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Relativistic NMR constants
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Theory and computation of nuclear magnetic resonance parameters
Juha Vaara*
Received 23rd April 2007, Accepted 25th May 2007
First published as an Advance Article on the web 6th July 2007
DOI: 10.1039/b706135h
The art of quantum chemical electronic structure calculation has over the last 15 years reached a
point where systematic computational studies of magnetic response properties have become a
routine procedure for molecular systems. One of their most prominent areas of application are
the spectral parameters of nuclear magnetic resonance (NMR) spectroscopy, due to the immense
importance of this experimental method in many scientific disciplines. This article attempts to give
an overview on the theory and state-of-the-art of the practical computations in the field, in terms
of the size of systems that can be treated, the accuracy that can be expected, and the various
factors that would influence the agreement of even the most accurate imaginable electronic
structure calculation with experiment. These factors include relativistic effects, thermal effects, as
well as solvation/environmental influences, where my group has been active. The dependence of
the NMR spectra on external magnetic and optical fields is also briefly touched on.
1. Introduction
Nuclear magnetic resonance (NMR) spectroscopy1,2 occupies
a central position in the experimental toolbox in many scien-
tific disciplines: chemistry, materials science, solid-state and
molecular physics, biosciences and medicine. It is based on
observing, in the radio-frequency range, the transitions be-
tween the Zeeman levels of nuclear magnetic moments con-
tained in a sample placed in the external magnetic field of the
NMR spectrometer. In order to possess a magnetic moment,
the spin quantum number IK of the nucleus has to equal at
least 1/2. While the paramagnetic nuclei are a necessity,
typically the systems investigated by NMR have a diamag-
netic, closed-shell electronic state. The spectra are analysed in
terms of a phenomenological energy expression for a limited
number of degrees of freedom, the NMR spin Hamiltonian3
HNMR ¼ hHBPðB0; frig; fsig; fRKg; fIKgÞifrig;fsig;fRKg
¼ � 1
2p
XK
gK IK � ð1� sKÞ � B0 þXKoL
IK
� ðDKL þ JKLÞ � IL þXK
IK � BK �IK : ð1:1Þ
Here, lK = gKIK is the magnetic moment of nucleus K
possessing the dimensionless spin IK and gyromagnetic ratio
gK, the latter factor determined by the nuclear structure.
HNMR is an effective Hamiltonian: of the complex total
interaction of matter consisting of charged and magnetic
nuclei and electrons with the magnetic field B0, dependencies
on only IK and B0 are explicitly retained. The complete
Hamiltonian, in eqn (1.1) represented by HBP, is a function
also of the electronic positions {ri}, electronic spins {si},
nuclear positions {RK} and possible external electric fields,
etc. These degrees of freedom are averaged over and the effects
of the additional variables are implicitly included in the
parameters remaining in HNMR.
The first NMR parameter is the nuclear shielding tensor, rK,
which changes the Zeeman interaction of bare nuclei with the
magnetic flux density, �lK �B0. This is a consequence of the
presence of the electron cloud and its modification induced by
the magnetic field. The shielding tensor is not the direct
experimental observable; instead it is the chemical shift
dK ¼sK;ref � sK1� sK;ref
ð1:2Þ
with respect to a suitably chosen reference compound contain-
ing a similar nucleus. sK is a scalar quantity arising from the
tensor rK via rotational averaging. In rotationally isotropic
media, sK = 1/3 (sK,xx + sK,yy + sK,zz), the nuclear shieldingconstant. As a result of rK, the magnetic field at the nuclear
position deviates from B0 both in its magnitude and direction.
Not only the magnetic field of the spectrometer but also the
fields from the other magnetic nuclei L influence the Zeeman
Juha Vaara studied micro-electronics and materials phy-sics at the University of Oulu,and obtained PhD in physicsin 1997. After postdoctoralstudies at Linkoping Univer-sity, Max-Planck-Institut furFestkorperforschung and Uni-versity of Helsinki, he wasappointed as university lec-turer in physical chemistry atHelsinki, in 2003. He holdsdocentships in physics at theUniversity of Oulu and in phy-
sical chemistry at the University of Helsinki. His researchinterest is theoretical and computational magnetic resonancespectroscopy.
Laboratory of Physical Chemistry, Department of Chemistry, P. O.Box 55 (A.I. Virtasen aukio 1), FIN-00014 University of Helsinki,Finland. E-mail: juha.t.vaara@helsinki.fi
This journal is �c the Owner Societies 2007 Phys. Chem. Chem. Phys., 2007, 9, 5399–5418 | 5399
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levels of nucleus K. This is reflected in the spin–spin coupling
term in HNMR, featuring the direct dipolar coupling tensor
DKL that represents the magnetic through-space interaction of
the bare nuclei. This parameter can be calculated using
classical electromagnetism from the knowledge of the nuclear
positions {RK} alone. Analogously to rK modifying the Zee-
man interaction with the external field, the (indirect) spin–spin
coupling tensor JKL furnishes a correction to the interaction of
naked nuclei, determined by the properties of the electron
cloud.
The final term present in the standard NMR spin Hamilto-
nian arises from the interaction of the nuclear electric quadru-
pole moment (NQM) of nuclei with IK Z 1, with the electric
field gradient (EFG) at the nuclear site. The fact that the
NQM tensor is cylindrically symmetric with respect to lKmakes it possible to express this electric interaction with a self-
interaction term bilinear in IK. The NMR parameter is in this
case the quadrupole coupling tensor BK which contains the
product of NQM and EFG. The spin–spin and quadrupole
couplings manifest themselves as splittings in the spectral lines
whose positions are determined by the shielding term.
The full Hamiltonian HBP contains also higher-order inter-
actions involving IK and B0. Experience gained in the 60 years
of practical NMR spectroscopy so far has limited the selection
of terms in HNMR to those presented in eqn (1.1). Interesting
exceptions to this rule of thumb have been investigated (see,
e.g., ref. 4), as also discussed in this article.
As already hinted at above, the two-index parameter tensors
rK, JKL and BK that represent microscopic interactions in the
molecule-fixed coordinate frame in eqn (1.1), correspond to
experimental observables after rotational averaging appropri-
ate to the conditions at hand, be it isotropic gas or liquid
phase, a partially oriented system such as liquid crystal, or
solid state in either a single crystal or powder form. Due to the
long timescale of NMR experiments, these static parameters
reflect the average structural characteristics. The spin–lattice
and spin–spin relaxation rates, T1 and T2, respectively, carry
information on the time dependence (dynamics) of the NMR
interactions in the molecular scale.
As the NMR parameters are intimately connected to the
electronic structure of the investigated atoms, molecules and
solids, their theory and calculation forms one natural target
for quantum chemistry. Besides the practical importance of
NMR, the field is phenomenologically rich and methodologi-
cally broad and consequently interesting also for many theo-
reticians. This article tries to provide an overview of the state-
of-the-art in the quantum chemical calculation of static NMR
parameters from the point of view of the interests of the
author’s research group over the past few years. As such, the
approach is naturally biased and I apologise in advance for the
certainly numerous omissions of references to authors who
have elaborated similar subjects and ideas, also prior to us. A
recommended, comprehensive review was published by
Helgaker, Jaszunski and Ruud.5 A recent compilation work6
also provides a wider scope than the present article.
The article first discusses the theoretical background and
electronic structure methods for NMR parameters, illustrat-
ing the strengths and weaknesses of the presently used quan-
tum chemical models. The impressive limits in terms of both
the system size and accuracy that can be reached, are
referred to.
We then carry on to factors that are ubiquitous in experi-
ment but necessitate taking modelling beyond the standard
quantum chemical approach, involving relativistic effects,
rovibrational (thermal) motion, intermolecular interactions,
as well as solvation effects. The more exotic phenomena of the
dependence of NMR parameters or spectra on external fields,
in our case magnetic and optical fields, are subsequently
treated. Two items of significant current interest in my group,
paramagnetic molecules as well as noble gases, are subse-
quently treated.
The article finishes with an outlook at the status and
development of the field in the near future.
2. Magnetic interactions
To formulate microscopic theories for NMR parameters, a
Hamiltonian including the electronic degrees of freedom is
necessary for a molecular system in the presence of both the
external and nuclear magnetic fields. Despite the well-known
shortcoming constituted by its variationally unstable relativis-
tic terms, the Breit–Pauli electronic Hamiltonian,7–9 HBP,
continues to be the basic building block of electronic structure
theory. In the presence of the vector potentials of the external
magnetic field, A0, and two nuclear magnetic moments, AK
and AL, the electronic momentum becomes10
pi ¼ �iri þ1
2B0 � riO þ a2gK
IK � riK
r3iK
þ a2gLIL � riL
r3iL; ð2:3Þ
where riO = ri � O and riK = ri � RK specify the location of
electron i with respect to the gauge origin O and the position
RK of nucleus K, respectively. a is the fine structure constant
that provides an expansion parameter the powers of which can
be used as a guideline when formulating a consistent theory.
The magnetic field terms are treated systematically for HBP
in ref. 11. Up to orders including both nonrelativistic (NR),
O(a2), and leading-order relativistic, O(a4), theories for rK, as
well as the corresponding theories for JKL that appear at
respective orders higher by two powers of a, several terms of
HBP can be omitted. One is left with, first, the kinetic energy
HNRkin ¼
1
2
Xi
p2i � hKE þHOZB0þHPSO
K þHDSKB0þHDSO
KL ;
ð2:4Þ
where the field-free term hKE is supplemented by magnetic
contributions either linear in the external field such as the
orbital Zeeman (OZ) interaction, linear in the nuclear spin
such as the orbital hyperfine (or paramagnetic nuclear spin-
electron orbit, PSO) interaction, and the bilinear diamagnetic
terms involving either the external field (DS) or two nuclei
simultaneously (DSO).
Second, the Zeeman energy involves the electronic spin
HNRZ ¼ 1
4iXi
si � pi � pi ¼ HSZB0þHFC
K þHSDK ð2:5Þ
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and leads to the spin–Zeeman (SZ) as well as the Fermi
contact (FC) and spin–dipole (SD) hyperfine interactions.
Finally, a large group of relativistic terms arises,11 including
the leading-order corrections to the kinetic energy and Zeeman
terms, HRkin and HR
Z (the former including the well-known
mass-velocity term), hmv, as well as the Darwin and spin–orbit
(SO) terms hDar and HSO that lack NR counterparts. For
example, ref. 11 can be consulted for the explicit functional
forms of all these operators.
3. NMR parameters
Expressions for the NMR parameters are obtained by com-
bining, in the sense of perturbation theory, the terms ofHBP to
various powers n in a2n. For example, when formulating
theory for rK, one considers the perturbed energy terms
E(lK, B0) that contain a bilinear dependence on IK and B0.
The resulting terms can be simplified by making use of time-
reversal symmetry reflected in the fact that the resulting energy
terms should be real, as well as the rules of angular momentum
coupling applied to the electronic spin. For example, a closed-
shell electronic state must be perturbed by at least two electron
spin-containing operators, to obtain a contribution to mole-
cular properties via triplet intermediate states.
With the perturbed energy expressions at hand, a compar-
ison with the form of HNMR in eqn (1.1) tells that the spectral
parameters are obtained as derivatives of the perturbed energy
terms (in the Cartesian tensor component notation)
sK ;et ¼ det þ@2EðlK ;B0Þ@mK ;e@B0;t
����lK¼0;B0¼0
ð3:6Þ
JKL;et ¼ �DKL;et þ1
2pgKgL
@2EðlK ; lLÞ@mK;e @mL;t
����lK¼0;lL¼0
ð3:7Þ
BK ;et ¼1
2p@EðIKIK Þ@IK;eIK;t
����IK IK¼0
: ð3:8Þ
In the expression for rK, the sometimes omitted unit matrix
term (det) takes care of the Zeeman interaction of the bare
nuclei, present in HBP. The spin–spin coupling expression
involves analogously the term DKL, the direct dipolar cou-
pling. BK can be cast as the first derivative with respect to the
bilinear product IKIK.2
4. Electronic structure methods
4.1 Perturbation theory
The external and internal magnetic fields that are relevant for
NMR spectroscopy only give rise to energetically very small
effects, as compared to the dominating energetics of the
nucleus–electron attraction and electron–electron repulsion.
Consequently, the use of perturbation theory is well-justified
in the calculation of the NMR parameters.
My group has been active in the application of particularly
the Dalton12 quantum chemical package as a property calcu-
lation toolbox, due to the possibility within this programme to
freely combine different terms of HBP up to fourth order of
perturbation theory and, hence, to obtain a rich variety of
static and dynamic molecular electronic properties. The basic
practice in NMR calculations is to apply the perturbations on
top of the spin-free NR reference state obtained with the
standard electronic structure Hamiltonian
hð0Þ ¼ � 1
2
Xi
r2i �
XiK
ZK
riKþ 1
2
Xij
0 1
rij; ð4:9Þ
treated at the different ab initio or density-functional theory
(DFT) levels. The former range from the uncorrelated
Hartree–Fock (HF) to the correlated multiconfiguration self-
consistent field (MCSCF), second-order Møller–Plesset many-
body perturbation theory (MP2), coupled-cluster singles and
doubles (CCSD), as well as CCSD with perturbational triples
[CCSD(T)] methods. This hierarchy of approximations
features a systematically increasing accuracy for solving the
problem of electron–electron interaction but at a rapidly
increasing computational cost. Consequently, DFT is used
as a pragmatic tool for including electron correlation effects in
the NMR calculations of large molecules.
The relevant perturbation operators represent different mag-
netic or relativistic interactions and are typically obtained from
HBP. Whereas the latter do not typically constitute small
perturbations (in contrast to the magnetic interactions) for
molecules containing heavy elements, in practice a leading-
order relativistic perturbational approach works surprisingly
well for rK.13,14 In the context of NMR parameters, mostly
static perturbations and, consequently, time-independent per-
turbation theory are considered, as magnetism is represented by
‘‘slow’’ interactions in the electronic time scale. It is convenient
to represent different orders in perturbation theory in terms of
response functions,15,16 where second-order Rayleigh–
Schrodinger expression is seen to correspond to the spectral
representation of the static limit of a linear response function
Eð2Þ0 ¼ h0jHð2Þj0i þ
Xn
0 h0jHð1ÞjnihnjHð1Þj0iE0 � En
ð4:10Þ
� Hð2Þ00 þ
1
2hhHð1Þ;Hð1Þiio¼0: ð4:11Þ
For example, the NR expressions of rK or JKL can be cast as
linear response functions and expectation values, involving
different choices of first- and second-order perturbations H(1)
andH(2), respectively. The interactions involve excited states |niof the unperturbed situation having either a singlet or triplet
spin symmetry, when the reference state is singlet. Both field
dependence of the parameters as well as relativistic corrections
require higher-order, nonlinear response functions.
The leading-order, NR theory of rK by Ramsey17 is stated as
rK ¼ rdK þ r
pK ð4:12Þ
sdK;et ¼1
2a2 0
Xi
detðriO � riKÞ � riO;eriK ;t
r3iK
����������0
* +ð4:13Þ
spK ;et ¼1
2a2
Xi
liK ;e
r3iK;Xi
liO;t
* +* +0
; ð4:14Þ
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where the diamagnetic shielding rdK is an expectation value of the
DS operator >eqn (2.4)] and the paramagnetic part rpK involves
the PSO and OZ operators in a singlet linear response function.
In the latter the liK and liO denote the electronic angular
momentum with respect to RK and O, respectively.
BK at the corresponding level of theory involves the
expectation value of the operator for EFG
BK ;et ¼ �QK
4pIK ð2IK � 1Þ 0Xi
3riK ;eriK ;t � detr2iKr5iK
����������0
* +
þ nuclearterms:
ð4:15Þ
Finally, the NR JKL equals
JKL ¼ JDSOKL þ JPSO
KL þ JFCKL þ JSD
KL þ JSD=FCKL ð4:16Þ
JDSOKL;et ¼
1
4pa4gKgL 0
Xi
detðriK � riLÞ � riL;eriK ;t
r3iK r3iL
����������0
* +ð4:17Þ
JPSOKL;et ¼
1
2pa4gKgL
Xi
liK ;e
r3iK;Xi
liL;t
r3iL
* +* +0
ð4:18Þ
JFCKL;et ¼
1
2p4p3
� �2
a4g2egKgLdet
Xi
dðriK Þsi;e;Xi
dðriLÞsi;e
* +* +0
ð4:19Þ
JSDKL;et ¼
1
2pa4
4g2egKgL
Xn¼x;y;z
Xi
3riK ;nriK;e � denr2iKr5iK
si;n ;
**
Xi
3riL;nriL;t � dtnr2iLr5iL
si;n
++0
ð4:20Þ
JSD=FCKL;et ¼
1
2p4p3
a4
2g2egKgL
Xi
dðriKÞsi;e;Xi
3riL;eriL;t � detr2iLr5iL
si;e
* +* +0
"
þXi
3riK ;triK ;e � detr2iKr5iK
si;t;Xi
dðriLÞsi;t
* +* +0
#;
ð4:21Þ
where contributions from the electron-spin-free DSO and PSO
operators appear in an expectation value term and a singlet
linear response function, respectively, in analogy with rK. In
addition, the FC and SD hyperfine operators [eqn (2.5)]
appear in triplet linear response functions. These operators
involve the free electron ge-factor.
Each linear response term corresponds to finding the first-
order perturbed wave function with respect to one of the two
perturbations involved. Table 1 summarises the number and
types of terms involved in the standard NR calculations. It is
worth noting that with nonvariational quantum chemical
models such as MP2 or coupled cluster (CC), the expectation
value term in reality involves a response calculation to find the
relaxed one-particle density matrix.
The different widely used quantum chemical methods in the
context of NMR calculations are characterised in Table 2 from
the point of view of computational cost or, inversely, the size
of systems that can be treated, their problems and merits. The
scaling with the number of basis functions N of the wave
function-based methods increases from N4 of the HF model to
N8 for CCSDT (CC up to iterative triples excitations) with the
concomitant, systematic increase in accuracy. MCSCF is
difficult to place on the same scale and its most important
application area is in systems featuring a large influence of
nondynamical correlation. DFT scales as well as HF or even
better when density fitting is used, and can be used to estimate
electron correlation contributions in large systems that are
beyond the capabilities of the MP2 method.
In properties involving triplet perturbation operators, such
as JFCKL, JSDKL, JSD/FCKL as well as relativistic corrections to rK, the
HF model suffers from the possibility of triplet instability.18,19
This results from the reference wave function being only
optimised with respect to singlet orbital rotations, causing
an erroneously small or even negative triplet excitation energy
and, consequently, unphysical values for triplet response
functions. This occurs near the equilibrium geometry particu-
larly in certain multiply bonded molecules. The problem of the
HF reference state carries over to MP2 and even CCSD(T), in
the latter case due to the noniterative triples corrections. On
the other hand, BK is a ground-state property and the second-
order property rK (at the NR level) only involves singlet
operators. Consequently, the HF-MP2-CCSD(T) succession
of methods performs very well for these properties.
The problem in practical DFT calculations is that it is not
possible in general to improve the calculations by selecting
different exchange–correlation functionals, without referring
to experiment. Presently DFT stands nevertheless as the only
applicable, correlated first principles method for NMR calcu-
lations beyond a few tens of atoms, and even much earlier for
transition metal systems. It is affected by the triplet instability
problem but to a lesser degree than HF,20 and consequently it
can be viewed as an entry-level method for JKL. The main
caveat in such calculations is the very different performance of
DFT for different nuclei: while, e.g., 13C13C and 13C1H
couplings are well-reproduced by DFT, couplings involving19F and in general centres with lone pairs have large systematic
errors.21,22 This is evidenced by Table 3 where some of our
own results for the coupling constants JCC and JCF are listed
as resulting from applying the three ‘‘generations’’ of DFT
methods: local density approximation (LDA), generalised
gradient approximation (GGA, represented by the BLYP
Table 1 Computational requirements for the calculation of nonrela-tivistic NMR parameters for a N-atom closed-shell molecule
rK JKL BK
Expectation value | | |First-order wave function: | |� with singlet intermediate states 3 3N 0� with triplet intermediate states 0 N + 6N 0Gauge-origin dependence |
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functional23) and hybrid functionals (represented by
B3LYP24).
The errors of the DFT calculations diminish in general from
LDA to GGA to hybrid functionals for these main-group
systems, and the accuracy of the ab initio (MCSCF) approach
is surpassed for JCC in the somewhat larger C6H6 system.
While much improved ab initio calculations using the coupled-
cluster paradigm are currently possible for molecules of this
size, the cross-over to the DFT realm still persists albeit at a
larger system size. The DFT accuracy degenerates for
carbon–carbon couplings over bonds of increasing order, seen
in the case of the triply bonded ethyne. Recently it has been
found31 that improved accuracy in spin–spin couplings can be
obtained by performing DFT calculations at geometries opti-
mised for that method instead of experimental or higher-level
geometries.
For JCF, DFT systematically overestimates the experimental
data and a detailed investigation of the error22,25 reveals that
both the PSO and FC contributions to JCF have problems.
The situation typically improves for couplings over a larger
number of bonds. The deficiency has been attributed to the
long-range behaviour of the exchange–correlation functionals,
coupled to the problem of self-interaction in standard
DFT.32,33 An attempt at reducing the self-interaction error
for NMR properties was made by Patchkovskii and co-work-
ers.34 The development of systematic exact-exchange DFT
methodology35,36 holds a significant promise for NMR prop-
erties.
Linear-scaling ab initio37 and DFT methods38,39 are being
developed by many research groups. NMR calculations also
benefit from the implementation of parallel algorithms, cur-
rently widely used in the context of the HF and DFT methods.
These factors can be expected to lift up the accessible system
sizes (Table 2) for many methods in the near future. DFT is
nevertheless likely to hold its premium position for systems
large enough, rendering it imperative to look for improved
functionals where the ‘‘fluorine’’ problem (vide supra) has been
solved. The more fundamental problem of standard DFT for
magnetic properties, namely the neglect of the current density
dependence in the exchange–correlation functional40 appears
to be less severe in practice for NMR parameters, based on the
small effect on r resulting from the available model calcula-
tion.41 Nevertheless, the matter deserves more investigation.
4.2 Basis sets
In a typical study of NMR parameters, a finite set {wm} of
Gaussian-type basis functions
wKm ðriÞ ¼ Ylmðyi;jiÞrli expð � zr2i Þ ð4:22Þ
centered at the nuclei, {RK}, is used to expand the one-electron
orbitals. The set of exponents {z} has to be large because, on
the one hand, a variety of physical interactions are involved in
NMR properties, some of which work essentially in the
valence space such as the OZ interaction. These operators
require diffuse (small-z) basis functions. Some of the relevant
operators obtain, in turn, significant contributions from the
region of space near the atomic nuclei, and require tight
(large-z) functions. For NMR properties a wider exponent
range42 is typically required to reach converged results than in
the case of molecular geometries and energetics. On the other
hand, electron correlation plays a significant role in NMR
properties implying increased basis set demands as compared
to properties well-described using a single Slater determinant.
Quantum chemical calculations of properties arising from
B0 suffer from spurious gauge-origin dependence due to the
use of finite basis set. An efficient way of eliminating the
problem is to use gauge-including atomic orbitals (GIAOs)43
oKm ðri;AK
0 Þ ¼ expð � iAK0 � riÞwKm ðriÞ
AK0 ¼
1
2B0 � ðRK �OÞ;
ð4:23Þ
that attach a field-dependent phase factor to the basis func-
tions, which effectively transfers the local gauge origin to the
basis set expansion centre RK, i.e., the optimum location. This
Table 2 Standard quantum chemical models in the calculation of nonrelativistic NMR parameters
Method Single-ref. Multiref. Empirical ScalingaNumberof atoms Problems Successes in NMR
HF | N4 B300 Triplet instability, no correlation Easy rMP2 | N5 B50 Triplet instability rCCSD | N6 B15 Expensive r, JCCSD(T) | N7 B10 Expensive, (triplet instability) Accurate rCCSDT | N8 B4 Very expensive Most accurate r and JMCSCF | exp Xb B10 Dynamical correlation, no ‘‘black box’’ Low-lying excited statesDFT | N3. . .4 B300 Non-systematic, (triplet instability),
‘‘fluorine problem’’Inexpensive, transitionmetals, (r and J)
a N is the number of basis functions. b Scaling of the, e.g., complete active space (CAS) model is exponential in the number of active orbitals.
Table 3 Performance of density-functional theory calculations ofcarbon–carbon and carbon–fluorine spin–spin coupling constants fora few simple moleculesa
Molecule Coupling LDA BLYP B3LYP MCSCF Expt.
C2H61JCC 18.6 29.5 32.6 38.8 34.527
C2H41JCC 48.4 66.9 70.5 70.2 67.527
C2H21JCC 169.9 195.1 198.5 181.2 185.029
C6H61JCC 42.4 58.2 61.6 70.9 55.8262JCC 0.5 �0.4 �1.9 �5.0 �2.526
CH3F1JCF �230.8 �250.1 �225.3 �156.6 �163.030
CH2F21JCF �324.8 �342.0 �309.9 �220.7 �233.930
CHF31JCF �372.9 �390.9 �354.1 �242.1 �272.230
p-C6H4F21JCF �371.2 �390.1 �358.2 �184.7 �242.628
a DFT results from ref. 25, MCSCF from ref. 26–28.
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technique was pioneered by Ditchfield43 and introduced to the
modern quantum chemical context by Wolinski et al.44 as well
as Helgaker and Jørgensen.45 The use of GIAOs improves also
the basis-set convergence of rK due to the facts that, on the
one hand, they provide the correct wave function to first order
in the one-center, one-electron problem and, on the other
hand, the effect of the exponential prefactor may be viewed as
adding higher l-value basis functions to the original set {wm}.5
Alternative distributed gauge-origin methods exist, most
notably the individual gauges for localised orbitals (IGLO)
ansatz by Kutzelnigg and co-workers.46
Standard basis set families have been optimised with ener-
getic properties in mind. This necessitates using quite extensive
standard basis sets in NMR calculations or, alternatively,
specially designed sets. In my group we have used particularly
the Huzinaga/Kutzelnigg46,47 (IGLO, HII–HIV) basis sets
that have favourable contraction pattern and choice of polar-
isation functions. For elements heavier than Ar, FII–FIV sets
similar in spirit to the IGLO sets have been constructed based
on the primitives of Fægri.48 These basis sets have been
systematically extended by addition of successive tight and
diffuse primitives with exponents in a geometric progression
with the ratio of successive exponents equalling the factor of
three, chosen empirically.
We developed a systematic way of generating primitive
Gaussian basis sets by completeness optimisation (CO),42
where the use of energy criteria in selecting the parameters
has been abandoned. CO embraces the idea of the range of
exponents being the key factor affecting the performance of a
basis set, not only in NMR calculations but for molecular
properties in general. In CO, exponent families are optimised
to cover the prescribed exponent range with the basis set
completeness profile49
YðaÞ ¼Xm
hgðaÞjwmi2 ð4:24Þ
maximised to the extent allowed by the number of functions in
a given l value.
In eqn (4.24), the square of the overlap of the investigated
orthonormal basis set {wm} with a ‘‘test Gaussian’’ g(a) with an
arbitrary exponent a, obtains values in the range from zero to
unity as a function of a. The latter limit means that the basis
set constructed from a finite set of exponents is able to fully
simulate the presence of arbitrary Gaussian function in the
basis. Fig. 1 illustrates the completeness profiles of various
correlation-consistent basis sets50 for fluorine.
Provided that a sufficiently wide exponent range is used and
the completeness profile is sufficiently close to unity in that
range, CO leads to element-independent basis sets. Ref. 42
demonstrated significant computational savings in magnetic
properties obtained by this paradigm, as compared to stan-
dard energy-optimised basis sets (Table 4). While the CO
concept is attractive as a systematic means of guaranteeing
that the Gaussian basis-set limit is reached, further work is
necessary to be able to employ contracted basis sets in
standard calculations that are further away from the basis
set limit.
Fig. 1 Effect of n-tuple valence description as well as augmentation and core polarisation on the completeness of basis sets: completeness profiles
of (a) cc-pVDZ, (b) cc-pVTZ, (c) cc-pVQZ, (d) cc-pV5Z, (e) cc-pV6Z, (f) cc-pCVTZ, (g) aug-cc-pVTZ and (h) aug-cc-pCVTZ basis sets of fluorine.
Reprinted from ref. 42 with permission of John Wiley & Sons, Inc.
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5. How far can one go?
First principles NR calculations of NMR properties can be
taken to impressive system sizes as evidenced by the linear-
scaling HF calculations of rK reported in ref. 51 for systems
consisting of more than 1000 atoms. Fig. 2 illustrates a static
supermolecule model of N-methyl nicotinamide in a molecular
clip in aqueous solution with 1003 atoms and 8593 basis
functions altogether. While such calculations inevitably are
performed at a compromise basis set level and at most DFT
level of correlation treatment, they are promising for super-
molecule studies of intermolecular interaction effects in NMR
as well as, e.g., converged cluster calculations of solid-state
models.
From another point of view, it is illustrative to look at the
limiting accuracy of correlation treatment in ab initio theory.
Table 5 lists some calculated shielding constants as well as the
demanding FC contribution to spin–spin coupling constants
carried out at levels up to full CCSDT, obtained by Gauss.52,53
Note that experimental results are not listed in the Table
because they also involve rovibrational and environmental
influences, and the accuracy is for the time being only assessed
by the convergence in the systematically improving ab initio
calculations. Hydrogen fluoride serves as an example of a
simple electronic structure, and the 1H and 19F shielding
constants therein as easy properties. Table 5 indicates total
correlation effect for these properties amounting to ca. 1
and 4.5 ppm, respectively. Consequently, convergence of
the correlation problem for r to the experimentally relevant
accuracy (about 0.01 and 0.1 ppm for 1H and 19F, respectively)
takes place at CCSD or CCSD(T) levels. It is worth
noting that typically chemical shifts between different systems
can be calculated to a better accuracy than absolute shieldings.
While JFC is very much off at the uncorrelated level for
this molecule, the correlation limit can be specified to within
1 Hz.
Hartree–Fock is a poor method for the multiply bonded N2
and CO molecules. CCSD provides results that are qualita-
tively correct, but either CCSD(T) or CCSDT is necessary for
converged r. Changes of the order of 1 ppm remain even
between the CCSD(T) and CCSDT data for the shielding
constants. For JFC that features triplet perturbations, both
molecules serve as examples of CCSD being a more balanced
level than CCSD(T), in the sense of providing results that are
closer to CCSDT.
Ozone is a particularly challenging system and the shielding
constants in Table 5 at the CCSD level are off from the
CCSDT data by up to ca. 200 ppm. The remaining deviation
between CCSD(T) and CCSDT is half as large. This molecule
would obviously necessitate higher excitation levels than
triples in the coupled cluster hierarchy, for r converged with
respect to the electron correlation problem.
Above we have considered two limits of NMR calculations,
either in terms of the size of systems that can be treated at
the uncorrelated HF level or the accuracy that one can
obtain using high-level ab initio electron correlation
methods. Linear-scaling post-HF approaches would combine
the two worlds and initial steps in this direction have
been taken in the implementation of the GIAO LMP2
method.37
Fig. 2 System subjected to an ab initio calculation of NMR shielding
constants in ref. 51: more than 1000 atoms! Reprinted from ref. 51
with permission of Wiley-VCH.
Table 5 Attainable accuracy in nonrelativistic calculations of thenuclear shielding constant s (in ppm)52 and the Fermi contactcontribution to the spin–spin coupling constant JFC (in Hz)53 in afew small molecules
Molecule Property HF CCSD CCSD(T) CCSDT
HF sH 28.67 29.54 29.61 29.62sF 413.5 417.8 418.1 418.0JFC1H19F
458.91 331.68 332.21 332.65N2 sN �112.7 �60.4 �54.9 �56.1
JFC14N14N
�8.16 1.21 2.67 1.09CO sC �25.5 3.2 7.9 7.4
sO �87.4 �52.0 �48.6 �49.0JFC13C17O
�8.63 7.01 8.29 7.28O3 sOterm
�2862.0 �1402.7 �1183.8 �1261.1sOcentr
�2768.1 �968.1 �724.2 �774.7
Table 4 Universality and applicability of the primitive completeness-optimised basis sets: the use of completeness-optimized primitive setsfor the magnetic properties of first-row hydrides.a Reprinted fromref. 42, with permission of Wiley InterScience.
Molecule Basis set sXb sH JHX
b xc Functions
H2O co-reduced 327.4 30.65 �23.11 �233.3 91aug-cc-pCV5Z 325.8 30.36 �21.85 �231.5 341
NH3 co-reduced 262.7 31.70 54.38 �289.8 115aug-cc-pCV5Z 261.6 31.50 53.73 �287.6 421
CH4 co-reduced 195.6 31.64 153.7 �317.5 139aug-cc-pCV5Z 194.6 31.54 150.2 �313.7 501
a Results from self-consistent field linear response calculations. The
primitive basis sets are the co-reduced sets of Table 3 in ref. 42; the set
of fluorine is used for all O, N and C. b X denotes O, N and C in water,
ammonia, and methane, respectively. c x is the isotropic magnetisa-
bility in a.u.
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6. Beyond standard calculations
The traditional quantum chemical approach has been to per-
form NR electronic structure calculations and taking care of
electron correlation and basis set effects for isolated molecules
in vacuo, at rest at their equilibrium geometry. From the
experimental point of view, such calculations are mostly help-
ful, but very often the deviation that the true experimental
conditions induce to the investigated properties, is of impor-
tance. First, experiments are always relativistic, and for any
property relativistic effects become increasingly important for
systems including heavy elements. In NMR this is pronounced
due to the nature of many of the involved interactions, which
probe the electron cloud in regions close to the atomic nuclei
where the electron speeds are high.
Experimental NMR spectra are always taken at finite
temperature, and both zero-point and thermal vibrations
influence the NMR parameters. Finally, while some experi-
mental activity takes place in low-pressure gases, the most
relevant conditions for NMR spectroscopy are liquid solution
and solid state. Intermolecular interactions influence the
NMR observables and should be included, along with
dynamic and relativistic effects, in realistic experiment-
oriented NMR modelling. These issues have been central in
much of the activity of my group.
7. Relativistic effects
The significance and growth with atomic number of the
relativistic effects on r are evidenced by atomic calculations
along the noble gas series in Table 6. Relativity is seen to
provide an increase of 25%, e.g., for the experimentally
relevant sXe. However, due to the fact that changes in the
chemical surroundings of a given nucleus leave the atomic core
region almost unaffected, relativistic effects on chemical shifts
are smaller than those on absolute shielding57 and d can be at
times calculated fairly well using NR theory even for the heavy
elements.
In molecules the electron cloud generally deviates from
spherical symmetry, rendering anisotropic shielding effects
observable provided that the molecular reorientational motion
is not isotropic. Again the innermost, almost spherical atomic
shells cancel out for the shielding anisotropy (with respect to
an arbitrary z axis)
Ds ¼ szz �1
2ðsxx þ syyÞ ð7:25Þ
and, consequently, relativistic influences can be qualitatively
different on Ds than on the absolute shielding constant s.The influence of relativity on the JKL and BK tensors is even
more pronounced than for rK due to the fact that only the
core-like hyperfine operators and no valence-like Zeeman
operators are involved. It is evident that in the NMR context,
relativity is anything but a curiosity of theoreticians!
7.1 Fully relativistic calculations
A range of relativistic quantum chemical methods are avail-
able for calculating NMR properties. The practical four-
component Dirac–Hartree–Fock (DHF) method has been
pioneered in the community around the Dirac58 program
package. The theory for the case of JKL was introduced by
Aucar and Oddershede59 and practical calculations for both
JKL and rK appeared in ref. 60. Calculations of BK were
reported in ref. 61. The treatment of relativity in the DHF
method can for most chemical purposes be considered exact,
albeit the question about the influence of the Breit correction
on NMR properties deserves further investigation. While a
marked downside of the DHF method is the large computa-
tional cost, the data obtained by this method are invaluable
for calibrating more approximate relativistic approaches.13,14
A lot of attention has been paid lately to the basis-set
dependence of the DHF calculations of rK.56,62,63 The basis
set of the small component of the wave function is larger than
that of the intuitively more important large component, and
different coupling schemes by kinetic balance exist for deriving
the former basis from the latter.62,63
Whereas a correlated fully relativistic ab initio formulation
of BK exists (see, e.g., ref. 61, 64 and 65), corresponding
calculations of rK and JKL are currently only emerging at
the DFT level.66,67 In particular for the spin–spin coupling this
is a very welcome advance, as DFT is much less susceptible to
triplet instability than HF, a tendency that has rendered DHF
calculations of quite limited applications interest for JKL. A
fully relativistic coupled-cluster implementation of the linear
response equations necessary for both rK and JKL would
provide the ultimate calibration level for other methods.
7.2 Approximate relativistic calculations
At the low end of the spectrum of relativistic methods for rKone finds the perturbational approaches, the Breit–Pauli
perturbation theory (BPPT)11,13,14 developed primarily in
my group and the linear response elimination of the small
component method68–70 due to Argentinian workers. These
methods employ the NR, spin-free reference wave function
and apply all magnetic and relativistic operators as perturba-
tions on equal footing. Relativistic contributions are included
up to complete leading order, and involve both passive and
active terms,11,13 where the former introduce relativistic influ-
ence on the wave function by magnetic-field-free mass-velocity
and Darwin operators, as well as by the SO interaction in the
well known SO shielding terms.71–76 The active terms, in
contrast, contain relativistically modified hyperfine or Zeeman
operators.
The BPPT analysis for r leads to a total of 16 relativistic
correction terms, 14 of which contribute to the isotropic s. The
Table 6 Atomic Hartree–Fock calculations of the nuclear shieldingconstants (in ppm) of the noble gases nonrelativistically (NR) andrelativistically (R)
Atom NRa R D/%
He 59.90 59.95b 0.1Ne 552.3 558.6b 1Ar 1238 1276b 3Kr 3246 3598b 11Xe 5642 7040b 25Rn 10 728 19 630c 83
a Ref. 54. b Ref. 55. c Ref. 56.
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terms can be formulated as expectation values as well as linear
and quadratic response functions. Characteristic of BPPT is
that each contribution finds a straightforward interpretation
in terms of analogous NR concepts. This can be seen as a
major benefit either in its own right or in the interpretation of
the more elaborate four-component data, where the relativistic
r only consists of a single linear response term. Another nice
feature of the method is that electron-correlated CC, MCSCF
or DFT models may readily be used as reference state for
BPPT, e.g., in the Dalton software.
Fig. 3 illustrates the relative error of HF-level BPPT calcu-
lations of the shielding constants s as calibrated against basis-
set converged DHF data in simple main-group systems.14 The
deviation remains acceptable up to the 5th row of the periodic
table. For systems containing heavier elements than Xe, the
perturbational approach breaks down, however, both for the
heavy center itself and for the 1H nucleus. Current investiga-
tions focus on the performance of BPPT for the experimentally
interesting 129Xe chemical shift both in molecules77,78 and
weakly bound complexes.79,80 Among the findings are that
the relative BPPT xenon shifts are more accurate than the
shielding constants, and that a significantly reduced set of
terms as compared to the full 16-term expansion, give the
essential dXe contributions due to the cancellation of the inner
shells.
Ref. 57 contains a recent discussion of the heavy-atom s and
d within the perturbational realm, focusing on the heavy-atom
effects on heavy-atom shielding (HAHA).81 Concerning hea-
vy-atom effect on the light atom (HALA) shieldings,72,75 the
dominating SO effects are known to arise from the (in the
perturbational sense) third-order terms where the coupling
between the OZ and FC or SD hyperfine interactions is
furnished by the SO operator:
sSO�IK / hhhFCK þ hSDK ; hOZB0; hSOii0;0: ð7:26Þ
An appealing qualitative argument relates this effect to the NR
JKL (ref. 75). The standard showcase in this context is the large
SO-induced 1H shielding in hydrogen halides, depicted in Fig.
4. Rovibrational effects thereto have been found to be respon-
sible for the majority of the remaining error at the MCSCF
level.83
Fig. 3 Difference in heavy-atom and 1H nuclear magnetic shielding
constants in H2X (X = O–Po), HX (X = F–At) and noble gas
(Ne–Rn) series, provided by the Breit–Pauli perturbation theory
approach applied at the Hartree–Fock level and basis-set converged
four-component Hartree–Fock calculations. Data from ref. 14.
Fig. 4 Calculated 1H nuclear shieldings in HX (X = F–I), including
spin–orbit (SO) interaction corrections. (a) Isotropic shielding con-
stant, (b) deviation from the experiment82 at various levels and
(c) shielding anisotropy Ds = s|| � s>. MCSCF data from ref. 76,
UHF and DFT from refs. 73 and 74, respectively. Reused with
permission from J. Vaara, K. Ruud, O. Vahtras, H. Agren and J.
Jokisaari, Journal of Chemical Physics, 1998, 109, 1212, Copyright
1998, American Institute of Physics.
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An important shortcoming of the BPPT method is that its
formulation for JKL leads to divergent operator combinations
in the Breit–Pauli/point-like nucleus realm.11 Another built-in
limitation is that relativistic effects can only be taken up to
leading order by BPPT. A family of variationally stable one-
and two-component approaches exist where these restrictions
are lifted: the zeroth-order regular approximation (ZORA)84
and the Douglas–Kroll–Hess (DKH)85 transformation. Parti-
cularly the former has found many chemical applications as
implemented in the ADF86 programme system. Currently
studies into an infinite-order formulation of the two-compo-
nent (IOTC) model are being pursued by different groups.87,88
This direction is likely to over time provide an appealing
alternative to both the demanding four-component calcula-
tions and the formally complicated transformed one- and two-
component models. Meanwhile, the ZORA and DKH meth-
ods combine relativistic effects with acceptable computational
effort at the DFT level. Correlated ab initio implementations
of magnetic properties at the IOTC level would be extremely
useful.
Concerning physics beyond Dirac relativity, quantum elec-
trodynamic effects on the NMR properties have been
discussed by Romero and Aucar.89 Tiny parity-violating
electroweak contributions to r and J arise in chiral systems.90
The results, e.g., in our work91 for CHFClBr and CHFBrI are
of the order of 2|s13C,19FPV| B 3 � 10�10 ppm and 2|J13C,19F
PV|
B 5 � 10�11 Hz. A recent methodological study92 assesses the
roles played by correlation, basis set and relativity therein.
8. Rovibrational effects
For a molecule undergoing small-amplitude motion only,
including the contributions of both zero-point vibrational
effects and thermally excited vibrational and rotational states
is conceptually rather straightforward. The property hypersur-
face can be averaged over the intramolecular motion of the
molecule as governed by the potential energy hypersurface,
hsiT ¼ se þXk
@s@Qk
� �e
hQkiT
þ 1
2
Xkl
@2s@Qk@Ql
� �e
hQkQliT þ . . . ; ð8:27Þ
where a Taylor series expansion of a property tensor compo-
nent (or a combination thereof, such as the shielding constant
s) in terms of a suitable set of internal coordinates, here the
vibrational normal coordinates {Qk}, has been thermally
averaged at T.
While the truncation error of the series at any given order
depends on the system, a general finding is that both the
first- and second-order terms should be retained.93 The latter
can to a good approximation be calculated using the harmonic
part of the intramolecular potential energy surface, but the
first-order term is obtained with the third-order force field
perturbing the harmonic oscillator wave function of the
normal modes.94,95 The contributions of rotational states can
be accommodated in this approach through centrifugal dis-
tortion terms.94 The use of a different coordinate system such
as symmetry coordinates or local valence coordinates is
facilitated by nonlinear transformations to the normal co-
ordinate basis,96 where the averages hQkiT and hQkQliTare readily obtained using established formulae.94
The fact that internal motion effects are relevant in accurate
quantum chemical calculations of NMR properties is demon-
strated by Table 7 reporting work by Auer et al.97 on 13C
shieldings in hydrocarbons. Improving the approximation for
the electronic structure problem by using more accurate
ab initio methods all the way up to CCSD(T) and increasingly
flexible basis sets, allows the statistical deviations from the
experimental results to decrease. It is not before considering
the zero-point vibrational (ZPV) contributions, however, that
a quantum leap to approaching the 1 ppm accuracy with
respect to gas-phase experimental data, is taken. This work
illustrates the necessity to maintain balanced approximations
for realistic computations of spectroscopic parameters. Taking
one part of the problem (such as the electron correlation
treatment) ad absurdum may be pointless unless the other
contributing physical effects are also accounted for.
We used eqn (8.27) to perform zero-point vibrational and
thermal rovibrational corrections to a gas-phase water mole-
cule using elaborate MCSCF property and potential energy
surfaces.98 Based on solely theoretical information, the work
improved the 17O absolute nuclear shielding scale to s= 324.0
� 1.5 ppm from the established 344.0 � 17.2 ppm by
Wasylishen et al.99 At 300 K, ZPV and thermal contributions
to sO in 1H217O amount to �11.66 and �0.38 ppm,98 respec-
tively, indicating that including these effects was mandatory
for the improved shielding scale. A new analysis of the
experimental data100 subsequently settled at 323.6 � 0.6
ppm, verifying our results.
To perform rovibrational averaging of property tensor
components which, in contrast to the isotropic shielding or
coupling constants, depend on the coordinate frame in which
they are represented, it is necessary to find the Eckart frame101
for each displaced geometry and use this frame for the
anisotropic property derivatives in eqn (8.27).98,102
Mapping out both the potential and property hypersurfaces
by single-point calculations at different geometries is rather
demanding due to the large amount of data to be handled. For
this reason, automated methods have been developed.97,103,104
The approach proposed by Ruud et al.103–105 involves finding
first the sc. ra(T) geometry (with the nuclei in their thermal
average positions) for the isotopomer and temperature in
question. The property value at this effective geometry
Table 7 Correlation, basis-set and zero-point vibrational effects inaccurate 13C shielding constant (in ppm) calculations on 20 hydro-carbons97
Method BasisMean dev.from expt.
Standard dev.from expt.
HF QZ2P �2.5 10.5MP2 QZ2P 6.1 2.0CCSD QZ2P 4.9 2.3CCSD(T) QZ2P 5.8 1.8
PZ 3d2f 4.7 1.313s9p4d3f 4.0 1.4
+ZPV corr. 13s9p4d3f 1.6 0.8
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includes the leading anharmonic vibrational contributions.95
The harmonic corrections are then obtained by performing
vibrational analysis and using a combination of analytic and
numerical differentiation to obtain the property derivatives at
this geometry. The method was used recently by us to settle the
long-standing question of the 2H quadrupole coupling
constant in deuterated benzenes.106 The use of the ra(T)
geometry instead of the equilibrium geometry as the expansion
point of the Taylor series (8.27) implies also reduced trunca-
tion error.
Besides the role in affecting the accuracy of quantum
chemical predictions, rovibrational effects also give rise to
particular phenomena and observables that cannot be
obtained to any accuracy using static calculations alone.
Isotopic substitution of molecules affects the rovibrationally
averaged NMR parameters. In particular the secondary iso-
tope effects, i.e., changes in the property of a nucleus induced
by isotopic substitution at another nucleus, are experimentally
feasible, and their modelling requires the equivalent of eqn
(8.27) to be applied for each distinct isotopomer. Fig. 5
compares the calculated secondary isotope effects on sC due
to substitution of different Se isotopes in CSe2 (ref. 107). While
the correct trend over the different isotopomers is obtained
using NR property surface, an accurate reproduction of the
experiment108 is only achieved after considering the relativistic
SO effect in the property derivatives. This is due to the fact
that the triplet excitation energy decreases with the bond
extension in this system,83 rendering the SO mechanism in
rC increasingly efficient. This serves as an example of a
coupling of relativistic and rovibrational effects.
Finally, large-amplitude intramolecular motions are beyond
the domain of applicability of Taylor series expansions of
the kind of eqn (8.27). Rovibrational averaging of NMR
parameters over such degrees of freedom can be performed
with the help of statistical sampling of configurations.
9. Solvation and intermolecular interaction effects
The methods for taking into account environmental influences
on NMR parameters include various continuum solvation
approaches109,110 where the investigated molecule is located
in a cavity immersed in a continuous medium. The electric
field arising from the multipole moments of the molecule
polarises the medium, causing secondary fields to be induced
back at the molecule and resulting in modification of its NMR
properties. While useful for many particularly ‘‘global’’ prop-
erties such as polarisability, the continuum approaches fail to
capture the short-range specific solvation effects that are
sensitively reflected in NMR quantities. Often the application
of the continuum models results in modifications of NMR
parameters that are much smaller than the observed changes
between different solvents, and the predictions may even have
the wrong sign. Hydrogen bonding is one problematic short-
range mechanism that is beyond the capabilities of the con-
tinuum models.
To remedy this shortcoming, supermolecule calculations
can be performed on the static complexes of the investigated
molecule and one or more suitably placed, explicit solvent
molecules. While useful NMR calculations can be performed
this way, a representative placement of the solvent molecules
requires chemical insight on the microscopic nature of solva-
tion. More importantly, dynamical effects are neglected.
The most satisfactory method for treating solvation is the
‘‘snapshot-supermolecule’’ approach where configurational
sampling is carried out by molecular dynamics (MD) or
Monte Carlo (MC) simulation and instantaneous configura-
tions are subjected to NMR property calculations. The meth-
od captures explicit short-range solvation including dynamic
effects. The instantaneous supermolecular system can even be
placed in a dielectric cavity to incorporate long-range electro-
static influences. This approach has been largely pioneered by
Huber and co-workers111 and can be pursued using empirical
interatomic potentials or first principles simulation methods,
such as the Car–Parrinello plane wave technique.112 The
NMR property calculation step may be performed in the same
software in which the snapshots are generated113,114 or can be
made into a distinct phase. Statistical averaging over snap-
shots allows error margins to be assigned to the calculated
observables.
Our own work in the field includes investigation of liquid
water at ambient conditions for which we performed a
Car–Parrinello MD simulation115 that allowed us to average
the full rK and quadrupole coupling tensors. The trajectory
was sampled by cutting from snapshots roughly spherical
clusters centred around a randomly chosen central molecule
(Fig. 6). These supermolecules were subjected to NMR calcu-
lation with continuum solvation. For averaging anisotropic
properties, the Eckart frame had to be used to represent the
instantaneous tensor components, as referred to above. Com-
parison with corresponding simulation of the gas-phase
molecule allowed the gas-to-liquid shifts of the full tensor to
be obtained in contrast to the rotationally invariant isotropic
observables only (Fig. 7).
In further work116 we combined the NMR properties of
instantaneous liquid water configurations with analysis of the
Fig. 5 Comparison of the calculated107 and experimental108 second-
ary one-bond isotope shifts on 13C shielding for CSe2 at 300 K. The
reference is the 76SeQ13CQ78Se isotopomer. The calculated shifts are
presented both at the nonrelativistic (NR) level and with spin–orbit
corrections (NR + SO). Reprinted with permission from ref. 107,
Copyright 2002, American Chemical Society.
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hydrogen-bonding situation. One resulting finding is that the
statistical property distributions for the differently hydrogen-
bonded water species have significant overlap. In such a
classification, the gradual evolution of the NMR tensor
components from the gas-phase situation to first two- and
finally all the way to six-coordinated water molecules can be
followed. The four-fold coordinated instances are the most
abundant as determined by the Car–Parrinello simulation
model.
Another application of supermolecule NMR calculations
has been to investigate 129Xe shielding in low-pressure gas
phase.117 In such conditions, the virial expansion118
sðr;TÞ ¼ s0 þ s1ðTÞrþ s2ðTÞr2 þ � � � ð9:28Þ
s1ðTÞ ¼ �4pZ 10
dðrÞexp½�VðrÞ=ðkTÞr2 dr ð9:29Þ
applies, where the one-body term s0 includes no temperature
dependence for an atomic monomer, and the following two-
and three-body terms etc. feature an increasing power of the
number density of the medium, r. Jameson et al.119 have
provided temperature dependence data for the second virial
coefficient s1(T) in low-pressure Xe(g), making it probably the
most accurately experimentally characterised intermolecular
interaction effect in NMR. Eqn (9.29) represents a semiclassi-
cal approximation to s1(T), justified as the quantum dynami-
cal effects of a system as heavy as Xe2 are very small at the
temperature range of the experiment (Fig. 8).
d(r) in the expression of s1(T) is the binary chemical shift
function
dðrÞ ¼ sðfree XeÞ � sXe2ðrÞ ð9:30Þ
and V(r) is the interatomic potential energy function. Using
CCSD theory for d(r) and either an empirical V(r) (ref. 121) or
one from our own CCSD(T) calculations featuring core-
polarisation potential corrections and mid-bond basis
functions,117 leads to a rather good agreement with the
experimental data. Ongoing work79 addresses higher-order
correlation contributions, relativistic effects and the cross-
coupling of the two, in d(r).Finally, the shortcomings of the snapshot-supermolecule
method include the necessity of using quite large clusters or,
Fig. 7 Average 1H nuclear shielding tensors (in ppm) of water at 300
K in gas and liquid states.115 The snapshot-supermolecule method was
used with a Car–Parrinello simulation trajectory and B3LYP-level
property calculation step.
Fig. 8 Experimental119,120 and computational117 temperature depen-
dence of the second virial coefficient of 129Xe nuclear shielding in
low-pressure Xe gas. The lower panel illustrates the deviation of
various computational approximations from the experiment. The
notation in the legend denotes methods used for binary shielding/
potential energy. CP denotes counterpoise correction. Hartree–Fock
(HF), complete active space self-consistent field (CAS), second-order
Møller–Plesset many-body perturbation theory (MP2) and coupled-
cluster singles and doubles (CCSD) theories were used for the binary
chemical shift. The potential energy surface is either the empirical
Aziz–Slaman potential (Aziz86)121 or from CCSD with noniterative
triples [CCSD(T)] calculations where relativistic effective core poten-
tials (RECP), core-polarisation potentials (CPP) and mid-bond basis
functions (BF) were used. Reused with permission from M. Hanni,
P. Lantto, N. Runeberg, J. Jokisaari and J. Vaara, Journal of Chemical
Physics, 2004, 121, 5908. Copyright 2004, American Institute of
Physics.
Fig. 6 Instantaneous water cluster sampled from a Car–Parrinello
molecular dynamics simulation.115 In the snapshot-supermolecule
method for calculating the solvation effects on NMR properties, such
snapshots are fed into a quantum chemical property calculation step
for the central molecule. The results are subsequently statistically
averaged over the selection of snapshot and central molecule.
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in the case of periodic NMRmethods,113,114 simulation cells as
required by explicit solvation. Typically a large number of
snapshots are needed for statistically meaningful results. This
restricts the choice of quantum chemical model typically to
HF or DFT that do not contain dispersion interactions in their
standard formulation. The advances in speeding up ab initio
correlated calculations are likely to improve the situation.
Finite systems, molecules and clusters, constitute a natural
application field for the snapshot-supermolecule method,
enabling rather accurate calculations. In the method, classical
nuclear dynamics is usually considered, causing particularly
the intramolecular dynamic effects to be only quite approxi-
mately captured. The use of path-integral simulations122 in the
snapshot-supermolecule context is thinkable, although quite
expensive.
10. Magnetic-field dependence of NMR
parameters
It is natural from the identification of HNMR with the corre-
sponding series expansion of the true molecular energy that
includes dependence on B0 to infinite order, to realize that also
NMR parameters are magnetic-field dependent.123 With the
increasing available field strengths, the question of the magni-
tude of the magnetic-field effects in NMR parameters becomes
timely.
We presented a theory for the effect in rK in ref. 124.
Requiring energy terms to be time-reversal invariant, lead-
ing-order expansion of rK obtains a quadratic dependence
on B0:
sK ;et ¼ sð0ÞK ;et þXgd
tK ;etgdB0;gB0;d þ . . . ð10:31Þ
s ¼ sdiaþ sdia�paraþ spara�diaþ spara; ð10:32Þ
where the four-index tensor setgd parametrising the field
dependence consists of four terms124 attributed as dia, dia-
para, para-dia and para, in analogy with the common desig-
nations employed in conventional rK or magnetisability. The
terms feature the PSO and DS operators [eqn (2.4)] with
additional powers of B0 arising in multiple occurrences of
the OZ operator. Whereas only the dia-term contributes in
spherically symmetric systems such as in noble gas atoms,125 in
the molecular case one has to use the full expression (10.32)
with many terms. In particular, the isotropic rotational aver-
age of s can be written as
t ¼ 1
15ð3tzzzz þ 3txxxx þ 3tyyyy þ tzzxx þ tzzyy
þ txxzz þ tyyzz þ txxyy þ tyyxx þ tzxxz þ tzyyzþ tzxzx þ tzyzy þ txzzx þ tyzzy þ txzxz þ tyzyzþ tyxxy þ txyyx þ tyxyx þ txyxyÞ: ð10:33Þ
Experimentally the dependence of s on B0 was reported by
Bendall and Doddrell126 for two cobalt complexes (such were
already suggested by Ramsey in his early paper123). These and
our calculated results124 for the same systems are listed in
Table 8. The calculations indicate one or two orders of
magnitude smaller a field dependence than the work of ref.
126, in part probably due to the fact that in these systems the
overwhelmingly dominating para term could only be calcu-
lated at the HF level due to software limitations at the time.
The question of the field dependence of s remains an open and
interesting one, and already the magnitude of the effect pre-
dicted by us raises hope for experimental verification.
The magnetic-field dependence of BK is also of the quad-
ratic leading-order type,127
BK ;et ¼ Bð0ÞK;et þ
1
2!
Xgd
Bð2ÞK ;etgdB0;gB0;d þ . . . ð10:34Þ
Bð2Þ ¼ Bð2Þ;diaþBð2Þ;para; ð10:35Þ
and this effect has already been experimentally observed for
atomic 131Xe in isotropic gas and liquid phases128 (Fig. 9).
Theoretical analysis as well as quantitative modelling were
published in ref. 129. While the effect in atomic Xe is rather
independent of electron correlation, a significant relativistic
influence is included in the theoretical result for the width of
the quadrupole splitting pattern (with only the dia term
contributing) 2nQ = 14.842 mHz T�2 to be compared with
the experimental result of �14 mHz T�2 (ref. 128). Predictions
for the other quadrupolar noble gases as well as for mole-
cules127 indicate that the phenomenon should be experimen-
tally observable in other systems in the near future.
Table 8 Experimental126 and computed124 magnetic-field-depen-dence (in 10�3 ppm T�2) of isotropic 59Co nuclear shielding constantin two cobalt complexes
Molecule Theory Expt.
[59Co(NH3)6]3+ �2.53 �70(30)
59Co(acac)3 �6.03 �80(50)
Fig. 9 Experimental128 and theoretical129 magnetic-field-dependent
width of the 131Xe quadrupole splitting pattern of atomic Xe in
isotropic gas and liquid phases. Reprinted from ref. 129 with permis-
sion of the American Physical Society.
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11. Laser-field-induced NMR splitting
Lasers can be used in NMR research to, e.g., indirectly spin-
polarise He or Xe nuclei and thereby obtain a much increased
signal-to-noise ratio.130 The effects of direct laser excitation of
the investigated system during the NMR experiment itself
constitute a theoretically exciting subject. Shining left (�) orright (+) circularly polarised laser (CPL) light causes a
magnetic field along the beam in all kinds of matter.131 At
the position of nucleus K the Cartesian a component of the
field can be written as
B�K ;a ¼1
2o
Xbg
bKabgðE�b _E�g �E�g _E
�b Þ: ð11:36Þ
Here, E and _E are the electric field component of the laser
beam and its time derivative, respectively, and o is the
frequency of the beam. The coefficient bK can be calculated
using time-dependent third-order perturbation theory as132
bKabg ¼1
2ImhhhPSOK ;a ; rb; rgiio;�o; ð11:37Þ
which suggests a phenomenological interpretation of the
effect: the beam interacts with the orbital motion of the
electrons by the electric dipole interaction �r to second order.
This effect is mediated to the nucleus by the orbital hyperfine
interaction.
The effect is quadratically dependent on the intensity I0 of
the beam and can be analysed with a new term in HNMR
(ref. 133)
H�NMR ¼ �1
4p½Eð0Þ2
XK
gKbKZXYIK ;Z; ð11:38Þ
where E(0) is the amplitude of the electric field component.
Upon rapid switching of the left and right CPL, H�NMR
corresponds to a splitting of all spectral lines by
D=I0 ¼1
4pce0gKb
KZXY : ð11:39Þ
We reported133 MCSCF calculations of D/I0 as a function of
the frequency o, for atomic Ne, Kr and Xe (Fig. 10). While the
calculated splitting is tiny with intensities (ca. 10 W cm�2 of
the conventionally attainable laser sources, the effect increases
towards the heavier, more polarisable systems. Furthermore,
there is a rapid increase of D by orders of magnitude when oapproaches an optical resonance. To facilitate experimental
observation, highly polarisable systems with low-lying excited
states should be considered.
12. Chemical shift in open-shell systems
NMR of paramagnetic, open-shell molecules (pNMR) is
widely used in studies of large systems particularly in bio-
sciences.134 pNMR often complements the information that is
available from electron spin resonance (ESR) spectroscopy,
particularly when the studied nuclei have small hyperfine
coupling and/or the system is of integer spin type.
Until lately, computations of pNMR d have been at a rather
elementary level when compared to the advances made in the
conventional NMR of closed-shell systems. The theory
involves in the pNMR case a thermally excited manifold of
the spin Zeeman levels of the unpaired electron, in contrast to
the pure state appropriate for the Ramsey theory17 for the
closed-shell situation. The thermal populations of the spin
Zeeman levels, and consequently the average spin polarisation
are determined by B0. On the other hand, hyperfine interac-
tions provide the required coupling with {IK}. Any low-lying
excited electronic states also have to be included.
For the doublet (S = 1/2) situation and in systems with no
thermally accessible excited states, ref. 135 arrives at the
expression
rK ¼ rorbK� 1
gK
mB3kT
SðS þ 1Þg �AK ; ð12:40Þ
where the orbital contribution is formally similar to the NR rK
in closed-shell systems, except that this time singly occupied
spin orbitals are considered. d arising from the orbital term is
often approximated by the experimental shift in a ‘‘corre-
sponding’’ diamagnetic compound. This approach may lead to
significant error.136
The second, temperature-dependent term in eqn (12.40)
includes the hyperfine shift contributions. In there, mB is the
Bohr magneton and use is made of the parameters of the ESR
spin Hamiltonian, the g and the hyperfine coupling tensors,
g and AK, respectively, to parametrise the Zeeman and
hyperfine interactions. Table 9 presents a break-down of the
product g �AK137 that we currently use in application calcula-
tions of pNMR d. The first two terms correspond to the
leading-order theory, and include the interaction of the NR
hyperfine coupling with the free-electron part of the g tensor.
The first systematic calculations of the pNMR shifts were
carried out by Rinkevicius et al.136 including these two terms
for the temperature-dependent part, as well as rorbK . DFT
results for a couple of nitroxide radicals are compared to the
experiment138 in Fig. 11. It is evident that useful predictions
are enabled by the method for main-group radical systems.
Fig. 10 Calculated133 circularly polarised laser-induced splitting of
NMR lines per unit of beam intensity in atomic 21Ne, 83Kr and 129Xe,
as a function of laser beam frequency. Reprinted from ref. 133 with
permission of Elsevier Science Publishers.
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The further terms in the expansion presented in Table 9
arise from relativistic SO contributions, either causing con-
tributions toA (as introduced in ref. 137) or deviation of the g
tensor from the free-electron value,
Dg ¼ g� ge1 ¼ Dgiso1� D~g; ð12:41Þ
where 1 is the 3 � 3 unit matrix. The further subdivision is to
the isotropic part Dgiso1 and the anisotropic part Dg. Among
the SO-induced terms is the anisotropic contact term DgAcon
and the isotropic pseudocontact term, Dg �Adip. The latter is
of particular experimental interest as the corresponding shift
contribution of light ligand nuclei can be used to approxi-
mately determine the distances of these nuclei from the metal
centre, in organometallic systems where the spin density is
thought to be localised at the metal.
Table 10 lists DFT results137 with the PBE functional139 for
the isotropic 13C and 1H shift as well as Ds for 13C in
metallocenes X(C5H5)2 (X = Co, Rh, Ir). The results indicate
significant shift contributions from the SO terms, with in
general increasing magnitude towards the heavier metallo-
cenes. Not only the metal-centred pseudocontact term
Dg �Adip is important, but there are marked contributions
from the geAcon and DgisoAcon terms as well. This points to
the inadequacy of the approximation of metal-centred spin
density in these systems.
While the accuracy of the calculations is somewhat limited
by the available exchange–correlation functionals, the method
is ready for applications on large doublet systems. Hrobarik
et al.145 presented an extension of the theory to axial systems
of higher than doublet spin multiplicity. The effects introduced
to the electronic Zeeman level structure by the zero-field
splitting Hamiltonian must naturally be considered in this
case.
13. Noble gas NMR
13.1 Guest noble gases in various host materials
Significant applications interest lies in the NMR of atomic
noble gases, 3He, 21Ne, 83Kr and 129/131Xe. These isotopes can
be used as inert NMR probes introduced to different host
materials, gases, liquids, liquid crystals (LCs) and solids.146
The sensitivity of particularly 129/131Xe spectra to the micro-
structure of the environment allows detailed studies of the
density, temperature, orientational and positional order, as
well as the nature of adsorption sites or cavities, including the
shape and size of the latter.
Electronic structure aspects such as electron correlation and
the treatment of relativistic effects determine only partially the
adsorption shifts and quadrupole couplings. The dynamics of
the guest atoms as well as that of the host material itself, are
essential for the reproduction of the experiment. Modelling the
implied large-amplitude motion necessitates sampling of the
configurational space typically either by MD or MC methods,
after which, e.g., the snapshot-supermolecule method can be
applied (section 9). The second and less costly alternative is to
preparametrise an ‘‘NMR force field’’ consisting of the effects
of solvent molecules or general structural elements of the host
material on the NMR property of the guest atom. In the
analysis of the simulation, such interaction contributions are
summed in a pairwise additive fashion to provide the total
interaction effect for a given snapshot. In this approach, the
analysis of the simulation only involves combining the NMR
force field with the positional information for the atoms. This
method has been pioneered in the context of Xe NMR by
Jameson et al.147
An example of a simple application of NMR force field is
the MD study of 129Xe chemical shifts when adsorbed in the
one-dimensional channels of the AlPO4-11 molecular sieve.148
In this system it is essential to consider the host dynamics, as
the access between adjacent cavities (each capable of housing
one Xe adsorbate at the time) of the channel requires dyna-
mical opening of the intermediate necks. The oxygen atoms
define the cavity as seen by the Xe guest, by pointing inwards
into the cavities. The NMR force field consists in this case of
only the chemical shift effect on the guest from the oxygen
atoms of the host cage. Quantum chemical calculations of the
isotropic dXe in interaction with the water molecule were used
to preparametrise the Xe–O NMR force field
dXeðrÞ ¼ sfree XeðrÞ � sXe���OH2ðrÞ; ð13:42Þ
with a corresponding form used for the Xe–Xe interactions
between adsorbates located in successive cavities. Fig. 12
illustrates the results for three distinct cases of dXe from the
Fig. 11 Correlation between calculated136 and experimental138 para-
magnetic 13C chemical shifts in nitroxide radicals. Reprinted from ref.
135, with permission of Wiley-VCH.
Table 9 Break-down of the product g �A appearing in the tempera-ture-dependent hyperfine part of the NMR shielding tensor in para-magnetic systems137
Term Interpretation
geAcon1 + geAdip Nonrelativistic A with thefree-electron g factor
+ geAPC1 + geAdip,2 + geAas Spin–orbit correction to A withthe free-electron g factor
+DgisoAcon1 + DgisoAdip Nonrelativistic A with theisotropic g shift
+DgAcon + Dg �Adip Nonrelativistic A with theanisotropic part of the g shift tensor
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point of view of the occupancy of the neighbouring cavities by
the adsorbates. The different first-principles parametrisations
of the NMR force field account well for the experimental trend
of ref. 149. While DFT with the GGA functional (BPW91)
agrees well with the experiment, LDA and HF-level parame-
trisations over- and underestimate dXe, respectively.
It is challenging to apply the concept of the NMR force field
to anisotropic NMR interactions necessary in the analysis of sor B for Xe dissolved in LCs.150 An additional complication
arises from the fact that a large number of LC molecules is
necessary for reliable simulation of the ordered phases. This
largely precludes using atomistic models for, e.g., the tempera-
ture and phase dependence of the NMR of solute atoms.
The coarse-grained Gay–Berne model151 consists of ellip-
soidal particles that interact by anisotropic repulsion–disper-
sion forces. Its particular parametrisation152 has been found to
reproduce the phase sequence Cr–Sm-A–N–I, i.e., molecular
crystal, smectic-A, nematic and isotropic phases, respectively.
While the N phase has orientational order with respect to an
optical axis (director), the lower-temperature smectic-A phase
also features positional order demonstrated by the appearance
of molecular layers in the direction perpendicular to the
optical axis. Finally, the Cr phase contains positional order
also within the layers. We applied the interaction between
Gay–Berne particles and spherical solutes153 in constant tem-
perature and pressure (NPT) MC simulations of the 129/131Xe
NMR of atomic Xe dissolved in a model LC.154
The single LC molecule–Xe interaction energy as well as
interaction effects on the components of rXe and BXe were
quantum chemically parametrised using supermolecule calcu-
lations of Xe interacting with an atomistic LC model, designed
to match the parametrisation of the Gay–Berne particle (Fig.
13). To enforce cylindrical symmetry, the calculated results
were Boltzmann-averaged around the long axis of the LC
Fig. 12 Calculated isotropic chemical shifts of 129Xe adsorbed in
AlPO4-11 molecular sieve at 300 K (ref. 148). The cases where the Xe
occupying the adsoption cavity has no, one or both neighbouring
cavities also occupied by a xenon atom, are considered separately.
Statistical error bars from the analysis of the simulation trajectory are
shown. The experimental data are from ref. 149. Reused with permis-
sion from J.-H. Kantola, J. Vaara, T. T. Rantala and J. Jokisaari,
Journal of Chemical Physics, 1997, 107, 6470. Copyright 1997,
American Institute of Physics.
Fig. 13 Atomistic and Gay–Berne models of a generic thermotropic
liquid crystal molecule. The local coordinate system appropriate to the
interaction with guest Xe atom, is also shown. Reprinted from ref. 154
with permission of the American Physical Society.
Table 10 Calculated137 13C and 1H chemical shifts (in ppm, wrt. TMS) and 13C shielding anisotropies Ds (ppm) for metallocenes X(C5H5)2(X = Co, Rh, Ir) at 298 K
Physical origin
13C chemical shift 13C shielding anisotropya 1H chemical shift
Co Rh Ir Co Rh Ir Co Rh Ir
Orb. 101.4 103.1 100.9 83.2 89.2 85.3 4.8 5.5 5.7geAcon 523.5 581.8 682.3 — — — �63.4 �93.9 �90.1geAdip — — �839.0 �1295.3 �1266.3 — — —geAPC �11.4 �3.9 �2.7 — — — �0.1 �0.1 �0.1geAdip,2 — — 2.7 �1.6 �2.3 — — —DgisoAcon �8.9 �29.0 �46.4 — — — �1.1 4.7 6.1DgisoAdip — — 14.3 64.5 86.1 — — —Dg �Adip �18.5 �50.7 �48.2 27.8 76.1 46.5 �0.1 �0.3 �0.4DgAcon — — — 52.0 102.6 81.3 — — —Total 586.1 601.4 685.9 �658.9 �964.6 �969.4 �57.8 �84.2 �78.7Expt. 541b–618c �443c B55cd
a Ref. 140. The anisotropy contributions listed in the original article137 contain slight numerical inaccuracies that are corrected here.b Ref. 141. c Ref. 142. d Ref. 143 and 144.
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molecule. The model for the parametrisation of the NMR
tensor components takes into account the anisotropic shape of
the Gay–Berne particle and features a dependence on the polar
coordinate y that enforces the correct site symmetry of the
shielding tensor. NMR interaction contributions evaluated in
a local coordinate frame of one solvent–solute pair were
transformed to and summed over in a global frame defined
by the optical axis of the phase.
The simulated hszzi is shown in Fig. 14 as a function of the
reduced temperature T* in the Gay–Berne unit system. Many
features found in the experimental studies150,155 are repro-
duced by the simulations and correspond to the phenomen-
ological interpretations155,156 in terms of the T-dependence of
medium density and orientational order: linear temperature
dependence in I, jump to larger interaction at the onset of
orientational order in N (with nonlinear shift evolution reflect-
ing the behaviour of the order parameter) and no discontinuity
but a change of trend at the Sm-A/N transition. The decrease
in the layered Sm-A phase results from a partial expulsion of
Xe from within the molecular layers to the less dense inter-
layer regions. Finally, a large jump is seen upon entering the
Cr phase.
Despite the qualitative success, the overall interaction-
induced shift is clearly underestimated as compared to the
experimental results that are about �190 ppm. At the same
time, the simulated changes upon phase transitions and within
the temperature ranges of the ordered phases are exaggerated
in comparison to real LCs, implying a need to further develop
the LC model.154
13.2 Noble gas compounds
Noble gas compounds represent a frontier of chemistry with
new paradigms of chemical bonding appearing.157,158 An
NMR modelling interest exists in providing predictions of
the spectral parameters to guide future attempts at NMR
characterisation. The chemical shift range of 129/131Xe in
compounds is large and typically both electron correlation
and relativistic effects are significant. Consequently, these
applications provide a challenging methodological test for
the computational tools.
We recently carried out a study of BXe in a series of XeMF
(M = Cu, Ag, Au) complexes64 that were earlier experimen-
tally characterised by Gerry et al.159 Fig. 15 illustrates the
results at different computational levels. It is seen that while
neither the trend or magnitude of the experimental data are
reproduced at the NR level, applying the relativistic DHF
method corrects for the trend and including correlation as well
is necessary to obtain the correct magnitude. The correlation
effect at the fully relativistic DMP2 level is very significantly
larger than at the NR MP2 level, which reveals the presence of
important non-additivity of relativistic and correlation effects.
In contrast to electric properties, for r and other magnetic
properties no fully relativistic, correlated ab initio methods are
currently available. In many NMR applications this is not a
serious problem due to the possibility of with useful accuracy
to carry out, e.g., ZORA DFT calculations.84 Ongoing studies
of xenon fluorides77 as well as the prototypic organo-xenon
compound HXeCCH78 reveal significant deficiencies in the
performance of DFT, however, in particular the NR part of
rXe, as judged against, e.g., the CCSD(T) theory. While
awaiting for better tools to become available for these difficult
cases, a pragmatic approach for realistic prediction of, say, dXe
may be found by piecewise approximations where terms
accounting for relativistic and correlation effects separately,
Fig. 14 Simulated temperature dependence of the observable nuclear
shielding hrzzi of atomic 129Xe guest atoms dissolved in a coarse-
grained model thermotropic liquid crystal. I, N, Sm-A and Cr corre-
spond to the isotropic, nematic, smectic-A and molecular crystal
phases of the system, respectively. Reprinted from ref. 154 with
permission of the American Physical Society.
Fig. 15 Calculated64 and experimental159 131Xe nuclear quadrupole
coupling constants in XeMF (M = Cu, Ag, Au). Results obtained at
various nonrelativistic [HF, MP2, CCSD(T)] levels of theory are
compared to relativistic uncorrelated (DHF) and correlated (DMP2)
calculations, as well as calculations where correlation effects evaluated
at the nonrelativistic level [DMP2 and DCCSD(T)] are added on top of
DHF data. Reused with permission from P. Lantto and J. Vaara,
Journal of Chemical Physics, 2006, 125, 174315. Copyright 2006,
American Institute of Physics.
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as well as their cross-coupling are summed, and the effects
of intramolecular dynamics and solvation are added at a
manageable level.
14. Outlook
The field of NMR modelling has undergone an explosive
growth since the early 1990s as nonrelativistic ab initio and
DFT methodologies have matured. The phenomenology and
partially also practical methods for taking the calculations of
static NMR parameters beyond the ‘‘standard’’ level of con-
sidering the electronic structure problem alone, i.e., to addi-
tionally consider relativistic effects, rovibrational and more
generally thermal motion, as well intermolecular interactions
have been advanced, and this Perspective has attempted to
give a highly personal viewpoint on these issues.
Considering the practical importance of NMR in various
disciplines, its rich phenomenology and the methodological
challenges it offers for computational science, it is well-moti-
vated to continue investing in fundamental methods develop-
ment to resolve the remaining deficiencies of the
computational NMR toolbox. Concerning highly accurate
calculations for heavy-element systems, it is imperative to
obtain fully relativistic, correlated ab initio methods for calcu-
lating magnetic properties. They are necessary for the calibra-
tion of less expensive approaches with a more approximate
treatment of relativistic and/or electron correlation effects,
e.g., by using the various quasirelativistic Hamiltonians and/
or DFT, respectively.
Investigations of intermolecular interaction effects such as
solvation in molecular sciences and particularly modelling the
NMR of continuous solid state materials, greatly benefits from
methods employing periodic boundary conditions. Such
approaches are emerging113,114 and, e.g., a method for recon-
structing the property contributions of the core electrons in the
context of periodic pseudopotential calculations has been
forwarded.160 Both the extensive technology that is available
in the quantum chemical community as well as the local
character of the hyperfine interactions responsible for NMR,
render it in the author’s opinion preferable to apply periodic
methods on the basis of localised Gaussian orbitals.
The application of correlated ab initio methods becomes
possible for molecules of increasing size due to both parallel
and linear-scaling algorithms. Nevertheless, DFT is likely to
remain the tool of choice for frontier research at least in terms
of the system size. The applicability of DFT is limited by the
lack of generally well-performing exchange–correlation func-
tionals for NMR calculations, on the one hand, of systems
containing different elements in the p-block of the periodic
table or, on the other hand, of both the main-group and
d-elements, or of both heavy and light elements. The vigorous
activity in the development of new functionals will undoubt-
edly materialise also in improved NMR applications in the
future.
Most NMR computations are concerned with the static
spectral parameters calculated from first principles. The data
are compared to the results of the analysis of experimental
spectra in terms of the spin Hamiltonian. This way the
experimentalists and computational scientists meet at no
man’s land, equally removed from each other’s home territory
of recording spectra on the one hand and applying electronic
structure methods on the other. An interesting deviation from
this modus operandi was demonstrated in the ESR framework
by Barone et al.161 They have combined electronic structure
studies as well as empirical modelling of the small- and large-
amplitude nuclear motion with Liouvillian techniques for
establishing the spectral density, and hence gone all the way
to the actual ESR spectra. Intimately connected to this spirit is
the modelling of NMR relaxation from first principles, a field
that is still in its infancy. Such work necessarily combines
electronic structure calculations for snapshots and/or quan-
tum chemical parametrisations of the NMR force field
(vide supra) with modelling of long time-scale dynamics as
well as taking into account cross-relaxation effects.
Despite this list of methodological and phenomenological
questions that still require theory and method development,
the current computational NMR toolbox already allows large-
scale materials and biochemical problems to be tackled.
Acknowledgements
I am thankful to all my past and present co-workers and
collaborators. The Academy of Finland is thanked for the
Academy Research Fellowship. My group belongs to the
Finnish Center of Excellence in Computational Molecular
Science (CMS). Further financial support from the Emil
Aaltonen Foundation is gratefully acknowledged. The com-
putational facilities have over the years been partially provided
by the Center for Scientific Computing (CSC, Espoo, Finland)
and the Materials Grid (M-Grid) consortium.
References
1 A. Abragam, The Principles of Nuclear Magnetism, Oxford Uni-versity Press, Oxford, 1961.
2 C. P. Slichter, Principles of Magnetic Resonance, Springer-Verlag,Berlin, 1990, 2nd edn.
3 HNMR is usually expressed in frequency units. SI-based atomicunits are employed throughout the article.
4 P. L. Corio, J. Magn. Reson., 1998, 134, 131.5 T. Helgaker, M. Jaszunski and K. Ruud, Chem. Rev., 1999, 99,293.
6 Calculation of NMR and EPR Parameters: Theory and Applica-tions, ed. M. Kaupp, V. G. Malkin and M. Buhl, Wiley-VCH,Weinheim, 2004.
7 R. E. Moss, Advanced Molecular Quantum Mechanics, Chapmanand Hall, London, 1973.
8 J. E. Harriman, Theoretical Foundations of Electron Spin Reso-nance, Academic, New York, 1978.
9 R. McWeeny, Methods of Molecular Quantum Mechanics, Aca-demic, London, 1992, 2nd edn.
10 The Coulomb gauge is used for the vector potential of the static,external magnetic field B0.
11 J. Vaara, P. Manninen and P. Lantto, in Calculation of NMR andEPR Parameters: Theory and Applications, ed. M. Kaupp, V. G.Malkin and M. Buhl, Wiley-VCH, Weinheim, 2004, p. 209.
12 DALTON, a Molecular Electronic Structure Program, Release2.0 (2005), see http://www.kjemi.uio.no/software/dalton/dalton.html.
13 P. Manninen, P. Lantto, J. Vaara and K. Ruud, J. Chem. Phys.,2003, 119, 2623.
14 (a) P. Manninen, K. Ruud, P. Lantto and J. Vaara, J. Chem.Phys., 2005, 122, 114107; (b) P. Manninen, K. Ruud, P. Lanttoand J. Vaara, J. Chem. Phys., 2006, 124, 149901E.
15 J. Olsen and P. Jørgensen, J. Chem. Phys., 1985, 82, 3235.
5416 | Phys. Chem. Chem. Phys., 2007, 9, 5399–5418 This journal is �c the Owner Societies 2007
Dow
nloa
ded
by M
cMas
ter
Uni
vers
ity o
n 10
/05/
2013
13:
25:0
4.
Publ
ishe
d on
06
July
200
7 on
http
://pu
bs.r
sc.o
rg |
doi:1
0.10
39/B
7061
35H
View Article Online
16 J. Olsen and P. Jørgensen, inModern Electronic Structure Theory,ed. D. R. Yarkony, World Scientific, New York, 1995, p. 857.
17 N. F. Ramsey, Phys. Rev., 1950, 78, 699.18 J. Ci�zek and J. Paldus, J. Chem. Phys., 1967, 47, 3976.19 R. H. Contreras and J. C. Facelli, Annu. Rep. NMR Spectrosc.,
1993, 27, 255.20 R. Bauernschmitt and R. Ahlrichs, J. Chem. Phys., 1996, 104,
9047.21 V. G. Malkin, O. L. Malkina and D. R. Salahub, Chem. Phys.
Lett., 1994, 221, 91.22 O. L. Malkina, D. R. Salahub and V. G. Malkin, J. Chem. Phys.,
1996, 105, 8793.23 (a) A. D. Becke, Phys. Rev. A, 1988, 38, 3098; (b) C. Lee, W. Yang
and R. Parr, Phys. Rev. B, 1988, 37, 785.24 (a) A. D. Becke, J. Chem. Phys., 1993, 98, 5648; (b) P. J. Stephens,
F. J. Devlin, C. F. Chabalowski and M. J. Frisch, J. Phys. Chem.,1994, 98, 11623.
25 P. Lantto, J. Vaara and T. Helgaker, J. Chem. Phys., 2003, 117,5998.
26 J. Kaski, J. Vaara and J. Jokisaari, J. Am. Chem. Soc., 1996, 118,8879.
27 J. Kaski, P. Lantto, J. Vaara and J. Jokisaari, J. Am. Chem. Soc.,1998, 120, 3993.
28 J. Vaara, J. Kaski and J. Jokisaari, J. Phys. Chem. A, 1999, 103,5675.
29 (a) R. D. Wigglesworth, W. T. Raynes, S. Kirpekar, J. Odder-shede and S. P. A. Sauer, J. Chem. Phys., 2000, 112, 3735; (b) R.D. Wigglesworth, W. T. Raynes, S. Kirpekar, J. Oddershede andS. P. A. Sauer, J. Chem. Phys., 2001, 114, 9192E.
30 P. Lantto, J. Kaski, J. Vaara and J. Jokisaari, Chem.-Eur. J.,2000, 6, 1395.
31 T. Helgaker, O. B. Lutnæs and M. Jaszunski, J. Chem. TheoryComput., 2007, 3, 86.
32 V. Polo, E. Kraka and D. Cremer, Mol. Phys., 2002, 100, 1771.33 R. Baer and D. Neuhauser, Phys. Rev. Lett., 2005, 94, 043002.34 S. Patchkovskii, J. Autschbach and T. Ziegler, J. Chem. Phys.,
2001, 115, 26.35 S. Ivanov, S. Hirata and R. J. Bartlett, Phys. Rev. Lett., 1999, 83,
5455.36 A. Gorling, Phys. Rev. Lett., 1999, 83, 5459.37 J. Gauss and H.-J. Werner, Phys. Chem. Chem. Phys., 2000, 2,
2083.38 S. Goedecker, Rev. Mod. Phys., 1999, 71, 1085.39 G. E. Scuseria and P. Y. Ayala, J. Chem. Phys., 1999, 111, 8330.40 G. Vignale and M. Rasolt, Phys. Rev. Lett., 1987, 59, 2360.41 A. M. Lee, N. C. Handy and S. M. Colwell, J. Chem. Phys., 1995,
103, 10095.42 P. Manninen and J. Vaara, J. Comput. Chem., 2006, 27, 434.43 R. Ditchfield, J. Chem. Phys., 1972, 56, 5688.44 K. Wolinski, J. F. Hinton and P. Pulay, J. Am. Chem. Soc., 1990,
112, 8251.45 T. Helgaker and P. Jørgensen, J. Chem. Phys., 1991, 95, 2595.46 W. Kutzelnigg, U. Fleischer and M. Schindler, in NMR Basic
Principles and Progress, ed. P. Diehl, P. Fluck, H. Gunther, R.Kosfeld and J. Seelig, Springer-Verlag, Berlin, 1990, p. 165.
47 S. Huzinaga, Approximate Atomic Functions, University ofAlberta, Edmonton, 1971.
48 K. Fægri, Jr and J. Almlof, J. Comput. Chem., 1986, 7, 396, seehttp://folk.uio.no/knutf/bases/one.
49 D. P. Chong, Can. J. Chem., 1995, 73, 79.50 (a) T. H. Dunning, Jr, J. Chem. Phys., 1989, 90, 1007; (b) R. A.
Kendall, T. H. Dunning, Jr and R. J. Harrison, J. Chem. Phys.,1992, 96, 6796; (c) D. Woon and T. H. Dunning, Jr, J. Chem.Phys., 1995, 103, 4572; (d) A. Wilson, T. van Mourik and T. H.Dunning, Jr, J. Mol. Struct. (THEOCHEM), 1997, 388, 339.
51 C. Ochsenfeld, J. Kussmann and F. Koziol, Angew. Chem., Int.Ed., 2004, 43, 4485.
52 J. Gauss, J. Chem. Phys., 2002, 116, 4773.53 A. Auer and J. Gauss, J. Chem. Phys., 2001, 115, 1619.54 G. Malli and C. Froese, Int. J. Quantum Chem., 1967, 1, 95.55 D. Kolb, W. R. Johnson and P. Shorer, Phys. Rev. A, 1982, 26,
19.56 J. Vaara and P. Pyykko, J. Chem. Phys., 2003, 118, 2973.57 P. Lantto, R. H. Romero, S. S. Gomez, G. A. Aucar and J.
Vaara, J. Chem. Phys., 2006, 125, 184113.
58 Dirac, a relativistic ab initio electronic structure program, ReleaseDIRAC04.0 (2004), written by H. J. Aa. Jensen, T. Saue and L.Visscher with contributions from V. Bakken, E. Eliav, T. En-evoldsen, T. Fleig, O. Fossgaard, T. Helgaker, J. Laerdahl, C. V.Larsen, P. Norman, J. Olsen, M. Pernpointner, J. K. Pedersen, K.Ruud, P. Salek, J. N. P. van Stralen, J. Thyssen, O. Visser and T.Winther, (http://dirac.chem.sdu.dk).
59 G. A. Aucar and J. Oddershede, Int. J. Quantum Chem., 1993, 47,425.
60 L. Visscher, T. Enevoldsen, T. Saue, H. J. Aa. Jensen and J.Oddershede, J. Comput. Chem., 1999, 20, 1262.
61 L. Visscher, T. Enevoldsen, T. Saue and J. Oddershede, J. Chem.Phys., 1998, 109, 9677.
62 G. A. Aucar, T. Saue, L. Visscher and H. J. Aa. Jensen, J. Chem.Phys., 1999, 110, 6208.
63 M. Pecul, T. Saue, K. Ruud and A. Rizzo, J. Chem. Phys., 2004,121, 3051.
64 P. Lantto and J. Vaara, J. Chem. Phys., 2006, 125, 174315.65 L. Belpassi, F. Tarantelli, A. Sgamellotti, H. M. Quiney, J. N. P.
van Stralen and L. Visscher, J. Chem. Phys., 2007, 126, 064314.66 Y. Xiao, D. Peng and W. Liu, J. Chem. Phys., 2007, 126, 081101.67 R. Bast, T. Saue, 2007, private communication.68 J. I. Melo, M. C. Ruiz de Azua, C. G. Giribet, G. A. Aucar and
R. H. Romero, J. Chem. Phys., 2003, 118, 471.69 M. C. Ruiz de Azua, J. I. Melo and C. G. Giribet, Mol. Phys.,
2003, 101, 3103.70 J. I. Melo, M. C. Ruiz de Azua, C. G. Giribet, G. A. Aucar and P.
F. Provasi, J. Chem. Phys., 2004, 121, 6798.71 (a) N. Nakagawa, S. Shinada and S. Obinata, The Sixth NMR
Symposium, Kyoto, 1967; (b) Y. Nomura, Y. Takeuchi and N.Nakagawa, Tetrahedron Lett., 1969, 10, 639.
72 P. Pyykko, A. Gorling and N. Rosch, Mol. Phys., 1987, 61, 195.73 H. Nakatsuji, H. Takashima and M. Hada, Chem. Phys. Lett.,
1995, 233, 95.74 V. G. Malkin, O. L. Malkina and D. R. Salahub, Chem. Phys.
Lett., 1996, 261, 335.75 M. Kaupp, O. L. Malkina, V. G. Malkin and P. Pyykko, Chem.-
Eur. J., 1998, 4, 118.76 J. Vaara, K. Ruud, O. Vahtras, H. Agren and J. Jokisaari, J.
Chem. Phys., 1998, 109, 1212.77 P. Lantto and J. Vaara, J. Chem. Phys., 2007, in press.78 M. Straka, P. Lantto, M. Rasanen and J. Vaara, in preparation.79 M. Hanni, P. Lantto, M. Ilias, H. J. Aa. Jensen and J. Vaara,
submitted.80 M. Straka, P. Lantto and J. Vaara, in preparation.81 U. Edlund, T. Lejon, P. Pyykko, T. K. Venkatachalam and E.
Buncel, J. Am. Chem. Soc., 1987, 109, 5982.82 W. G. Schneider, H. J. Bernstein and J. A. Pople, J. Chem. Phys.,
1958, 28, 601.83 B. Minaev, J. Vaara, K. Ruud, O. Vahtras and H. Agren, Chem.
Phys. Lett., 1998, 295, 455.84 (a) S. K. Wolff, T. Ziegler, E. van Lenthe and E. J. Baerends, J.
Chem. Phys., 1999, 110, 7689; (b) J. Autschbach and T. Ziegler, J.Chem. Phys., 2000, 113, 936; (c) J. Autschbach and T. Ziegler, J.Chem. Phys., 2000, 113, 9410; (d) H. Fukui and T. Baba, J. Chem.Phys., 2002, 117, 7836.
85 (a) C. C. Ballard, M. Hada, H. Kaneko and H. Nakatsuji, Chem.Phys. Lett., 1996, 254, 170; (b) H. Fukui and T. Baba, J. Chem.Phys., 1998, 108, 3854; (c) T. Baba and H. Fukui, J. Chem. Phys.,1999, 110, 131; (d) R. Fukuda, M. Hada and H. Nakatsuji, J.Chem. Phys., 2003, 118, 1015; (e) R. Fukuda, M. Hada and H.Nakatsuji, J. Chem. Phys., 2003, 118, 1027.
86 ADF2006.01, SCM, Theoretical Chemistry, Vrije Universiteit,Amsterdam, The Netherlands, 2006, http://www.scm.com.
87 K. Kudo, H. Maeda, T. Kawakubo, Y. Ootani, M. Funaki andH. Fukui, J. Chem. Phys., 2006, 124, 224106.
88 M. Ilias and T. Saue, J. Chem. Phys., 2007, 126, 064102.89 (a) R. H. Romero and G. A. Aucar, Phys. Rev. A, 2002, 65,
053411; (b) R. H. Romero and G. A. Aucar, Int. J. Mol. Sci.,2002, 3, 914.
90 (a) A. L. Barra, J. B. Robert and L. Wiesenfeld, Phys. Lett. A,1986, 115, 443; (b) A. L. Barra, J. B. Robert and L. Wiesenfeld,Europhys. Lett., 1988, 5, 217.
91 V. Weijo, P. Manninen and J. Vaara, J. Chem. Phys., 2005, 123,054501.
This journal is �c the Owner Societies 2007 Phys. Chem. Chem. Phys., 2007, 9, 5399–5418 | 5417
Dow
nloa
ded
by M
cMas
ter
Uni
vers
ity o
n 10
/05/
2013
13:
25:0
4.
Publ
ishe
d on
06
July
200
7 on
http
://pu
bs.r
sc.o
rg |
doi:1
0.10
39/B
7061
35H
View Article Online
92 V. Weijo, R. Bast, P. Manninen, T. Saue and J. Vaara, J. Chem.Phys., 2007, 126, 074107.
93 C. J. Jameson, in Theoretical Models of Chemical Bonding. Part 3.Molecular Spectroscopy, Electronic Structure and IntramolecularInteractions, ed. Z. B. Maksic, Springer-Verlag, Berlin, 1990.
94 M. Toyama, T. Oka and Y. Morino, J. Mol. Spectrosc., 1964, 13,193.
95 J. Lounila, R. Wasser and P. Diehl, Mol. Phys., 1987, 62, 19.96 A. R. Hoy, I. M. Mills and G. Strey, Mol. Phys., 1972, 24, 1265.97 A. A. Auer, J. Gauss and J. F. Stanton, J. Chem. Phys., 2003, 118,
10407.98 J. Vaara, J. Lounila, K. Ruud and T. Helgaker, J. Chem. Phys.,
1998, 109, 8388.99 R. E. Wasylishen, S. Mooibroek and J. B. Macdonald, J. Chem.
Phys., 1984, 81, 1057.100 R. E. Wasylishen and D. L. Bryce, J. Chem. Phys., 2002, 117,
10061.101 C. Eckart, Phys. Rev., 1935, 47, 552.102 P. W. Fowler and W. T. Raynes, Mol. Phys., 1981, 43, 65.103 K. Ruud, P.-O. Astrand and P. R. Taylor, J. Chem. Phys., 2000,
112, 2668.104 K. Ruud, J. Lounila and J. Vaara, unpublished.105 P.-O. Astrand, K. Ruud and P. R. Taylor, J. Chem. Phys., 2000,
112, 2655.106 A. M. Kantola, S. Ahola, J. Vaara, J. Saunavaara and J.
Jokisaari, Phys. Chem. Chem. Phys., 2007, 9, 481.107 P. Lantto, J. Vaara, A. Kantola, V.-V. Telkki, B. Schimmelpfen-
nig, K. Ruud and J. Jokisaari, J. Am. Chem. Soc., 2002, 124, 2762.108 J. Lounila, J. Vaara, Y. Hiltunen, A. Pulkkinen, J. Jokisaari, M.
Ala-Korpela and K. Ruud, J. Chem. Phys., 1997, 107, 1350.109 P.-O. Astrand, K. V. Mikkelsen, P. Jørgensen, K. Ruud and T.
Helgaker, J. Chem. Phys., 1998, 108, 2528.110 R. Cammi, J. Chem. Phys., 1998, 109, 3185.111 D. B. Searles and H. Huber, in Encyclopedia of Nuclear Magnetic
Resonance Volume 9: Advances in NMR, ed. D. M. Grant and R.K. Harris, John Wiley & Sons, Chichester,2002.
112 R. Car and M. Parrinello, Phys. Rev. Lett., 1985, 55, 2471.113 F. Mauri, B. G. Pfrommer and S. G. Louie, Phys. Rev. Lett.,
1996, 77, 5300.114 D. Sebastiani and M. Parrinello, J. Phys. Chem. A, 2001, 105,
1951.115 T. S. Pennanen, J. Vaara, P. Lantto, A. Sillanpaa, K. Laasonen
and J. Jokisaari, J. Am. Chem. Soc., 2004, 126, 11093.116 T. S. Pennanen, P. Lantto, A. J. Sillanpaa and J. Vaara, J. Phys.
Chem. A, 2007, 111, 182.117 M. Hanni, P. Lantto, N. Runeberg, J. Jokisaari and J. Vaara, J.
Chem. Phys., 2004, 121, 5908.118 A. D. Buckingham and J. A. Pople, Discuss. Faraday Soc., 1956,
22, 17.119 C. J. Jameson, A. K. Jameson and S. M. Cohen, J. Chem. Phys.,
1975, 62, 4224.120 C. J. Jameson, A. K. Jameson and S. M. Cohen, J. Chem. Phys.,
1973, 59, 4540.121 R. A. Aziz and M. J. Slaman, Mol. Phys., 1986, 57, 825.122 H. Kleinert, Path Integrals in Quantum Mechanics, Statistics,
Polymer Physics, and Financial Markets, World Scientific, Singa-pore, 3rd edn, 2004.
123 N. F. Ramsey, Phys. Rev. A, 1970, 1, 1320.124 P. Manninen and J. Vaara, Phys. Rev. A, 2004, 69, 022503.125 J. Vaara, P. Manninen and J. Lounila, Chem. Phys. Lett., 2003,
372, 750.126 M. R. Bendall and D. M. Doddrell, J. Magn. Reson., 1979, 33,
659.127 P. Manninen, J. Vaara and P. Pyykko, Phys. Rev. A, 2004, 70,
043401.128 T. Meersmann and M. Haake, Phys. Rev. Lett., 1998, 81, 1211.129 J. Vaara and P. Pyykko, Phys. Rev. Lett., 2001, 86, 3268.130 B. M. Goodson, J. Magn. Reson., 2002, 155, 157.
131 (a) A. D. Buckingham and L. C. Parlett, Science, 1994, 264, 1748;(b) A. D. Buckingham and L. C. Parlett, Mol. Phys., 1997, 91,805.
132 M. Jaszunski and A. Rizzo, Mol. Phys., 1999, 96, 855.133 R. H. Romero and J. Vaara, Chem. Phys. Lett., 2004, 400, 226.134 I. Bertini, C. Luchinat and G. Parigi, Solution NMR of Para-
magnetic Molecules, Elsevier, Amsterdam, 2001.135 S. Moon and S. Patchkovskii, in Calculation of NMR and EPR
Parameters: Theory and Applications, ed. M. Kaupp, V. G.Malkin and M. Buhl, Wiley-VCH, Weinheim, 2004.
136 Z. Rinkevicius, J. Vaara, L. Telyatnyk and O. Vahtras, J. Chem.Phys., 2003, 118, 2550.
137 T. O. Pennanen and J. Vaara, J. Chem. Phys., 2005, 123, 174102.138 H. Heise, F. H. Kohler, F. Mota, J. J. Novoa and J. Veciana, J.
Am. Chem. Soc., 1999, 121, 9659.139 (a) J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett.,
1996, 77, 3865; (b) J. P. Perdew, K. Burke and M. Ernzerhof,Phys. Rev. Lett., 1997, 78, 1386E.
140 Teemu O. Pennanen, private communication, 2007.141 F. H. Kohler and W. Prossdorf, J. Am. Chem. Soc., 1978, 100,
5970.142 H. Heise, F. H. Kohler and X. Xie, J. Magn. Reson., 2001, 150,
198.143 F. H. Kohler, J. Organomet. Chem., 1976, 110, 235.144 H. Eicher and F. H. Kohler, Chem. Phys., 1988, 128, 297.145 P. Hrobarik, R. Reviakine, A. V. Arbuznikov, O. L. Malkina, V.
G. Malkin, F. H. Kohler and M. Kaupp, J. Chem. Phys., 2007,126, 024107.
146 (a) J. Jokisaari, Prog. Nucl. Magn. Reson. Spectrosc., 1994, 26, 1;(b) D. Raftery and B. F. Chmelka, in NMR Basic Principles andProgress Vol. 30, ed. P. Diehl, E. Fluck, H. Gunther, R. Kosfeldand J. Seelig, Springer, Heidelberg, 1994; (c) C. I. Ratcliffe, Annu.Rep. NMR Spectrosc., 1998, 36, 123.
147 (a) C. J. Jameson and A. C. de Dios, J. Chem. Phys., 1992, 97,417; (b) C. J. Jameson, A. K. Jameson, B. I. Baello and H. Lim, J.Chem. Phys., 1994, 100, 5965; (c) C. J. Jameson, A. K. Jameson,B. I. Baello and H. Lim, J. Chem. Phys., 1994, 100, 5977; (d) C. J.Jameson, D. N. Sears and S. Murad, J. Chem. Phys., 2004, 121,9581.
148 J.-H. Kantola, J. Vaara, T. T. Rantala and J. Jokisaari, J. Chem.Phys., 1997, 107, 6470.
149 J. A. Ripmeester and C. I. Ratcliffe, J. Phys. Chem., 1995, 99, 619.150 J. Jokisaari, in NMR of Ordered Liquids, ed. E. E. Burnell and C.
A. de Lange, Kluwer, Dordrecht, 2003.151 (a) B. J. Berne and P. Pechukas, J. Chem. Phys., 1971, 56, 4231;
(b) J. G. Gay and B. J. Berne, J. Chem. Phys., 1981, 74, 3316.152 (a) M. A. Bates and G. R. Luckhurst, J. Chem. Phys., 1999, 110,
7087; (b) E. J. de Miguel, E. M. del Rio and F. J. Blas, J. Chem.Phys., 2004, 121, 11183.
153 (a) H. Fukunaga, J. Takimoto and M. Doi, J. Chem. Phys., 2004,120, 7792; (b) D. J. Cleaver, C. M. Care, M. P. Allen and M. P.Neal, Phys. Rev. E, 1996, 54, 559.
154 J. Lintuvuori, M. Straka and J. Vaara, Phys. Rev. E, 2007, 75,031707.
155 J. Lounila, O. Muenster, J. Jokisaari and P. Diehl, J. Chem.Phys., 1992, 97, 8977.
156 M. Ylihautala, J. Lounila and J. Jokisaari, J. Chem. Phys., 1999,110, 6381.
157 L. Khriachtchev, M. Pettersson, N. Runeberg, J. Lundell and M.Rasanen, Nature, 2000, 406, 874.
158 L. Khriachtchev, H. Tanskanen, J. Lundell, M. Pettersson, H.Kiljunen and M. Rasanen, J. Am. Chem. Soc., 2003, 125, 4696.
159 J. M. Michaud and M. C. L. Gerry, J. Am. Chem. Soc., 2006, 128,7613, and references therein.
160 T. Charpentier, S. Ispas, M. Profeta, F. Mauri and C. J. Pickard,J. Phys. Chem. B, 2004, 108, 4147.
161 V. Barone, M. Brustolon, P. Cimino, A. Polimeno, M. Zerbettoand A. Zoleo, J. Am. Chem. Soc., 2006, 128, 15865.
5418 | Phys. Chem. Chem. Phys., 2007, 9, 5399–5418 This journal is �c the Owner Societies 2007
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