shieldings but may contribute to the heavy-atom shieldings. In the case of ~H shielding in hydrogen halides, the o'~(~>-'and o"~>-'have been found to be of similar N ,uv N ,uv magnitude and opposite sign, thus cancelling each other to a large extent. In contrast, for ol3c in the methyl halides, the two terms had the same sign and thus were not altogether negligible anymore [61,62].

563 As pointed out already by Cheremisin and Schastnev [55] and emphasized again recently by Fukui et al. [63], a gauge-invariant perturbational inclusion of SO effects requires also to take into account the field dependence of the SO operators, leading to the following second-order contributions (FFC0)-n .~_ ._ec(2-n) Jr,,. --N,,. ON~ + -u

~

v

i~" SD(I-U) - - i V ,UV

+0"

sD(2)-n

(14)

JV ,uv

that may be expressed as SO-I1-K__ ~2 I~o(O[l'IKlnr)(nrllts~176176 crr.~ -~t.tr.u~Bo.~ (E 0 - E ~ )

in which H K represents again the FC or SD operators, HS~176

the field-

dependent one- or two-electron SO operators (see expressions in [58,62,63]), and the summation rims over triplet excited states. While these second-order terms have been found to be vanishingly small for shieldings of hydrogen near heavy halogen or chalcogen atoms, they are as important as the third-order SO contributions when discussing heavy-atom shieldings [62] (the heavy-atom effect on the heavy a t o m - HAHA [64]; note that for spherical closed-shell atoms the third-order terms vanish and the second-order terms are the only remaining SO contributions to nuclear shieldings [62]). The recent further development of the BP-based perturbation formalism has uncovered even more relativistic contributions [23,27a,58,59,60,63]. Some of these have been implemented and evaluated numerically [23,27a,58,59,60], whereas others still lack numerical estimates. Some of the terms arise from SO coupling, some are SFR terms, but interestingly others may not be classified that easily, as they are spin-dependent but do not necessarily require SO operators in their derivation. It turned out that the most important cross term of this type (eq. 16) is an isotropic contribution arising from the coupling of the relativistic kinetic energy (KE) correction to the spin-Zeeman term (SZ-KE term, eq. 17, related to a first-order contribution to the electronic g-tensor [2]) to the FC operator in second order (eq. 16). This seems to be the largest individual contribution to the HAHA effect (cf. section 5). This contribution to heavy-atom shielding has received various labels, such as FC(IV) [63], MVEF-FC [27a], or FC/SZ-KE [58]. We will adher to the latter notation throughout this article.

564 FC

02

FC / SZ-KE

In)utnl"s:-lO)v

+ C.C.

O'uv

@K,uOBo,v

(E o - E nr )

(16)

with HSZ_KE _

Bo,u

1

4C2

geESiu

V2 ,

i

I.

(17)

In [57], a classification of the individual relativistic corrections into 'hctive" and ']3assive" operators has been proposed (similar to the classification into 'tlirect" or '~ndirect" effects within a scalar relativistic ZORA framework in [39]). Active relativistic operators are those that carry a dependence on the magnetic fields (e.g. the field-dependent SO operators in eq. 15 or the SZ-KE term in eq. 17), whereas passive relativistic operators are field independent and act via a change of the wavefunction of the system (as, e.g., the field-flee SO operators in eqs. 9 and 10 entering the SO-I contributions), whereas the additional contributions from the fields are in their nonrelativistic form. Let us now come to some of the more recent implementations of the pertm'bation formalism into computer codes. The first implementation of the o s~ contributions at DFT level by Malkin et al. [65,66] included H l~c by FPT (neglecting It sD) and evaluated the H s~ and H B0 contributions by finite difference of second-order expressions for the remaining perturbations, based on the Kohn-Sham orbitals spin-polarized by the FC tenn. While the initial implementation [66] included only H s~176and thus evaluated o~.., -~")-' , the method was later extended to include the two-electron SO terms either explicitly at the Breit-Pauli level, in an all-electron atomic meanfield approximation [67], or within the framework of SO pseudopotentials (effective-core potentials, ECPs) [68,69] 9 The second-order ,,vc(,~ ~ term was also explicitly considered in [69] V N,uv Inclusion of the FC operator by FPT avoids complex orbitals that arise, e.g., in the FPT-UHF treatment of Nakatsuji et al., who include H s~ and HB0within an FPT framework [44] (cf. 2.3), but it is more difficult to include the SD terms.

565 Both approaches share the properties of an FPT approach, i.e. the fact that the numerical accuracy has to be controlled carefully. This is avoided by a completely analytical but computationally more demanding approach like the quadratic response treatment of Vaara et al. at RHF and MCSCF levels of theory [61,62] (see also [27a]). Regardless of the actual implementation, the third-order perturbation theory framework will serve us in the following as a basis to qualitatively interpret SO-I 'heavy-atom" effects on chemical shifts. We note in passing, that the contributions from the H s~176operator (eq. 9), which arise from the coupling of electronic spin with the orbital momentum of the same electron under the influence of the potential of the naked nuclear charge of the heavy atom, are to some extent diminished by contributions from H s~ (eq. 10), which are smaller and have opposite sign. The first term in eq. 10 corresponds to the classical shielding of nuclear charge by the core electrons. The second of the two-electron terms is the nonclassical, so-called %pin-other-orbit" term, which arises from the relativistic Breit correction to electron-electron interactions. Notably, the abovementioned approaches deal with these terms in different ways and to different extents. This is less important for systems, where the SO contributions come from very heavy atoms, as the importance of the twoelectron terms relative to the one-electron terms is typically small. For lighter atoms the SO effects are overall smaller, but the relative importance of the twoelectron terms is larger. For example, the tsFc(2)-~ contributions to the 1H shieldings in hydrogen halides HX have been calculated to amount to ca.-33%,-20%,-13%, and-7% of tsvc(1)I fox: X = F, C1, Br, I, respectively [68]. The spin-other-orbit term has been found to be important for quantitative calculations of, e.g., g-tensors of organic radicals [70]. Before leaving the theoretical formalism section, it is important to note that perturbation theory for relativistic effects can also be done at the fourcomponent level, i.e. before elimination of the small component by a FoldyWouthuysen (FW) or Douglas-Kroll transformation. This is best done with direct perturbation theory (DPT) [71]. DPT involves a change of metric in the Dirac equation and an expansion of this modified Dirac equation in powers of c 1.

The

four-component

Lrvy-Leblond

equation

is

the

appropriate

nonrelativistic limit. Kutzelnigg [72] has recently worked out in detail the simultaneous DPT for relativistic effects and magnetic fields (both external and

566 nuclear dipole fields). The great advantage of DPT is that the singularities of the FW-transformation are avoided, and operators like the FC operator arise naturally. A first implementation and application to, e.g., hyperfme and electricfield terms of H, H2+, and H2 has been reported very recently [73]. Undoubtedly, the DPT treatment is an interesting alternative to numerically less stable approaches, and it provides unique insights into the origins of certain operators in the more traditional treatments. Currently, DPT implementations of NMR parameter calculations are lacking, probably due to the relatively complicated formalism involved when going to higher orders of perturbation theory.

3. SO EFFECTS ON NUCLEAR SHIELDINGS OF NEIGHBOR ATOMS

We will now attempt to extract qualitative models for the chemical interpretation of NMR chemical shifts from the admittedly sometimes quite complicated theoretical formalisms outlined above. And we will provide examples of explicit quantum chemical calculations that corroborate the proposed models. The most widely studied field, in which much insight has been obtained recently, are SO effects of atoms neighboring some 'heavy-atom SO center". A significantly revised view of 'heavy-atom effects" on chemical shifts has arisen from the theoretical studies. Let us thus first concentrate on these HALA SO effects that heavy-atom substituents have on the shieldings of other atoms in the system. HALA effects may also be important for heavy nuclei neighboring other heavy atoms, but then things may be complicated by HAHA effects (cf. section 5). As mentioned already in 2.4, in the 'heighbor-atom" HALA case the third-order SO effects dominate by far, sometimes augmented by SFR effects (see section 4). Interestingly, the origin of the relativistic effects is now near the nucleus of one heavy atom, whereas the consequences are probed by the chemical shift of a different nucleus! As eq. 13 shows, in a sum-over-states picture the magnitude o f the individual 6 s~

terms depends on various products of three matrix

elements over singlet ground state and two intermediate excited states (which may be one singlet and one triplet, or two triplet states), and on the inverse of the product of the two relevant excitation energies. The main perturbation

567 operators acting together in this framework are H s~ (both one- and two-electron terms, see above), H Fc (H sD is less important), and H B0, which provides the coupling to the external magnetic field. While the situation thus seems quite complicated at first sight, it may be visualized reasonably well by Figure 1, which shows a useful analogy to the FC mechanism of nuclear spin-spin coupling [74].

Soin-Orbit Chemical Shifts 9electronic spin polarization

'~NMR

a)

,,

ucl. B

HB. FC+SD Mechanism of Nuclear Spin-Spin Coupling electronic spin pola(,!,zation

b)

NMRN~

cl.B H Fc+sD

H Fc+sr

nucl. ~

Figure 1. Analogy between third-order SO corrections to chemical shift of neighbor atoms and the (nonrelativistic) FC+SD mechanism of indirect spin-spin coupling. Adapted from [74] with modifications.

This analogy had already been used by Nakagawa et al., prior to the availability of numerical calculations, to rationalize qualitatively heavy-atom effects on nuclear shieldings in halogen-substituted benzenes [16]. More recently, it has been demonstrated in detail by DFT calculations, that the analogy holds, and that it provides remarkable insights into the mechanisms of "SO chemical shiRs" on %pectator NMR atoms" in the vicinity of heavy-atom centers [74]. Let us start on the fight hand side of Figure l a: Spin-orbit coupling (H s~ mixes some

568 triplet character into the closed-shell singlet ground-state wavefunction and thereby spin-polarizes the density distribution in the core and valence shells of the heavy %pin-orbit" atom (interaction with the external magnetic field B0 completes the third-order perturbation mechanism). This spin polarization extends throughout the system, such that nonnegligible spin density arises also near the other atoms of the molecule. This affects now the energies of the nuclear spin states of these atoms (atom B in Figure l a) via hyperfme interactions (mainly via I-Irc, to a lesser extent via I-IS~, thus providing an additional contribution (SO-correction oso-I, eq. 8) to the nuclear shielding ON at a given nucleus N. The left hand side of Figure l a is completely identical to the lett hand side of Figure lb. The latter Figure depicts schematically the well-known (nonrelativistic) FC- and SD-mechanisms of indirect spin-spin coupling, which are completed on the right hand side by a second hyperfine interaction to another nucleus A (orbital terms are not considered here). In this case the system is modified by the additional interactions, so that a splitting of the nuclear energy levels by the indirect spin-spin coupling arises, which becomes visible in the NMR spectrum. The usefulness of the analogy between spin-spin coupling and %pin-orbit chemical shifts" arises from the much larger database available on the former property compared to the latter. The analogy suggests that modifications of molecular or electronic structure near atom B should influence both properties in a comparable way. Both quantitities should also share the same dependence on pathways of spin polarization throughout the molecule. Indeed, similar analogies may be drawn between nuclear spin-spin coupling and EPR hyperfme coupling [75]. The contact shitts of NMR in paramagnetic compounds may be mentioned as a more remotely related property [76]. The analogy of Figure 1 may be illustrated nicely by DFT calculations of o s~ (using the formalism of [66]) for 13C and IH nuclei in iodobenzene [74]. Figure 2 shows both reduced spin-spin coupling constants nKFC(I-C) (n = 1, 2, 3, 4), calculated using the FPT approach of [77], and third-order SO corrections to the carbon shieldings. In contrast to [74], the solid curve in Figure 2 includes both o Fc(1)~ and G'FC(2)-I in eq. 8 within the atomic meanfield approximation, as described in [68,69].

569 para

40-

~

meta

~",-~

ortho "--+L"-"I,,j -

~',

30"

-400

-~[~

KFc(C,I )

. . ~ lpso

-300

~"

I ~,,L.

* w,,,t

20

9 D 10"

-200

=

-100

Z>

r/2

O" (i.+2).7I.......... ~ . . ................................ ~ --- . .. : .- --~i- ....... ............ :* ......... I

Cipso

I

Cortho

I

Cmeta

0

I

Cpara

2. Demonstration of the analogy in Fig. 1 by computed reduced coupling constants K(C,I) and a s~ corrections to 13C shieldings in iodobenzene. Reduced coupling constants taken from [74], which provided c rc0)I corrections to shieldings. Here, the complete o vcO+2)I corrections are provided. They have been computed within the atomic meanfield approximation, as in [68,69] (other parameters as in [74]).

Figure

The reduced

coupling constants

K Fc follow the expected

'tlamped

oscillation" behavior known for spin-spin coupling constants in substituted benzenes. Thus, a large negative 1K(I-Cipso) contrasts to a small positive 2K(ICo~tho), and the couplings to Cmeta or Cpara are already too small to extract reliably their signs from the calculations. Looking now at the curve for os~

we see

that the 'SO shieldings" follow exactly the same pattern: A large shielding effect of ca. 33 ppm for Cip~o contrasts with a small deshielding (-2 ppm) for Cortho, and very small effects arise for Cmeta or Cpara. This remarkable result indicates that the transfer of the SO-induced spin polarization from the heavy iodine substituent to the carbon nuclei follows exactly the same pathways as we know them for the FC mechanism of spin-spin coupling. Correction of

570 nonrelativistic shieldings by o s~ and comparison with experiment shows that the inclusion of SO effects is mandatory to obtain reasonable agreement for C~pso, whereas SO-corrections are already quite small for the other carbon nuclei [74]. Notably, however, the crs~ contributions have been found absolutely necessary to describe the pattern of 1H shifts for Hortho, Hmeta, Hpara [74]. This indicates a particularly large dependence of proton shifts on SO effects, which will be explained timber below. Further numerical checks for the validity of the analogy shown in Figure 1 have included the discovery of a Karplus-type dependence of the o s~ contribution to the 13-~H shifts in iodoethane on the I-C%CILH dihedral angle [74].

o.FC(1)-I - -700

70

-650

65 s S

60

KFC ( c , i)

-600

, s

55

s S

-55o , ,

..= 50 L)

-500 s ~

45

s ~ s

9"T r~

40

~0 ~ g o

35 30

9

|-, 7.",,

-450

,' ,"

-400

,'"

.l"

-350

~

o.FC( 1+2)-I

-300

>

t~

~'

-250

25 20

~"

9 I

H3C-CH2-I ,,sp 3''

,

, I

H2C=CH-I ,,sp 2''

I

-200

HC-C-I ,~p"

Figure 3. Increase of nK(C,I) and of O"SO'I corrections to 13C(1) shieldings with formal hybridization of C(1). Reduced coupling constants and o Fc(l)I corrections to shieldings taken from [74]. The complete o Fc(l+2)I corrections have been added. They have been computed within the atomic meanfield approximation, as in [68,69] (other parameters as in [74]).

Probably the most important consequence of the analogy is, however, illustrated by Figure 3" As is well known for spin-spin couplings or EPR

571 hyperfine couplings, the dominant FC mechanism operates via the spherical parts of the density around the nucleus in question. It requires thus spin density in s-type valence or core orbitals at this atom. Therefore, large spin-spin coupling constants are found in particular, when the s-orbitals on both centers are involved significantly in bonding (or when they become spin-polarized to a significant extent by secondary interactions [78]), and hyperfme coupling constants also tend to increase with increasing s-character of the singly occupied orbitals. Figure 3 shows clearly that not only the magnitude of the computed reduced coupling constants 1K(C,I) but also the 3rd-order SO shifts of the orcarbon increase considerably from sp 3 in iodoethane to sp 2 in iodoethylene to sp in iodoacetylene [74]. These fmdings are the basis for an important, generally applicable rule: significant third-order SO-shifts on atoms in the vicinity of heavy 'SO atoms" require that the NMR atom under observation uses significant s-character in its bonds. This is particularly obvious when the bond is directly to the heavy substituent atom that carries the SO-induced spin polarization. But it holds also when two or more bonds separate the NMR atom and the '~SO atom". When there is no s-character, there are no SO shitts! This holds even when the NMR atom is connected to more than one heavy '5;0 atom" which otherwise leads to particularly large SO shifts. A whole range of consequences arises from this simple rule, which allow many qualitative predictions to be made about the expected magnitude of SO contributions to chemical shifts in different regions of the periodic table and in different bonding situations. For example, the observation of 'hormal" or 'inverse halogen dependence" (NHD vs IHD [78]) of NMR chemical shifts throughout the periodic table may be understood to a large extent from this rule. Halogen-substituted main group compounds experience particularly large SO effects, provided that the NMR atom is in its maximum formal oxidation state (e.g. B m, A1m, Ga m, In m, T1m, CrY, siiV, GeiV, SniV, pbiv, pV, Sb v, etc .... ). This holds even more so for hydrogen (see below). Only then the valence s-orbitals of the NMR atom participate fully in bonding to the heavy substituent 'SO atoms" As the SO effects are shielding with halogen substituents (see below for discussion) and increase with increasing atomic number and thus increasing SO coupling constant of the halogen, the overall shielding increases (the shift decreases) from

572 C1 to Br to I. This is the well-known NHD behavior. Indeed, in all the many examples

studied

computationally

up

to

now

(see,

e.g.

[4,5,26,27,38,28,48,53,54,55,62,65,66,68,61,74,77,79,80,81,82,83,84]), NHD in main group chemistry has been found to result from SO shifts.

90 9,..,

~176176

NR

80

70-

ff~' o

....

~176176176176176

d

50

m

40

,~

30

~

20,

~

10

~176

9 99 9

.

.A..... "''~ ." m___ ----.,,,tw.~. t ~ . , ~

j

~

,,.~.+ Fco+2)-I

exp.

-10 i

BF 3

!

Be! 3

!

BBr 3

1, I

BI 3

Figure 4. Comparison of computed and experimental 11B shifts in boron trihalides G FC(I+2)'I SO corrections were computed within the atomic meanfield approximation, using the approach in [68,69] (other parameters as in [79]).

As an example that had not yet been reported, Figure 4 shows computed and experimental '~B shifts in the boron trihalides. Note that in this case, and even more so for the 13C shitts in the isoelectronic trihalomethyl cations, CX3+ (X = F, C1, Br, I) [79], the t~p contributions would favor IHD much more strongly than in saturated compounds, due to large contributions from couplings to low-lying unoccupied MOs. Nevertheless, already from X = C1 to X = Br, NHD is observed, due to the overriding importance of the SO contributions. In case of the trihalomethyl cations, these results showed that previous interpretations based on halogen electronegativity and backbonding [85] do not hold [79]. Further applications include the prediction of a 'l'ecord" low-frequency 31p shift in PI4+, subsequemly confirmed in solid-state NMR studies [81a], and the computationally assisted first identification of the mixed cations PBr3I+ and

573 PBr212+ [81b].

An enormous

(yso-i of ca.-8000 ppm on the Pb shift in the

unknown PbI4 has been predicted by ZORA-DFT calculations [80]. In lower oxidation states, the s-character of p-block main-group elements is concentrated to a large extent in nonbonding orbitals, and the bonds tend to have mostly p-character. This leads to a very inefficient FC mechanism, and thus SO shifts are expected to be small. In this case, the o p contributions (eq. 7) dominate the overall shielding, and IHD may result. This is illustrated in Figure 5, where the 31p shifts of pin halides and of pV oxohalides are compared (relative to [4], the plot has been extended by results including oFC(1+2)']). In the pV oxohalides (as for other pV halogen compounds [81]), SO effects are large and lead to a pronounced NHD behavior. For the pm halides, SO effects are small, and very weak IHD prevails. Experimentally, IHD has also been found for Sn n and Pb u [86] or A11 halides [87], whereas the corresponding Sn TM, Pb TM or A1m halides exhibit NHD [78].

- - = - - exp. ,. A.,

"~ 300m2502oooo 150"

NR

-,v-,

NR+oFC(1)-I

-

NR+ (3" F C ( 1 ) ' I +

~-

pX _ O "FC(2)'I

"

E~ loo-

X=

'"

~

POX 3

50 9~ 0 -5O -100-150

3

~"W f

F

h

ir

Figure 5. Comparison of computed and experimental Sip shifts ill phosphorus(III)trihalides and phosphorus(V)oxytrihalides. The nonrelativistic and o rc(l)I results are as reported in [4], whereas the a Fc(1§ SO corrections were computed within the atomic meanfield approximation, as in [68,69] (other parameters as in [4]).

574 Along the same lines, we may now easily understand why proton chemical shifts are particularly sensitive to SO effects, even when removed by several bonds from the '~;O atom" (see above [74]; we have recently also found remarkably large

1H SO shit's even for methylene protons in amine ligands

coordinated to cobalt, with significant conformational dependencies [88]). Why, for example, are the SO shifts so dramatical for the hydrogen halides? The answer relates to the fact that hydrogen is the atom that uses most predominantly its valence s-orbitals for chemical bonding. This is the perfect prerequisite for an efficient FC mechanism, and particularly large SO effects arise. Other examples with particularly high s-character in bonding would be dicoordinate Hg or Au. While the latter is not a good NMR nucleus, Hg shifts in HgX2 complexes are clearly affected dramatically by SO effects from halogen substituents [40,48,82a,83]. Similarly, cadmium shifts in cadmium iodide species come at relatively high fields [89]. On the other hand, fluorine tends to use very little scharacter in bonding (the s-character is concentrated in nonbonding orbitals).

Consequently, 19F shifts tend to be rather insensitive to o s~ effects [51,52,90]. Similar arguments hold, e.g., for 170 shifts. Indeed, SO effects on oxygen shieldings appear to be small in transition metal d o tetraoxo complexes [37,39,91,92], and even in uranyl complexes [51,52]! The s-character argument may be extended to transition metal chemical shifts: Halides of early transition metals like Ti, Zr, Hf, Nb, Ta, Mo, or W exhibit strong IHD. Calculations give very small SO effects on the metal shifts in

TiX4 and NbX5 [93,94] (and also for tungsten complexes [80,82b]), and the

IHD is due to very large oP contributions (presence of low-lying unoccupied orbitals) that increase for the heavier halogen substituents. Nakatsuji et al. argued that "Soft d-orbitals adsorb the SO effect, leaving only a small net spinorbit effect."[ 94]. A simpler explanation is provided by the very small metal scharacter in the M-X bonds that allows no efficient FC mechanism [93]. Interestingly, NHD takes over again later in the transition metal series [78]. We wondered to what extent SO effects are important in this case. Calculations on complexes [Mn(CO)sX] (X = halogen) gave very small SO contributions to the metal shifts, and the NHD was due to the oP term [95]. Closer analysis indicated that the highest occupied molecular orbitals are Mn-X antibonding and thus have larger coefficients at the metal for the lighter, more electronegative

575 halogens. This leads to enhanced o p (in spite of larger energy denominators in eq. 7) and thus to larger deshielding [95]. Similar conclusions were drawn in a recent DFT study of 195pt chemical shifts in platinum(II)halides and substituted derivatives [96]. The larger crp for the heavier halogens was attributed to better matching of occupied and unoccupied MOs. In this case, SO effects were notable and contributed to the NHD, although the crp trend was the dominant one [96]. This suggests that metal s-character in bonding increases as we move to the right side of the transition metal rows. Indeed, when we finally come to d 1~ systems like the mercury halides, the NHD is again dominated by SO shifts, as in the main group case [38,40,48] (see also above). In an FPT-UHF study, Nakatsuji and coworkers have recently examined why the

13C shifts of halogen-substituteI methanes cn4_nXn (X--Br, I) exhibit

'laonlinear NHD" with increasing n, whereas

the

corresponding

mixed

complexes CYa.nXn (e.g. Y = Br, X = I) show essentially a linear decrease [84]. Figures 6a and 6b show SOS-DFPT-IGLO results for the 13C shifts in the CH4.nln and CBr4_nln series, respectively. They agree substantially better with experiment than the previous UHF results, as the lack of electron correlation in [84] leads to a significant overestimate of the crs~ contributions. The strongly nonlinear trend for CH4_,I~ and the linear trend for CBra_nln are both apparent. Nakatsuji et al. explained the nonlinear dependence of o s~ on n for CH4.nXn by a pulling out of the ,,..spin density cloud in the direction of the electronegative [halogen] atom, and the maximum point of spin density moves out slightly from the carbon nucleus." Figure 7 shows that the above discussion of s-character affords a more transparent explanation. Both the average p/s hybridization ratio on the carbon atom from natural population analysis [97], and the corresponding p/s ratio of the hybrids used for the C-I bonding natural localized MOs (NLMOs [98]) in both series of systems are shown. It is clear that the electronegative iodine substituents in the CHa_nla series lead successively to larger carbon s-character. This is due to an enhancement of hybridization defects [99] on the carbon atom by the iodine substituents [100], related to an increasingly different radial extent of carbon s- and p-orbitals. More specifically, the relative s-character of the C-I bonds is enhanced dramatically with increasing n. The FC mechanism for the transfer of SO-induced spin density to the carbon nucleus is obviously improved with increasing n.

576

,,m

10

r,r

NR

..

9 m 9 ,

III

r,r

El

. . . . . . .

l'"

-5o-, -I00-

''

'

~

=~~'~~.

NR+SO

_ so

rj

,

-200 ~ -25ff

-300" i

I

CH 4

I

CH3I

m

CH2I 2

f

CHI 3

CI 4

b)

150r.~ r.i3

;~

E

NR m

........

m

.........

9

.........

9

.........

9

100 50 O-5o

".

exp

".

NR+SO

-100 r,.)

-150-200 -250

~O

exp "

-300 I

CBr 4

I

I

CBr3I

Figure 6. Comparison of computed SOS-DFPT-IGLO level, a Fc0§

CBr2I2

I

CBrI 3

I

CI4

13C shifts in halogen-substituted methanes. NR results at

SO corrections were computed within the atomic meanfield

approximation, as in [68,69]. Basis sets and functionals as in [79]. Experimental data as cited in [84]. (a) 'Nonlinear NHD" in the CH 4-nln series. (b) 'Linear NHD" in the CBr 4-nIn series. See also [84].

577

This explains readily the nonlinear NHD. In contrast, the hybridization on carbon in the CBr4_~I~ series is affected in a very similar manner by I or Br substituents. In particular, the hybridization ratio of the C-I bond changes very little with increasing n (Figure 7). In consequence, a linear NHD arises, corresponding to essentially additive t~s~

contributions from the halogen

substituents (a similar 'linear" NHD has been computed for the mixed cations PBra_~In§ [81 b]). The enhancement effect of electronegative substituents on the scharacter and thus on oso-i has already been demonstrated earlier by DFT calculations of the 13C shifts in CF3I vs CH3I [74].

5.0 o

MO

4.5 O 4.0 N

9~

3.5

~

3.0

r~

~

. ~ o t a l

~

CBr. nln C=I NLMO 2.5

x-

~ ~ , , _ _ ~ ,,,,"

CBr4_nI n total

.0

,

,

,

,

,

0

1

2

3

4

Figure 7. Average NPA p/s hybridization ratio [97] on carbon and corresponding NPANLMO hybridization ratio [98] of C-I bonds in the CHn-nln and CBr4.nln series.

Chalcogen substituents X = - E R typically give rise to significantly smaller SO effects than their direct neighbor halogens (e.g. -SeR vs -Br o r - T e R vs -I).

Apart from the slightly smaller SO coupling constant, this is mainly due to the fact that only one n-type nonbonding electron pair is left on the heavy atom, in

578 contrast to two for the halogens. Already the early relativistic four-component extended-Hiickel calculations of Pyykk/5 et al. [26] suggested that the n-type lone-pair MOs on the halogen are responsible for the shielding SO effect in hydrogen halides, whereas the H-X ~-bond made a deshielding contribution. The signs have been rationalized based on the phases of the orbitals involved and based on the diagonalization of the SO matrix [26]. The role of the n-type MOs has been confirmed by SO-corrected DFT calculations of 13C (and 19F) chemical shifts in a series of iodine compounds CF3IFn (n = 0, 2, 4, 6) [90] (see below for the closely related discussion of SO effects on shielding anisotropy). Shielding SO effects have also been found for the 13C shi~s in a number of d 6 hexacarbonyl complexes (scalar relativistic DFT calculations with the frozencore Pauli formalism [ 101 ], frozen-core Pauli-DFT calculations with SO effects added [38], and SO-ECP calculations [69] were used). Closer analysis indicated that the t2g HOMOs were mainly responsible for the shielding 6so effects [38]. Indeed, these MOs are mainly of metal d-character but rr-backbonding relative to the M-CO bond. The situation resembles [69] thus very much that with halogen or chalcogen substituents. This leads to the conclusion that 6 s~ coming from occupied MOs with predominantly 7r-character (relative to the bond between heavy atom and NMR atom) on the heavy-atom substituent is shielding, be it due to a heavy transition metal or a main group atom. The computed increase of 6so with increasing nuclear charge along the isoelectronic series from [Hf(CO)6] 2" through [Ir(CO)6] 3+ was attributed to shorter metal-carbon

bonds and to a consequently enhanced FC mechanism [38]. Further, striking examples of shielding SO effects are found for the 1H shitts in d 6 or d 8 transition metal hydrides [102]. A main reason for the pronounced shifts to high fields (low frequencies) are actually 6P contributions due to ring currents in the metal d-shell that are off-center at the position of the metal-bound hydrogen atom [103], as had already been suggested by Buckingham and Stephens [104]. However, for heavier metals like Ir or Rh, where the shifts correspond to the most negative 1H values known for diamagnetic compounds [78], calculations indicate a large shielding c s~ contribution as well [102]. Again, protons are particularly sensitive to SO effects, due to their large s-character in bonding (see above).

579 Deshielding SO effects have also been found in a variety of cases. They appear to always arise from o-type occupied MOs on the heavy-atom substituent, consistent with the early extended-Hiickel-based analysis on HI by Pyykk6 et al. [26]. Examples are 13C shifts a) in HgX-substituted hydrocarbons [69,105], b) in [Au(CO)2] + and [Hg(CO)2] 2+ [38,69], c) in alkylindium(I) compounds [106]. Particularly striking examples are the large high-frequency (de shielding) IH shifts of the metal-bound hydrogen atoms in organomercury hydrides [105]. MO analyses of o s~ in these cases suggest that a Oa-type MO, which provides coupling to/_/ec, is coupled to zc*-type, mainly metal-centered vacant MOs by angular momentum operators as present in H s~ and t t B~ (cf. eq. 15). Notably, the 13C-deshielding due to crs~ is larger in CH3In than in CH3HgX, in spite of the much higher atomic number and SO coupling constant of mercury vs indium (Z = 80 vs Z = 49). This is due to the fact that the Ou-type MO is bonding for the alkylmercury systems but largely nonbonding for the indium(l) alkyls. Very large deshielding SO effects may thus be expected for TI(I) species. We recall also (see above) that computed long-range SO effects on 1H or 13C shifts in iodobenzene o r [~-IH shifts in iodoethane have been found to be shielding or deshielding, depending on position and conformation [74]. A particularly interesting example of long-range, deshielding o s~ is provided by olH of the methoxy hydrogens in [F6U(OCH3)], which was computed at the ZORA-DFT level to be reduced by ca. 7 ppm (averaged over the three methoxy hydrogen atoms) [51]. It should be emphasized that the oso-~ contributions are usually far from isotropic, and that they may therefore affect the overall shielding anisotropies appreciably. This had already been predicted by early relativistic extendedHiickel calculations [26]. More recent, quantitative examples come from both perturbational treatments at MCSCF level and from Dirac-Fock calculations of SO effects on shieldings in the hydrogen halides [27a,61]. The shielding anisotropies of the hydrogen nuclei were affected dramatically, increasingly so with the heavier halogens, thus confirming the earlier predictions. This is due to the fact that the SO contributions affect mainly cr• i.e. the component perpendicular to the H-X bond. As discussed above, this is due to the dominant interaction of the n-type nonbonding MOs centered on iodine with a ~u*-orbital

580 of the HX bond [26], an idea that has been confirmed and extended recently by DFT calculations of t~s~ corrections to 13C (and 19F) shieldings in a series of iodine compounds CF3IFn (n = 0, 2, 4, 6), in which the nonbonding MOs on iodine have been successively ,~removed by oxidation" with increasing n, i.e. with increasing number of fluorine atoms bonded to iodine [90]).

a) ~'~,

(~

~5zz=141.8ppm

J ' ~

833=141.4ppm

b) ~ 522=135.1ppm '"~' /

J

8,''1=147.Oppm

"~F "~-"~

533=70.9ppm

Figure 8. Orientation and magnitude of the computed 13C shielding tensor principal components in CF3IF2. (a) Nonrelativistic result. (b) Result with third-order SO correction added. Adapted from [90].

581 The o s~ contributions are ca. 57, 27, 0, and 0 ppm for n = 0, 2, 4, and 6, respectively.

A reduction of high-field SO shifts account well for the

experimentally observed low-field trend [107] from n = 0 to n -

2 and n = 4.

Together with a detailed MO analysis, these results confirmed nicely [90] the importance of the x-type lone pairs on iodine: CF3I exhibits two x-type and one o-type lone pair, CFalF2 has one x- and one o-type lone pair, and CFaIF4 has only a o-type lone pair left, which makes only a very small SO contribution. Moreover, the calculations showed clearly that the SO contributions arise for the components of the shielding tensor perpendicular to the C-I bond, and for each lone pair also perpendicular to its own orientation, due to coupling in third-order perturbation theory with a o*(C-I) antibonding MO [90]. The change in tensor components and orientation is shown for CFalF2 in Figure 8. We expect that the crs~ contributions arising from heavy chalcogen substituents are also highly asymmetric, due to the presence of only a single free electron pair on the chalcogen, similar to the results for CF3IF2. A dramatic reduction of 199Hg shielding anisotropy in mercury halides by SO contributions from heavy halogens has been computed in DFT-ZORA calculations [83]. The o s~ effects on 13C shit, s in methylmercury complexes CH3HgX were also found to affect

almost exclusively the perpendicular components of the shielding tensor. In this case, the SO effects were de shielding, and the anisotropy was thus increased. The closer analysis of structural effects on o s~ is still in its infancy. Examples include the Karplus-type relation for I3-1H shieldings in iodoethane described above [74], and a similar conformational behavior of the 1H shifts in polyamine alcohol complexes of cobalt(III) [88]. Another example is provided by a comparison of the alp shifts in two Sn-P cage compounds [108], where larger Sn-P-Sn angles lead to larger phosphorus s-character in the P-Sn bonds, and thus to a more effective FC mechanism and larger shielding SO effects due to the free electron pairs on the heavy Sn(II) substituents. In their ZORA-GIAODFT study of 195pt shifts, Gilbert and Ziegler [96] found that SO-effects due to halogen substituents in [PtXE(PMe3)2] complexes were generally larger in the trans than in the cis isomers (Table 1). This was attributed to the shorter Pt-X

bonds (and thus to better transfer of SO-induced spin polarization to the Pt nucleus) in the trans complexes. In all of the cases mentioned here, the structural dependence was clearly connected to H ec. We may well assume, however, that

582 different relative orientation of the orbitals at the heavy substituent atoms, which are relevant for t/so (of eq. 13), could also contribute to structural dependencies of crs~ More investigation is needed in this area.

Table. Relative t~s~ contributions to platinum shieldings in PtX2(NH3)2 a

cis isomer

trans isomer

C1

335

220

Br

497

342

I

710

404

X

~DFT-ZORA results in ppm relative to the computed values for

cis-PtC12(SMe2)2

[96].

In a recent application of DFT to experimentally noted solvent effects on the 13C shifts in iodoalkynes, it was found that charge transfer complexes from DMSO solvent molecules to the iodine substituent enhanced crs~ for the C1 carbon atom [109]. Nevertheless, the overall solvent effects were found to be deshielding, due to changes in crp. Further application of the same method allowed the identification of the first iodo-substituted cumulene in different solvents [ 110]. The t~s~ contributions depend also strongly on bond length. This has a number of consequences. Two recent studies evaluated ro-vibrational effects on the shielding constants in hydrogen halides, based on MCSCF [111] or DFT [112] calculations, respectively, including SO corrections by third-order perturbation theory. For example, in the case of HBr the dependence of the SO correction to shielding on bond length r was just the opposite (more positive with increasing r) than that of cd+~ p (less positive with increasing r). This leads to a U-shaped dependence of the overall shielding fimction, and to unusual rovibrational corrections to shielding. For example, in HBr and HI, SO effects were found to change the sign of the zero-point vibrational contributions to ~, whereas in HC1 they only reduced the absolute value somewhat [111]. SO

583 effects may in this way be expected to be responsible for unusual temperature dependencies of chemical shifts, as well as for unusual (inverse) isotope effects. While the temperature dependence and primary isotope effects are difficult to measure accurately, secondary isotope effects are more easily accessible. Lantto et al. [113] have recently studied computationally ro-vibrational corrections and secondary isotope effects on 13C shieldings in CX2 systems (X = O, S, Se, Te). DFT and MCSCF calculations with third- and second-order SO corrections included were used. The SO corrections (mainly the ~FC(1)-~ term) reduce significantly the nonrelativistic results for the secondary isotope effects on shielding (by ca. 65% for CTe2) and thus bring the computational results into much better agreement with experiment. The temperature dependence of these isotope effects was also influenced significantly by SO contributions, but in this case the experimental data were considered to be somewhat doubtful [113]. More work is clearly needed generally on the interplay between relativistic and ro-vibrational effects on NMR and EPR parameters. Another underdeveloped area of research are cross effects between environmental influences and relativistic contributions (see, e.g., [40] for solvent effects on 199Hg shifts).

4.

SPIN-FREE

RELATIVISTIC

(SFR)

EFFECTS

ON

NUCLEAR

SHIELDINGS OF N E I G H B O R A T O M S

While the treatment of the by far dominant 'passive" part (cf. 2.4) of SFR effects on neighboring atom shieldings is comparatively straightforward, e.g. with ECPs or at ZORA or Douglas-Kroll levels, significantly less work has been invested into their study than into those of SO effects. This may be related to two facts: a) the SFR effects start to become important somewhat lower in the periodic table than SO effects, and b) at least the ']9assive" SFR effects discussed in this section do not involve fundamentally new mechanisms. As the core shells of heavy atoms do not play a role for the shielding of neighboring atoms (see, e.g. [15]), SFR effects on light nuclei in the vicinity of heavy atoms must reflect changes in the valence orbitals. It is of course well known that SFR effects may change the overall electronic structure of heavy-element compounds significantly, at least with 6th period elements. Nonneglible effects may be

584 found even earlier [114]. It should thus not be surprising that SFR effects on nuclear shieldings of neighbor atoms may be significant for 5d or 6p elements and beyond. SFR effects on lighter neighbor atoms may be treated adequately by using a scalar relativistic ECP on the heavy atom (see, e.g. [3,4,69,91,92,105,115]). In contrast, treatment of SFR effects on shieldings of heavy nuclei (discussed in section 5) requires some type of all-electron approach. In a two- or one-component framework, this requires also inclusion of 'hctive" oontributions arising from coupled relativistic-magnetic operators (cf. section 2). While SO effects influence molecular structures significantly only in relatively extreme cases (see, e.g. [114]), SFR effects are usually appreciable for structures when they become effective for shieldings of neighbor atoms. We may thus distinguish a) 'Indirect" SFR effects due to structural changes and b) 'ttirect" SFR effects at a given structure. It has been found that these two contributions can be both significant, and they may either reinforce each other [91,92] or act in opposite directions [105]. The first explicit computational demonstration of SFR effects on shieldings were DFT calculations of 170 shitts in a series of d o transition-metal tetraoxo complexes [91,92], in which results with SFR ECPs on the transition metal were compared with results obtained with nonrelativistic (NR) ECPs. As Figure 9 shows, the suggested SFR effects are almost negligible for the 3d metals, still relatively small for the 4d metals, and significant for the 5d metals (note that SO effects are expected to be small here, due to the small oxygen s-character in bonding, cf. section 3). For example, the relative order of 8170 from [MoO4] 2" to [WO4] 2 is in disagreement with experiment at the NR//NR level (that is, in a nonrelativistic ECP calculation at the nonrelativistically optimized structure) but gets in line with experiment upon inclusion of both direct and indirect SFR effects (i.e. SFR//SFR level). The structural effects and the SFR effects at a given structure both increase the 170 shielding (in particular for the 5d metals), mainly due to an increase of the energy denominator for o p by the relativistically enhanced bonding between the relativistically expanded metal acceptor dorbitals with the oxygen donor orbitals [92]. We note fi~hermore that the SFR effects influence both Gj_ and GII component of the ~70 shielding tensors, in particular for the 5d metals. Interestingly, the changes are such that the absolute

585 value of the shielding anisotropy was lowered in all cases (it became less positive for OSO4, less negative for WO42, and changed sign for the almost isotropic shielding in ReO4-). The relatively large SFR effects in these systems were rationalized as being due to the cooperative effect of both enhanced energy denominators and enhanced charge transfer from metal to oxygen [92]. The same systems have later been studied at different relativistic levels (using either the QR-BP Hamiltonian with frozen cores [37] or the ZORA Hamiltonian [40]), where available [37], with very similar results for the 170 shifts and for the magnitude of the SFR effects.

.~, 1300" 9e q

1200

Q

1100

;>

1000"

''A'

,1,,-i

900-

SFR//NR

800"

1,.

700 9=9 -~

NR//NR ~_

s

, , \'.'/

/:t

V ' " ' "..... "": ' "

600

-exp.

500 SFR//SFR

400 J

" l ....

t

I

i

WO42- MoO42- CRO42- ReO 4- ToO 4-

i

"

i

MnO 4- OsO 4

I

RuO 4

Figure 9. Comparison of computed and experimental 170 chemical shills in do transition metal tetraoxo complexes. SFR//SFR = scalar relativistic metal ECP in the shift calculations and in the structure optimization. SFR//NR = scalar relativistic metal ECP in the shift calculations and nonrelativistic ECP in the structure optimization. NR//NR = nonrelativistic metal ECP in the shift calculations and in the structure optimization. Chemical shills were obtained by converting computed absolute shieldings with an absolute shielding value of 307.9 ppm for liquid water.

In contrast to the situation for the oxo complexes discussed above, ECPDFT calculations on 13C shieldings in organomercury complexes indicated an

586 opposite influence of structural (indirect) and direct electronic SFR influences [105]: As for the abovementioned oxo-complexes, the relativistic bond length contraction enhanced shielding. However, the direct SFR effect at a given structure was found to be clearly deshielding, partly due to charge transfer to the relativistically contracted mercury 6s shell [105]. Differences relative to the above case of the oxo complexes are mainly due to the dominance of metal 6sorbital in bonding to mercury compared to the dominance of 5d orbitals in the oxo complexes. The SFR effects were larger for ~• than for ~11 components of the 13C shielding tensor. As the overall SFR effects were deshielding, the anisotropy of the tensor was computed to increase. In case of the alkylmercury complexes, SFR and SO effects were found to be of comparable magnitude [68,105]. SFR effects were found to be much smaller than SO effects for the ~3C shiRs in a series of 5d transition metal carbonyl complexes. That series has also served as a benchmark example for various computational methods. For example, SFRECP calculations with subsequent SO corrections by SO-ECPs and third-order perturbation theory [69] provided good agreement with earlier SO-Pauli calculations [38]. SFR effects on 13C shifts were slightly shielding for the d 1~ dicarbonyl complexes Au(CO)2 + and Hg(CO)22+, in contrast to deshielding crs~ contributions [38,69]. SFR effects on 199Hg shifl:s ill Hg(EH3)2 complexes due to Sill3 or GeH3 relative to CH3 substituents were considered to be nonnegligible in a recent DKH-UHF study [ 116] (cf. section 5 for relativistic effects arising from the heavy center itself). In a comparative study of ZORA, Pauli and ECP-based DFT methods for ligand chemical shiRs in uranium complexes, Schreckenbach et al. [51] concluded that ECP-based methods are beyond their limit for such a heavy atom as uranium. However, this may be more a limitation of the particular ECP parameterization used in that study. Even in cases where the SFR effects alone are negligible, their inclusion may be warranted to obtain a good description of SO effects. This is due to the fact that the all-electron SO operators probe the inner tails of the valence orbitals, and SFR effects on the position of the radial nodes of these orbitals are thus potentially important [114]. This is apparent in some of the computational results obtained, e.g., from a number of quasirelativistic all-electron HartreeFock studies of heavy-atom shieldings [82,116]. In contrast, in SO-ECP

587 calculations these effects are transferred to regions spatially more remote from the nuclei, near the radius where pseudo-valence and true valence orbitals have to coincide [68].

5. R E L A T I V I S T I C HEAVY-ATOM E F F E C T S AT THE HEAVY-ATOM C E N T E R ( ' ~ A H A E F F E C T ")

Pyykk6 coined the 'HAHA" label [ 64] for relativistic heavy atom effects that are probed by the shielding of the heavy atom itself (in contrast to the heavy atom effect at the light atom, 'HALA" effect, cf. section 3). The motivation to study heavy-atom shifts came from a lack of proportionality between the l l9Sn and 2~ shifts in a series of analogous di- and tricoordinate compounds - the metal shifts were generally considerably more downfield for lead than for tin. Early qualitative calculations at the four-component extended-Hiickel level suggested that it is mainly SO effects coupled to Pb 6s hyperfme effects, that modify the shift of the heavy nucleus [64], a proposition that has not yet been evaluated by more quantitative methods. A somewhat more complicated picture of HAHA effects is obtained from perturbational treatments (see below). As discussed in section 2, reliable calculations of heavy-atom shieldings require relativistic all-electron methods (however, cf. [42]). Moreover, it appears that the HAHA effects are much less spectacular on relative chemical shifts than the HALA

ones.

This restricts experimental observation of HAHA

effects

significantly, as absolute shieldings are difficult to measure. It took therefore somewhat longer for HAHA than for HALA effects to get scrutinized by quantitative quantum chemical methods. It should be noted, that not in all studies that investigated heavy-atom chemical shifts, the individual contributions were analyzed in detail. We will in the following summarize the main points that have been established during the past few years. We will keep in mind that HALA effects may sometimes be superimposed on the HAHA effects. In the case of SO-I contributions, this may in principle be studied, for example, by using appropriate one-center approximations to the SO operator (e.g. via the atomic meanfield approximation [67]), and by then switching on only SO operators on specific centers.

588 Let us start with nuclear shieldings in heavy atoms, where only the HAHA effects are present. Moreover, due to the spherical symmetry, those SO effects that we have labelled ers~ (eq. 8, section 2) in the perturbation theoretical framework, and which dominate the HALA effects (section 3), vanish for atoms [62]. Earlier four-component DHF-based results for the shieldings of the noble gases He-Rn gave rather different values [21,117]. More recent DHF and twocomponent DKH results [28] were closer to the RRPA results of [117] than to the FPT-DHF results of [21]. The most accurate calculations available were at RPA-DHF level by Vaara and Pyykk6 [24] using very large basis sets. The results were shown to be extremely basis-set dependent and thus provided an explanation for discrepancies to earlier calculations. As correlation effects are thought to be of relatively little importance for these atomic cases [24], the new results were suggested to provide a good basis for absolute shielding scales of the noble gases. For the heavier atoms, a significant dependence on the chosen nuclear model was found in the relativistic case [28,24], and a point nuclear model is not adequate in these types of studies. The overall relativistic effects on shielding were comparable to the nonrelativistic shielding constant for Rn, and still on the order of about 20% thereof for Xe [28,24]. In a quadratic and linear response study of HX and C H 3 X (X - F, C1, Br, I) and H2X (X = O, S, Se, Te), Vaara et al. [62] used HF and MCSCF methods (based on nonrelativistic reference wavefunctions) to investigate not only the light-atom shieldings but also SO effects on the heavy-atom shieldings in HX and H2X. It turned out that the second-order r s~ contributions (eq. 14) were of opposite sign and larger than the third-order o s~ terms (eq. 8), so that the overall SO effect for X was net deshielding (while the dominant o s~ effects for the light neighboring nuclei were shielding, cf. section 3). The second-order SO effects were less correlation dependent than the third-order contributions, and they provided a practically isotropic contribution to the heavy-nucleus shielding (the third-order SO effects tended to reduce the shielding anisotropy [62]). This suggests that the HAHA SO effects arise largely from SO-induced spin polarization in the core shells of the heavy atoms, and that they should largely cancel out in relative chemical shifts, as do the scalar relativistic contributions (see below). Similar inferences were made in the context of the first ZORA-DFT calculations of uranium chemical shitts [51,52].

589 Notably, however, the SO-II and SO-I effects appear to account only for a relatively modest fraction of the overall HAHA effect. This may be estimated when comparing perturbational treatments to four-component results. In their comparison of RPA-DHF and perturbational calculations on the hydrogen halides, Visscher et al. found that it is the FC/SZ-KE term (eq. 16, termed FC-IV in [63], MVEF-FC in [27a]), which provides the dominant HAHA contribution [27a]. As discussed in 2.4, this term is an active, spin-dependent contribution [58]. As discussed above and in [58,60], SO-I and SO-II terms make further, smaller contributions to the HAHA effect in HI (the SO-I effects are shielding, whereas the SO-II effects are larger and deshielding [60,62]). 'Passive" SFR effects make further contributions that partly cancel each other, and a number of ft~her 'hctive non-SO" oontdbutions, are also nonnegligible [23,58,60]. The overall picture of HAHA shielding effects becomes thus quite complicated when analyzed from a perturbation theoretical point of view. Recent DKH-UHF results also confirmed the overriding importance of interaction terms analogous to the FC/SZ-KE contribution [48]. In that work the contribution was assigned to a ers~ term. This underlines the difficulties in defining the exact nature of the term. Common features of most of the important HAHA contributions appear to be the following: a) they arise from the core shells of the heavy atom rather than from core tails of the valence orbitals like the dominant HALA SO-I effects discussed in section 3 (this has been confirmed also by a localized MO analysis of the FC/SZ-KE contributions to heavy-atom shieldings in various small hydrides in [59]). b) the contributions are predominantly isotropic [23,27a,58,59,60]. In agreement with our above discussion, this suggests that HAHA effects are essentially atomic in nature and should cancel to a large extent when looking at relative chemical shifts. This appears to be bourne out by the available computational studies. In a quasirelativistic Hartree-Fock study of 125Te shifts in a number of tellurium compounds [47], SFR effects were relatively small even for the absolute shieldings and further cancelled out in the relative shifts. The same was found in scalar relativistic DFT calculations [5,42,118]. Third-order SO effects, which were also considered in [47], were significant for shieldings but again largely cancelled for relative shifts. This confirms that HAHA SO effects on

590 shieldings are atomic, core-shell-like in nature (as in the atomic systems considered above) and may cancel in relative shifts. It also appears that the 'l-IAHA"-type relativistic corrections are relatively little influenced by electron correlation, compared to significant correlation effects on the c s~ contributions to light-atom shielding (cf. section 3) [23,61]. While the workers in [47] tended to support the existing experimental absolute shielding scale for Te [119], others preferred a reduction by ca. 900 ppm and pointed to problems a) with the nonrelativistic formalism used to extract the paramagnetic shielding from spinrotation constants (cf. section 2), and b) with the influence of relativistic effects on the computed shielding results [5,42,118]. In DKH-based Hartree-Fock calculations [36], and in a comparison of fourcomponent relativistic and perturbational SO-corrected calculations [27] the scaling of the relativistic effects on heavy-atom shieldings in the hydrogen halides with nuclear charge was studied. The scaling was roughly proportional to Z 3~ for non-SO and to Z 3'5 for SO effects [27] (but Z 35 for non-SO effects in [36]). Dirac-Fock calculations on the group 15 and 16 hydrides gave similar exponents of ca. 3.2, corresponding to an average of non-SO and SO effects [30]. A recent DFT study based on both the scalar relativistic Pauli method (augmented by 'hctive" non-SO terms) and the ZORA approach indicates nonnegligible 'laon-SO" effects on the relative tungsten shills in WO4 2- relative to W(CO)6 [39]. More detailed analysis at the Pauli level suggested that both ']gassive" and '~ctive" non -SO effects (following the terminology of [58]) play a role. In [39], the most important effects were included in the erp contribution. We may probably place at the frontier of applications of relativistic methods to heavy-nucleus shieldings the recent SO-ZORA and SO-Pauli based study of uranium shit, s by Schreckenbach [51,52]. Experimentally, 235U shifts are as yet unknown. A shielding range of more than 21000 ppm was predicted for this I = 7/2 nucleus [52], and the computed shielding trends for differently substituted complexes were rationalized mainly by changes up to 20000 ppm in 6P, which were largely controlled via the energy gap between occupied and unoccupied orbitals with significant metal 5f-character. SO effects were on the order of 7000-9000 ppm but had a much smaller (yet nonnegligible) effect on relative shifts [52]. No particularly heavy substituent atoms were present in the examples studied, and it was suggested that SO effects might well be the dominant ones

591 when heavy atoms are bonded to uranium. However, as typically the uranium 7s-orbitals play a relatively small role in bonding compared to 4f or 5d orbitals, we would anyway not expect a too dramatic dependence of uranium shifts on SO-I effects due to heavy neighboring atoms. Appreciable differences were found between Pauli- and ZORA-based computational results, and in the absence of experimental data, the latter were favored for their better theoretical justification [51,52]. This is clearly an area of research where theory is currently ahead of experiment.

6. C O N C L U D I N G R E M A R K S

It should be clear from the examples provided in this article, that relativistic effects cannot be ignored when one wants to understand NMR chemical shifts throughout the periodic table. While the local heavy-atom effects on the heavy atoms ('HAHA" effects) can be very large for absolute shield ings, they tend to cancel to a large extent in relative shifts and are thus probably less important for the interpretation of the observed shifts for different compounds. HAHA effects are nevertheless of interest, not only for the development of reliable relativistic computational methods but also, for example, when deriving absolute shielding scales for heavy nuclei (section 5). The more dramatic phenomenon from an experimental point of view are undoubtedly the heavy-atom effects on neighboring 'light" ato ms ('HALA" effects), which we have discussed here in considerable detail. The HALA effects feature both spin-orbit (SO) and spin-free (SFR) contributions, of which the former are the more unusual ones. The SFR-HALA effects can be important for the 5d and 6p elements and beyond and may be included more or less conveniently into calculations, e.g. by ECP approaches (section 4). In the largest individual section 3, we have analyzed and interpreted the SO-HALA effects based on the perturbational formalism outlined in 2.4. We have discussed an analogy between the crs~

effects and the Fermi-contact (and spin-dipolar)

mechanisms of indirect spin-spin coupling. This provides a wealth of insight and helps to understand many of the experimentally observed trends. We saw that both SFR and SO effects may be either shielding or de shie lding, and

592 interpretations were also provided in each case. The shielding anisotropies are frequently also affected significantly by relativistic influences. We have discussed the relations between the relativistic contributions to chemical shifts and molecular structure, a field that will see considerable further development in the

futt~e.

Possible

consequences

include

also

unusual

temperature

dependencies or isotope effects, as well as a large coupling of relativistic and environmental effects on nuclear shieldings. Many of these aspects require further study, and the field of relativistic computations of NMR parameters is currently a very active one. Most importantly, it is now neccessary that the principles outlined here will also find their way into NMR textbooks and into the interpretational toolbox of NMR spectroscopists.

ACKNOWLEDGMENTS. I am very grateful to Vladimir G. Malkin and Olga L. Malkina (Bratislava), as well as to Juha Vaara and Pekka Pyykk6 (Helsinki) for illuminating discussions and helpful remarks on the manuscript. Our own work in this field over the past years has been based on a close and fruitful collaboration with the Bratislava group.

Further

important

contributors

were

Juha

Vaara,

Bernd

Schimmelpfennig, Roman Reviakine, and Pekka PyykkS. Our research has been funded by Deutsche Forschungsgemeinsehaft, Fonds der Chemischen Industrie, Universit/it Wiirzburg, Universitgt Stuttgart (Graduiertenkolleg 'Magnetische Resonanz'), and Max-Planck Gesellschaft.

593

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