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 |

5402 | Phys. Chem. Chem. Phys., 2007, 9, 5399–5418 This journal is �c the Owner Societies 2007

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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.

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