Spin–spin coupling tensors as determined by experiment and computational chemistry

Spin–spin coupling tensors as determined by experiment

and computational chemistry

Juha Vaaraa,1, Jukka Jokisaarib,*, Roderick E. Wasylishenc,2, David L. Brycec,3

aDepartment of Chemistry, P.O. Box 55 (A. I. Virtasen aukio 1), FIN-00014 University of Helsinki, Helsinki, FinlandbNMR Research Group, Department of Physical Sciences, P.O. Box 3000, FIN-90014 University of Oulu, Oulu, Finland

cDepartment of Chemistry, University of Alberta, Edmonton, Alberta, Canada T6G 2G2

Accepted 2 September 2002

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

1.1. Scope of the review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

1.2. NMR spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

1.3. Symmetry aspects and tensorial properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

1.4. Nonrelativistic theory of the spin–spin coupling tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

2. Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

2.1. High field approximation in NMR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

2.2. NMR in isotropic media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

2.3. Liquid crystal NMR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

2.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

2.3.2. Liquid crystal solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

2.3.3. J tensor contribution to Dexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

2.3.4. Vibration and deformation effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

2.3.5. Limitations in the quantitative determination of J tensors. . . . . . . . . . . . . . . . . . . . . . . . 247

2.3.6. Qualitative determination of Janiso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

2.3.7. Results derived from LCNMR experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

2.4. Solid-State NMR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

2.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

2.4.2. Solid-State NMR determination of J tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

2.4.3. Results from single crystal studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

2.4.4. Results from studies of stationary powders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

2.4.5. Results from spinning powder samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

0079-6565/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved.

PII: S0 07 9 -6 56 5 (0 2) 00 0 50 -X

Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304

www.elsevier.com/locate/pnmrs

1 Tel.: þ358-9-191-50181; fax: þ358-9-191-50169.2 Tel.: þ1-780-492-4336; fax: þ1-780-492-8231.3 Also at: Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada, B3H 4J3.

* Corresponding author. Tel.: þ358-8-553-1308; fax: þ358-8-553-1287.

E-mail addresses: [email protected] (J. Jokisaari), [email protected] (J. Vaara), [email protected] (R.E.

Wasylishen), [email protected] (D.L. Bryce).

http://www.elsevier.com/locate/pnmrs

2.5. High-resolution molecular beam spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

2.6. NMR relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

3. Quantum chemical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

3.1. Correlated ab initio methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

3.2. Density-functional theory methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

3.3. Basis set requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

3.4. Effects of nuclear motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

3.5. Relativistic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

3.6. Solvation and intermolecular forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

3.7. Couplings for large systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

3.8. Quantum chemical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

3.8.1. Symmetric components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

3.8.2. Antisymmetric components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

1. Introduction

1.1. Scope of the review

Our aim is to examine the recent experimental and

theoretical research involving the nuclear spin–spin

coupling tensor, the indirect coupling mediated by

the electronic structure, JMN ; between general

magnetic nuclei M and N in closed-shell molecules.

The principal experimental techniques in the field are

NMR spectroscopy of molecules dissolved in liquid

crystalline media (liquid crystal NMR, LCNMR) or

solid samples either as powders or single crystals.

Interpretation of hyperfine data taken from molecular

beam experiments is also discussed in this context.

Quantum chemical electronic structure calculations

provide a theoretical means to study this property.

We focus on the developments since the previous

review on LCNMR and computational methods,

which was written in 1982 [1]. The solid state

NMR literature prior to 1990 has, in turn, been

reviewed in Ref. [2]. We have omitted many

references to classic papers as they were given in

Refs. [1,2]. Of the new material, we include only

references reporting properties of the spin–spin

coupling tensor as opposed to solely the isotropic

coupling constants, i.e. 13

of the trace of J.

Concerning quantum chemical data, only results of

non-empirical work carried out either by ab initio

or density-functional theory will be included. The

list of relevant, yet omitted semiempirical papers

includes Refs. [3–9]. Despite the fact that corrections

for relativistic effects, rovibrational motion, and

environmental (solvent) effects have not been

extensively applied to the tensorial properties of J,

we devote some space to these issues as they are

likely to be subjects of increased interest in the near

future.

We have tried to be comprehensive but it is

inevitable that some important papers have been

overlooked. We apologize for these oversights. Our

review covers literature published prior to autumn

2001.

1.2. NMR spin Hamiltonian

The NMR spin Hamiltonian for spin- 12

nuclei is

written in its general form (in frequency units) as

HNMR ¼ 21

2p

XM

gMIM·ð1 2 sMÞ ·B0

þX

M,N

IM·ðD0MN þ JMNÞ·IN : ð1Þ

HNMR is a phenomenological, effective energy

expression designed to reproduce the transition

energies between the Zeeman states of nuclear

magnetic dipole moments

mM ¼ gM"IM ð2Þ

placed in the external magnetic field B0: Here "IM is

the spin angular momentum of nucleus M and

J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304234

Nomenclature

B3LYP three-parameter Becke – Lee –

Yang–Parr

CAS(SCF) complete active space (self-con-

sistent field)

CC coupled cluster

CCSD coupled cluster singles and

doubles

CCSDPPA coupled cluster singles and

doubles polarization propagator

approximation

CCSDT coupled cluster singles, doubles

and triples

CCSD(T) coupled cluster singles, doubles

and non-iterative triples

CDFT current density-functional theory

CI configuration interaction

CISD configuration interaction singles

and doubles

CLOPPA contributions from localized

orbitals within the polarization

propagator approximation

CP cross polarization

CPMAS cross polarization magic-angle

spinning

DFT density-functional theory

DHF Dirac–Hartree–Fock

DNA deoxyribonucleic acid

DSO diamagnetic nuclear spin-elec-

tron orbit

DZP double-zeta plus polarization

EFG electric field gradient

EOM equations-of-motion

EOM-CC equation-of-motion coupled

cluster

FC Fermi contact

FCI full configuration interaction

FOPPA first-order polarization propaga-

tor approximation

FPT finite perturbation theory

FWHH full width at half height

GGA generalized gradient approxi-

mation

GIAO gauge-including atomic orbital

HFA high-field approximation

IPPP inner projections of the polariz-

ation propagator

LC liquid crystal

LCNMR liquid crystal nuclear magnetic

resonance

LDA local density approximation

LR linear response

MAS magic-angle spinning

MBER molecular beam electric reson-

ance

MBMR molecular beam magnetic reson-

ance

MCLR multiconfiguration self-consist-

ent field linear response

MCSCF multiconfiguration self-consist-

ent field

MO molecular orbital

MP2 second-order Møller – Plesset

perturbation theory

MQMAS multiple quantum magic-angle

spinning

NMR nuclear magnetic resonance

NOE nuclear Overhauser enhancement

NQR nuclear quadrupole resonance

PAS principal axis system

PES potential energy surface

PPA polarization propagator approxi-

mation

PSO paramagnetic nuclear spin-elec-

tron orbit

QCISD(T) quadratic configuration inter-

action singles, doubles and non-

iterative triples

RAS(SCF) restricted active space (self-con-

sistent field)

REX relativistic extended Huckel

RHF restricted Hartree–Fock

RPA random phase approximation

SCF self-consistent field

SD spin dipole

SOPPA second-order polarization propa-

gator approximation

SOPPA(CCSD) second-order polarization propa-

gator approximation with

coupled cluster singles and

doubles amplitudes

J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 235

the magnetogyric ratio, gM ; is a nuclear property. In

addition to the interaction with B0; the Hamiltonian of

Eq. (1) contains coupling terms describing the

interaction of mM with the fields arising from

the dipole moments of other magnetic nuclei N of

the system.

Conceptually HNMR is obtained by averaging the

full molecular Hamiltonian H over all its degrees of

freedom apart from B0 and the set of nuclear spins

{IM} as

HNMRðB0; {IM}Þ

¼ kHðB0;EE; {ri}; {si}; {RM}; {IM}ÞlEE;{ri};{si};{RM};

ð3Þ

where the effects of external electric fields EE, the

positions of the nuclei {RM} and electrons {ri}; as

well as the spins of the latter {si} are absorbed in the

parameters sM ; JMN ; and D0MN of HNMR: The

functional form of HNMR can be seen from an

expansion of the energy appropriate to H in terms of

the small perturbations caused by B0 and the mM ;

around B0 ¼ mM ¼ 0;

EðB0; {mM}Þ

¼ E0 þ EB0·B0 þ

XM

EmM·mM þ

1

2B0·EB0;B0

·B0

þXM

mM·EmM ;B0·B0 þ

XM,N

mM·EmM ;mN·mN þ · · ·;

ð4Þ

where the nomenclature Ea ¼ ›E=›ala¼0; etc. is used.

There are thus, in principle, terms linear, quadratic,

cubic, etc. in mM (IM ; by Eq. (2)) in the expansion.

The properties EB0and EmM

are related to the

permanent magnetic moment of the molecule and

the hyperfine coupling tensor of the nucleus M. Both

properties vanish for a closed-shell system. EB0;B0is

related to magnetizability (susceptibility). Higher

order dependencies on B0; appearing as a magnetic

field dependence in the parameters of HNMR have been

speculated upon [10,11] and even found [12–14]. The

influence of terms in HNMR higher than quadratic in Ihas not been experimentally observed, although the

forms in which the terms would appear have been

investigated [15]. The diagonal ðM ¼ NÞ occurrences

in the coupling term correspond to either the (true)

nuclear quadrupole coupling between the nuclear

electric quadrupole moment and the electric field

gradient tensor at the nuclear site [16,17], or

pseudoquadrupole coupling where magnetic hyper-

fine operators produce, to second order, energy terms

bilinear in IM [18,19]. We will not consider these

properties here.

Comparing Eqs. (1) and (4), the parameters sM and

JMN are obtained by searching for energy terms with

particular functional dependencies on B0 and IM : The

terms bilinear in the two correspond to the nuclear

shielding tensor, the Cartesian et component of

which is

sM;et ¼ det þ›2EðmM ;B0Þ

›mM;e ›B0;t

��mM¼0;B0¼0

: ð5Þ

sM corresponds to the modification, caused by the

presence of the electron cloud, of the Zeeman

interaction of bare nuclei with B0;

HZ ¼ 21

h

XM

mM·B0 ¼ 21

2p

XM

gMIM·B0; ð6Þ

expressed in frequency units. The det in Eq. (5) takes

HZ into account and makes the definition of shielding

consistent with HNMR of Eq. (1).

SOS sum-over-states

SOS-CI sum-over-states configuration

interaction

SOS-DFPT sum-over-states density-func-

tional perturbation theory

SS solid state

TEPS tetraethyldiphosphine disulfide

TLC thermotropic liquid crystal

TMPS tetramethyldiphosphine

disulfide

TZP triple-zeta plus polarization

VAS variable angle spinning

ZORA zeroth-order regular approxi-

mation


The terms bilinear in IM and IN correspond to the

spin–spin coupling tensor,

JMN;et ¼ 2D0MN;et

þ"

2pgMgN

›2EðmM ;mNÞ

›mM;e ›mN;t

��mM¼0;mN¼0

: ð7Þ

In analogy with the nuclear shielding vs. the nuclear

Zeeman interaction, J constitutes a (usually but not

always) small electronic perturbation to the direct

through-space magnetic dipole–dipole interaction of

bare nuclei, HDD ¼P

M,N IM·D0MN·IN ; where

D0MN ¼ 2

"

2p

m0

4pgMgN

3RMNRMN 2 1R2MN

R5MN

: ð8Þ

The direct dipolar coupling tensor D0MN

contains information about the internuclear vectors

RMN ¼ RM 2 RN ; which makes it an important tool in

investigating molecular structure by NMR spectra

obtained in anisotropic media, as well as molecular

beam experiments, as will be discussed below. D0MN is

traceless and symmetric, in particular axially sym-

metric with respect to the direction of RMN in the

absence of motion (for asymmetry induced by motion,

see Refs. [20,21]).

The electronic, indirect coupling tensor is often

discussed using the related reduced coupling tensor

KMN;et ¼2p

"

1

gMgN

JMN;et ð9Þ

to remove the parametric dependence on the magne-

togyric ratios. This enables studies of trends in

indirect spin–spin coupling between various elements

and/or isotopes without the need to take into account

the nuclear factors.

1.3. Symmetry aspects and tensorial properties

In general, J is described by a 3 £ 3 matrix,

expressable as a sum of zeroth-, first-, and second-

rank tensors,

JMN ¼ JMN1 þ JAMN þ JS

MN : ð10Þ

The rank-0 contribution, J1, corresponds to the

isotropic spin–spin coupling constant,

JMN ¼ 13

TrJMN ¼ 13ðJMN;xx þ JMN;yy þ JMN;zzÞ: ð11Þ

The general symmetric and antisymmetric Cartesian

components of J are

JSMN;et ¼

12ðJMN;et þ JMN;te Þ2 JMNdet ð12Þ

and

JAMN;et ¼

12ðJMN;et 2 JMN;te Þ; ð13Þ

corresponding to the rank-2 and -1 contributions,

respectively.

Whereas the nuclear site symmetry in a given

molecular system determines which components of

sM are non-vanishing, the local symmetry about the

internuclear vector determines the situation for JMN :

The number of independent components in J for a

number of point group symmetries was reported in

Ref. [22]. Ref. [3] revisited the problem concerning

coupled nuclei that are exchanged through a local

symmetry operation. The paper contains an explicit

listing of independent components of both JS and JA

in most important point group symmetries (see also

[23]). In the general case, JMN and JNM differ only in

their antisymmetric components [3]:

JSMN;et ¼ JS

NM;et; JAMN;et ¼ 2JA

NM;et: ð14Þ

In particular, JA has a non-vanishing component only

if it generates the totally symmetric representation of

the local point group. To first order JA does not affect

NMR spectra; however in strongly coupled systems

perturbations have been predicted [23–25], but not

observed experimentally so far. In principle, JA

contributes to the relaxation rates T1 and T2; as

discussed in Section 2.6. Other mechanisms are

typically much more efficient, however. Examples

where JS and/or JA influence T1 or T2 have not been

reported [26]. For a recent ab initio calculation of JA;

see Ref. [27].

1.4. Nonrelativistic theory of the spin–spin

coupling tensor

We limit ourselves to the case of molecules with

a closed-shell singlet electronic ground state.

Ramsey’s paper on the non-relativistic theory of J

appeared in 1953 [28]. It, among his other classic

works on molecular magnetic properties, was

recently treated in a perspective article [29]. Here


we choose to restate the theory in modern response

theory [30] notation that is both compact and lends

itself naturally for discussions of many of the

practical methods of approximate electronic struc-

ture calculation of J. The latter are thoroughly

reviewed in Ref. [31].

The standard basis for the non-relativistic

treatment of molecular electromagnetic properties

is provided by the Breit–Pauli Hamiltonian, HBP

[32,33], correct to order a2 in the fine structure

constant a ¼ e2=ð4pe0"cÞ: The assumptions under-

lying HBP break down for systems with heavy

nuclei, in which case a genuinely relativistic theory

must be applied. Comments on recent research in

relativistic calculations of J will be included in

Section 3.5. Ramsey’s theory is obtained from HBP

by looking for the energy terms of the required

hIM·JMN·IN form and including all the contributions

up to the order a4: This involves three Oða2Þ

quantum mechanical operators that contribute

through second-order expressions, and one Oða4Þ

operator that gives a first-order (expectation value)

term. We list the operators below, using the atomic

unit system for simplicity.4 It is useful to divide the

operators into singlet and triplet operators depend-

ing on whether or not, respectively, they include a

dependence on the electron spin si:

The singlet operators are the diamagnetic and

paramagnetic nuclear spin-electron orbit operators,

Hð2ÞDSOðMNÞ ¼

1

2a4gMgNIM·

Xi

1ðriM·riNÞ2 riNriM

r3iMr3

iN

·IN

ð15Þ

and

Hð1ÞPSOðMÞ ¼ a2gMIM·

Xi

liMr3

iM

; ð16Þ

respectively. Here, riM ¼ ri 2 RM is the position

vector of electron i with respect to the position of

nucleus M, and liM ¼ 2iriM £ 7i is the (field-free)

angular momentum with respect to the same reference

point. Eqs. (15) and (16) are obtained from the

gauge-invariant expression for the electronic kinetic

energy, including the contributions to the momentum

from the vector potential

AMðriÞ ¼ a2gM

IM £ riM

r3iM

; ð17Þ

corresponding to the magnetic field from the nuclear

magnetic dipole moment of nucleus M, in the point

dipole approximation. The triplet operators relevant in

the present context arise from the electronic spin

Zeeman interaction with the magnetic field from the

point dipole nucleus [34]. They are the Fermi contact

and spin–dipole interactions,

Hð1ÞFCðMÞ ¼

4p

3a2gegM

Xi

dðriMÞsi·IM ð18Þ

and

Hð1ÞSDðMÞ ¼

1

2a2gegM

Xi

si·3riMriM 2 1r2

iM

r5iM

·IM ; ð19Þ

respectively. Here, ge is the free electron g-

value for which the latest standard value is

2.0023193043737(82) [35] and dðriMÞ is the Dirac

delta function at nucleus M.

From Eq. (7), limiting ourselves to the electronic

terms only, we obtain five contributions to the indirect

coupling tensor

JMN;et ¼1

2p

›2EðIM;e ; IN;tÞ

›IM;e ›IN;t

��IM;e¼IN;t¼0

¼ JDSOMN;et þ JPSO

MN;et þ JFCMN;et þ JSD

MN;et þ JSD=FCMN;et :

ð20Þ

The diamagnetic coupling is obtained from the

bilinear operator of Eq. (15) as a ground state

expectation value

JDSOMN;et ¼

1

4pa4gMgNk0l

£X

i

detðriM·riNÞ2 riN;eriM;t

r3iMr3

iN

l0l: ð21Þ

The DSO terms generally contribute to the trace as well

as symmetric and antisymmetric parts of J. Typically,

JDSO is either numerically small in comparison with the

other contributions to the coupling constant, or

occasionally (in couplings involving hydrogen, par-

ticularly JHH) largely cancelled by the PSO term to be

4 In the a.u. system, the numerical values of the following

constants are equal to unity: "; e, me; and 4pe0: Then, the speed of

light in vacuum, c ¼ 1=a and the permeability of vacuum m0 ¼

4pa2; in a.u.


discussed below. In the past, the efficient calculation of

the molecular integrals of the DSO operator was a

practical bottleneck; currently, however, the method in

Ref. [36] is often used.

The remaining coupling tensor contributions

involve second-order perturbation theory expressions.

For these, two first-order perturbations Hð1ÞM ¼ hM;e IM;e

and Hð1ÞN

0 ¼ h0N;tIN;t give rise to a general term

In the final identity, the spin–spin coupling tensor

is expressed as a linear response function [30].

The subscript zero indicates that the static limit,

corresponding to time-independent perturbations, is

taken. By definition, RA;BS0 is symmetric with

respect to the order in which the operators A and B

occur, and it includes contributions of both the AB

and BA successions. No double-counting nor

associated numerical prefactors occur in the case

of J due to the fact that the operators hM;e and h0N;t

refer to different nuclei.

Depending on the spin rank of the perturbations,

singlet or triplet, the singlet closed-shell ground state

l0l is coupled to singlet or triplet excited electronic

states, lnl ¼ lnSl or lnT l; respectively, in the

sum-over-states expression of Eq. (22). Fig. 1

illustrates the different couplings allowed by the

electronic spin symmetry.

The singlet operators referring to the two nuclei,

PSO(M ) and PSO(N ), couple to each other,

whereas the triplet operators FC(M ), FC(N ),

SD(M ), and SD(N ) couple among themselves.

Operators from the two different spin ranks do

not mix, unless electronic spin-orbit coupling is

allowed for in third-order perturbation theory [28,

37,38]. These relativistic contributions are Oða6Þ;

however.

Like the DSO term, the PSO term

JPSOMN;et ¼

1

2pa4gMgN

Xi

liM;e

r3iM

;X

i

liN;t

r3iN

* +* +0

; ð23Þ

contributes to all parts of the J tensor listed in Eq.

(10). JPSO is often the second-most important

contribution to the coupling constant after JFC: For

example, in systems where the valence s-electrons of

a certain atom contribute little to bonding, the JPSO

contribution to coupling involving this atom may

exceed JFC: This happens particularly in couplings to

the 19F nucleus as well as other halogen nuclei.

Computationally, the calculation of the PSO

term involves solving for the first-order wave

functions, e.g. through solving linear response

equations [30], with respect to the three imaginary

operators hPSOðNÞt ðt ¼ x; y; zÞ; corresponding to the

Cartesian components of the liM vector operator

in Eq. (16). The situation thus resembles that

encountered in the calculation of s, where first-

order wave functions with respect to the com-

ponents of the orbital Zeeman operator are usually

solved for. In contrast to s, however, the ‘natural’

gauge origin for the nuclear magnetic dipole field

is at the nucleus in question. Hence, there is no

need to apply special techniques such as the gauge-

including atomic orbital (GIAO) ansatz in the

calculation of the spin–spin coupling tensors [31].

JMN;et ¼1

2p

›2

›IM;e ›IN;t

Xn–0

k0lðhM;e IM;e þ h0N;tIN;tÞlnlknlðhM;e IM;e þ h0

N;tIN;tÞl0lE0 2 En

��IM;e¼IN;t¼0

¼1

2p

Xn–0

k0lhM;e lnlknlh0N;tl0lþ k0lh0

N;tlnlknlhM;e l0lE0 2 En

;1

2pRhM;e ; h0

N;tS0: ð22Þ

Fig. 1. Schematic illustration of the second-order processes

contributing to the nuclear spin–spin coupling interaction.


The Fermi contact term

JFCMN;et ¼

1

2p

4p

3

!2

a4g2egMgNdet

£

**Xi

dðriMÞsi;e ;X

i

dðriNÞsi;e

++0

ð24Þ

is isotropic, JFC ¼ 1JFC; and thus only contributes to

the isotropic coupling constant. It is often the dominant

term. It can be shown that the Cartesian components of

the two electronic spin vectors occurring in Eq. (24), as

well as in other second-order expressions involving

two triplet operators, must be the same [31]. Due to the

isotropic spatial structure of the FC operator, one then

has to solve only one linear response equation, e.g. with

respect to dðrNÞsz; to obtain JFC: This term poses

otherwise heavy computational requirements, how-

ever, through the need (a) to describe electron

correlation (the N-electron problem) very accurately

and (b) to use a good one-electron basis set flexible

enough in the atomic core region. These matters will be

discussed in Section 3.

The contribution of the spin-dipolar term

JSDMN;et ¼

1

2p

a4

4g2

egMgN

£X

n¼x;y;z

RXi

3riM;nriM;e 2 denr2iM

r5iM

si;n ;

Xi

3riN;nriN;t 2 dtnr2iN

r5iN

si;nS0ð25Þ

is often small although a priori non-negligible in the

general case. JSD can be broken into contributions

with tensorial ranks 0, 1, and 2. It is computationally

the most demanding mechanism, generally requiring

solutions to six response equations corresponding to

the six independent Cartesian components of the riN

riN operator that appears in the nominator of hSDðNÞ:

Finally, the traceless and symmetric [22] cross-term

JSD=FCMN;et ¼

1

2p

4p

3

a4

2g2

egMgN

�X

i

dðriMÞsi;e;X

i

3riN;eriN;t2detr2iN

r5iN

si;e

* +* +0

"

þX

i

3riM;triM;e2detr2iM

r5iM

si;t;X

i

dðriNÞsi;t

* +* +0

#

ð26Þ

of the FC and SD operators contributes to JS only. Often

the SD/FC mechanism dominates numerically the

anisotropic properties of J. It is a sum of two response

functions, where the FC and SD interactions refer to both

nuclei in turn. The two responses may be physically

different in a coupling tensor between non-equivalent

nuclei. Interestingly, the separation of JSD=FCMN into the

SD(M )/FC(N ) and SD(N )/FC(M ) contributions has only

been investigated in a few papers [39–42] to the authors’

knowledge.

The information already gathered when calculating

the other contributions to J is sufficient to evaluate the

SD/FC terms as well. Often the wave function

responses necessary for JSD are not calculated at the

highest possible level. Then, the FC and SD/FC terms

that give the often dominant contributions to J may be

obtained from solving the first-order wave functions

with respect to the two FC perturbations involved, at

the best available theoretical level.

Summarising the contributions from the terms

discovered by Ramsey to the different-rank tensorial

properties of JMN ;

JMN ¼ JDSOMN þ JPSO

MN þ JSDMN þ JFC

MN

JSMN;et ¼ JDSO;S

MN;et þ JPSO;SMN;et þ JSD;S

MN;et þ JSD=FCMN;et

JAMN;et ¼ JDSO;A

MN;et þ JPSO;AMN;et þ JSD;A

MN;et:

ð27Þ

As mentioned above, the related direct dipolar

coupling tensor also consists only of the symmetric

contribution, i.e. D0 ¼ D0 S:

2. Experimental methods

2.1. High field approximation in NMR spectroscopy

The magnetic field of the NMR spectrometer is

generally taken to coincide with the z0 axis of

the laboratory coordinate system ðx0; y0; z0Þ : B0 ¼ B0

z0: When the Zeeman interaction of the bare nuclei,

Eq. (6), is large compared with the other interactions,

it is sufficient to treat the energy of the nuclear spin

system by first order perturbation theory,

ENMR ¼ EZ þXM

kmM lHslmMl

þX

M,N

kmMmN lHD0 þ HJ lmMmNl; ð28Þ


where the lmMl and lmMmNl denote the unperturbed

one- and two-spin Zeeman states. This high field

approximation (HFA) is broken by strong quadrupolar

interactions [43] (experimental examples are given

later for this situation). Collecting the terms of Eq. (1)

that contribute to ENMR under the HFA, gives

HHFANMR ¼ 2

B0

2p

XM

gMIM;z0 þ Hs þ HD0 þ HJ ; ð29Þ

where the operator for the shielding interaction is

Hs ¼B0

2p

XM

gMsM;z0z0 IM;z0 ; ð30Þ

and for the coupling interactions

HD0 ¼X

M,N

D0MN;z0z0 IM;z0 IN;z0 2

1

4ðIMþIN2þ IM2INþÞ

�ð31Þ

HJ ¼X

M,N

JMN;z0z0 IM;z0 IN;z0 21

4ðIMþIN2þ IM2INþÞ

�

þ3

4

XM,N

JMNðIMþIN2þ IM2INþÞ; ð32Þ

where the tracelessness of D0 has been used. Eqs. (31)

and (32) involve the ladder operators for nuclear

spins,

IMþ ¼ IM;x0 þ iIM;y0 IM2 ¼ IM;x0 2 iIM;y0 : ð33Þ

The spectral observables in the HFA correspond to the

time average of the components of the NMR tensors,

T¼s; D0, and J, along the direction of the external

magnetic field, kTz0z0 l: Transformation of T between

any two sets of Cartesian axes, ðe;t;nÞ and ða;b;cÞ;

may be accomplished using

Tab ¼Xet

cos uea cos utbTet; ð34Þ

where uea is the angle between the e and a axes. It

then follows that

Tz0z0 ¼Xet

cos uez0 cos utz0Tet

¼1

3

Xe

Tee þ2

3

Xet

1

2ð3 cos uez0 cos utz0 2detÞTet

ð35Þ

and for the time average,

kTz0z0 l; T þTaniso ¼1

3

Xe

kTeelþ2

3

Xet

ksetTetl; ð36Þ

where e and t denote any of the molecule-fixed

coordinates ðx;y;zÞ; and

set ¼12ð3 cos uez0 cosutz0 2detÞ: ð37Þ

The assumption

ksetTetl¼ SetkTetl ð38Þ

corresponds to neglecting correlation between

rotation and internal (vibrational) motion of the

system. It defines the traceless and symmetric

orientation tensor, S [44]:

Set ¼ ksetl¼ 12k3 cos uez0 cos utz0 2detl: ð39Þ

S carries information on the probability distribution of

molecular orientation with respect to B0: As discussed

below, the assumption expressed by Eq. (38) has been

abandoned in the modern LCNMR determination of

D0 [45–47].

Eq. (36) defines the isotropic and anisotropic parts

of the NMR tensors,

T ¼ 13

TrT ¼ 13ðkTxxlþ kTyylþ kTzzlÞ ð40Þ

and

Taniso ¼2

3S : kTl ¼

2

3

Xet

SetkTetl; ð41Þ

respectively, using Eq. (38) for the latter equality. In

these equations, the time averaging has been

explicitly indicated using the angular brackets k l.This notation will be dropped in most of the

following. It should be remembered, however, that

the NMR parameters to be discussed are time-

averaged quantities.

2.2. NMR in isotropic media

In the gas phase or in ordinary liquids, the

molecules have no orientational order to a first

approximation. Consequently, Set ¼ 0 and the static


spin Hamiltonian takes the form

HisoNMR ¼ 2

B0

2p

XM

gMð1 2 sMÞIM;z0 þX

M,N

JMNIM·IN :

ð42Þ

The spectral observables are reduced to their isotropic

parts, Eq. (40), the shielding constant

sM ¼ 13

TrsM ; ð43Þ

and the spin–spin coupling constant J defined in Eq.

(11) (time-averaged). While sM determines the

chemical shift of M, the JMN are responsible for the

fine structure of the spectra [48–50].

2.3. Liquid crystal NMR spectroscopy

2.3.1. Introduction

In 1963, Saupe and Englert [51] proposed the use

of LCs as a medium to create an anisotropic

orientation distribution for solute molecules. In a LC

environment, solute molecules undergo translational

and rotational motion sufficiently fast that intermole-

cular dipole–dipole interactions vanish. On the other

hand, intramolecular dipolar interactions, the aniso-

tropic contributions of the s and J tensors, as well as

the quadrupole coupling tensors, average to non-zero

values. Consequently, LCNMR can be used for the

determination of molecular structures, components of

s, J and quadrupole coupling tensors (for nuclei with

spin $ 1).

For spin systems consisting only of spin- 12

nuclei,

the Hamiltonian of Eq. (29) becomes [52]

HLCNMR ¼ 2

B0

2p

XM

gMð1 2 sM 2 sanisoM ÞIM;z0

þX

M,N

JMNIM·IN þX

M,N

DMN þ1

2Janiso

MN

�

� ð3IM;z0 IN;z0 2 IM·INÞ; ð44Þ

where

DMN ¼ 12

D0anisoMN ð45Þ

is commonly denoted the direct dipolar coupling.

One should, however, note that sometimes another

definition, D ¼ D0 aniso; is also used. Furthermore, in

the solid state context the related quantity

RDD ¼m0"gMgN

8p2

1

R3MN

* +ð46Þ

is used.

A noteworthy feature of LCNMR is the fact that

peak widths (FWHH) of a few Hz and even better

than 1 Hz in 1H NMR spectra are possible in

favorable circumstances. Because of the high NMR

receptivity of 1H nuclei, a good signal-to-noise

ratio may be obtained with short accumulation

times. As a result, peak positions and spectral

parameters may be determined with a high degree

of accuracy. Spectral analysis is very similar to that

of isotropic systems except that spectra of mol-

ecules in LCs are very rarely first order, since

dipole–dipole couplings are typically on the order

of kHz [53]. The 1H NMR spectrum of benzene

and 13C6-benzene in an isotropic solution and in a

LC, shown in Fig. 2, illustrates the superb

resolution available for solute molecules in LCs.

The spectrum of 13C6-benzene also reveals one of

the limitations of the method; when the number of

interacting nuclei increases, the spectrum becomes

very complicated, and consequently its analysis may

be difficult or impossible. In practice, systems

consisting of up to 10–12 spin- 12

nuclei may be

analyzed, depending upon the symmetry of the

system.

The LCNMR method as a means to derive J

tensors is in principle quite straightforward. How-

ever, in order to obtain reliable, solvent-indepen-

dent results, molecular vibrations and the

correlation between vibrational and reorientational

motion must be properly taken into account. A

comprehensive review article on the anisotropies of

s and J as determined using LCNMR appeared in

1982 [1]. Since then, however, remarkable progress

has taken place [54], particularly in the character-

ization of J tensors. One should regard old data,

particularly those that report small anisotropies,

with caution.

2.3.2. Liquid crystal solvents

The most important LC solvents in studies of the

structure of low molar mass molecules as well as

the characterization of s, J, and quadrupole coupling

tensors are those known as thermotropic (TLC) that


feature liquid-crystalline mesophases within certain

temperature ranges. In a few cases, lyotropic LCs

have also been used. The disadvantage in applying

lyotropics is the generally small orientational order of

solute molecules, which leads to correspondingly

small anisotropic contributions to the Dexp and

chemical shifts.

Investigations of J tensors have generally been

performed on solute molecules dissolved in nematic

phases of TLCs. The nematic phase exists at

temperatures immediately below the isotropic

phase. In nematic phases, LC molecules possess

only a short range positional order. However, they

tend to align with their long axes5 parallel to a

common axis which defines the director, n, of the

LC phase. Nematic phases are in almost all known

cases uniaxial, meaning that there exists cylindrical

symmetry around n, and that the directions n and

2n are equivalent, i.e. the phase is apolar. For a

more detailed description of the physical properties

of LCs, see Ref. [55]. Fig. 3 gives a schematic

illustration of a nematic phase.

When a LC sample is placed in an external

magnetic field, B0, the sample becomes magnetized.

The magnetization M is given by [56]

Me ¼1

m0

Xt

xetBt; ð47Þ

where m0 is the permeability in vacuo and the xet are

the components of the diamagnetic (volume) suscep-

tibility (magnetizability) tensor, xd; which is diagonal

in the uniaxial phase.The energy density, rB; due to

the magnetization can be represented as

rB ¼ 2ðB0

0M·dB

¼ 2B2

0

2m0

xd þ2

3DxdP2ðcos uBnÞ

�; ð48Þ

where xd ¼ 13

Trxd is the isotropic diamagnetic

susceptibility, Dxd is the anisotropy of the

susceptibility tensor, and P2ðcos uBnÞ ¼12ð3

cos2uBn 21Þ is the second-order Legendre poly-

nomial (uBn being the angle between B0 and n).

Eq. (48) determines the orientation of the director

with respect to the external magnetic field (note

Fig. 2. Top: 1H NMR spectrum of benzene in an isotropic solution.

Middle: 400 MHz 1H NMR spectrum of benzene oriented in a

liquid-crystalline solution. Bottom: 400 MHz 1H NMR spectrum of13C6-benzene oriented in a liquid-crystalline solution.

Fig. 3. Molecular orientational order in a nematic phase. Reprinted

with permission from Ref. [55]. Copyright (1998) Wiley–VCH.

5 In most applications described in this context, the LCs consist of

elongated molecules. Such LCs are called calamitic.


that xd , 0 always):

1. when Dxd . 0; rB reaches a minimum with

P2ðcos uBnÞ ¼ 1, i.e. uBn ¼ 08 and thus n places

itself parallel with B0, and

2. when Dxd , 0; rB is a minimum with P2ðcos

uBnÞ ¼ 2 12; i.e. uBn ¼ 908 and thus n places

itself perpendicular to B0.

TLCs with both positive and negative Dxd

values exist and have been used in studies of J

tensors within solute molecules. Furthermore,

appropriate mixtures of these two kinds of LCs

have been utilized.

2.3.3. J tensor contribution to Dexp

As seen from Eq. (44), the experimental aniso-

tropic coupling,

DexpMN ¼ DMN þ 1

2Janiso

MN ; ð49Þ

includes a contribution from the J tensor. From Eq.

(41), this contribution can be represented as

JanisoMN ¼

2

3P2ðcos uBnÞ

Xet

SDetJMN;et; ð50Þ

where JMN;et is the component of J in the

molecule-fixed coordinate system. SDet in turn is

the component of the Saupe orientational order

tensor (with respect to n) [44] as defined in

Eq. (39). The factor P2ðcos uBnÞ changes the

reference direction from B0 to n. Fig. 4 illustrates

the different coordinate systems involved.

The number of independent components of S is

determined by molecular symmetry (Table 1). The

values of the components of S are obtained from the

Dexp provided that at least one internuclear distance

within the solute molecule is known or assumed.

Particularly in the theoretical description of orienta-

tional order, Wigner matrices [57] are used because of

their convenient transformation properties. In the

early LCNMR literature, the so-called Snyder

motional constants [58] were also used. Table 2

gives the relationships between the order parameters

in the various representations. In this review,

however, the Saupe orientational order tensor is

used exclusively.

In the most general case, Eq. (50) can be written as

JanisoMN ¼ 2

3JMN;zz 2

12ðJMN;xx þ JMN;yyÞ

h iSD

zz

nþ 1

2ðJMN;xx 2 JMN;yyÞðS

Dxx 2 SD

yyÞ

þ2JSMN;xySD

xy þ 2JSMN;xzS

Dxz þ 2JS

MN;yzSDyz

�

� P2ðcos uBnÞ: ð51Þ

The factor

DJMN ¼ JMN;zz 212ðJMN;xx þ JMN;yyÞ ð52Þ

defines the anisotropy of J with respect to the

molecular z axis. In practically all studies of J, the

solute molecules possess high symmetry so that their

orientation can be described with two or only one

orientational order parameter (Table 1). For solute

molecules with C2v; D2; or D2h symmetry, Eq. (51)

reduces to

JanisoMN ¼ 2

3DJMNSD

zz þ12ðJMN;xx 2 JMN;yyÞðS

Dxx 2 SD

yyÞh i£P2ðcos uBnÞ ð53Þ

and for molecules with at least a 3-fold symmetry axis

to an even simpler form

JanisoMN ¼ 2

3DJMNSD

zzP2ðcos uBnÞ: ð54Þ

Fig. 4. The coordinate systems used in the determination of the

anisotropic properties of NMR tensors in uniaxial liquid crystals. In

the illustrated example the director of the liquid crystal phase, n, is

perpendicular to the external magnetic field B0: The laboratory-

fixed axis system is ðx0; y0; z0Þ where B0 ¼ ð0; 0;B0Þ: The molecule-

fixed axis system is ðx; y; zÞ:


In Eqs. (53) and (54), the z-axis is chosen to be along

the n-fold ðn $ 2Þ symmetry axis of the molecule.

The molecule-fixed ðx; y; zÞ frame is not in the

general case the principal axis system (PAS) of either

the S tensor or J. If we assume ða; b; cÞ to be

the PAS(J ) of the J tensor, Eqs. (53) and (54)

transform to

JanisoMN ¼1

3DcJMN SD

zz½ð3cos2ucz21Þn

þhcðcos2uaz2cos2ubzÞ�

þðSDxx2SD

yyÞ cos2ucx2cos2ucyþ13hc

h�ðcos2uax2cos2uay2cos2ubxþcos2ubyÞ

io�P2ðcosuBnÞ ð55Þ

and

JanisoMN ¼1

3DcJMN½S

Dzzð3cos2ucz21ÞþðSD

xx2SDyyÞ

�ðcos2ucx2cos2ucyÞ�P2ðcosuBnÞ; ð56Þ

respectively. In Eqs. (55) and (56),

DcJMN¼JMN;cc212ðJMN;bbþJMN;aaÞ ð57Þ

is the J tensor anisotropy in its PAS, and

hc¼3

2

JMN;aa2JMN;bb

DcJMN

� ð58Þ

is the asymmetry parameter. ucz; for example, is the

angle between the c axis of PAS(J ) and the z axis

of the molecule-fixed frame. These equations show

that in order to determine the J tensor components

in its PAS by applying LCNMR, it is necessary to

know the angles between the principal axes and the

axes of the molecule-fixed frame. This information

is typically not available.

Another revealing way to look at the problem

is to choose the coordinate system in which the S

tensor is diagonal, i.e. in the principal axis system,

PAS(S ), of S, (1,2,3). In this frame, the molecular

orientational order is determined by two indepen-

dent order parameters, S33 and S11 2 S22: Conse-

quently, if the rotation – vibration correlation

effects (see Section 2.3.4) are neglected, any

dipolar coupling and the anisotropic indirect

contribution can simply be written as

DMN ¼2 12

FMN½SD33ð3 cos2 uMN;3 21Þ

þ ðSD11 2SD

22Þðcos2 uMN;1 2 cos2 uMN;2Þ�

�P2ðcosuBnÞ ð59Þ

Table 2

Relations between the components of the Saupe orientational order

tensor [44], averages of Wigner rotation matrix elements [57], and

Snyder motional constants [58]

Saupe

tensor

component

Wigner matrix

element

Snyder

motional

constant

Szz kD20;0l

�15

�1=2C3z22r2

Sxx 2 Syy

�32

�1=2ðkD2

0;2lþ kD20;22lÞ

�35

�1=2Cx22y2

Syz i�

38

�1=2ðkD2

0;2l2 kD20;22lÞ

�3

20

�1=2Cxy

Sxz 2�

38

�1=2ðkD2

0;1lþ kD20;21lÞ

�3

20

�1=2Cxz

Syz i�

38

�1=2ðkD2

0;1lþ kD20;21lÞ

�3

20

�1=2Cyz

Table 1

Independent second-rank orientational order parameters for mol-

ecules of various symmetry in uniaxial liquid crystal phases (From

Ref. [57])

Molecular

point group

Number of

independent

order

parameters

Order parameters

C1, Ci 5 Szz, Sxx 2 Syy, Sxy, Sxz, Syz

Cs, C2, C2h 3 Szz, Sxx 2 Syy, Sxy

C2v, D2, D2h 2 Szz, Sxx 2 Syy

C3, S6 1 Szz

C4, C4h, S4

C4v, D2d, D4h, D4

C5, C5h, C5v

D4d, D5, D5h, D5d

C3h, C6, C6h, C6v

D3h, D6, D6h, D6d

C1, C1v, C1h, D1h


and

JanisoMN ¼ 2

3D3JMNSD

33 þ12ðJMN;11 2 JMN;22Þ

h� ðSD

11 2SD22ÞiP2ðcosuBnÞ: ð60Þ

In Eq. (59), the factor F is defined as

FMN ¼m0"gMgN

8p2

1

R3MN

* +¼RDD

MN : ð61Þ

One should note that uMN;a ða¼ 1;2;3Þ is now the angle

between the RMN vector and the a axis of PAS(S ),

D3JMN as well as JMN;11 2 JMN;22 are given in the

PAS(S ), and, consequently, are not the same as those in

the other coordinate frames. Eqs. (59) and (60) clearly

illustrate the fact that although D and Janiso possess

similar dependence on the orientational order par-

ameters they generally do not vanish under the same

conditions; the zero condition for the former is

S11 2S22

S33

¼23 cos2 uMN;3 21

cos2 uMN;1 2 cos2 uMN;2

; ð62Þ

i.e. determined by the molecular geometry alone,

whereas Janiso vanishes when

S11 2S22

S33

¼22D3JMN

JMN;11 2 JMN;22

: ð63Þ

with S33 – 0: Thus, the experimentally determined

Dexp ¼Dþð1=2ÞJaniso may have a (sizable) non-zero

value even though the dipole–dipole coupling, D, is

vanishingly small [59,60].

In papers dealing with NMR spectra of biomacro-

molecules partially oriented in dilute LC solutions,

Eq. (59) is generally written in the form [61]

DMN ¼2 12

FMNSD33 ð3 cos2a21Þh

þ 32hS

3sin2a cos 2biP2ðcosuBnÞ; ð64Þ

where hS3 ¼ ðSD

11 2SD22Þ=D3SD is the asymmetry and

D3SD ¼ SD33 21=2ðSD

11 þSD22Þ ¼ ð3=2ÞSD

33 the anisotropy

of the S tensor, and the angles a¼ uMN;3 and b (angle

between the 1-axis and the projection of RMN in the

12-plane) are polar angles defining the orientation of

the internuclear vector RMN in PAS(S ).

In order to take into account the internal motion of

the RMN vector, a scaling factor is generally

introduced [61]; however, in the present case this

factor is omitted. Similarly,

JanisoMN ¼2SD

33½3D3JMNþhSðJMN;112JMN;22Þ�P2ðcosuBnÞ:

ð65Þ

Thus, in the situation where DMN¼0;

JanisoMN ¼ 2SD

33 3D3JMN 22

3

3 cos2a21

sin2acos2b

"

�ðJMN;11 2 JMN;22Þ

#P2ðcosuBnÞ ð66Þ

can be non-vanishing.

2.3.4. Vibration and deformation effects

Since the Dexp values are averages over internal

molecular vibrations, it was recognized in the late

1960s that they should be corrected for the vibrations

[62]. However, more than ten years passed before

Sykora et al. [63] published a theory and a general

computer program, VIBRA, became available to

correct dipolar couplings for harmonic vibrations.

Later, another program (AVIBR) was developed to

compute the effects of anharmonic vibrations [64].

In 1966 Snyder and Meiboom [65] and a few

years later Ader and Loewenstein [66] recorded

NMR spectra of tetramethylsilane and methane in

LC solutions, respectively, and detected a small

dipolar splitting despite the fact that the molecules

should not be oriented because of their high

symmetry. This observation was ascribed to a slight

distortion of the solute molecule by the anisotropic

force exerted by the solvent. The anisotropic

interactions visible in the spectra arise in this case

from correlation between internal molecular

vibrational and reorientational motion with respect

to the anisotropic solvent frame. Consequently, the

separation ksMNR23MNl ¼ SMN kR23

MN l; Eq. (38), is not

strictly valid. Here, sMN ¼ P2ðcosuMN;BÞ is the

component of set; Eq. (37), along RMN : Then,

the dipole–dipole coupling must be represented as

DMN ¼ 2m0"gMgN

8p2

sMN

R3MN

* +P2ðcosuBnÞ: ð67Þ

In 1984, a general theory was presented to take the

deformation effect into account [45,47]. Five years

later, the computer program MASTER [67] was

published; this program computes the vibrational


and deformation contributions to the dipole–dipole

couplings. For small-amplitude motion, the various

contributions to the dipolar coupling can be

separated as follows:

DMN ¼ DeqMN þ Dah

MN þ DhMN þ Dd

MN : ð68Þ

Deq is the dipole-dipole coupling corresponding to

the equilibrium structure of the molecule, Dah arises

from the anharmonicity of the vibrational potential,

Dh is the contribution from the harmonic vibrations,

and Dd is the deformation contribution.

The experimental couplings thus are

DexpMN ¼ D

eqMN þ Dah

MN þ DhMN þ Dd

MN þ 12

JanisoMN : ð69Þ

It has been experimentally realized that undistorted

structures of solute molecules may be derived

through the use of proper mixtures of LCs. Some

LCs lead to positive structural deformations whereas

others cause negative deformations. When mixed in

a proper molar ratio, they produce an environment

that does not distort solute molecular structure. The

methane molecule is a suitable deformation refer-

ence [68]. The method can be expected to work for

solutes that interact with the solvent LCs in a

manner qualitatively similar to that of CH4. There

exists some evidence that not only molecular

structure deformation but also the apparent defor-

mations of the s and J tensors are cancelled (or at

least reduced) in such LC mixtures [69]. Table 3

lists the importance of each contribution to Dexp in

benzene.

2.3.5. Limitations in the quantitative determination

of J tensors

In order to obtain a reliable value for Janiso; which

is often small and in many cases comparable in

magnitude with the vibrational and deformation

correction terms (see Table 3), the molecular structure

should be determined as completely as possible with

the aid of Dexp values. This necessarily means a full

analysis of data taking into account the molecular

vibrations and rotation-vibration correlation. In such

an analysis, the number of unknown parameters may

exceed the number of Dexp couplings obtainable from

one experiment, i.e. the problem becomes under-

determined. Occasionally this can be overcome by

carrying out experiments in several LC solvents

and performing a joint analysis of the couplings

[69,71,72].

In order to keep the problem of a solute molecule in

one LC solvent (such as those listed in Table 4)

overdetermined, one must have a sufficient number of

Dexp couplings in which the Janiso contribution can be

considered negligible. If the spin system consists of N

interacting nuclei, the total number of available Dexp

couplings is NðN 2 1Þ=2: If the molecule has no

symmetry, the number of orientational order par-

ameters is five and the number of coordinates 3N:

However, a basic property of dipolar couplings is that

they do not define absolute but only relative inter-

nuclear distances, i.e. the shape of a molecule. For the

determination of absolute order parameters and

distances, one internuclear distance has to be

assumed. This means that the number of adjustable

coordinates is 3ðN 2 2Þ: Consequently, in this general

case N has to fulfill the condition

NðN 2 1Þ=2 2 5 2 3ðN 2 2Þ $ 0: ð70Þ

This means that the derivation of atomic coordinates

and orientational order parameters necessitates N $ 7;

in other words, 21 Dexp couplings with negligible

anisotropic contribution are necessary for this purpose.

Thus, the determination of some of the J tensors

becomes feasible only if the number of couplings

exceeds 21. For a planar molecule with, e.g. C2v

symmetry (two independent order parameters), N has

to satisfy the inequality

NðN 2 1Þ=2 2 2 2 2ðN 2 2Þ $ 0 ð71Þ

from which N $ 5: The condition of insignificant

ð1=2ÞJaniso (as compared to D ) is usually valid for X1H

(X ¼ 1H, 13C, 14N, 15N, 19F, etc.) couplings.

Uncertainty in the determination of Janiso may

appear in the case of the spin system consisting of

different kinds of nuclei, for instance I and S spins.

Namely, the NMR spectrum renders it possible to

determine only the sum l2DexpIS þ JISl: If JIS cannot

be determined in the same experimental conditions

(the same solvent, temperature, concentration, etc.)

as the sum, additional uncertainty may be introduced to

Dexp; and consequently to Janiso as well. In principle, JIS

can be determined in LC phases by performing,

e.g. variable angle spinning (VAS) experiments [73].

The anisotropic contribution, Janiso; for molecules

with high symmetry depends exclusively on the


orientational order parameter of the symmetry axis, as

shown in Eq. (54). Therefore, the determination of DJ;

once Szz and the molecular structure are known, is

relatively straightforward. In contrast, in less sym-

metric molecules DJ and other components of J, such

as Jxx 2 Jyy in Eq. (51), can be derived only if the ratio

of the orientational order parameters, such as ðSxx 2

SyyÞ=Szz; can be changed by choosing another LC

solvent (then, of course, one has to assume that the J

tensor is independent of solvent). If the ratio remains

constant or changes only by a small amount, only the

combination of the order parameters and J tensor

components can be determined.

2.3.6. Qualitative determination of Janiso

Molecular symmetry can constrain the ratios of D

couplings. This can be used to reveal whether Dexp

includes a significant contribution from ð1=2ÞJaniso

or not. If the examination of the ratio DexpMN =D

expOP

reveals a deviation from the corresponding ratio of

direct dipolar couplings (D ¼ Dexp 2 ð1=2ÞJaniso ¼

Deq þ Dah þ Dh þ Dd; see Eq. (69)), the couplings

(or at least some of them) may be affected by

ð1=2ÞJaniso:

Benzene is a good example of a molecule in which

symmetry completely defines the ratios of the D

couplings. The order parameter of each MN-direction

in the ring plane is equal to 2ð1=2ÞSzz (Szz is the order

parameter of the 6-fold symmetry axis), consequently

the following equation is valid (in the first order

approximation)

DexpMN

DexpOP

¼gMgN

gOgP

kR23MNl

kR23OPl

£ 121

3DJMN

kR23MNl

21

F0MN

2DJOP

kR23OPl

21

F0OP

!" #

ð72Þ

where F0MN ¼FMNkR23

MN l: The expression in the square

brackets equals 1, i.e. the ratio of the D couplings

Table 3

Various contributions (in Hz) to D exp in benzene

Couplinga D eq D h D d D calc D exp b

3DHH 2701.16 9.78 21.99 2693.37 2693.368(7)4DHH 2134.94 0.98 20.27 2134.23 2134.220(9)5DHH 287.65 0.40 20.14 287.39 287.417(12)1DCH 22108.96 158.55 214.23 21964.64 21964.637(14)2DCH 2269.33 5.08 20.58 2264.83 2264.790(9)3DCH 268.43 0.48 20.08 268.03 268.131(9)4DCH 246.20 0.18 20.01 246.02 246.022(12)1DCC 2248.79 1.83 20.44 2247.40 2248.217(21)2DCC 247.88 0.03 0.05 247.80 247.569(20)3DCC 231.10 20.05 0.08 231.07 231.613(32)

Coupling ratioc Experimental valueb Theoretical value

DoHH=D

mHH 5.166(5) 5.1962

DoHH=D

pHH 7.932(5) 8.0000

DmHH=D

pHH 1.535(6) 1.5396

DoCC=D

mCC 5.218(4) 5.1962

DoCC=D

pCC 7.852(11) 8.0000

DmCC=D

pCC 1.505(2) 1.5396

The difference between D calc and D exp for the CC couplings is due to ð1=2ÞJanisoCC [70]. On the bottom are also shown the ratios of the

experimental 1H1H and 13C13C couplings.a The number in the upper left corner indicates the number of bonds between the interacting nuclei (analogous to the notation for the indirect

spin–spin coupling), although dipolar coupling is a through-space interaction.b The figure in parentheses gives the experimental error in units of the last digit(s).c The ortho, meta, and para couplings are denoted by o, m, and p, respectively.


corresponds to the ratio of the rovibrational averages

(derived from the dipolar couplings corrected for

vibrations and deformation effects) of the inverse

cube distances between interacting nuclei, if (1) DJMN

and DJOP vanish simultaneously, or (2) DJMN =

DJOP ¼ðgMgN =gOgPÞkR23MNl=kR23

OPl: For the cases

M;N;O;P all equal to either 1H or 13C, Eq. (72)

reduces to the form

DexpMN

DexpOP

¼kR23

MNlkR23

OPl

� 121

3F0ðDJMNkR

23MNl2DJOPkR

23OPlÞ

�; ð73Þ

where F0 ¼F0MN ¼F0

OP and the condition (2) above

becomes DJMN =DJOP ¼ kR23OPl=kR

23MNl:

It follows from the hexagonal symmetry of

benzene that Rm ¼ffiffi3

pRo and Rp ¼ 2Ro (Ri is the

distance between protons or carbons in ortho (o ),

meta (m ) or para ( p ) positions with respect to each

other). Thus, for the interacting nuclei of the same

isotopic species, Do : Dm : Dp ¼ 1 :ffiffi3

p=9 : 1=8 < 1 :

0:1925 : 0:1250: It has been found that the 1H1H

couplings in benzene indeed fulfill these ratios

and thus the JanisoHH contributions obviously are

negligible. In contrast, significant deviations have

been detected in the ratios of the 13C13C couplings

(see Ref. [70] and Table 3).

Other interesting and illustrating cases are pro-

vided by linear solute molecules. The ra structure, i.e.

the structure determined from the D couplings

corrected for harmonic vibrations, is internally

consistent. Consequently, there is no shrinkage effect

and the internuclear distances are additive. For

example, in ethyne (C2H2), RCC ¼ 2R0CH 2 RHH;

where RHH is the distance between the average

positions of the hydrogen atoms, and RCC and R0CH

are the corresponding one- and two-bond distances

between hydrogen and carbon positions. The use of

this relation renders possible the derivation of the

following equation [72]:

DJCC ¼3

Szz

pCCDexpCC 2 2

gH

gC

� 13

24

8<:

� 2pCH2D

expCH 2

1

3D2JCHSzz

� �2 13

2gH

gC

� 23

pHHDexpHH

� �2 13

35239=

; ð74Þ

where the anisotropy of JHH is assumed to be

negligible, and pMN ¼ 1þDhMN =DMN is the harmonic

correction factor (which is independent of molecular

orientation for molecules with at least a 3-fold

symmetry axis). One should emphasize in this context

that the vibrational corrections have to be calculated

exclusively for the purely dipolar part of the exper-

imental coupling, i.e. for D¼Dexp 2 ð1=2ÞJaniso:

Eq. (74) can be approximated by a linear equation

DJCC ¼AD2JCH þB ð75Þ

where experiments gave average values of 6.107

and 2112:7 Hz for A and B, respectively [72]. The

ab initio calculated point [72], (D2JCH ¼ 28:2 Hz,

DJCC ¼ 47:5 Hz), is relatively close to the above-

mentioned straight line. Consequently, this finding

can be regarded as the first experimental evidence

of the anisotropy of a 13C1H spin–spin coupling

Table 4

Liquid crystals used in studies of J tensors by their code name and

composition

Code name Composition

EBBA N-( p-ethoxybenzylidine)-p-n-butylaniline

HAB p,p0-di-n-heptylazoxybenzene

Phase 4 Eutectic mixture of p-methoxy-p0-n-

butylazoxybenzenes

Phase 5 Mixture of Phase 4 and p-ethoxy-p0-n-

butylazoxybenzenes

Phase 1221 Mixture of phenylcyclohexanes,

biphenylcyclohexane and phenylcyclohexane

esters

ZLI 997 Mixture of azoxy compounds and a biphenyl ester

ZLI 1132 Mixture of trans-4-n-propyl-(4-cyanophenyl)

-cyclohexane (24%), trans-4-n-pentyl

-(4-cyanophenyl)-cyclohexane (36%),

trans-4-n-heptyl-(4-cyanophenyl)-cyclohexane

(25%), trans-4-n-pentyl-(40-cyanobiphenyl-4)-

cyclohexane (15%)

ZLI 1167 Mixture of 4-n-trans,trans-bicyclohexyl-40

-carbonitrile (36%), 4-n-propyl-trans,trans-

bicyclohexyl-40-carbonitrile (34%), 4-n-heptyl

-trans,trans-bicyclohexyl-40-carbonitrile (30%)

ZLI 1982 Mixture of alkylphenylcyclohexanes,

alkylcyclohexanebiphenyls, and

bicyclohexanebiphenyls

ZLI 2806 Mixture of alkylbicyclohexanes and

alkyltricyclohexanes


tensor. In principle, a similar procedure can be

applied to the one-bond JCH but in practice it is

restricted by the uncertainty (a few percent) of the

vibrational correction factor 1pCH:

Another interesting example of a linear molecule is

carbon diselenide (CSe2). In this case the harmonic

corrections to the D couplings are presumably small,

and thus are omitted. Furthermore, omitting the

deformation contributions we obtain for the ratio of

the experimental couplings

DexpSeC

DexpSeSe

¼ 8gC

gSe

1 2 13DJSeCR3

SeC=F0SeC

1 2 83DJSeSeR3

SeC=F0SeSe

ð76Þ

(a series expansion cannot be applied here because DJ

may be relatively large) where the numerical values of

F0SeC and F0

SeSe are 5780:3 and 4398:6 A3 Hz, respect-

ively. If neither JSeC nor JSeSe is anisotropic, the ratio

should equal 8gC=gSe < 10:513: The experimentally

determined ratio is^3:66 ^ 0:05 : the sign of the DexpSeSe

coupling cannot be determined from the experimental

spectra without knowing DJSeSe: The coupling DexpSeC

can be deduced to be negative on the basis of good

agreement between experimental and calculated

nuclear shielding tensor anisotropies, DsC and DsSe;

DexpSeC was used to solve for the orientational order

parameter that has to be positive [8]. Consequently,

either one or most likely both of the two tensors possess

a sizable anisotropy. Relativistic extended Huckel

(REXNMR) [7,74] calculations estimated for the ratio

DJSeC=DJSeSe the value of , 0:2: Using this result one

obtains the following anisotropies:DJSeSe < 2654 Hz

and DJSeC < 2131 Hz or DJSeSe < 1212 Hz and

DJSeC < 242 Hz with the positive and negative sign

of the ratio of the experimental D couplings,

respectively. Which one of the two possible solutions

is closer to the truth, cannot be determined from the

experimental data. REXNMR calculations favor the

latter solution since they yieldþ330 andþ1330 Hz for

DJSeC and DJSeSe; respectively [8]. It is evident that the

anisotropic contribution ð1=2ÞJanisoSeSe ¼ ð1=3ÞDJSeSeSzz

dominates in DexpSeSe and determines the sign of the

coupling.

The above procedure can be applied even more

generally. Namely, if SMN for the nuclei M and N

is the same as SOP for the nuclei O and P in any

molecule, in other words the axes passing through

the nuclear pairs MN and OP are parallel, the ratio

DMN =DOP is independent of the orientation. If the

corresponding ratio of the experimental couplings is

found to deviate from the ratio of the purely

dipolar couplings, it suggests a Janiso contribution to

at least one of the couplings.

2.3.7. Results derived from LCNMR experiments

The LCNMR results for J tensors derived for a

number of ‘model systems’ since 1982 are shown in

Table 5.

The molecules investigated possess high sym-

metry (only one or two orientational order

parameters are needed to describe their orientation)

and in most cases they contain hydrogen atoms.

The importance of hydrogen atoms follows from

the fact, as stated above, that the indirect

contribution ð1=2ÞJanisoXH to D

expXH can generally be

neglected, and consequently the determination of

molecular structure and orientational order par-

ameters can be based on these couplings. Much

emphasis is given to the investigation of X13C

(X ¼ 13C, 14N, 15N, 19F) coupling tensors. There

are two reasons for this; first, the tensors can be

theoretically computed with reasonable effort and

good accuracy allowing for comparison between

experimental and calculated results, and second, the

DexpXC couplings are used to determine orientational

order parameters of LC molecules [82–85] and

biomacromolecules dissolved in dilute liquid-

crystalline solvents [61]. In order for the couplings

to be applicable in the latter cases it is necessary

to know the size of the indirect contribution

as compared to the respective DexpXC or DXC: As

pointed out above, D and ð1=2ÞJaniso do not vanish

simultaneously, and therefore in certain circum-

stances the anisotropic contribution may even

dominate in Dexp:

There are only a few LCNMR studies dealing

with couplings between heavy nuclei. The only

examples since 1982, as shown in Table 5, are the77Se13C and 77Se77Se coupling tensors in carbon

diselenide [8] and the 199Hg13C coupling tensor in

dimethyl mercury [81]. Earlier on, results for111,113Cd13C [86], 29Si13C and 119Sn13C [87] and77Se31P [88,89] were published but they can be

regarded as more or less qualitative as compared to

what is achievable today with all the necessary

corrections to Dexp:


Table 5

Indirect spin–spin coupling tensors as determined experimentally by LCNMR

Coupling and molecule Results and commentsa

Reference

13C1H coupling

Ethyne C2H2 A relation, based on the additivity of the ra distances, between DJCC and D2JCH was derived.

The relation was found to be linear: DJCC ¼ 6.107 £ D2JCH 2 112.7 Hz. MCLR

calculations predict the point (D2JCH ¼ 28.2 Hz, DJCC ¼ 47.5 Hz) which is fairly close to

the straight line; substitution of 47.5 Hz for DJCC in the above relation leads to D2JCH of

26.2 Hz, in perfect agreement with the calculations. Thus this finding may be considered as

the first experimental evidence of a non-negligible 13C1H spin–spin coupling tensor

anisotropy.

[72]

Values from 49.20 ^ 0.03 to 49.26 ^ 0.03 Hz were obtained for 2JCH in the isotropic state

of the LCs (ZLI 1167 and in three mixtures of ZLI 1167 and Phase 4) used in the

determination of the tensor anisotropy. For the details, see text.13C13C coupling

Acetonitrile CH3CN DJCC values of 115 ^ 24 and 112 ^ 25 Hz were determined for acetonitrile dissolved in

ZLI 1132 and ZLI 1167 LCs, respectively. In both cases, experimental dipolar couplings

were corrected only for harmonic vibrations, i.e. correlation between vibrational and

reorientational motion was neglected.

[75]

For JCC, values of 58.0 ^ 0.2 and 57.5 ^ 0.3 Hz were determined in acetone-d6 solution

and in the isotropic state of the ZLI 1167 LC, respectively.

A new method was developed in order to take into account correlation between

vibration and rotation. The method is based on considering only torques acting on the bonds

between light atoms of a molecule. The NMR data obtained for acetonitrile in five LCs

(ZLI 1132, ZLI 1167, Phase 4, EBBA, and a mixture of ZLI 1167 and Phase 4) were treated

applying the above procedure and a joint analysis of the dipolar couplings. This leads to

kDJCCl ¼ 30 ^ 33 Hz:

[71]

For the JCC, see above.

Benzene C6H6 The values of D exp were corrected for both vibrational and deformation effects.

Experiments performed in three LCs gave the following anisotropies for the nJCC tensors:

[70]

LC D1JCC D2JCC D3JCC

ZLI 1167 21.2 25.2 8.7

Phase 4 17.5 22.5 10.7

MIX 13.8 23.9 9.1

The respective average values are: 17.5, 23.9 and 9.5 Hz. (MIX is a 58:42 wt% mixture of

ZLI 1167 and Phase 4.)

The coupling constants, determined in the isotropic state of the LCs, range as follows:1JCC: 55.811 ^ 0.004…55.98 ^ 0.01 Hz2JCC: 22.519 ^ 0.009… 2 2.434 ^ 0.007 Hz3JCC: 10.090 ^ 0.006…10.12 ^ 0.02 Hz.

Ethane C2H6 DJCC was determined separately in five LC solvents (ZLI 1167, ZLI 1982 and three mixtures

of ZLI 1167 and Phase 4). Vibrational and deformation contributions were taken into

account and the internal rotation around the CC bond was treated quantum mechanically.

[72]

The anisotropy values range from 49 to 61 Hz, the average being 56 Hz.

JCC ranges from 34.498 ^ 0.015 to 34.558 ^ 0.006 Hz in the isotropic state of the LCs

used.

Ethene C2H4 Due to the D2h point group symmetry of ethene, two order parameters, Szz and Sxx 2 Syy, are

needed to describe its orientation. Therefore, the anisotropic contribution, JanisoCC ; is given by

Eq. (53). As the experiments in the six LCs (ZLI 1167, ZLI 1982, ZLI 2806, and three

mixtures of ZLI 1167 and Phase 4) do not yield independent information to determine both

DJCC and JCC,xx 2 JCC,yy, the asymmetry parameter (JCC,xx 2 JCC,yy)/JCC,zz was constrained

to be the same in different solvents. Least-squares fit of the mean value of the anisotropy

(DJCC was allowed to change from one LC to another) and the tensor asymmetry parameter

led to (with dipolar couplings corrected both for vibrational and deformation effects):

kDJCCl ¼ 11 Hz; and kJCC;xx 2 JCC;yyl ¼ 244 Hz: There is, however, quite a large variation

in the individual anisotropy values, from 3 to 21 Hz, when determined in different LCs.

[72]

JCC varies between 67.45 ^ 0.02 and 67.62 ^ 0.01 Hz in the isotropic state of the LCs used.

(continued on next page)


Table 5 (continued)


Reference

Ethyne C2H2 Fixing D2JCH to the ab initio calculated value of 28.2 Hz and applying the experimentally

determined relationship between DJCC and D2JCH gives DJCC ¼ 59.5 Hz.

[72]

JCC ranges from 169.63 ^ 0.02 to 169.819 ^ 0.014 Hz in the LCs heated to the isotropic

state.

For the details, see text and 13C1H coupling/ethyne14,15N13C coupling

Acetonitrile CH3CN A similar procedure as described above in determination of DJCC was applied to the

determination of D2J15NC The analysis resulted in D2J15NC ¼ 218 ^ 7 Hz:

[71]

A value of 2.9 ^ 0.2 Hz was measured for D2J15NC in CDCl3. [75]

Methylisocyanide CH3NC Experiments were performed in five LC solvents (ZLI 1167, ZLI 1132, Phase 4, EBBA, and

a 58:42 mixture of ZLI 1167 and Phase 4). Besides the correction of D couplings for

vibrational and deformation effects, also the deformation contribution to the JNC coupling

tensors was taken into account using an adjustable parameter. The DJ from a joint analysis

of five sets of DexpNC (altogether 10 couplings and four parameters) led to the following results:

[76]

kDJC14Nl ¼ 8:7 ^ 1:7 Hz; and kDJ14NCl ¼ 42:8 ^ 2:8 Hz:

The first coupling is over the single bond whereas the latter is over the triple bond.

Scaling of these results to correspond to the 15N13C couplings leads to 212.2 and

260.0 Hz, respectively.

JC14N ¼ 7:63 ^ 0:04 Hz and J14NC ¼ 6:30 ^ 0:09 Hz when determined in a CDCl3 solution. [77]19F13C coupling

Difluoromethane CH2F2 Experiments were carried out at several temperatures in three LCs (ZLI 1132, ZLI 1167 and

Phase 5). The experimental dipolar couplings were corrected for harmonic vibrations and

deformation effects. Furthermore, the contribution of the anharmonicity of the vibrational

potential was partially considered by estimating the diagonal cubic stretching force

constants from the respective harmonic ones. For symmetry reasons, the orientation of

CH2F2 is described by two independent orientional order parameters, Szz and Sxx 2 Syy, and

consequently the anisotropic contributions, JanisoFC ; in principle allow the determination of

both DJFC and JFC,xx 2 JFC,yy. In practice, however, experiments did not yield enough

independent data, and therefore, in performing a joint analysis of the experimental data, the

ratio DJFC/(JFC,xx 2 JFC,yy) was fixed to the corresponding ab initio value. Then

DJFC ¼ 13.5 Hz and JFC,xx 2 JFC,yy ¼ 2360 Hz. JFC ¼ 2236:01 ^ 0:05… 2 236:186 ^

0:006 Hz in the isotropic state of the LCs, whereas in the gas phase a value of

2233.91 ^ 0.11 Hz was measured.

[78]

Fluoromethane CH3F Experiments were carried out in eight LCs (ZLI 1167, EBBA, Phase 4, Phase 1221, and four

mixtures of ZLI 1167 and EBBA). When correcting the dipolar couplings only for harmonic

vibrations, DJFC ranges from 24955.3 ^ 260.2 to þ689.8 ^ 62.5 Hz in the individual LCs.

Performing a joint fit to 44 couplings corrected for both harmonic vibrations and

deformation effects, and taking into account the deformation contribution to JanisoFC ; leads to

kDJFCl ¼ 404 ^ 31 Hz:

[69]

JFC ¼ 2161.62 ^ 0.26… 2 161.20 ^ 0.24 Hz in the isotropic state of the LCs used.

In a recent paper, the kDJFCl was determined by using spectra recorded at eight temperatures

in one LC (ZLI 1132) and applying a joint analysis of the set of dipolar couplings corrected

for both vibrational and deformation effects. The resulting kDJFCl ¼ 350 Hz:

[78]

JFC ¼ 2161.30 ^ 0.04 Hz in the isotropic state of the LC.

p-Difluorobenzene

p-C6H4F2

Experiments were performed in five LCs (ZLI 1167, ZLI 1132, ZLI 1695, Phase 4, and a

mixture of ZLI 1167 and Phase 4 in which DexpCH of methane is vanishingly small). Only

harmonic vibrations were considered. Due to the C2v symmetry of the molecule, the nJanisoFC

depend on both DnJFC and nJFC,xx 2nJFC,yy, see Eq. (53).

[79]

In these particular LCs the ratio of the orientational order parameters, (Sxx 2 Syy)/Szz, varies

so that the two nJFC tensor properties could be determined for n ¼ 3 and 4:

Coupling tensor DJ Jxx 2 Jyy3JFC 8 ^ 9 Hz 25 ^ 10 Hz4JFC 111 ^ 17 Hz 2130 ^ 18 Hz

The z axis of the molecule-fixed frame lies along the CF bond.


Table 5 (continued)


Reference

Experiments were performed in five mixtures of various LCs (ZLI 1167 þ Phase 4, ZLI

2806 þ Phase 5, ZLI 1132 þ EBBA, ZLI 997 þ ZLI 1982, and ZLI 997 þ ZLI 1167). The

D exp were corrected for both harmonic and anharmonic vibrations, apart from the

deformation effects. In calculating the anharmonic corrections, a similar approximation was

applied as described for difluoromethane.

[60]

The experiments did not yield enough independent information to resolve the two tensorial

properties (see above), and therefore the ratio DnJFC/(nJFC,xx 2nJFC,yy) was fixed to the

value given by the ab initio calculation.

The results are shown below:

Coupling tensor DJ Jxx 2 Jyy1JFC 400 ^ 90 13 ^ 32JFC 239 ^ 2 220.5 ^ 1.13JFC 17.6 ^ 0.2 13.7 ^ 0.14JFC 220.0 ^ 0.9 235 ^ 2

The z axis of the molecule-fixed frame lies along the CF bond.

The nJFC couplings were found to be independent of the LC solvent in the isotropic state (at

355 K): 1JFC ¼ 2242.61 Hz, 2JFC ¼ 24.29 Hz, 3JFC ¼ 8.18 Hz, and 4JFC ¼ 2.67 Hz.29Si13C coupling

Methylsilane CH3SiH3 The harmonic force field was calculated at the semiempirical level with two

parametrizations (AM1 and PM3) and at the ab initio MP2 level. The anharmonic vibrations

were treated as described for difluoromethane. A quantum mechanical approach was applied

to average couplings over the internal rotation. The analysis of the set (obtained from

experiments in the ZLI 1167 and ZLI 2806 LCs) of corrected ‘best’ experimental dipolar

couplings led to an average value of 289 Hz for DJSiC. The use of harmonic force fields

derived from calculations at various levels results in DJSiC values that range from 286 to

2108 Hz. JSiC is 251.59 ^ 0.03 and 251.55 ^ 0.02 Hz in the isotropic state of the LCs

used.

[80]

77Se13C coupling

Carbon diselenide CSe2 The ratio of the experimental anisotropic couplings, lDexpSeC=D

expSeSel ¼ 3:66; was found to

deviate from 10.513, which is the value of the ratio for the case that DJSeC and DJSeSe are

simultaneously vanishingly small. Utilization of the REXNMR calculations in the analysis

of the experimental data leads to the value of either 2131 or þ242 Hz, depending upon

whether the sign of the ratio of the experimental couplings is positive or negative,

respectively. For details, see text.

[8]

JSeC ¼ 2226.59 ^ 0.36 Hz in CDCl3 solution and 2226 ^ 6 Hz in the isotropic state of

the ZLI 1132 LC.199Hg13C coupling

Dimethylmercury (CH3)2Hg The DJHgC ranges from 655 ^ 56 to 864 ^ 15 Hz when determined in four LCs (ZLI 1167,

Phase 4, and two mixtures of ZLI 1167 and Phase 4). Only a harmonic force field was taken

into account when calculating corrections for D exp.

[81]

JHgC varies from 690.3 to 693.8 Hz in the isotropic state of the LCs used.19F19F coupling

Trifluoromethane CHF3 In this particular case, the deformation contribution to 1DexpCH appeared to be exceptionally

large for the two LCs used (ZLI 1132 and ZLI 1167). Experiments were performed at

several temperatures and the set of corrected D exp was analysed using D2JFF as a free

parameter but keeping the DJ of the other coupling tensors fixed to their ab initio values.

[78]

The resulting D2JFF is 2200 Hz.2JFF could not be determined experimentally because of the chemical equivalence of the 19F

nuclei.

p-Difluorobenzene

p-C6H4F2

For details, see 19F13C coupling/p-difluorobenzene

Coupling tensor DJ Jxx 2 Jyy5JFF 230 ^ 15 Hz 236 ^ 15 Hz [79]

236.5 ^ 0.5 Hz 238.4 ^ 0.5 Hz [60]5JFF ¼ 17.445 Hz in the isotropic phase of the ZLI 997 (32.1 wt%)/ZLI 1982 (67.9 wt%) LC

mixture.



2.4. Solid-State NMR spectroscopy

2.4.1. Introduction

In the case of crystalline solids, the constituent

molecules are oriented in certain directions deter-

mined by the crystal structure. Large-amplitude

reorientational motion is usually not possible and

only small-amplitude lattice vibrations contribute to

the motional effects on NMR properties6. All the

interactions incorporated in HNMR can in principle

contribute to solid-state spectra, thus the correspond-

ing peaks are generally broad compared to those of

liquid and gaseous samples.

In a static sample, the angles in Eq. (35)

between the z0 direction of observation and the

molecule-fixed axes are constant. There is no need

for rotational averaging of Tz0z0 as in isotropic

media and LCs. The Hamiltonian corresponding to

spin-1=2 nuclei in the solid state thus takes the

form specified in Eqs. (29)–(32), HSSNMR ¼ HHFA

NMR:

The solid-state NMR observables vary depend-

ing on the nature of the NMR sample. A single

crystal sample is one coherent block of matter.

Apart from inevitable defects and vibrational

motion, the lattice vectors remain constant through-

out the sample. It often is difficult to grow large

enough single crystals for NMR experiments, and a

powder sample must be used. The latter consists of

randomly oriented crystallites that are small single

crystals themselves.

The single crystal samples are studied using a

goniometer whose rotation axis, z00 in the goniometer-

fixed frame ðx00; y00; z00Þ; is at the angle of u with respect

to B0: The components of the NMR tensors along the

direction of the field can again be obtained from the

transformation Eq. (35) as

Tz0z0 ¼ c0 þ c1coswþ s1sin wþ c2cos 2wþ s2sin 2w;

ð77Þ

where w is the turn angle of the goniometer and

c0 ¼ 12ðTx00x00 þ Ty00y00 Þsin2uþ Tz00z00cos2u;

c1 ¼ TSx00z00sin 2u; s1 ¼ TS

y00z00sin 2u;

c2 ¼ 12ðTx00x00 2 Ty00y00 Þsin2u; s2 ¼ TS

x00y00sin2u:

ð78Þ

The different coordinate systems used are illustrated

in Fig. 5.

The spectrum is a periodic function of w; thus

the five constants in Eq. (77) are available by using

one known angle u: The spectral observables are

the six (five if T ¼ D0) independent components of

T1 þ TS in the goniometer-fixed frame. By using

either several u values or different mounting

directions of the sample to the goniometer, the

full T1 þ TS is available. For discussions of

the effect of TA; for which a second-order treatment

has to be adopted, see Refs. [23,24,25]. So far,

there is no evidence from solid-state NMR for TA:

Finally, to obtain the NMR observables from

solid powder samples, Eq. (35) may be written with

e ; t ¼ a; b; or c, i.e., the transformation is between

the laboratory-fixed frame and PAS(T ),

T þ TSz0z0 ¼ sin2u cos2wTaa þ sin2u sin2wTbb

þ cos2uTcc; ð79Þ

Table 5 (continued)


Reference

77Se77Se coupling

Carbon diselenide CSe2 For the details, see 77Se13C coupling/carbon diselenide and text. [8]

Depending upon the sign of the ratio of the D exp, DJSeSe is either 2654 or þ1212 Hz. Both

of these are so large that they lead to the ð1=2ÞJanisoSeSe contributions that are much larger than

the purely dipolar contribution DSeSe in DexpSeSe. JSeSe was not determined experimentally

because of the magnetic equivalence of the 77Se nuclei

a For the code names and composition of LC solvents, see Table 4.

6 This is not strictly true, e.g. for guest species in molecular sieves

that have large enough cavities to allow large-amplitude rotation

and/or translation. The situation from the point of view of the NMR

observables of the guest then resembles that in LCs or isotropic

systems, depending on how hindered the motion is.


where u and w are the spherical coordinates that now

specify the orientation of B0 in the ða; b; cÞ frame of one

crystallite. The distribution of u and w; due to the

differently oriented crystallites, gives rise to a powder

pattern from which the principal values of the tensors

can be identified. The Tii (i ¼ a; b; c) are thus the NMR

observables of powder samples. The tensor T1 þ TS is

completely specified in PAS(T ) by the principal values

Taa; Tbb and Tcc: Alternatively, T, DcT ; and hc (Eqs.

(40), (57), and (58)) can be used. Practical details of

the analysis of single-crystal NMR spectra may be

found in Refs. [90–93].

2.4.2. Solid-State NMR determination of J tensors

As discussed above, solid-state NMR spectroscopy

offers the potential of providing a wealth of

information on anisotropic NMR interaction tensors.

Solid-state NMR techniques for the characterization

of J may be divided into three categories, based on the

nature of the sample and whether it is examined as a

stationary sample or a spinning sample. Stationary

samples may be either a single crystal or a powder

sample. NMR measurements on single crystals are

performed as a function of the orientation of the single

crystal in the applied B0: Under special circumstances

it may also be beneficial to spin a single crystal. NMR

measurements on powder samples can be performed

on a stationary powder sample or a sample that is spun

about some axis relative to B0: In principle, samples

may be spun at a rate that is relatively fast or slow

with respect to all internal NMR interactions. Most

often the angle between the spinning axis and B0 is the

so-called magic angle, but spectra may be acquired for

spinning at any angle b:

Each of the methods for characterizing J has its

own advantages and drawbacks; however, as with all

methods for determining DJ; the accuracy of the final

results depends strongly on knowledge of the direct

dipolar coupling constant, RDD; Eq. (61). In the solid-

state literature it is customary to define the exper-

imentally measured dipolar coupling constant as

Reff ¼ RDD 2DJ

3ð80Þ

for coincident dipolar and J tensors. In cases where

Reff is very similar in magnitude to RDD; corrections

for motional averaging become very important. It is

relatively straightforward to correct RDD for rovibra-

tional effects for diatomic molecules in the gas

phase, given the availability of high-quality exper-

imental data. Similarly, more complicated correc-

tions may be made for small molecules in liquid

crystal media (vide supra ). However, for molecules

in the solid state, how to carry out such corrections is

not obvious.

In total, there are a very limited number of accurate

and precise measurements of the complete J tensor

available from solid-state NMR due primarily to the

large number of parameters that are involved in

the analysis. To obtain reliable experimental J

tensors, the molecule, spin system, and type of

experiment to be carried out must be very carefully

chosen such that the number of assumptions that must

be made is minimized. The following discussion of

the available data delineates some of the assumptions

that are commonly made in the analysis of NMR

spectra for the extraction of DJ; and provides an

evaluation of the reliability of several of the reported

results.

The results to be discussed will generally be

restricted to the period 1990–2001, as literature on

the experimental measurement of J by solid-state

NMR methods has been covered in the review of

Power and Wasylishen [2] (see also Refs. [23,94]).

It is important to emphasize that the values of DJ

Fig. 5. The coordinate systems used in the determination of the

anisotropic properties of NMR tensors in single crystalline solids. u

is the angle between the rotation axis z00 of the goniometer-fixed

frame ðx00; y00; z00Þ and the external magnetic field B0: The laboratory-

and molecule-fixed frames are as in Fig. 4.


that we report in the solid-state context are defined

as DcJ ¼ Jk 2 J’; or in the more general case,

DcJ ¼ Jcc 2 ðJaa þ JbbÞ=2: Some literature uses a

so-called reduced anisotropy dJ ¼ ð2=3ÞDcJ:

2.4.3. Results from single crystal studies

The limited data on J available from single-crystal

NMR experiments are summarized in Table 6.

While single-crystal NMR experiments offer the

potential to provide some insight into the orientation

and asymmetry of J, in all reports to date these

properties have been dictated by symmetry and J has

been found to be axially symmetric within exper-

imental error.

One of the most frequently cited references on the

subject of anisotropic J is the 31P single-crystal NMR

experiment on tetraethyldiphosphine disulfide (TEPS)

of Tutunjian and Waugh [101]. This study, and their

subsequent one on the structurally similar compound

tetrabutyldiphosphine disulfide [102], reported par-

ticularly large values for DJPP; e.g. 2.2 or 8.8 kHz for

TEPS. A reinvestigation of the same coupling in

TEPS via single-crystal NMR by Eichele et al., in

1995 [92] revealed that the inadvertent neglect of a

factor of 3/2 in the analysis of Tutunjian and Waugh

was the likely cause of the apparently substantial

values of DJ: The value determined by Eichele et al.,

DJ ¼ 462 Hz, represents an upper limit and is more in

line with known 1JPP: A 31P NMR study of a single

crystal of the related compound, tetramethyldipho-

sphine disulfide (TMPS) resulted in an upper limit of

450 Hz for DJPP [96]. In any case where Reff and RDD

are of the same sign and Reff is less than RDD; the

resultant value of DJ must be an upper limit since the

effects of motional averaging are not known

accurately.

The most convincing evidence for DJ comes from

two single-crystal NMR investigations carried out by

Lumsden et al., on 1:1 and 1:2 mercury phosphine

complexes [99,100]. The large values of DJ199Hg31P; on

the order of 4–5 kHz, provide conclusive evidence for

non-Fermi contact coupling mechanisms. It is poss-

ible that the anisotropy arises solely from the

anisotropic SD/FC term, and J is nevertheless

dominated by the FC term. It seems very unlikely,

however that while the SD/FC mechanism would be

active, the SD and/or PSO mechanisms would not

contribute in a substantial way to both J and DJ:

Table 6

Indirect nuclear spin–spin coupling tensors determined from single-crystal NMR spectroscopy

Coupling and molecule Results and comments Reference

207Pb19F coupling

PbF2 (cubic) DcJ ¼ 8130 ^ 300 Hz (preferred) or 210 ^ 300 Hz. [95]

J ¼ ^(2150 ^ 50) Hz (negative sign preferred).31P31P coupling

Tetraethyldiphosphine disulfide DcJ ¼ 462 Hz (preferred) or 10362 Hz. [92]

J is axially symmetric within experimental error.

Tetramethyldiphosphine disulfide DcJ # 450 Hz [96]

113,115In31P coupling

InP DcJ ¼ 1220 ^ 75 or 2600 ^ 75 Hz [97]

J ¼ ^(225 ^ 10) Hz115In31P coupling

InP J ¼ ^(170 ^ 40) Hz [98]

B(pseudodipolar) ¼ 2300 ^ 70 or 2990 ^ 70 Hz199Hg31P coupling

HgPCy3(NO3)2 (Cy ¼ cyclohexyl) DcJ ¼ 5404 ^ 150 Hz (site 1) and 5385 ^ 150 Hz (site 2).

J ¼ 8199 ^ 25 Hz

[99]

J is axially symmetric within experimental error, with the unique

component coincident with the unique component of D0

Hg(PPh3)2(NO3)2 DcJ ¼ 4000 ^ 500 Hz, J ¼ 5550 Hz [100]

J is axially symmetric within experimental error, with the unique

component coincident with the unique component of D0.


Sears et al., have analyzed 19F NMR spectra of a

single crystal of cubic lead fluoride [95]. The crystal

was first oriented such that [100] axis was along the

direction of B0; for which both the 207Pb19F direct

dipolar and anisotropic J coupling are zero. By

examining spectra acquired for other crystal orien-

tations, JPbF was characterized, with preferred values

of J ¼ 22:15 ^ 0:05 kHz and DJ ¼ 8:13 ^ 0:3 kHz.

The 31P NMR spectrum of a stationary single

crystal of InP was analyzed by accounting for

contributions to the second moment, M2; from direct115In31P dipolar coupling, 31P31P homonuclear dipolar

coupling, isotropic J coupling, and anisotropic J

coupling [98]. Employing a similar strategy to what

was used for PbF2, a single crystal of InP was first

oriented such that [100] axis was along the direction

of B0 and subsequent moment analysis of the 31P free

induction decay as a function of crystal orientation

provided values for the isotropic and anisotropic parts

of J115In31P:

Tomaselli et al., also carried out a study of

J113;115In31P in undoped InP [97]. Triple-resonance

NMR experiments were performed on both powder

and single crystal samples, under both stationary and

magic-angle spinning (MAS) conditions. One of the

key experimental methods was to cross-polarize (CP)

from 113In nuclei to 31P, and acquire the 31P spectra

while decoupling 115In nuclei. The value of

J113;115In31P; ^ð225 ^ 10Þ Hz, was determined using

this triple-resonance technique on a powder sample,

from the splitting induced by 31P coupling to the

spin-9=2 113In nuclei. Insight into the magnitude of

DJ113;115In31P was afforded by a 113In31P CP experiment

on a single crystal spinning at the magic angle. As

shown in Fig. 6, the signal build-up upon cross-

polarization at the þ1 sideband matching condition

was simulated to successfully yield the value of

Reff ¼ ^�230 ^ 25

�Hz.

In combination with the known In–P bond length,

the two possible values of DJ113;115In31P were found to

be þ1220 ^ 75 Hz and þ2600 ^ 75 Hz. Of the

numerous studies of the JInP in InP [98,103,104], the

study of Tomaselli et al., provides the most convinc-

ing results. Furthermore, it is one of the most reliable

determinations of DJ for the case where lReff l is not

larger than lRDDl:The problems that arise in the interpretation of

spin–spin coupling tensors in situations where the D0

and J tensors are non-coincident have been discussed

for MAS powdered samples (vide infra). For exper-

iments involving either powdered or single crystal

samples, an important point is that only an effective

dipolar coupling tensor may be measured. If the D0

and J tensors are not forced to be in the same PAS by

symmetry, there is generally no unambiguous way to

analyze the dipolar couplings to gain information on

Fig. 6. (a) Total intensity of the 161.196 MHz 31P NMR signal

under conditions of cross-polarization from 113In to 31P in an indium

phosphide single crystal spinning at the magic angle at a rate of

10 kHz, as a function of contact time tCP: The circles represent data

obtained under J CP conditions and the crosses represent data

obtained for the þ1 sideband matching condition. The fit to these

data corresponds to lJ113In31Pl ¼ 225 Hz: (b) Same as part (a), with

an expansion in the region tCP ¼ 0–1 ms: The solid line fit to the

data points represented by crosses corresponds to an effective

dipolar coupling constant of 230 Hz, with error limits of ^25 Hz

denoted by the dashed lines. Reprinted with permission Ref. [97].

Copyright (1998) by the American Physical Society.


the orientation, anisotropy, or asymmetry of the D0

and J tensors individually. In practice, the only way to

confidently access information on J is to carefully

choose a spin system of appropriate symmetry, for

which the assumption of coincident D0 and J tensors is

well-founded. In such cases, the asymmetry of J will

likely be close to zero as a result of the requisite

symmetry.

As discussed in Section 2.3.3 and below, evidence

exists that the largest component of J does not always

lie along the internuclear vector, where the largest

component of D0 lies.

2.4.4. Results from studies of stationary powders

The J tensors characterized by analysing NMR

spectra of stationary powder samples are presented in

Table 7.

The major difficulty in determining precise values

of DJ from stationary powder samples lies in

the uncertainties associated with motional averaging

of D0: This is exemplified by the 1981 study of CH3F

in an argon matrix by Zilm and Grant [116], where

after correcting RDD19F13C

by about 3% for motional

averaging, a value of DJ19F13C ¼ 1200 ^ 1200 Hz

resulted, i.e. one cannot state with confidence that

DJ is non-zero.

It is clear that in order for reliable, precise, non-

zero values of DJ to be determined, the values of Reff

and RDD must differ significantly, beyond the point

where the difference could be attributed to motional

averaging. The absolute minimum difference in Reff

and RDD in order for a credible value of DJ to be

determined may be stated as approximately 10%,

though convincing and careful arguments should be

presented to convince the reader that such a difference

is not due solely to motional averaging of Reff : One

must also always bear in mind that the effect of DJ is

reduced by a factor of three when it is manifested as

part of Reff (see Eq. (80)). This fact further

complicates the extraction of very accurate and

precise values of DJ:

Three papers have reported on unexpectedly large

values of DJPP for 1,2-bis(2,4,6-tri-t-butylphenyl)

diphosphene and tetraphenyldiphosphine and DJPC for

2-(2,4,6-tri-t-butylphenyl)-1,1-bis(trimethylsilyl)

phosphaethene and 2-(2,4,6-tri-t-butylphenyl)phos-

phaethyne determined from analyses of stationary

powder samples [105,107,109]. For all of these

compounds except 1,2-bis(2,4,6-tri-t-butylphenyl)

diphosphene, it is likely that the relatively small

discrepancies in the measured Reff and the RDD

calculated from known bond lengths are due to

motional averaging rather than due to substantial

values of DJ: For example, for the 31P and 13C nuclei

involved in the double bond in 2-(2,4,6-tri-t-butyl-

phenyl)-1,1-bis(trimethylsilyl)phosphaethene, an

assumption of negligible anisotropy in J results in

an NMR-derived bond length of 1.72 A, which is only

3% longer than the X-ray value, 1.665 A [105].

Motional averaging is known to account for differ-

ences of approximately 1–4% between NMR-derived

bond lengths and those determined from X-ray

crystallography [117–120]. The large values of

DJPP and DJPC reported in Refs. [105,109] are clearly

suspect.

In the case of 1,2-bis(2,4,6-tri-t-butylphenyl)

diphosphene [107], however, the value of Reff ;

2800 ^ 100 Hz, was found to be greater than RDD;

2345 Hz. The difference of 455 Hz cannot be

attributed to motional averaging since such averaging

serves to reduce Reff to a value less than RDD: The

preferred value of DJ; 21380 Hz, is certainly

unexpectedly large for a phosphorus spin pair. It is

important to note that this system was treated as an A2

spin system (i.e. where the 31P are magnetically

equivalent) while subsequent studies indicated that it

is in fact an AB spin system [108]. It is conceivable

that the assumption of an A2 system could introduce

considerable error into the value of Reff determined

from the 2D spin-echo experiment, since J ¼ 580 Hz,

would contribute to the observed splitting in the F1

dimension. The actual value of Reff could be as low as

2220 Hz, which is 5% less than RDD:

Several reliable values of DJ199Hg31P are known

from 31P NMR spectroscopy of stationary powder

samples [113–115]. For the series [HgPR3(NO3)2]2

(see Table 7), J199Hg31P ranges from 8008 to 10566 Hz,

and DJ199Hg31P is on the order of 5 kHz, with errors of

less than 10%. In these systems, DJ=3 (<1670 Hz)

makes a larger contribution to Reff than does RDD

(<645 Hz). Since the magnitude of Reff is larger than

RDD for all of these mercury–phosphorus compounds,

one can be confident that the source of the difference

is due to DJ: Presented in Fig. 7 is an example of the

rotation plots generated in the 31P NMR analysis of a

single crystal of Hg(PPh3)2(NO3)2.


In this case, the maximum possible splitting

Dn2 Jiso in the absence of an anisotropic J tensor is

1200 Hz. The experimental measurement of larger

splittings provides unambiguous evidence for a

significant DJ:

A series of cis and trans platinum phosphines of the

type Pt(PR3)2Cl2 has been investigated by 31P NMR

[111]. This study has provided several large values of

DJ195Pt31P; on the order of 1–2 kHz (Table 7). As for the

HgP couplings, Reff differs substantially from RDD; thus

providing convincing evidence for the existence of large

anisotropy in J. For example, in the case of trans-

Pt(PCy3)2Cl2, RDD is 822 Hz from the Pt–P bond length

of 2.337 A [111], while Reff is only 35% of this value.

Table 7

Indirect nuclear spin–spin coupling tensors determined from NMR spectroscopy of stationary powder samples (results in Hz)


31P13C coupling

2,4,6-t-Bu3C6H2PyC(SiMe3)2 DcJ ¼ 777 or 15 117, J ¼ 91. See text for discussion. [105]

2,4,6-t-Bu3C6H2PyC(SiMe3)2 DcJ ¼ 1008 or 10 638, J ¼ 90. Coupling to aryl 13C. See text for

discussion.

[105]

2,4,6-t-Bu3C6H2CP DcJ ¼ 1233 or 19 821, J ¼ 59. See text for discussion. [105]199Hg13C coupling

K2Hg(CN)4 DcJ ¼ 950 ^ 60, J ¼ 1540 ^ 2 [106]31P31P coupling

1,2-Bis(2,4,6-tri-t-butylphenyl)diphosphene DcJ ¼ 21380 or 15 420 (J ¼ 580 ^ 20 [108]). See text for

discussion of possible errors in the spectral analysis for this

compound.

[107]

Tetraphenyldiphosphine DcJ ¼ 300, J ¼ 2200 ^ 100. See text for discussion. [109]115In31P coupling

Br3InP(4-(CH3O)C6H4)3 DcJ ¼ 1178 ^ 150 (preferred) or 22558 ^ 150 [110]

J ¼ 1109 ^ 9195Pt31P coupling

trans-Pt(PPh3)2Cl2 DcJ ¼ 1865 ^ 250, J ¼ 2624 ^ 25 [111]

trans-Pt(PCy3)2Cl2 DcJ ¼ 1602 ^ 250, J ¼ 2420 ^ 25 [111]

trans-Pt(PEt3)2Cl2 DcJ ¼ 1536 ^ 250, J ¼ 2392 ^ 25 [111]

cis-Pt(PPh3)2Cl2 DcJ ¼ 2184 ^ 600 or 3282 ^ 600 (site 1) [111]

DcJ ¼ 1968 ^ 600 or 3498 ^ 600 (site 2)

DcJ ¼ 2037 ^ 600 or 3429 ^ 600 (site 3), J ¼ 3727 ^ 25 (site 1),

3910 ^ 25 (site 2), 3596 ^ 25 (site 3)

cis-Pt(Et3)2Cl2 DcJ ¼ 1356 ^ 600 or 4104 ^ 600 [111]

J ¼ 3448 ^ 25

Cl2Pt(PPh2CH2PPh2) DcJ ¼ 2130, J ¼ 3064 [112]

Cl2Pt(PPh2CH2CH2PPh2) DcJ ¼ 1660, J ¼ 3591

Cl2Pt(PPh2CH2CH2CH2PPh2) DcJ ¼ 840, J ¼ 3354199Hg31P coupling

(EtO)2P(O)Hg(OOCCH3) DcJ ¼ 2700 ^ 250, J ¼ 13 324 ^ 15 [113]

(EtO)2P(O)HgI DcJ ¼ 1500 ^ 250, J ¼ 12 623 ^ 15 [113]

(EtO)2P(O)Hg(SCN) DcJ ¼ 1600 ^ 250, J ¼ 12 119 ^ 15 [113]

[HgPR3(NO3)2]2 [114]

R ¼ phenyl DcJ ¼ 4545 ^ 500, J ¼ 9572 ^ 15

m-tolyl DcJ ¼ 5235 ^ 200, J ¼ 9165 ^ 15

p-tolyl DcJ ¼ 5470 ^ 200, J ¼ 9144 ^ 15

mesityl DcJ ¼ 5560 ^ 500, J ¼ 10 468 ^ 15 (site 1)

DcJ ¼ 5560 ^ 500, J ¼ 10 566 ^ 15 (site 2)

p-MeOPh DcJ ¼ 4765 ^ 250, J ¼ 9327 ^ 15 (site 1)

DcJ ¼ 3740 ^ 375, J ¼ 9309 ^ 15 (site 2)

cyclohexyl DcJ ¼ 5525 ^ 200, J ¼ 8008 ^ 15

[HgP(o-tolyl)3(NO3)2]2 DcJ ¼ 5170 ^ 250, J ¼ 9660 [115]


Phosphorus-31 dipolar-chemical shift NMR exper-

iments carried out on a series of metal compounds

featuring cyclic phosphino ligands yielded three

values of DJ195Pt31P [112]. For the series Cl2Pt(PPh2-

CH2PPh2), Cl2Pt(PPh2CH2CH2PPh2), Cl2Pt(PPh2-

CH2CH2CH2PPh2), DJ195Pt31P was found to decrease

with increasing ring size, with a maximum value of

2130 Hz for Cl2Pt(PPh2CH2PPh2). The reported

values are reliable in that the measured Reff195Pt31P

are

significantly different from the predicted dipolar

coupling constants, well beyond any reasonable

differences due to motional averaging. For example,

for Cl2Pt(PPh2CH2PPh2), Reff is approximately 25%

of the value of RDD:

Anisotropy in J199Hg13C in partially 13C-enriched

K2Hg(CN)4 was determined from 199Hg spectra of

stationary samples [106]. The symmetry of the

tetracyanomercurate anion guarantees axial symmetry

of J. The measured value of Reff was found to be 60%

less than the value of RDD obtained from the Hg–C

bond length of 2.152 A [121]; such a large difference

clearly cannot be accounted for by considering

motional averaging effects alone. The value of DJ

obtained, 950 ^ 60 Hz, is in good agreement with the

value obtained for dimethylmercury in a LC solvent,

864 Hz [81,122].

Wasylishen and co-workers analysed the 31P NMR

spectra of MAS and stationary samples of solid

Br3InP(4-(CH3O)C6H4)3 (Fig. 8) and obtained values

of J115In31P ¼ 1109 ^ 9 Hz and DJ ¼ 1178 ^ 150 Hz

[110].

The presence of a 3-fold symmetry axis about the

indium–phosphorus bond guarantees axial symmetry

of D0 as well as J. Analysis of the spectrum of a

stationary sample, shown in Fig. 8(b), provided a

value for Reff of 230 ^ 50 Hz, which differs signifi-

cantly from RDD ¼ þ623 Hz; determined from the

bond length. The analysis also demonstrates the

different effects of the direct and indirect spin–spin

coupling interactions on each of the ten 31P subspectra

arising because of the allowed indium spin states

(Fig. 8(c)).

2.4.5. Results from spinning powder samples

Values of DJ determined by analysing NMR

spectra of spinning powdered samples are summar-

ized in Table 8.

In this section, we will discuss selected

representative examples in detail. As with all

methods for determining reliable values of DJ;

experiments where powdered solid samples are

spun at an angle with respect to B0 rely on a priori

knowledge of RDD: Reliable estimates of RDD may

be calculated from a relevant internuclear distance

determined from a diffraction experiment. Many

efforts to measure DJ have involved MAS;

however, by spinning the sample about an axis

off the magic angle, one can in principle access a

value of Reff which is scaled by ð3 cos2b2 1Þ=2;

where b is the angle between the rotation axis and

B0: Of course all anisotropic interactions will be

scaled by rapid sample spinning. By obtaining

high-quality NMR spectra at several angles b, the

Fig. 7. Variation in the 199Hg31P effective dipolar coupling obtained

at 81.03 MHz for rotation of a single crystal of Hg(PPh3)2(NO3)2

about the crystal-holder X-axis. The signals due to two crystal-

lographically distinct sites are represented by squares and circles.

The horizontal line at 1200 Hz represents the maximum possible

coupling due solely to direct dipolar coupling and in the absence of

DJ. Since the experimental data points surpass this limit, DJ199Hg31P

is non-zero for this compound. Reprinted with permission from Ref.

[100]. Copyright (1995) American Chemical Society.


spectrum of the stationary sample may be inferred

by extrapolating the frequencies of shoulders and

singularities to b ¼ 08: This could be advantageous

if the NMR spectrum under consideration consists

of several peaks which overlap as b! 0: Regard-

less of the angle, it is desirable to spin very fast to

reduce the number of spinning sidebands, and

concentrate the signal intensity in the centreband.

In cases where this is not possible, the intensity

from the sidebands must be added to the centre-

band after acquisition, using spectral processing

software.

We may divide the methods for obtaining DJ from

spinning samples into three general categories based

on the types of nuclei involved: (i) a spin-1/2 nucleus

coupled to a quadrupolar nucleus; (ii) a heteronuclear

spin-1/2 pair; (iii) a homonuclear spin-1/2 pair. In

general, the methods rely on the assumption that the

spectrum may be successfully analysed as an isolated

spin pair. In principle, solid-state NMR experiments

on a spin system consisting of a pair of either

homonuclear or heteronuclear quadrupolar nuclei

will, in favourable cases, also yield information on

DJ: For example, Wi and Frydman [140] have

outlined the methodology for extracting DJ from

multiple quantum MAS (MQMAS) [141,142] spectra

by carrying out experiments involving 14N11B,11B11B, and 55Mn55Mn spin pairs; however, non-zero

values of DJ obtained using these methods have not

been reported.

To date, there have been some general assump-

tions regarding the orientation and symmetry

properties of J that have been required to facilitate

the determination of DJ in spinning samples. First,

it is inevitably assumed that the asymmetry

parameter of J tensor, h ¼ hc; is zero. This is a

valid assumption for geometrical arrangements of

high symmetry, i.e. C3v or higher. For systems of

lower symmetry, h may fortuitously be close to

zero. However, one must be aware that in general

there is no requirement for h to be zero. The

second assumption, which is almost always made,

is that J is coincident with D0: For 1J; this implies

that the largest components of both J and D0 are

along the bond axis. One of the major reasons for

this assumption has been the lack of evidence to

the contrary, although recent high-level ab initio

and DFT calculations have provided this evidence for

several systems, in particular many one-bond inter-

halogen couplings, where the largest component of J

is perpendicular to the bond axis [27,143–146]. In

general, an incorrect assumption of coincident J and

Fig. 8. (a) 81.033 MHz 31P CPMAS NMR spectrum of solid Br3In-P(4-(CH3O)C6H4)3, depicting splittings due to the isotropic J115In31P coupling

constant of 1109 ^ 9 Hz. (b) 31P CP NMR spectrum of a stationary powder sample of Br3In-P(4-(CH3O)C6H4)3, with splittings due to the

effective dipolar coupling between 115In and 31P evident. (c) Simulation of the spectrum shown in part (b), from which a value of

DJ115In31P ¼ 1178 ^ 150 Hz is extracted. Shown in part (d) are each of the ten 31P subspectra arising due to the allowed indium spin states.

Reprinted with permission from Ref. [110]. Copyright (1994) American Chemical Society.


Table 8

Indirect nuclear spin–spin coupling tensors determined from solid-state NMR of spinning samples


199Hg14,15N coupling

K2Hg(CN)4 Dc2J199Hg14N ¼ 69 ^ 15 Hz; 2J199Hg14N ¼ 20:6 ^ 2:0 Hz [106]

Dc2J199Hg15N ¼ 297 ^ 15 Hz; 2J199Hg15N ¼ 29:0 ^ 2:0 Hz

199Hg MAS NMR119Sn19F coupling

Me3SnF Reff ¼ 24020 ^ 350 Hz, 1J ¼ ^1300 ^ 10 Hz [123]

(i-Bu)3SnF Reff ¼ 23740 ^ 350 Hz, 1J ¼ ^1260 ^ 10 Hz

Ph3SnF Reff ¼ 23260 ^ 350 Hz, 1J ¼ ^1530 ^ 10 Hz119Sn MAS NMR. The precise extraction of DcJ from Reff is hampered by the

lack of knowledge of the precise and accurate Sn–F bond lengths for these

compounds

Mes3SnF Dc2J ¼ 23150 and 22950 Hz (2 sites), J ¼ ^2275 Hz [124]

(n-Bu)3SnF Reff ¼ 24000 ^ 400 Hz, DcJ ¼ 240 to 21320 Hz

The large range of possible values for DcJ arises due to the lack of knowledge of

the precise Sn–F bond lengths for this compound.55Mn31P coupling

Mn2(CO)9PPh3 DcJ ¼ 1027 or 5400 Hz, J ¼ ^297 Hz [125]31P MAS NMR. The two possible values of DcJ arise because the sign of C

Q55Mn

is not known. hQ is assumed to be zero.

PhCH2C(O)Mn(CO)4(PPh3) DcJ ¼ 678 ^ 42 Hz, J ¼ 216 ^ 4 Hz [126]

MesCH2C(O)Mn(CO)4(PPh3) DcJ ¼ 589 ^ 24 Hz, J ¼ 233 ^ 2 Hz

PhCH2C(O)Mn(CO)4[P(C6H11)3] DcJ ¼ 639 ^ 41 Hz, J ¼ 220 ^ 2 Hz

MesCH2C(O)Mn(CO)4[P(C6H11)3] DcJ ¼ 495 ^ 10 Hz, J ¼ 232 ^ 2 Hz

PhCH2Mn(CO)4(PPh3) DcJ ¼ 412 ^ 13 Hz, J ¼ 202 ^ 2 Hz

PhCH2Mn(CO)4[P(tolyl)3] DcJ ¼ 508 ^ 22 Hz, J ¼ 196 ^ 3 Hz

PhCH2Mn(CO)4[P(PhF)3] DcJ ¼ 538 ^ 30 Hz, J ¼ 204 ^ 1 Hz31P MAS NMR.

See text for discussion of these data, in particular the small reported errors.63Cu31P coupling

Several DcJ ¼ 600 Hz, J ¼ 900–2000 Hz [127]

Triphenylphosphine-copper(I)

complexes

31P MAS NMR. In some cases, the sign of CQ63Cu

and magnitude of hQ are

estimated by the method of Vega [128].63,65Cu31P coupling

[(PBz3)2Cu][CuBr2] DcJ ¼ 750 ^ 50 Hz, J ¼ 1535 ^ 10 Hz [129]

[(PBz3)2Cu][PF6] DcJ ¼ 720 ^ 50 Hz, J ¼ 1550 ^ 10 Hz31P MAS NMR. Interaction tensor orientations are dictated by symmetry to be

coincident. Similarly, the value of hQ for copper is zero by symmetry.77Se31P coupling

(Me)3PSe DcJ ¼ 640 ^ 260 Hz, J ¼ 2656 Hz from 31P MAS NMR,

DcJ ¼ 550 ^ 140 Hz, J ¼ 2656 Hz from 77Se CPMAS NMR

[130]

(Ph)3PSe DcJ ¼ 590 ^ 150 Hz, J ¼ 2735 Hz from 31P CPMAS NMR [130]113Cd31P coupling

Cd(NO3)2·2PMe2Ph DcJ ¼ 22600 or 21101 Hz, J ¼ 2285 Hz [131]

From 113Cd-31P rotary resonance MAS spectra.

Two possible values arise due to lack of knowledge concerning the absolute

sign of Reff (Reff ¼ 0 ^ 250 Hz)119Sn35Cl coupling

(Benzyl)3SnCl DcJ ¼ 2438 Hz with ‘substantial possible error’, J ¼ 227 Hz [132]

SnCl2(acac)2 DcJ ¼ 2740 Hz, J ¼ ^276 Hz119Sn MAS NMR. C

Q35Cl

is assumed to be negative, and hQ is assumed to be

zero. For SnCl2(acac)2, the magnitude of CQ35Cl

is estimated.

Ph3SnCl DcJ ¼ 2350 Hz. 119Sn MAS NMR. CQ35Cl

is assumed to be negative, and hQ is

assumed to be zero.

[133]


D0 tensors will have a large impact on the

resulting value of DJ; however for many systems

this is likely to be a valid assumption. Less

general assumptions which are sometimes invoked

to extract DJ will be discussed as appropriate in

the sections below.

Coupled spin-1/2 and quadrupolar nuclei. Most

MAS studies of DJ have involved a spin-1/2 nucleus

Table 8 (continued)


119Sn55Mn coupling

( p-XC6H4)3SnMn(CO)5 [134]

X ¼ CH3 DcJ ¼ 622 ^ 42 Hz, J ¼ 132 ^ 2 Hz

X ¼ H (a,b) DcJ ¼ 354 ^ 4 Hz, J ¼ 135 ^ 2 Hz

X ¼ H (c) DcJ ¼ 515 ^ 33 Hz, J ¼ 141 ^ 3 Hz

X ¼ H (d) DcJ ¼ 352 ^ 4 Hz, J ¼ 141 ^ 2 Hz

X ¼ OCH3 DcJ ¼ 398 ^ 13 Hz, J ¼ 149 ^ 1 Hz

X ¼ F (a) DcJ ¼ 401 ^ 13 Hz, J ¼ 165 ^ 2 Hz

X ¼ F (b) DcJ ¼ 566 ^ 16 Hz, J ¼ 151 ^ 2 Hz

X ¼ Cl DcJ ¼ 305 ^ 12 Hz, J ¼ 160 ^ 2 Hz

X ¼ SCH3 DcJ ¼ 501 ^ 21 Hz, J ¼ 170 ^ 1 Hz

X ¼ SO2CH3 DcJ ¼ 584 ^ 15 Hz, J ¼ 250 ^ 3 Hz119Sn MAS NMR

(X ¼ H: four molecules (a–d) in the unit cell; X ¼ F: two (a,b).)

Ph3SnMn(CO)5 (3 sites) DcJ ¼ 353 ^ 8 Hz, J ¼ 135 ^ 1 Hz [135]

DcJ ¼ 345 ^ 8 Hz, J ¼ 142 ^ 2 Hz

DcJ ¼ 507 ^ 55 Hz, J ¼ 141 ^ 1 Hz119Sn MAS NMR

[Mn(CO)5]2SnPh2 RDD ¼ 2560 ^ 4 Hz, Reff ¼ 223.6 ^ 0.6 Hz, J ¼ 139 ^ 1 Hz [136]

DcJ not reported. 119Sn MAS NMR207Pb55Mn coupling

Ph3PbMn(CO)5 Recrystallized from octane: [135]

Reff ¼ 47.4 ^ 12.4 Hz, J ¼ 250 ^ 4 Hz (site A),

Reff ¼ 190 ^ 15 Hz, J ¼ 253 ^ 4 Hz (site B),

Reff ¼ 267 ^ 22 Hz, J ¼ 275 ^ 8 Hz (site C),

Reff ¼ 52.9 ^ 4.3 Hz, J ¼ 274 ^ 7 Hz (site D).

DcJ not reported. 207Pb MAS NMR

Recrystallized from benzene–octane:

Reff ¼ 49.7 ^ 0.4 Hz, J ¼ 251 ^ 1 Hz (site A),

Reff ¼ 65.8 ^ 2.3 Hz, J ¼ 247 ^ 1 Hz (site B),

Reff ¼ 97.0 ^ 6.5 Hz, J ¼ 273 ^ 3 Hz (site C),

Reff ¼ 52.7 ^ 2.2 Hz, J ¼ 274 ^ 1 Hz (site D)

DcJ not reported. 207Pb MAS NMR

[Mn(CO)5]2PbPh2 RDD ¼ 293 ^ 3 Hz, Reff ¼ 5.6 ^ 0.4 Hz, J ¼ 228 ^ 1 Hz [136]

DcJ not reported. 207Pb MAS NMR119Sn117Sn coupling

(benzyl3Sn)2O Dc2J ¼ 1263 ^ 525 Hz (preferred), or 429 ^ 525 Hz [137]

2J ¼ ^950 Hz. 119Sn off-MAS NMR. The two tin atoms are

crystallographically equivalent. Spinning off the magic angle (e.g. 568)

reintroduces the effective dipolar coupling between 119Sn and 117Sn nuclei in

the linear Sn–O–Sn fragment. The value of 1263 Hz is preferred if the

supposition that 2J is positive holds. Note that a ‘reduced anisotropy’, dJ ¼

ð2=3ÞDcJ; is reported in Ref. [137].125Te123Te coupling

(Me4N)2Te2 DcJ ¼ 24270 ^ 800 Hz, J ¼ ^2960 ^ 5 Hz [138]123Te MAS NMR. The two Te atoms are crystallographically equivalent.

Information on the tellurium chemical shift tensor is extracted from the

spinning sideband manifold of uncoupled 123Te nuclei, via the method of

Herzfeld and Berger [139]

All results rely on the assumption that J is axially symmetric and coincident with the direct dipolar tensor, unless otherwise stated.


coupled to a half-integer spin quadrupolar nucleus, S

(Table 8). The spin angular momentum of the

quadrupolar nucleus is not completely quantized by

B0; rather, the largest component of the electric field

gradient tensor competes with B0 to determine the

final direction of quantization. As a result, MAS does

not completely average Reff to zero for a spin pair

involving a quadrupolar nucleus. This phenomenon

has been extensively discussed in the literature; the

perturbation approach of Olivieri is most useful [133,

147,148]. According to the perturbation approach,

under conditions of rapid MAS, the spectrum of the

spin-1/2 nucleus depends on d, the residual dipolar

coupling constant,

d ¼3CQ

20nS

SðS þ 1Þ2 3m2

Sð2S 2 1Þ½RDDð3 cos2bD 2 1

þ hQ sin2bD cos 2aDÞ21

3DJð3 cos2bJ 2 1

þ hQ sin2bJ cos 2aJÞ�: ð81Þ

This form of d accounts fully for the relative

orientations of the electric field gradient (EFG), D0;

and J tensors, and is valid in the regime where CQ ,

4Sð2S 2 1ÞnS: Here, CQ is the quadrupole coupling

constant of the quadrupolar nucleus S, nS is the

Larmor frequency of spin S, hQ is the asymmetry

parameter of the EFG tensor for S, bD and aD are the

polar angles that describe the orientation of D0 in the

PAS of the EFG tensor of the quadrupolar nucleus.

Analogously, bJ and aJ are the corresponding angles

which describe the orientation of J in PAS(EFG).

When CQ is of the same order of magnitude as

4Sð2S 2 1ÞnS; the perturbation treatment which yields

Eq. (81) is no longer valid and complete diagonaliza-

tion is required [149].

Clearly, there are several parameters which must

be determined accurately in order to obtain convin-

cing evidence for anisotropy in J. CQ and hQ may be

measured independently via an NMR experiment on

the quadrupolar nucleus. In many cases, however, the

value of CQ is prohibitively large for NMR exper-

iments at moderate field strengths and in these

situations, nuclear quadrupole resonance (NQR)

experiments may provide CQ: It is important to note,

however, that the sign of CQ is not provided by direct

observation of the quadrupolar nucleus. Knowledge of

the relative signs of CQ and Reff are nevertheless

extremely important when attempting to determine

DJ from a MAS spectrum of the spin-1/2 nucleus.

Furthermore, a relatively straightforward analysis

via Eq. (81) is only feasible if some knowledge of the

relative orientations of the three interaction tensors is

available. In particular, the relative orientations of the

EFG, D0; and J tensors may confidently be assigned

only in cases where high symmetry dictates the

orientations of these tensors. In situations of lower

symmetry, e.g. where hQ is not zero, it is extremely

difficult to make statements concerning the relative

orientations of the three interaction tensors. For

example, one cannot state with certainty that J is

coincident with D0: This requires that all parameters in

Eq. (81) be considered independently, and thereby

renders the already formidable task of extracting DJ

even more daunting. In general, therefore, the most

reliable values of DJ which are determined by

observing the MAS NMR spectrum of a spin-1/2

nucleus coupled to a quadrupolar nucleus are those for

which hQ is zero.

Most of the values of DJ which have been

extracted from analysis of the MAS spectrum of a

spin-1/2 nucleus coupled to a quadrupolar nucleus

involve 119Sn or 31P. For example, analysis of the 31P

CPMAS spectra of two linear bis(tribenzylphosphine)

cuprate(I) salts, wherein 31P is coupled to 63Cu and65Cu, both spin-3/2 nuclei, yielded precise values of

þ720 ^ 50 Hz and þ750 ^ 50 Hz for DJ [129].

Shown in Fig. 9 is a demonstration of the sensitivity of

the simulated 31P MAS NMR spectra of [(PBz3)2-

Cu][CuBr2] to the magnitude as well as the sign of DJ:

In these cases, molecular symmetry of the

P–Cu–P fragment [150] guarantees that the EFG

and J tensors are axially symmetric, and also provides

strong indications that the EFG, J, and D0 tensors will

be coincident. The values of CQCu were obtained via

NQR experiments. 31P CPMAS measurements were

made at three applied field strengths, and the analysis

involved a complete diagonalization of the Hamil-

tonian, rather than the perturbation approach dis-

cussed above.

Christendat et al. have provided several values of

DJ for 55Mn31P and 119Sn55Mn spin pairs in a series

of compounds [126,134,135]. For several complexes

of relatively low symmetry involving J207Pb55Mn and

J119Sn55Mn; it was recognized that in cases where


symmetry does not dictate the relative orientations of

the EFG, D0; and J tensors, quantitative information

concerning DJ may not be extracted with confidence

[135,136]. For other systems of higher symmetry,

analyses of the high-quality 31P MAS NMR spectra

for a series of tertiary phosphine substituted alkyl- and

acyltetracarbonylmanganese(I) complexes relied on

assumptions concerning the sign of CQ55Mn

; the value of

hQ (approximately zero), and the relative orientations

of the EFG, D0; and J tensors. These assumptions,

which are based on crystal symmetry, calculations on

model systems, symmetry arguments, and prior data

on similar compounds are generally well founded.

However, the CQ55Mn

are not known independently.

Given the large number of approximations which

must be made for these systems, the very small errors

on DJ which are reported seem optimistic. For

example, DJ55Mn31P for MesCH2C(O)Mn(CO)4[P(C6-

H11)3] is reported as 495 ^ 10 Hz. Such a small error

would imply that the error in Reff is only 3.3 Hz; this is

implausible considering that X-ray structures are not

available for many of the complexes. An error in the

manganese–phosphorus bond length of just 0.002 A

will lead to an uncertainty in RDD of more than 3.3 Hz.

Additionally, the reported values of Reff were not

corrected for vibrational averaging, a procedure

which would be required to claim such a small error

in DJ: Errors as small as 4 Hz were reported for

C6H4SnMn(CO)5 [134].

The quadrupolar nucleus involved in J does not

have to be of half-integer spin; the possibility of

anisotropic coupling to 14N ðI ¼ 1Þ has been dis-

cussed by Olivieri and Hatfield [151]. However, for

silicon nitride and associated compounds involving

SiN spin pairs, no conclusive evidence for DJ was

found. Anisotropic coupling between the spin-1/2

nucleus 199Hg and 14N was found by analysing

the 199Hg MAS NMR spectra of K2Hg(CN)4, where

D2J199Hg14N was found to be þ69 ^ 15 Hz [106].

The tetrahedral geometry of the tetracyanomercurate

anion once again provides the symmetry necessary for

a confident analysis of the spectra. This is the first D2J

to be determined for a solid. The scarce previous

reports of two-bond coupling anisotropies are from

LCNMR results (vide supra).

Coupled heteronuclear spin-1/2 pairs of nuclei.

Several values of DJ have been determined for

heteronuclear spin-1/2 pairs of nuclei (Table 8).

Many studies have employed slow MAS followed

by either a Herzfeld–Berger [152] analysis or analysis

by the method of moments [153] for the individual

subspectra arising from the two possible spin states of

the coupled spin-1/2 nucleus, as described by Harris

et al. [123,124,154]. These methods of analysis are

typically only valid for an isolated spin-1/2 nucleus;

however if each subspectrum is treated independently,

then ‘effective’ tensor components may be extracted

and interpreted to provide a value of Reff ; since

the relative intensities of each sideband for the two

subspectra are dependent on Reff : An obvious

prerequisite to this analysis is that J is large enough

for the subspectra to be resolved. This method has

been employed, for example, by Grossmann et al. to

extract DJ77Se31P ¼ þ590 ^ 150 Hz from the 31P

CPMAS NMR spectrum of triphenylphosphine sele-

nide [130]. The value determined in this manner for

trimethylphosphine selenide, DJ77Se31P ¼ þ640 ^

260 Hz [130], is in excellent agreement with the

Fig. 9. Simulated 31P CPMAS NMR spectra of [(PBz3)2Cu][CuBr2]

at 4.7 T, demonstrating the influence of the sign and magnitude of

DJ65=63Cu31P ¼ 750 Hz: Reprinted with permission from Ref. [129].

Copyright (1998) Academic Press.


value determined in a nematic LC phase,

þ680 ^ 60 Hz [89,94]. Although the molecules

possess only C1 symmetry in the solid state, their

pseudo-C3 symmetry lends support to the assumption

that J is nearly axially symmetric, and coincident with

D0: It should be cautioned that this slow MAS method

is critically dependent on good signal-to-noise ratios

for as many spinning sidebands as possible. Reports

by Harris and co-workers on SnF couplings have

provided several precise values of Reff ; however,

the determination of reliable values of DJ has been

hampered by a lack of precise Sn–F bond lengths

from which RDD may be calculated [123,124].

Marichal and Sebald have presented an interesting

determination of a substantial two-bond J coupling

anisotropy D2J119Sn117Sn ¼ þ1263 ^ 525 Hz for solid

(benzyl3Sn)2O [137]. This is in contrast to the negli-

gible value which was reported for (cyclohexyl3Sn)2S

[155]. The linear arrangement of the Sn–O–Sn

moiety in (benzyl3Sn)2O allows for the usual

assumptions concerning the symmetry and orientation

of J. In this case, off-magic-angle spinning (e.g. 568)

was employed to determine DJ for the two crystal-

lographically equivalent tin atoms (Fig. 10).

By spinning off the magic angle, Reff is reintro-

duced with a scaling factor of 20:0619: Since the

chemical shift parameters may be determined before-

hand using standard MAS, the only parameter to be

optimized in simulating the off-angle spectra is Reff ;

with the usual assumptions regarding the relative

tensor orientations. It is critical in these types of

experiments to have independent accurate knowledge

of the spinning angle b. In the case of tin, this is

facilitated by simulating the spectrum arising solely

due to an isolated, uncoupled 119Sn nucleus.

There are several potential methods for the

determination of DJ for heteronuclear spin-1/2 pairs

which remain almost entirely unexploited. The area of

dipolar recoupling under MAS conditions has been

the focus of intense research in solid-state NMR

for several years [156,157]. Such experiments are

available for both heteronuclear and homonuclear

spin pairs. Ideally, dipolar recoupling experiments

selectively reintroduce a direct dipolar interaction of

interest while suppressing chemical shift interactions

and unwanted additional dipolar interactions. The

direct dipolar coupling is then interpreted to provide

distance information. What is often ignored in these

experiments is that the measured quantity is Reff rather

than RDD: Thus, the opportunity exists in all dipolar

recoupling experiments to measure DJ: Of course, the

same limitations apply to all experimental measure-

ments of DJ; e.g. the need for an accurate independent

measurement of RDD: To our knowledge, the only DJ

which has been measured via a dipolar recoupling

technique is DJ113Cd31P in Cd(NO3)2·2PMe2Ph, which

was found to be 21200 ^ 700 Hz via rotary

resonance recoupling [158]. It is clear that

the potential exists to apply heteronuclear dipolar

recoupling experiments in order to determine DJ for a

wider variety of spin-1/2 pairs.

Coupled homonuclear spin-1/2 pairs of nuclei. As

with heteronuclear spin-1/2 pairs, there are several

dipolar recoupling experiments which may be applied

to homonuclear spin-1/2 pairs [156,159,160].

Although the experiments themselves involve differ-

ent pulse sequences depending on whether the spin

Fig. 10. (a) Off magic-angle (568) spinning 119Sn NMR spectrum of

(benzyl3Sn)2O acquired at 4.7 T. Simulation (part (b)) of the scaled

powder patterns denoted by asterisks allows for the extraction of an

effective 119Sn, 117Sn dipolar coupling constant, from which a value

of DJ119Sn117Sn may be determined. Reprinted with permission from

Ref. [137]. Copyright (1998) Elsevier.


system is heteronuclear or homonuclear, the end result

is the same: Reff between the spins is reintroduced

under conditions of MAS. Although no precise

experimental determinations of non-zero values

of DJ exist from MAS experiments for homonuclear

spin pairs, the potential certainly exists, especially for

relatively heavy spin-1/2 nuclei such as 119Sn.

Dusold et al. have investigated the potential of

using an iterative fitting technique for the extraction

of DJ from MAS NMR spectra of homonuclear spin-

1/2 pairs [131,161]. For example, the possibility of

anisotropic J between the phosphorus nuclei in

Cd(NO3)2·2PPh3 was addressed by carrying out a

full iterative optimization of all parameters involved.

The conclusion reached is that there is no significant

anisotropy when coincident D0 and J tensors are

assumed; however, if non-coincident tensors are

considered, no definite conclusions may be made

about the magnitude of the anisotropy in J. This work

demonstrates the importance of considering the

relative orientations of D0 and J.

2.5. High-resolution molecular beam spectroscopy

A less well-recognized source of J is the hyperfine

structure in molecular beam and high-resolution

microwave spectra [143,162–167]. The case of

diatomic molecules is particularly simple, and in

favourable cases both the isotropic and anisotropic

portions of J may be extracted with a high degree of

precision [143,168]. The high-resolution spectra of

diatomics also provide information on the quadrupo-

lar, spin–rotation, and s tensors [163,165,169]. Since

these experiments are performed on gaseous samples

at very low pressures, intermolecular effects on the

interaction tensors are negligible. This has the

advantage of providing very accurate experimental J(or K) tensors which may be used to establish the

reliability of first-principles calculations. In addition,

due to the simplicity of the molecules which are

studied, the hyperfine data allow for particularly

meaningful interpretations of J and K in terms of the

local electronic structure.

Molecular beam spectroscopy allows for the

investigation of rotational transitions (e.g.

J ¼ 1 ˆ 0), and more importantly, the investigation

of so-called hyperfine structure within a single

rotational state (Fig. 11).

There are several versions of ‘molecular beam’

spectroscopy, including molecular beam electric

resonance (MBER), molecular beam magnetic reson-

ance (MBMR), molecular beam maser spectroscopy,

molecular beam absorption spectroscopy, and mol-

ecular beam deflection measurements. Much of the

reliable information on J has come from MBER and

MBMR, and in recent years almost all of the highly

precise data have come from MBER measurements in

the laboratory of Cederberg [171]. There, a spec-

trometer built by Norman Ramsey in 1970 is still used

to provide extremely high-quality data on diatomics.

Molecular beam spectroscopy differs from most other

forms of spectroscopy in that a beam of molecules is

detected rather than electromagnetic radiation of

some type. The MBER spectrometer is composed of

five main parts: the beam source, the A state selector,

the C transition region, the B state selector, and

the detector. MBER relies on the second-order Stark

effect to carry out rotational state selection in the A

and B regions, by applying an inhomogeneous electric

field to alter the trajectories of molecules with

differing rotational angular momentum quantum

numbers M. Only molecules with permanent dipole

moments will experience the Stark effect and there-

fore only these molecules are suitable for MBER

spectroscopy. In practice, a very weak electric field is

applied such that the results may be extrapolated to

zero field. Additionally, at least one of the nuclei must

be quadrupolar in order to split the energy levels such

that information on J is accessible. MBMR may be

used to investigate the magnetic hyperfine structure in

molecules which lack a permanent dipole moment,

such as homonuclear diatomics, e.g. iodine [172]. One

of the key features of molecular beam spectroscopy is

the very high resolution and narrow lines which may

be obtained; positions of the lines in the spectra may

be measured with uncertainties of less than 1 Hz

[173]. The hyperfine Hamiltonian in the absence of

external fields for a diatomic molecule such as

potassium monofluoride, 39K19F, may be written as

h21Hhf ¼ VK : QK þ c1IK·J þ c2IF·J

þ c3IK·dT ·IF þ c4IK·IF ð82Þ

If we are interested in the J ¼ 1 rotational level, higher-

order terms such as the nuclear magnetic octupole and

nuclear electric hexadecapole interaction are zero. The


first term describes the interaction of the potassium

nuclear electric quadrupole moment with the EFG

tensor and the next two terms describe the K and F spin–

rotation interactions. The last two terms describe the

sum of the direct and indirect nuclear spin–spin

coupling interactions between K and F, with dT

denoting here the traceless part of the interaction tensor.

Shown in Fig. 11 is an energy-level diagram for a 1S

diatomic molecule composed of a spin-3/2 nucleus and a

spin-1/2 nucleus, such as 39K19F or 87Rb19F [170]. The

levels all exist within a single rotational–vibrational

state, in this case n ¼ 0; J ¼ 1: In this diagram, the pure

J ¼ 1 state is first perturbed by the quadrupolar and

spin–rotation interaction associated with the spin-3/2

nucleus. When the spin-1/2 nucleus is considered, its

spin–rotation constant as well as the spin–spin

coupling tensors cause further splittings of the energy

levels. Measurement of the allowed transitions provides

enough data to solve for quadrupole coupling, c1; c2; c3;

and c4: The quantum numbers F1 and F are defined as

F1 ¼ I1 þ J and F ¼ F1 þ I2; where I1 is the angular

momentum quantum number of the spin-3/2 nucleus

and I2 is the angular momentum quantum number of the

spin-1/2 nucleus. The selection rules for the electric

dipole transitions are DF1 ¼ 0;^1;^2; DF ¼

0;^1;^2; and DMF ¼ 0;^1: If one notes the form

of the Hamiltonian, the parameter c4 is readily identified

with the isotropic J. The parameter c3 provides the

tensor part (D0 and J) of the total spin–spin coupling

tensor [143]:

c3 ¼ RDD 2DJMN

3; ð83Þ

where RDD is the direct dipolar coupling constant. c3

may thus be described as an effective dipolar coupling

constant, Reff : The NMR interaction tensors must be

axially symmetric for a 1S diatomic molecule, and

therefore the complete J is entirely described by J and

DJ (or equivalently, by c4 and c3). The parameter c3 is

frequently written as a sum of the direct and indirect

contributions, i.e. c3 ¼ c3 (direct) þ c3 (indirect) or

c03 þ c003 [165,174]. The relationship between c3 and J

familiar to NMR spectroscopists was originally

Fig. 11. Energy level diagram for a molecular beam electric resonance experiment on a diatomic molecule composed of a spin- 12

and a spin- 32

nucleus such as 87Rb19F. Adapted from the diagram for 87Rb19F shown in Ref. [170]. Second-order quadrupolar effects were evaluated

numerically and are shown for the F1 ¼ 3=2 and 5/2 levels.


described by Ramsey [165]. It should be noted that other

symbols are also used to represent c3 and c4:Reff is equal

to c3; which is also sometimes denoted dT [175], or S

[176–178], or simply d [179,180]. The c4 ¼ J; is also

denoted d [165,179,180] (which is also used for

chemical shifts in the NMR literature!) and dS [181].

The reader should be aware that in some of the older

literature, c3 and c4 are defined differently, i.e. with extra

factors.7 The relationship between c3 and the vaguely

defined ‘tensor part of the electron-coupled spin–spin

interaction’ has been alluded to in the literature [74,163,

182], but only very rarely has it been explicitly stated

that c3 is equal to RDD 2 DJ=3: The relationship is not

widely appreciated by NMR spectroscopists. Certainly

some of the molecular beam literature extracts c3

(indirect) from the full value of c3 and the rovibration-

ally averaged value of c3 (direct); however interpret-

ation of these data in the language of NMR J has been

lacking.

While it is certainly true in many NMR exper-

iments and some molecular beam experiments that the

contribution from the anisotropic J is swamped by the

contribution of D0; there are just as certainly many

cases where valuable information concerning J may

be extracted by a careful analysis of the data. English

and Zorn [174] provided a summary of the available

values of c3 and c4 for alkali fluorides in 1967, and

also extracted c3 (indirect). Had there been interest in

converting c3 (indirect) to DJ; much of the 1967 data

on c3 had such large relative errors that in many cases

it would have been difficult to determine the sign of

DJ; let alone the precise value. As with other

experimental methods for determining accurate and

precise values of DJ; D must be known with high

precision. Bond lengths in diatomics are frequently

determined to more than five significant figures, thus

providing very precise direct dipolar coupling con-

stants, RDD: Over the past few decades since the

summary of English and Zorn [174], very precise

values of c3 and c4; e.g. to five significant figures, have

become available for a wide variety of diatomics

(Table 9).

Muller and Gerry separated the direct and indirect

portions of c3 for five monofluorides; however the

values of c3 (indirect) were not discussed in terms of

DK [178]. Bryce and Wasylishen have extracted

several reliable values of DK from the very high-

resolution hyperfine data which are now available

[143] (Table 10).

In combination with high-level ab initio calcu-

lations, these high-quality experimental data have

provided some insight into periodic trends in K. For

example, periodic trends are clearly evident for both

the isotropic and anisotropic portions of K for the

thallium halides. The reduced K coupling is negative

and increases in magnitude as the atomic number of

the halogen increases. The reduced anisotropic

coupling is positive, and increases in magnitude as

the atomic number of the halogen increases. The

thallium halides are also interesting in that the ratio of

DJ to D is very large; in TlI this ratio is nearly 1500!

This clearly demonstrates the potential hazards of

neglecting the 2DJ=3 term in the interpretation of

measured Reff in NMR experiments. It is also

interesting to note that in many cases shown in

Table 10, the magnitude of DK is greater than K. Ref.

[143] provides further investigations of the periodic

trends in K in diatomic molecules.

Two recent studies of cesium fluoride [193] and

lithium iodide [186], for example, provide striking

demonstrations of the sensitivity of the molecular

beam method to the value of J. The values of c3

and c4 are clearly sensitive to the vibrational state

of the molecule (Table 9). Thus, molecular beam

experiments on diatomics provide a unique oppor-

tunity to learn about the rotational–vibrational

dependence of J [200]. We also emphasize that

the sign of c3 and c4 are determined in molecular

beam and microwave experiments. To our knowl-

edge, almost no information on J (c3 and c4

parameters) has been extracted for polyatomic

molecules. Even for some 1S diatomics it is

difficult to precisely determine c3 and c4; e.g.,

GaF [201] and see Table 9, simply due to poor

resolution, signal-to-noise, or relatively small

values of these parameters. The Hamiltonian

described by Dyke and Muenter for polyatomic

molecules neglects the effects of J [163]. One

polyatomic molecule for which c3 has been

measured is methane [202,203]. However, the

two-bond value, c3(H,H) ¼ 20.9 ^ 0.3 kHz, does

not provide any useful information on anisotropy

in J, since c3 may be accounted for fully in this7 See, for example, footnote b in Table III of Refs. [176,177].


Table 9

Magnetic hyperfine spin–spin coupling tensors available from molecular beam and microwave experiments

Molecule c3 (kHz) c4 ¼ J (kHz) Year Ref. to original

hyperfine

literature

H19F 143.45(3) 0.50(2) 1987 [176,177]7LiH 11.346(7)(n ¼ 0, J ¼ 1) 0.135(10)(n ¼ 0, J ¼ 1) 1975 [183]7LiH 11.03(8)(n ¼ 1, J ¼ 1) 0.17(4)(n ¼ 1, J ¼ 1) 1975 [183]7LiH 11.329(12)(n ¼ 0, J ¼ 2) 0.160(5) (n ¼ 0, J ¼ 2) 1975 [183]7LiD 1.7430(70) 0.005(10) 1975 [183]7Li19F 11.4292(42) 2 0.2122(86)ðnþ ð1=2ÞÞ þ

0.0039(29)ðnþ ð1=2ÞÞ20.1744(21) 2 0.0042(21)ðnþ ð1=2ÞÞ 1992 [184]

7Li81Br 1.1789(78) 0.0711(89) 1972 [185]7Li79Br 1.0710(61) 0.0604(70) 1972 [185]7Li127I 0.62834(68) 2 0.0050(11)ðnþ ð1=2ÞÞ 0.06223(36) þ 0.00041(26)ðnþ ð1=2ÞÞ 1999 [186]23Na19F 3.85(25) 0.150(250) 1964 [187]23Na19F 3.7(2) 20.2(2) 1965 [188]23Na81Br 0.4269(15) 2 0.0042(2)ðnþ ð1=2ÞÞ 2

0.00021(9)[J(J þ 1)]

0.0859(18) 1987 [189]

23Na79Br 0.3922(16) 2 0.0029(5)ðnþ ð1=2ÞÞ þ

0.00014(11)[J(J þ 1)]

0.078(3) 1987 [189]

39K19F 0.4749(27) 2 0.0065(10)ðnþ ð1=2ÞÞ 0.0578(13) 1988 [173]41K19F 0.2606(15) 2 0.0035(5)ðnþ ð1=2ÞÞ 0.0317(7) 1988 [173]39K35Cl 0.035(12) 0.009(6) 1984 [190]87Rb19F 2.45(37) 0.86(40) 1972 [191]85Rb19F 0.79681 ^ 0.00036 2 (0.00642 ^

0.00027)ðnþ ð1=2ÞÞ

0.23766 ^ 0.00032 2 (0.00245 ^

0.00022)ðnþ ð1=2ÞÞ

2002 [192]

133Cs19F 0.92(12)(n ¼ 0, J ¼ 1) 0.61(10)(n ¼ 0, J ¼ 1) 1967 [174]133Cs19F 0.92713(53) 2 0.00917(93)ðnþ ð1=2ÞÞ þ

0.00097(29)ðnþ ð1=2ÞÞ20.62745(30) 2 0.00903(22)ðnþ ð1=2ÞÞ 1999 [193]

133Cs35Cl 0.028(2) 2 0.000(3)ðnþ ð1=2ÞÞ þ

0.0002(7)ðnþ ð1=2ÞÞ20.060(4) þ 0.002(5)ðnþ ð1=2ÞÞ þ

0.0006(12)ðnþ ð1=2ÞÞ21977 [194]

23Na39K Not determined 0.306(30) 1976 [181]23Na2 0.3026(50) 1.0667(65) 1985 [180]35Cl19F 2.859(9) 0.840(6) 1977 [195]79Br19F 7.15a (n ¼ 0, J ¼ 1) 4.86(28)(n ¼ 0, J ¼ 1) 1995 [178]79Br19F 6.93(26)(n ¼ 1, J ¼ 1) 6.47(84) (n ¼ 1, J ¼ 1) 1995 [178]127I19F 5.202(146)a 5.73(105) 1995 [178]115In19F 2.62(3)(n ¼ 0, J ¼ 1) 22.15(3)(n ¼ 0, J ¼ 1) 1972 [196]115In19F 2.62(1)(n ¼ 0, J ¼ 2) 22.11(1)(n ¼ 0, J ¼ 2) 1972 [196]205Tl19F 3.50(15) 213.3(7) 1964 [182]203Tl35Cl 20.09(10)(n ¼ 0, J ¼ 2) 21.52(10) 1969 [197]205Tl35Cl 20.13(10)(n ¼ 0, J ¼ 2) 21.54(10) 1969 [197]203Tl37Cl 20.56(50) 21.11(50) 1969 [197]205Tl37Cl 20.13(15) 21.28(15) 1969 [197]203Tl79Br 21.55(8)(n ¼ 0, J ¼ 2)b 26.39(8)(n ¼ 0, J ¼ 2) 1970 [198]205Tl79Br 21.65(5)(n ¼ 0, J ¼ 2) 26.35(5)(n ¼ 0, J ¼ 2) 1970 [198]203Tl81Br 21.68(8)(n ¼ 0, J ¼ 2) 26.91(8)(n ¼ 0, J ¼ 2) 1970 [198]205Tl81Br 21.77(5)(n ¼ 0, J ¼ 2) 26.84(5)(n ¼ 0, J ¼ 2) 1970 [198]203Tl127I 22.59(2)(n ¼ 0, J ¼ 3) 26.57(1)(n ¼ 0, J ¼ 3) 1970 [199]205Tl127I 22.48(10)(n ¼ 0, J ¼ 3) 26.67(5)(n ¼ 0, J ¼ 3) 1970 [199]127I2 1.58(5)(n ¼ 0, J ¼ 13) 3.66(3)(n ¼ 0, J ¼ 13) 1980 [172]127I2 1.528(18)(n ¼ 0, J ¼ 13) 3.708(22)(n ¼ 0, J ¼ 13) 1999 [179]127I2 1.519(18)(n ¼ 0, J ¼ 15) 3.701(23)(n ¼ 0, J ¼ 15) 1999 [179]

Values for which no rovibrational dependence is given are for the n ¼ 0, J ¼ 1 state unless otherwise indicated.a The value for c3 reported in Ref. [178] is of opposite sign due to use of a different convention.b Extensive data are available for TlBr, e.g. for five vibrational states and two rotational states.


case by the DHH interaction, zero-point vibration,

and centrifugal stretching effects [203].

Finally, we note that there has been a recent report

of the c3 and c4 parameters for molecular iodine using

stimulated resonant Raman spectroscopy. This is not a

rotational spectroscopic technique; however, the

results complement those discussed in this section.

Wallerand et al. [179] have improved the precision in

the values reported by Yokozeki and Muenter using

MBMR [172] and also detected a slight rotational

dependence of the parameters. The Raman data

provide the following values for I2 : K ¼ ð763 ^ 5Þ

£ 1020 NA22 m23 and DK ¼ ð2785 ^ 11Þ NA22 m23

for the n ¼ 0; J ¼ 13 state, and K ¼ ð761^ 5Þ£ 1020

NA22 m23 and DK ¼ ð2779^ 11Þ£ 1020 NA22 m23

for the n¼ 0; J ¼ 15 state. Clearly, it would be of

interest to further investigate the rotational depen-

dence of this coupling tensor for a larger range of

rotational states.

Judging by the quality of much of the data

discussed in this section, it is clear that molecular

beam experiments provide information on J tensors

that is extremely valuable to NMR spectroscopists

and theoreticians. The accuracy and precision to

which the molecular beam data are determined,

especially the recent results from Cederberg et al.,

provide unique opportunities to study and interpret the

rotational–vibrational dependence of J.

2.6. NMR relaxation

Nuclear spin–spin and spin–lattice relaxation may

in principle occur via an anisotropic spin–spin

coupling ðDJÞ mechanism [26,204]. Equations

describing this phenomenon were outlined by

Blicharski in 1972 [204]; although, to date relaxation

by the DJ mechanism has not been identified

experimentally. As with all experimental methods

for the determination of DJ; complications arise due

to the identical transformation properties of the direct

dipolar and anisotropic J coupling Hamiltonians. In a

system where relaxation of a particular nucleus may

be predicted to arise solely from spin–spin coupling

interactions (direct and indirect), the known geometry

allows for the calculation of the spin relaxation rate

based solely on the direct dipolar coupling relaxation

mechanism. In such an ideal system, deviations from

the predicted rate would be attributed to contributions

from DJ; and as such, relaxation measurements on

carefully chosen systems represent a means to

characterize DJ experimentally. It is important to

note that this is the sole method which offers the

potential to measure DJ in an isotropic solution. In

principle the method may be applied to oriented

phases as well.

Precise measurements of DJ by relaxation studies

will pose several challenges, however, due to the

difficulty in choosing an appropriate spin system and

due to the assumptions which must be invoked

concerning the orientation and asymmetry of J. Of

course, in environments of high symmetry, e.g. linear

molecules, the orientation and asymmetry of J are

dictated by the local molecular symmetry. It is

important to note that relaxation by DJ can either

increase or decrease the rate of relaxation which

would be predicted based solely upon the direct

dipolar coupling mechanism, depending on the sign of

DJ: This is exemplified in the following equation

given by Blicharski for the spin–lattice relaxation rate

Table 10

Summary of precise indirect nuclear spin–spin coupling tensor data

available from hyperfine structure in high-resolution rotational

spectra of diatomic molecules [143]

Molecule K/1020

NA22 m23

DK/1020

NA22 m23

Ref. to original

hyperfine data

LiH 2.89 21.22 [183]

LiF 3.92 3.94 [184]

LiBr 5.15 18.1 [185]

LiI 6.65 18.4 [186]

NaBr 9.76 43.9 [189]

KF 10.9 23.8 [173]

CsF 41.8 46.5 [193]

CsCl 39.4 67.9 [194]

Na2 127 25.71 [180]

ClF 75.7 281.8 [195]

BrF 171 2206 [178]

IF 252 2257 [178]

InF 286.4 89.9 [196]

TlF 2202 173 [182]

TlCl 2224 262 [197]

TlBr 2361 448 [198]

TlI 2474 664 [199]

I2a 763 2785 [179]

All results are for n ¼ 0, J ¼ 1 rotational–vibrational state

except TlBr: n ¼ 0, J ¼ 2; TlI: n ¼ 0, J ¼ 3; and I2: n ¼ 0, J ¼ 13.a Data are from stimulated resonant Raman spectra.


in the rotating frame:

1

T1r

¼1

T1r

!dip

1 21

3x

� 2

: ð84Þ

Here, the total rate of relaxation is equal to the

relaxation due the pure dipolar interaction, ð1=T1rÞdip;

multiplied by a factor involving the ratio of DJ to RDD;

represented by x. Depending on the value of x, relaxa-

tion by DJ may either increase the total rate of

relaxation (x , 0 or x . 6), decrease the total rate of

relaxation ð0 , x , 6Þ; or completely interfere with

the direct dipolar coupling mechanism such that the

total rate is effectively zero ðx ¼ 3Þ: If x ¼ 6

fortuitously, no effect on the total rate of relaxation

will be observed.

Unambiguous experimental identification of con-

tributions from DJ to the spin–spin or spin–lattice

relaxation rate is an interesting and formidable

challenge and will require a unique and carefully

chosen spin system which is structurally well-

characterized and for which a particular spin pair

may be predicted to have a substantial DJ and a small

but accurately-known direct dipolar coupling con-

stant. Additionally, we note that in principle nuclear

spin relaxation may also arise from the antisymmetric

part of J. Identifying contributions to relaxation by

such a mechanism would no doubt be at least as

daunting as identifying contributions from DJ:

Finally, one must realize that despite the practical

difficulties associated with isolating and identifying

contributions to nuclear spin relaxation arising due to

anisotropic and antisymmetric J coupling, every

spin–spin and spin–lattice relaxation time constant

reported in the literature which has been measured

based on the assumption of relaxation exclusively by

the direct dipole–dipole mechanism contains contri-

butions from the J (except in cases where J is forced

to be perfectly isotropic by symmetry). This fact is

inescapable, and relates back to the similar forms of

the direct dipolar and indirect spin–spin coupling

Hamiltonians. It is similarly true that all measured

nuclear Overhauser enhancements (NOE) contain

contributions from DJ which may not be related to

internuclear distance in any straightforward manner;

of course many NOEs of interest involve proton–

proton couplings for which DJ=3 will be negligible

compared to RDD:

3. Quantum chemical methods

3.1. Correlated ab initio methods

A recent comprehensive review article discusses

the technical aspects of quantum chemical calculation

of spin–spin coupling and nuclear magnetic shielding

tensors [31].

In general, calculations of J have rather different

computational requirements compared to those of s:As already mentioned, there is no gauge origin

problem; however, there are more mechanisms

contributing to J than to s; as discussed earlier.

This makes the number of necessary first-order wave

functions much larger for J than in the s case. For

couplings, ten responses are needed for each nucleus,

whereas three suffice for all the sM regardless of the

size of the system.

Additional differences arise from the nature of the

perturbation operators involved in the calculation of J.

The fact that the FC and SD interactions couple the

singlet ground state to triplet excited states, makes

the restricted Hartree–Fock (RHF) reference state

unsuitable as it may be unstable towards triplet

perturbations [205]. This results in unphysically

large magnitudes of the triplet terms in J (for a

discussion, see, e.g. Refs. [206,207]). For example,

the calculation of JCC in ethene (C2H4) is a well-

known failure case, where a Hartree–Fock linear

response (SCF LR, equivalent to the random phase

approximation, RPA) calculation based on the RHF

reference state leads to values in the range of

thousands to tens of thousands of Hz. The exper-

imental result in solution is about 67.5 Hz [72]. Thus,

in contrast to calculations of s; the simplest ab initio

quantum chemical level, RHF, does not provide a

meaningful starting point even for qualitative work.

For the same reason, electron-correlated post-Har-

tree–Fock methods based on the RHF reference state

may be suspect. In practice, multiconfiguration self-

consistent field (MCSCF) linear response (MCLR)

[208] and coupled cluster (CC) [209–212] methods

(the latter without explicit orbital relaxation, i.e. only

including relaxation implicitly through the CC

amplitudes corresponding to the single excitations)

have been found to be stable in this respect.

Even systems that do not exhibit triplet instability

at or close to their equilibrium geometries may suffer


from a near- or quasi-instability, which may lead to

gross overestimation of the contributions from the

triplet mechanisms.

Another qualitative difference as compared to

the shielding theory is that a better treatment of

dynamic electron correlation is essential for reliable

calculations of J tensors. The four hyperfine

operators involved in the calculation of J,

Eqs. (15),(16),(18),(19), sample different spatial

regions of the electron cloud and couple to excited

states of different spin symmetry. Due to the need

to be able to accurately describe more physical

features of the system than in the case of s; error

cancellation has less room to operate in the

calculation of J. Satisfactory results are in practice

obtained at the coupled cluster singles and doubles

(CCSD) excitation level as well as MCLR with

large active molecular orbital (MO) space. A rule-

of-thumb in the latter case is that about 95% or

more of the total occupation of virtual MOs, based

on the natural orbital occupation numbers obtained

using, e.g. second-order Møller–Plesset (MP2) or

configuration interaction singles and doubles

(CISD) one-particle density matrices, should be

included in the chosen active space.

Despite the apparent challenge that J poses to

computational methods, it is possible to reach

quantitative agreement with experiment at least for

small main-group systems. A prime example of this is

a recent MCLR application [213] on the coupling

constants of ethyne, Table 11.

A brief list of the different implementations of J

calculations introduced or relevant in the review

period follows.

† The sum-over-states (SOS) method [216] fea-

tures an uncoupled property calculation using

ab initio wave functions. As the response of the

electron–electron interaction to the magnetic

field perturbation is neglected, the method is

physically not well-justified. The calculated

results have to be scaled for comparison with

experiment.

† Finite perturbation theory (FPT) calculations of

JFC have been carried out at various levels of

ab initio theory (see, e.g. Refs. [217–219]).

This is physically motivated, but the approach

lacks the remaining spin–spin coupling terms

and is unable to provide the anisotropic

properties. A major drawback of FPT is that

numerical instabilities may arise when supple-

menting the basis set with large-exponent

functions to better describe the FC perturbation

(see Section 3.3).

† Ab initio implementation of contributions from

localized orbitals within the polarization propa-

gator-inner projections of the polarization

propagator approach (CLOPPA-IPPP) has been

presented by Contreras and co-workers [220].

The RPA level method operates with localized

occupied and virtual orbitals and allows inves-

tigation of contributions to coupling from

different localized MOs, as well as coupling

pathways. The knowledge of MO contributions

in principle makes it possible to reduce the

dimension of the virtual space in calculations

of second-order properties, without losing much

quality in the results. Ref. [220] extends earlier

semi-empirical [9,221,222] and ab initio [223]

work.

† The equations-of-motion (EOM) method [224]

is an intermediate ab initio approach between

RPA and MP2.

† Analytic derivatives of the MP2 energy have

been used by Fukui and co-workers [225],

extending the earlier work by the same group

based on FPT [226,227]. The computational

cost of MP2 scales as N5 where N is the

number of basis functions.

† The second-order polarization propagator

approach (SOPPA) of Oddershede and co-

workers [228,229] is another analytic second-

order method roughly at or better than the MP2

level. While SOPPA (scaling as N5) is still

somewhat subject to the triplet instability

problem, its accuracy is very useful for

Table 11

Calculated (MCLR) spin – spin coupling constants in C2H2

compared to the experimental results extrapolated to the equili-

brium molecular geometry (results in Hz)

Method 3JHH1JCH

2JCH1JCC

MCLR [213] 10.80 244.27 53.08 184.68

Experimental [214,215] 10.89 242.70 53.82 185.04


qualitatively correct couplings in large systems

that are not affected by the triplet instability.

† The use of SOPPA with correlation amplitudes

taken from a CCSD calculation forms the

SOPPA(CCSD) method [230], a simpler version

of which was earlier called CCSDPPA [231,

232]. SOPPA(CCSD) provides generally

improved results as compared to SOPPA,

although it is strictly consistent only to

second-order. The CCSD amplitude calculation

scales as N6; while property calculations scale

as N5:

† Analytical implementation of the CCSD method

has been performed by Bartlett and co-workers

[210,211], replacing the earlier FPT version

[209,233]. The different models include the so-

called quadratic one, meaning unrelaxed ana-

lytic second derivatives of the CCSD energy, as

well as the equation-of-motion coupled cluster

(EOM-CC) approximation featuring unrelaxed,

configuration interaction (CI)-like SOS formu-

lation. The latter method is not size-extensive.

The results are compared in Ref. [234]. More

efficient versions of the CI-like method are

discussed in Ref. [235]. The full CCSD linear

response ðN6Þ is currently the most accurate

black-box model for systems where static

electron correlation is of little importance.

Analytic CCSD(T) ðN7Þ featuring perturbative

inclusion of triple excitations has been reported

by Auer and Gauss [212], as well as the

following methods using FPT [212]: full

CCSDT with explicit triples ðN8Þ and the CC3

model of Ref. [236] with approximate triples

ðN7Þ:

† The MCLR method of Vahtras et al. [208]

including both the complete active space

(CASSCF) and restricted active space

(RASSCF) models, has the possibility of

extending the active space in principle all the

way up to full configuration interaction (FCI).

MCLR can be expected to be successful

particularly for systems affected by static

correlation. The convergence of the treatment

of dynamical correlation is slow, however,

exemplified by the JFH coupling constant in

the HF molecule as a function of the size and

treatment of the virtual active space in Table

IV of Ref. [237]. The influence of correlating

the semicore and core molecular orbitals has

been investigated [238–241], as well as higher

than singles and doubles excitations in the

RASSCF model [239,241,242]. In contrast to

the other approaches listed here, the need to

choose the active molecular orbital space

renders MCLR a non-black-box method, requir-

ing insight in the electronic structure of the

system under investigation. The scaling of the

CASSCF model is factorial in the number of

active molecular orbitals.

To date, most applications are carried out using the

SOPPA, SOPPA(CCSD), EOM-CCSD, and MCLR

methods. Examples of their performance with respect

to experimental J coupling constants can be found in

Ref. [31]. The typical accuracy of state-of-the-art

calculations for small molecules composed of light

elements is 5–10%, with additional provisos for the

presence of rovibrational and solvent effects. Hence,

there is still room for improvement even in the

treatment of small model systems. In particular,

tractable CC models beyond CCSD are desirable, as

they are both black-box methods and likely to be more

easily extended for larger molecules than MCLR.

Despite not having yet been applied to J, the linear-

scaling CC approaches [243] are promising in this

respect. The CCSD(T) model has proven to be very

successful for calculations of s [244]. However, this

particular method for triples seems to reintroduce, in

the J case, problems related to the triplet instability

[212]. Instead, the performance of the numerical CC3

model was found promising by Auer and Gauss in

Ref. [212], and an analytic derivative implementation

of the method would be of substantial interest.

3.2. Density-functional theory methods

Density-functional theory (DFT) [245] has

become very popular in quantum chemistry due to

the fact that it allows the inclusion of electron

correlation effects roughly at the cost of the

uncorrelated ab initio Hartree–Fock level methods,

N3–4: The drawback of DFT is that there exists no sys-

tematic way of improving the exchange-correlation

functional of electron density Exc½r� that lies at the

heart of DFT, from one calculation to another. The


three ‘generations’ of Exc½r� functionals in use are

the local density approximation (LDA), generalized

gradient approximations (GGA), and various hybrid

functionals, with results generally improving in this

order. Whereas LDA is parametrized, in principle in

an ab initio manner, based on the exchange-

correlation energy per particle in a uniform electron

gas, the GGAs also parametrize density gradients

semi-empirically. The hybrid functionals incorporate

some specific fraction of the exact Hartree–Fock

exchange. There is considerable research activity

devoted to developing exchange-correlation func-

tionals, with emerging hope for systematic progress

[246,247].

In the presence of a magnetic field, Exc should

not only be a functional of the electron density as

in the field-free case, but it also should refer to

the current density [248–250], coining the name

current DFT (CDFT). A local model of CDFT has

been tested in the context of calculating s [251].

The effect of including the current dependence

was found to be very small in comparison with

the remaining errors of DFT calculations. Most

likely the same situation prevails for calculations

of J, for which CDFT has not been applied so far.

Since the RHF method is unsuitable for the

computation of J, DFT holds a different status in

calculations of this property as compared to s:DFT seemingly does not suffer from the triplet

instability [252–254] in J calculations, making it

by far the least computationally demanding method

by which qualitatively correct J values may be

calculated. Furthermore, its scaling with the system

size currently makes it the only practical method

for calculating J in large molecules.

DFT calculation of s as well as JPSO using

pure (i.e. non-hybrid) functionals (as well as

omitting any current dependence of Exc) can be

carried out non-iteratively, in an uncoupled

fashion, in contrast to the wave function methods.

This is due to the imaginary character of the

relevant perturbations, causing the corresponding

first-order density change to vanish. Calculation of

J by DFT necessitates additionally a coupled or

response procedure due to the real FC and SD

perturbations, unless FPT is used for these

interactions. In the hybrid DFT framework, the

presence of the exact Hartree–Fock exchange term

makes the coupled procedure necessary also for

JPSO:

A discussion of relevant DFT implementations for

spin–spin coupling calculations follows:

† Ref. [255] described an implementation of an

uncoupled SOS procedure for all the second-

order terms in J, i.e. including also JFC and JSD

for which SOS is not applicable. As hybrid

functionals were used to obtain the Kohn–

Sham orbitals [255], even the uncoupled

calculation of the PSO term is inconsistent.

The results are unsatisfactory and in poor agree-

ment with experiment.

† Malkin et al. [256,257] presented a combined FPT

and sum-over-states density-functional pertur-

bation theory (SOS-DFPT) method, with the

possibility of using pure DFT LDA and GGA

functionals. In the calculation of the FC and SD/FC

terms, Eqs. (24) and (26), the FC operator on the

chosen nucleus is applied as a finite perturbation in

the spin-polarized unrestricted Kohn–Sham cal-

culation. JSD is neglected because of the more

complicated finite perturbations that would be

necessary in this case. While not warranted a

priori, the omission is in practice often a justified

approximation on the basis of results obtained.

JPSO is calculated using a sum-over states

expression after a converged restricted Kohn–

Sham calculation:

JPSOMN;et/

Xocc

k

Xvirt

a

kfkllM;e =r3M lfalkfallN;t=r

3N lfkl

1k 21a 2DExck!a

ð85Þ

where the orbital energy denominators 1k 21a

have been subjected to the ‘Malkin correction’,

DExck!a [258]. This has been viewed as an a

posteriori attempt to model the current dependence

of Exc or merely an ad hoc correction for the

deficiencies in the orbital energy denominators. In

any case, the modified PSO terms are in good

agreement with correlated ab initio calculations

[256,257]. Earlier, Dickson and Ziegler [259]

reported a similar implementation without the

Malkin corrections, with the PSO contributions

apparently overestimated [144]. Slater-type basis

functions were used in Ref. [259]. Ziegler and


co-workers [260] made an interesting connection

of DExck!a to an approximate correction for the

unphysical electronic self-interaction effect on the

orbital energy differences.

Excellent performance has been reported [256,

257] for this approach for JCC and JCH using the

semi-empirical choice of the GGA functional,

namely Perdew86 [261,262], known to give

reliable isotropic electron spin resonance hyperfine

couplings for non-p radicals [263]. However, the

performance has been found to deteriorate for JFC

towards group 17 elements, particularly for19F. Apparently the quantitative description of the

FC-induced spin density becomes increasingly

difficult for systems with lone pairs. However,

the errors appear to be rather systematic, based on

comparison with ab initio calculations. Despite the

problems, this DFT method remains one of the

most popular computational approaches to J in

recent literature, with particularly many appli-

cations to large molecules.

† Ref. [264] described a FPT implementation of both

the SD and FC terms in the Gaussian suite of

programs, enabling the use of hybrid functionals.

† Full DFT implementations including also the JSD

terms and using analytical derivative theory were

reported in Refs. [252,253]. Both programs are

capable of using also the hybrid functionals.

Whereas the Hartree–Fock level of theory typi-

cally leads to overestimated spin density and,

hence, FC contributions, the GGA functionals tend

to underestimate the same quantities. Somewhat

expectedly then, the quality of results improves

significantly in the succession LDA ! GGA !

hybrid functionals, in the main-group systems

investigated so far [252,253]. The problem with19F is not, however, solved by the hybrid

functionals.

Presently this method holds the greatest

promise for solving chemical problems in

large main-group systems. The performance of

the popular B3LYP hybrid functional [265,266]

for the anisotropic properties of J has recently

been tested [42]. While not validated for J, in

transition metal systems hybrid functionals do

not appear to offer systematic improvement for

other properties.

† Autschbach and Ziegler have implemented the

relativistic zeroth-order regular approximation

(ZORA) for calculations of J tensors [144,

267]. This method includes both scalar relativistic

[267] and electronic spin–orbit effects [144], with

JSD calculated in connection with the latter. ZORA

leads to modified hyperfine operators that can be

interpreted in non-relativistic terms, however.

Analytical derivative techniques were used. The

method allows qualitatively accurate calculations

for spin–spin couplings also involving heavy

nuclei. The software used is limited to pure DFT

functionals. The Xa approximation is used for the

first-order exchange-correlation potential necessary

in the coupled DFT calculation.

In the pioneering study of Ref. [260], an approxi-

mate self-interaction correction at the LDA level was

not found to lead to a systematic improvement of the

total J, albeit JPSO as well as s were clearly improved.

One reason for these mixed observations might be the

fact that the magnetic field response of the potential

term corresponding to the self-interaction correction

was neglected. We note that this term is not present

in the above-mentioned successful cases where

the perturbation operators are purely imaginary,

in contrast to JSD and JFC: In any case, further

investigations along the direction of Ref. [260] would

be very interesting.

Concerning the problematic couplings to 19F, the

application of methods providing localized orbital

contributions to the calculated couplings in combi-

nation with DFT [220,221,268] might give increased

insight [269].

3.3. Basis set requirements

The treatment of the many-body problem as

well as the basis set requirements are demanding

issues that must be addressed in calculations of J.

The reason is 2-fold. First, the need for highly

correlated wave functions places the corresponding

demands on the basis set. JDSO has been found to

be remarkably easy to calculate, with SCF wave

functions and double-zeta plus polarization (DZP)

basis sets giving accurate values [270–272]. The

generally small magnitude of these terms contrib-

utes to the favourable situation as the error in the

total J is dominated by the second-order


contributions. For them, correlated calculations are

mandatory and the TZP basis set can be viewed as

the entry level. The basis set convergence in the

valence region may be expected to be faster with

DFT as compared to the correlated ab initio

methods.

Second, the description of the hyperfine pertur-

bations that sample the immediate vicinity of the

atomic nuclei, necessitates more basis functions in

the atomic core region than what is necessary for

standard energetic properties. Particularly, the FC

operator is difficult to represent using a small number

of Gaussian functions. Hence, the standard basis sets

used in quantum chemistry most often need to be

supplemented with high exponent, tight, basis functions

at least of s-type [273], if converged values of J are to

be obtained. The tight functions typically increase the

coupling constants by 5–10% as compared to state-

of-the-art basis sets for valence properties.

A few systematic studies of the basis set require-

ments for J at ab initio level have been carried out.

† The polarization propagator calculations for JHD

were performed for the hydrogen molecule [274].

The need for tight s-functions was established, all

the way up to exponents such as asðHÞ ¼ 150; 000

whereas in ordinary basis sets max½asðHÞ� ¼

Oð100Þ:

† The CASSCF LR results for JFH in HF [275]

recommended systematically converging, although

relatively expensive cc-pVXZ-sun basis sets that

are based on the correlation consistent paradigm

[276–279], decontracted in the s-function space,

and supplemented with n tight primitives of this

type. See also Ref. [280] for a related study.

† System-dependent basis set prescription has been

proposed, through using contraction coefficients

from MO coefficients for the molecule under study

[242,281,282]. This was slightly generalized,

based on simple model hydrides containing the

nuclei of interest, in Ref. [283].

† A pragmatic procedure more easily adopted in

large systems has been followed in Refs. [284,

285]. There, use has been made of sets that build on

the decontracted Huzinaga/Kutzelnigg (‘IGLO’)

basis sets [286,287] commonly denoted BII–BIV

or HII–HIV. These basis sets have been shown to

perform very well for their size in Refs. [242,275].

Uncontracting and supplementing them with n sets

of tight s-type primitives (in some cases also p- and

d-type) provides nice convergence behaviour of

the properties that depend on the FC perturbation.

These basis are designated as, e.g. HIVun.

† In the locally dense basis set concept [288,289],

a large basis with tight primitives is only used

for the interesting part of the molecule, possibly

only at the centres with the coupled nuclei,

while the rest of the system is treated more

approximately. This method was applied to 3JHH

in C2H5X (X ¼ H, F, Cl, Br, I) in Ref. [282] at

the SOPPA level with encouraging results.

Changes of the order of 0.3 Hz, or 3% of the

total magnitude of the coupling, were observed

due to the locally dense approximation.

While the use of an all-electron basis set is

normally necessary for the nuclei for which couplings

are calculated, the reconstruction of the core response

to hyperfine operators in a pseudo-potential frame-

work [290,291] would be interesting in the context of

J, as well.

Nair and Chandra [292] have used energy-

optimized bond-centred s- and p-primitive functions

to significantly improve calculated coupling constants

at the SCF level, with otherwise very modest basis

sets. A systematic study at a correlated level would be

in order.

Interesting initial work has been carried out by

Rassolov and Chipman [293,294] (see also the earlier

paper by Geertsen [295]) where the delta function

sampling of the wave function at the nucleus is

replaced through integration by parts by a global

operator covering an extended area (r < 0.1 a.u.) in

space around the nucleus. This procedure eliminates,

at least partially, the need for tight s-functions. Results

are identical to those obtained with the d-function

operator for the exact wave function. For approximate

wave functions, errors are smaller than with the d-

function operator. Apart from the initial trials, the

performance of the method has not been investigated

in detail.

3.4. Effects of nuclear motion

Zero-point and thermal motion of the nuclei,

as well as the presence of a medium, affect


the parameters of HNMR: Ultimately, the comparison

of accurate theoretical calculations with experimental

results should take into account both. This has seldom

been the case in the spin–spin coupling literature.

Fig. 12 illustrates the different steps in the

comparison.

J depends on the geometrical parameters such as

bond lengths (see, e.g. Ref. [297] for N2 and CO

molecules) and angles [298]. Particularly the triplet

coupling mechanisms exhibit large geometry depen-

dence, the prototypical example being the increase of

JFC in HD by orders of magnitude as the bond length

is extended [299]. The origin of the effect is in the

shared dissociation limit of the singlet ground state

and the triplet excited state, and the consequently

decreasing triplet excitation energy as the bond is

extended (see also Ref. [300]). The dependence of3JHH on the dihedral angle, giving rise to the

well-known Karplus plot, is exemplified for ethane

(C2H6) in Ref. [301].

A comprehensive review on the rovibrational

averaging of molecular properties was given in

Ref. [302]. Computational modelling of rovibrational

effects involves determining the J coupling hypersur-

face

J ¼ Je þX

k

›J

›Qk

� eQk

þ1

2

Xkl

›2J

›Qk›Ql

!e

QkQl þ · · · ð86Þ

where J is now a component of the J tensor, Je its

equilibrium geometry value, and the Qk are some

nuclear displacement coordinates: either, for example

local coordinates such as Dr ¼ r 2 re; symmetry

coordinates, or normal coordinates. The derivatives

are the parameters of the property hypersurface. When

determining the surfaces for tensorial quantities such

as the components of JA and JS using molecular

geometries displaced from the equilibrium, it is

necessary to ensure that the Eckart conditions

[303,304] are fulfilled by the coordinate represen-

tation used for the property tensors [305,306].

The property surface is averaged over the nuclear

motion as

kJlT ¼ Je þX

k

›J

›Qk

� ekQkl

T

þ1

2

Xkl

›2J

›Qk›Ql

!e

kQkQllT þ · · ·; ð87Þ

where the nomenclature kAlT specifies either the

temperature average of A or its average in a particular

rovibrational state, occupied with a certain tempera-

ture-dependent probability. The averages are, in turn,

determined by the potential energy surface (PES) of

the system. In a normal coordinate expansion, the

second-order (harmonic) terms arise due to the

quadratic potential surface, while the leading anhar-

monic contributions are due to the semi-diagonal

components of the cubic force field. The effect of

vibrational anharmonicity can be covered, to a good

approximation, by carrying out a single-point calcu-

lation at the thermally averaged ra geometry, where

kJlT < Ja þ1

2

Xk

›2J

›Q2k

!e

kQ2kl

T: ð88Þ

The expansion of Eq. (87) is usually truncated after

the harmonic terms, causing typically only a small

error.

Once the property hypersurface is mapped out,

there are different ways to perform the averaging. In

the widely used perturbational method (see, e.g.

Ref. [307]), the thermal vibrational averages kQklT

and kQkQllT ; as well as the rotational contribution to

the former, are calculated based on the formulae given

in Ref. [308]. For diatomics, properties averaged in

individual rovibrational states are conveniently avail-

able by solving the rovibrational Schrodinger

equation numerically [309]. Recently, an approach

based on sampling geometries accessible to

the rovibrational motion by semi-classical path

integral simulation, has been advanced [310].

Fig. 12. Schematic diagram of the various factors affecting the

comparison of experimentally and theoretically determined NMR

parameters. Drawn after Ref. [296].


The process of rovibrational averaging is computer

and human resource-intensive, as many single-point

calculations are needed to map the property and

potential energy hypersurfaces. Refs. [311,312]

describe an automated procedure for carrying out

zero-point vibrational corrections by first finding the

average geometry rz ¼ ra (0 K) and then performing

the harmonic vibrational corrections at that point. The

method features numerical derivatives of analytic

single-point gradients and properties. Besides auto-

mation, its principal merit as compared to expansions

at re is the smaller truncation error. Generalization to

rovibrational averaging at finite temperatures is in

progress [313]. Another automated implementation of

the zero-point vibrational corrections to molecular

properties, based on expansion about the re geometry,

has been carried out [314].

Applications of rovibrational corrections to J

include HD [208,274], HF [237], FHF2 [315], N2

[297,316], CO [297,316], OH2 [317], H2O [241,318],

H3Oþ [317], CH4 [319,320], C2H2 [214,215], and

SiH4 [321]. The MCLR, EOM-CCSD, SOPPA, and

SOPPA(CCSD) methods have been used. Calcu-

lations at an a priori inadequate level where only the

first-order Taylor expansion of the coupling constants

is used, are reported for CH4 in Refs. [281,322] and

for XH4 (X ¼ C, Si, Ge, Sn) in Ref. [323]. A FCI

study of the FC contribution in H2 was carried out in

Ref. [299].

The effect of thermal motion on the tensorial

properties of J appears to have garnered almost no

attention in recent literature, apart from the CASSCF

study of diatomic molecules by Bryce and Wasylishen

[143]. Table 12 compares their calculated results for

equilibrium geometry and in specific rovibrational

states.

The calculated (ro)vibrational corrections both for

J and DJ are quite small in these systems. The

accuracy of the calculations is not yet sufficient to

assess their significance in comparison with the

experiment. Ref. [78] reports an estimate of the

rovibrational effect on D1JFC in CH3F with essentially

the same result. Ref. [324] reported J and DJ in HCN

and HNC as a function of the length of the triple bond,

but did not carry out averaging over nuclear motion.

The magnitudes of all of the DJ increase with

increasing bond length. The changes are, in most

cases, smaller than those of the corresponding

coupling constants, implying smaller rovibrationally

induced changes for the anisotropic observables than

for J. Galasso [325] reported a large dependence of

D1J (defined with respect to the direction of the

internuclear axis between the heavy atoms) on

the dihedral angle for N2H4, P2H4, and PH2NH2.

The case of P2H4 had been studied earlier by Pyykko

and Wiesenfeld in Ref. [74]. Further studies on

nuclear motion and rovibrational averaging effects on

the tensorial properties of J would be of interest.

3.5. Relativistic effects

Classic reviews on the effects of special relativity

in chemistry have been given by Pyykko [326,327].

Relativistic effects on atomic and molecular

Table 12

Comparison of calculated 1J for diatomic molecules at equilibrium geometries and in specific rovibrational states. Results from Ref. [143]

(results in Hz)

Molecule Coupling re value Rovib. state Rovib. average Exp.a

LiH 7Li1H J 152.47 n ¼ 0, J ¼ 1 151 135(10)

DJ 212.39 213 257(21)

LiF 19F7Li J 193.10 n ¼ 0, J ¼ 0 199.0 172.3(32)

DJ 177.43 176.9 173.2(28)

KF 39K19F J 76.59 n ¼ 0, J ¼ 0 78.2 57.8(13)

DJ 109.22 109.5 125.7(51)

Na223Na23Na J 1243.6 n ¼ 0 1245 1067(7)

DJ 229.88 230 248(15)

ClF 35Cl19F J 832.24 n ¼ 0, J ¼ 1 829 840(6)

DJ 2805.68 2800 2907(27)

a Experimental microwave spectroscopic results for the specified rovibrational states. For references, see Ref. [143].


electronic structure can be categorized into scalar

relativistic and spin–orbit effects. The rough conse-

quence of the former is a contraction of the atomic s-

and p-shells as well as expansion of the d- and f-shells,

whereas the latter causes spin polarization even in

closed-shell systems by mixing triplet excited states

with the ground state. Generally, relativistic effects

are larger for systems with heavy nuclei.

The NMR and hyperfine properties, in general, are

susceptible to relativity, even for light elements, as the

quantum mechanical operators involved probe the

region of the electron cloud close to the nuclei, where

the electron velocities are large. In the context of spin-

Hamiltonian parameters such as J, relativistic effects

enter first through modification of the wave function

due to (in the Pauli language) the mass–velocity,

Darwin, and spin–orbit interactions. Second, com-

pletely new terms or combinations of non-relativisti-

cally uncoupled mechanisms may appear, such as the

FC/PSO cross-terms [37,38] or second-order spin–

orbit terms [38]. Third, the relativistic hyperfine

operators themselves are different from their non-

relativistic limits [328–330]. For J, the leading

relativistic correction terms are Oða6Þ; two powers

of a higher than the basic non-relativistic theory.

A brief list of the currently available methods that

include relativity in the calculation of J is as follows.

† A posteriori multiplicative correction factors are

obtained as the ratio of the Dirac–Fock and

Hartree–Fock hyperfine integrals [328]. This is

a semi-empirical correction, the applicability of

which depends on the dominance of the FC

contribution. Ref. [331] applies the idea in the

DFT framework by borrowing the electron density

at one of the coupled nuclei from a scalar

relativistic atomic calculation.

† Relativistic extended Huckel (REXNMR) [7,74] is

a semi-empirical method based on the relativistic

parametrization (obtained by Dirac–Fock atomic

calculations) of the extended Huckel method.

While the results are at best qualitatively correct,

this is the relativistic method by which the largest

number of studies of J have been carried out so far

[3–5,7,8,74]. Among the obtained results, the

increase of the relative anisotropy DK=K due to

relativistic effects [74,332] seems to be a general

feature.

† The CLOPPA RPA method with relativistic

semi-empirical parametrization [9,221,222] in a

formally non-relativistic framework features

localized MO contributions.

† Breit–Pauli corrections for the spin–orbit effect

have been carried out through third-order pertur-

bation theory [37,38,333]. A second-order correc-

tion was added in Ref. [38]. The method requires

scalar relativistic effects for comparison with

experiment.

† The Pauli Hamiltonian in a scalar relativistic

frozen core DFT framework has been used by

Khandogin and Ziegler [331]. This is theoretically

somewhat incomplete as relativistic modification

of the wave function by the mass–velocity and

Darwin interactions is used with non-relativistic

hyperfine operators. The approach features FPT for

JFC and neglects JSD: Results seem to be worse

than in the simple modification of the FC

contribution, discussed earlier.

† The four-component Dirac–Hartree–Fock (DHF)

LR model has been implemented and applied

[334–337]. Both scalar relativistic and spin–orbit

effects are included in a fully relativistic frame-

work. The method has in principle a simple

structure due to only one relativistic hyperfine

operator, with the diamagnetic term in particular

arising from rotations between occupied electronic

and virtual positronic states [336]. DHF LR needs

to be extended beyond the RPA level for direct

comparison with experiment, however.

† The ZORA DFT method, already mentioned in

Section 3.2, includes both scalar relativistic and

spin–orbit effects. This is, for practical problems,

the most applicable of the presently available

methods, with potential for large systems as well.

Applications already include the tensorial proper-

ties of J [144–146]. Autschbach and Ziegler [267]

found the spatial origin of the relativistic increase of

JFC in the distance range up to 1022 a.u. from the

heavy nucleus. A comparison with frozen core

calculations points out that the core tails of valence

electrons are mainly responsible.

The DHF method is a useful benchmark for more

approximate methods of including relativity. More

practical applications are to be expected from the

transformed Hamiltonian methods, notably ZORA, but


most likely the Douglas – Kroll – Hess approach

[338,339] and direct perturbation theory [340,341]

will be implemented and applied in the context of J in

the near future. There, one has to account for the

picture change effects on the hyperfine operators [342,

343]. The relativistic pseudo-potential route can also

be followed for couplings between light nuclei in

systems where heavy elements are represented by

scalar relativistic [344–346] and spin–orbit pseudo-

potentials. For implementations and applications of

this method to s and the electron spin resonance

g-tensor, see Refs. [347,348], respectively.

Examples of calculated relativistic effects on

specific coupling tensors are deferred until Section

3.8.1.

3.6. Solvation and intermolecular forces

Medium and solvation effects on J and other

molecular properties have their origins in intermole-

cular interactions, repulsion, dispersion, electrostatic,

and induction forces, as well as hydrogen bonding

effects [349]. There are both indirect effects mani-

fested through changes of the molecular geometry as

well as direct electronic structure modifications that

already have an effect at the gas-phase structure.

Two approaches to account for medium effects are

used in the recent J coupling literature (for older

references see Ref. [350]).

† The molecule can be placed in a cavity within a

homogeneous, linear dielectric medium character-

ized by its dielectric constant, and subjected to the

reaction field caused by the response of

the environment to the charge distribution of the

molecule. This was first reported for J in Ref. [351]

within the spherical cavity model. This is an

analytic derivative method covering the long-range

electrostatic forces. In particular, hydrogen bond-

ing is not within reach in a pure reaction field

model.

† The supermolecule method has been applied,

where parts of the immediate molecular sur-

roundings of the system under study are

explicitly included in the finite field spirit. In

principle one would want to include parts of the

environment at least up to the first solvation

shell, but in practice that is a difficult require-

ment in the J coupling context. So far

applications have been limited to including

a few molecules of the environment. The

supermolecule method accounts for the short-

range intermolecular interactions, and can be

combined with the reaction field model for long-

range electrostatics by placing the supermolecule

in a cavity. Proper treatment of the dispersion

interaction is difficult in a supermolecule

calculation. An additional complication of the

method is caused by the basis set superposition

error, for which the counterpoise method [352]

is a pragmatic solution.

Applications for J include C2H2 [353,354], C2H3F

[238], H2O [355], CH3OH [356], CH3NH2 [356],

HCN [357], H2S [357], and H2Se [351] using the

reaction field method, and the first two molecules of

the list using the supermolecule or combined method

[354,355]. The effects on J caused by the solvation by

one water molecule were investigated in Ref. [358]

for CH2O, C2H2, and CH3OH. The hydrogen-bonded

complexes formamide dimer and formamidine–for-

mamide dimer were investigated in Ref. [359]. The

effect of dimerization on the couplings in HCOOH

was also studied. An earlier study of formamide

solvated by four water molecules was reported in Ref.

[360].

Transition metal compounds were studied at the

supermolecule level by Autschbach and Ziegler [361,

362]. Ref. [363] reports a Hartree–Fock study of the

effects on 1JFCNLi in LiNH2, LiN(CH3)2, and (LiNH2)2

due to explicit solvation by one to four water

molecules. Ref. [364] used IPPP-CLOPPA to inves-

tigate the effect of the electric field due to a solvent

water molecule on 1JCH in CH4 and HCN. The 1JCH in

the CH4–FH and H2O–HCN supermolecular systems

was investigated in Ref. [365]. Ref. [354] used the

concept of intermolecular coupling constant surfaces,

which is likely to be of large qualitative value in the

interpretation of medium effects on J.

Many recent applications on J mediated through

hydrogen bonds [355,358,359,366–376] can be

considered to fall into the supermolecule category,

although the goal there is in calculating actual

intermolecular couplings. Ref. [377] is a pioneering

report on J mediated by the van der Waals inter-

action in He2. Ref. [378] gives an estimate of


the corresponding FC contribution in Xe2. Ref. [376]

considers JNN over the hydrogen bond in a methyl-

eneimine dimer and reports both the coupling constant

and anisotropy as functions of the hydrogen bond

geometry. The only study of intermolecular J

featuring configurational sampling that we are aware

of, is that reported in Ref. [379]. In that work, Monte

Carlo simulations were used to estimate the coupling

constant between the nucleus of the F2 ion and the

nuclei in a water solvent. The intermolecular coupling

surface was initially parametrized using quantum

chemistry and the simulation was analysed assuming

pairwise additivity of the interactions. The present-

day computational resources should facilitate further

research in this direction.

Little is known about the medium effects on the

tensorial properties of J. Generally, the effects on J

are of sufficient magnitude to warrant investigation of

the first and second-rank contributions as well.

3.7. Couplings for large systems

To date, computational studies of J have mainly

concentrated on small molecules of prototypical value

for chemical purposes. Furthermore, the isotropic

coupling constants have almost exclusively been the

focus. The limited number of studies for large systems

reflects the unfavourable computational scaling of the

current correlated ab initio methods with the size of

the system, discussed in Section 3.1. Despite this, a

few computational tricks are already available to

facilitate studies of medium- to large-sized molecules.

† Chemically motivated model molecules. Cluster

models of the environment.

† Locally dense basis sets. Pseudo-potentials for

inter- and intra-ligand couplings.

† Tailor-made contraction of the basis set according

to the molecular orbital coefficients of the system

under study.

† Bond-centred basis functions [292].

† Calculations only of the most demanding

contributions (FC and SD/FC) at the highest

level; lower-level methods for DSO, PSO, SD.

These tools are naturally also available for DFT

calculations of J for which several application papers

have already appeared. These include one-bond

metal-ligand couplings at the quasi-relativistic

[331,380] and ZORA [144,267] levels, as well as

solvation effects in the couplings in a coordinatively

unsaturated transition metal compound [361,362]. In

these works, explicit solvation was found to be

absolutely necessary, along with the contribution of

scalar relativistic effects, to produce a qualitatively

correct description of the J coupling patterns.

Through-space FF couplings were studied in

different polycyclic organic fragments [264,268,

381]. It is noteworthy that through-space FF couplings

can be calculated with DFT much more accurately

than what would be expected on the basis of through-

bond couplings in small molecules. Bryce and

Wasylishen [382] investigated the XF (X ¼ H, C, F)

couplings using the MCLR method and HF–CH4 and

HF–CH3F complexes as model systems. Also coup-

ling anisotropies were reported; in particular the FF

coupling anisotropy was found to be large at small

inter-fragment distances. JCH and JCC in some

prototypical hydrocarbons, e.g. pyridine, were calcu-

lated in Ref. [256]. The dependence of 1JCaCb and1JCaHa on the backbone conformation of a model

dipeptide [383,384], the dependence of 3JCC on the

conformation of an open chain natural product

fragment [385], the HH and CH couplings in Me a-

D-xylopyranoside [386], 1JCH in cyclohexane-related

systems [387], 1JFH in (HF)n clusters [388], inter-

nucleotide JNH and JNN between DNA base pairs

[368], and 1JHD in Os(II)-dihydrogen complexes [346]

have also been investigated. In most of these

applications, JSD has been neglected.

An earlier investigation of JHD in Os(II)-dihydro-

gen complexes was carried out at the SCF and MP2

levels in Ref. [344].

Refs. [389–391] report a hybrid DFT (B3LYP)

level study of nJCC and nJCH in 2-deoxy-b-D-

ribofuranose and related systems using a FPT

calculation of JFC: In particular, the work at the

DFT level can be carried out without having to resort

to the awkward procedure of scaling low-level ab

initio results based on benchmark calculations for

smaller systems at a higher level [392–394].

Ref. [255] reports an uncoupled DFT study of nJHH

in terpenes. The results are hampered by serious

methodological deficiencies as discussed before. In

Ref. [216], the uncoupled SCF method was used for

JFCHH; JFC

CH; and JFCCC in a number of light main-group


molecules. In this context, FPT calculations for JFC at

levels ranging from SCF to quadratic configuration

interaction [QCISD(T)], carried out in Refs. [219,

395], are more consistent and superior in quality.

EOM-CCSD was applied to the CC, CH, and HH

coupling constants in the 2-norbornyl carbocation

with excellent results [396].

The EOM method was applied to the HH, CH, and

CC coupling constants in the cyclodecyl cation and

related systems in Ref. [397]. The 1JCC in bicyclo

butane, tricyclopentane, and tricyclohexane, as well

as octabisvalene were studied by this method in

Ref. [398]. For related, earlier work, the reader is

referred to Refs. [399–401].

A Hartree–Fock level study of 1JFCNLi in model

systems exhibiting the LiN-bond was carried out in

Ref. [363]. 1JFCCH in five- and six-membered hetero-

cyclic compounds were studied in Ref. [402]. JHH;

JCH; and JCC in bicyclo[1.1.1]pentane have been

investigated at the SCF LR level [403].

3.8. Quantum chemical results

3.8.1. Symmetric components

Tables 13–24 display the results for components of

J1 þ JS:

In cases where the principal values of the tensors

are specified, we do not report the directions of the

corresponding principal axes, for reasons of space.

The reader is asked to refer to the original publications

where this information can in most cases be found.

When converting between reduced coupling units,

1019 T2J21 and Hz, or between different isotopes of

the same nucleus, use has been made of magnetogyric

ratios tabulated in Ref. [408].

We divide the discussion into parts according to

the methodology used. The early work is character-

ized by inadequate electron correlation treatment

and/or, by today’s standards, modest basis sets.

Following that, the more recent papers with up-to-

date methods are commented upon.

Early work using Hartree–Fock-level methods.

Lazzeretti et al. carried out a coupled Hartree–Fock

study of PH22, PH3, and PH4

þ [40] as well as NH22,

NH4þ, and BH4

2 [41] using reasonable basis sets but

neglecting the DSO contribution. The authors should

be commended for reporting full information on their

coupling tensors, including JA; in contrast to most of

the work in the field. The calculated J were in

qualitative agreement with the experiment, although

error cancellation between both the tensor components

and the different mechanisms may make the individual

numbers not very trustworthy. In these papers, the

individual SD(M )/FC(N ) and SD(N )/FC(M ) contri-

butions to the SD/FC cross-term have been presented

separately. For 1J; roughly equally large contributions

appear with opposite signs and the total SD/FC value

of each tensor component is smaller in absolute terms

than either the SD(X)/FC(H) or SD(H)/FC(X) values.

The lighter the element X is (i.e., from P to N to B), the

more the term with the FC interaction at H dominates

the total SD/FC contribution.

Lazzeretti et al. performed first-order polarization

propagator (FOPPA) studies, equivalent to SCF or

RPA level, on AlH42 and SiH4, with decent basis sets

[406]. The full tensors were reported. The lack of

electron correlation limits the reliability of the

tensorial results, as can be seen from a comparison

with the later SOPPA(CCSD) calculation for SiH4

[321]. The isotropic J values are reasonable, but are

not in quantitative agreement with the available

experimental data. The relatively large importance

of the DSO contribution to DJ as compared to

coupling constants is evident from this and other

early work.

Galasso [325] used the SOS-CI method of

Nakatsuji [409] for nine dihydrides containing B, N,

or P as heteroatoms, in a study reporting 1J and D1J

with respect to the direction of the vector joining the

heavy nuclei. The SOS-CI method comprises a non-

iterative calculation with all singly and some doubly

excited configurations, and the results are roughly of

Hartree–Fock quality. A combination of the modest

6-31G and 4-31G basis sets was employed. The

available experimental J values are reproduced

qualitatively. These systems have not been subjected

to a modern study. For example, the SD contribution

to JPP and DJPP P2H4 is not negligible.

Pioneering work using correlated wave functions.

Geertsen and Oddershede compared SOPPA calcu-

lations with lower-order methods for water in Ref.

[228]. As the basis sets used were reasonably good,

electron correlation was included, all the physical

contributions were calculated, and results for the full

tensor reported, the work remains as one of the most

complete early papers on J. For 2JHH; the later


Table 13

Quantum chemically calculated components of the symmetric part J1 þ JS of the 1H1H spin–spin coupling tensors

System Bonds Theorya Values Reference

BH42 2 SCF/(s ) Jaa ¼ 212.10, Jbb ¼ 219.96, Jcc ¼ 220.67 [41]

CH4 2 SOPPA(CCSD)/(s ) Jaa ¼ 28.25, Jbb ¼ 29.48, Jcc ¼ 225.63 [321]C2H4 2 RAS/HIV DJ ¼ 5.3, Jxx 2 Jyy ¼ 15.3b, Jaa ¼ 4.5, Jbb ¼ 6.8, Jcc ¼ 28.5 [72]C2H6 2 RAS/HIII DJ ¼ 28.3, Jaa ¼ 28.0, Jbb ¼ 28.8, Jcc ¼ 225.5 [72]NH2

2 2 SCF/(s ) Jaa ¼ 8.96, Jbb ¼ 215.21, Jcc ¼ 234.89 [41]NH4

þ 2 SCF/(s) Jaa ¼ 25.13, Jbb ¼ 221.83, Jcc ¼ 235.75 [41]CH3CN 2 RAS/HIII DcJ ¼ 218.97, J ¼ 222.91c,d [324]CH3NC 2 RAS/HIII DcJ ¼ 217.65, J ¼ 219.05c,d [324]H2O 2 SOPPA(CCSD)/(s ) Jaa ¼ 1.21, Jbb ¼ 6.38, Jcc ¼ 233.33 [318]H2O 2 CAS/(s ) Jaa ¼ 0.56, Jbb ¼ 5.48, Jcc ¼ 234.84 [318]H2O 2 CAS þ SOe/HIVu4 Jaa ¼ 0.75, Jbb ¼ 5.25, Jcc ¼ 234.54 [38]H2O 2 CAS/HIVu4 Jaa ¼ 0.73, Jbb ¼ 5.19, Jcc ¼ 234.60 [38]H2O 2 SOPPA/(s ) Jaa ¼ 1.16, Jbb ¼ 5.66, Jcc ¼ 234.22 [280,318]H2O 2 SOPPA/(s ) Jaa ¼ 21.14, Jbb ¼ 1.69, Jcc ¼ 235.93 [228]H2O 2 MP2/(s ) Jaa ¼ 20.93, Jbb ¼ 25.12, Jcc ¼ 248.93 [227]HCONH2 2 RAS/HIII Jaa ¼ 11.2, Jbb ¼ 211.8, Jcc ¼ 11.9 [404,405]CH3F 2 RAS/HIII DJ ¼ 210.53, Jaa ¼ 25.37, Jbb ¼ 26.58, Jcc ¼ 222.61f [78]CH2F2 2 RAS/HIII DJ ¼ 6.06, Jxx 2 Jyy ¼ 16.65g, Jaa ¼ 3.35, Jbb ¼ 5.61, Jcc ¼ 211.04 [78]AlH4

2 2 FOPPA/(s ) Jaa ¼ 25.55, Jbb ¼ 27.19, Jcc ¼ 27.96 [406]SiH4 2 SOPPA(CCSD)/aug-cc-pVTZ(m ) Jaa ¼ 0.01, Jbb ¼ 2.43, Jcc ¼ 5.34 [321]SiH4 2 FOPPA/(s ) Jaa ¼ 0.42, Jbb ¼ 21.71, Jcc ¼ 25.48 [406]CH3SiH3 2h RAS/HIII DJ ¼ 27.33, Jaa ¼ 29.50, Jbb ¼ 29.73, Jcc ¼ 226.49 [80]CH3SiH3 2i RAS/HIII DJ ¼ 21.96, Jaa ¼ 0.15, Jbb ¼ 3.69, Jcc ¼ 3.72 [80]PH2

2 2 SCF/(s ) Jaa ¼ 5.92, Jbb ¼ 28.38, Jcc ¼ 29.42j [40]PH3 2 SCF/(s ) DJ ¼ 4.26, Jaa ¼ 27.79, Jbb ¼ 223.52, Jcc ¼ 224.56 [40]PH4

þ 2 SCF/(s ) Jaa ¼ 5.04, Jbb ¼ 26.09, Jcc ¼ 26.93 [40]H2S 2 CAS þ SOe/HIVu4 Jaa ¼ 26.52, Jbb ¼ 215.73, Jcc ¼ 224.67 [38]H2S 2 CAS/HIVu4 Jaa ¼ 26.70, Jbb ¼ 215.80, Jcc ¼ 224.85 [38]H2S 2 MP2/6-31G(**) Jaa ¼ 214.92, Jbb ¼ 221.07, Jcc ¼ 226.18 [227]H2Se 2 CAS þ SOe/HIVu4 Jaa ¼ 213.38, Jbb ¼ 214.14, Jcc ¼ 224.81 [38]H2Se 2 CAS/HIVu4 Jaa ¼ 213.74, Jbb ¼ 214.91, Jcc ¼ 225.67 [38]H2Te 2 CAS þ SOe/HIVu3 Jaa ¼ 213.26, Jbb ¼ 217.02, Jcc ¼ 223.03 [38]H2Te 2 CAS/HIVu3 Jaa ¼ 214.05, Jbb ¼ 218.59, Jcc ¼ 224.70 [38]C2H2 3 SOPPA(CCSD)/(s ) DJ ¼ 3.20, J ¼ 11.31 [214,215]C2H2 3 RAS/HIV DJ ¼ 3.4, J ¼ 10.8 [72]C2H2 3 ZORA DFT(GGA)/Slater DJ ¼ 12, J ¼ 10 [144]C2H4 3k RAS/HIV DJ ¼ 4.0, Jxx 2 Jyy ¼ 21.2b, Jaa ¼ 8.4, Jbb ¼ 9.7, Jcc ¼ 13.1 [72]C2H4 3l RAS/HIV DJ ¼ 5.0, Jxx 2 Jyy ¼ 20.8b, Jaa ¼ 14.4, Jbb ¼ 15.8, Jcc ¼ 20.9 [72]C2H6 3m RAS/HIII DJ ¼ 2.2, J ¼ 7.2 [72]HCONH2 3k RAS/HIII Jaa ¼ 0.7, Jbb ¼ 20.8, Jcc ¼ 2.4 [404,405]HCONH2 3l RAS/HIII Jaa ¼ 10.5, Jbb ¼ 10.6, Jcc ¼ 14.2 [404,405]CH3SiH3 3n RAS/HIII DJ ¼ 1.23, J ¼ 3.80 [80]

Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaal # lJbbl # lJccl), specified Cartesian components (xx, xy, etc.),or the anisotropy with respect to a certain axis (e.g. DzJ ¼ Jzz 2 1=2ðJxx þ JyyÞ), is indicated. Unless otherwise noted, in systems possessing aunique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz.

a (u ) designates the uncontracted form of the indicated standard basis, (m ) a more substantial modification of the indicated standard basis, and(s ) designates custom basis. Please refer to the original papers for complete details of the basis sets used.

b Anisotropy with respect to the z direction of the CC bond, with the molecule in the xz plane.c Anisotropy defined in the principal axis frame of the tensor.d Insufficient information given for obtaining principal values.e Including corrections for the relativistic spin–orbit interaction.f Error in the original paper [78]. Jaa along the internuclear axis and Jcc makes an angle of 48 with the normal of the local HXH plane, towards

the F atom.g Anisotropy with respect to z direction bisecting the FCF angle, with the F atoms in the xz plane.h Coupling between the CH3 group protons.i Coupling between the SiH3 group protons.j At the optimized geometry [40].k cis-Coupling.l trans-Coupling.

m Parameters averaged over trans and gauche positions.n Average coupling between the methyl and silyl groups.


SOPPA calculation of Ref. [318] produced a different

ordering for the two smallest principal values of the

tensor, the reason for the difference being probably

the basis sets used. The data for 1JOH [228] has stood

the test of time remarkably well.

Galasso and Fronzoni applied the EOM method

using the small 6-31G** basis set on a variety of

simple organic molecules [407]. Information on the

anisotropic properties of a number of 1J was given,

but limited to DJ with respect to the direction of the

bond only. The results for J are in qualitative

agreement with the available experimental data,

although, for triple bonds the agreement is worse

than for double bonds, and the couplings over

Table 14

Quantum chemically calculated components of the symmetric part J1 þ JS of the 13C1H spin–spin coupling tensors


CH4 1 SOPPA(CCSD)/(s ) DJ ¼ 225.6, J ¼ 123.8b [321]

CH4 1 MP2/(s ) DJ ¼ 64.44, J ¼ 130.63b [227]

C2H2 1 SOPPA(CCSD)/(s ) DJ ¼ 263.41, J ¼ 254.95 [214,215]

C2H2 1 RAS/HIV DJ ¼ 262.4, J ¼ 232.1 [72]

C2H2 1 ZORA DFT(GGA)/Slater DJ ¼ 238.7, J ¼ 262 [144]

C2H4 1 RAS/HIV DJ ¼ 2.6, Jxx 2 Jyy ¼ 228.8c, Jaa ¼ 123.9, Jbb ¼ 158.0,

Jcc ¼ 161.2

[72]

C2H6 1 RAS/HIII DJ ¼ 6.0, Jaa ¼ 102.7, Jbb ¼ 128.3, Jcc ¼ 128.5 [72]

C6H6 1 CAS/(s ) DJ ¼ 28.0, Jaa ¼ 144.0, Jbb ¼ 190.8, Jcc ¼ 195.4 [70]

HCN 1 RAS/HIV DJ ¼ 263.34, J ¼ 249.27 [324]

CH3CN 1 RAS/HIII DcJ ¼ 228.82, J ¼ 142.43d [324]

CH3NC 1 RAS/HIII DcJ ¼ 224.70, J ¼ 143.50d [324]

HCONH2 1 RAS/HIII Jaa ¼ 161.5, Jbb ¼ 192.9, Jcc ¼ 195.2 [404,405]

CH3F 1 RAS/HIII DJ ¼ 6.10, Jaa ¼ 122.00, Jbb ¼ 149.84, Jcc ¼ 152.62 [78]

CH2F2 1 RAS/HIII DJ ¼ 27.02, Jxx 2 Jyy ¼ 212.26e, Jcc ¼ 186.00, Jbb ¼ 184.15,

Jaa ¼ 156.88

[78]

CHF3 1 RAS/HIII DJ ¼ 231.19, J ¼ 236.79 [78]

CH3SiH3 1 RAS/HIII DJ ¼ 7.10, Jaa ¼ 96.73, Jbb ¼ 124.85, Jcc ¼ 125.63 [80]


C2H2 2 RAS/HIV DJ ¼ 28.2, J ¼ 50.1 [72]

C2H2 2 ZORA DFT(GGA)/Slater DJ ¼ 39.3, J ¼ 52.3 [144]

C2H4 2 RAS/HIV DJ ¼ 5.2, Jxx 2 Jyy ¼ 6.0c, Jaa ¼ 21.1, Jbb ¼ 23.0, Jcc ¼ 28.1 [72]

C2H6 2 RAS/HIII DJ ¼ 21.8, Jaa ¼ 22.5, Jbb ¼ 26.6, Jcc ¼ 27.0 [72]


HNC 2 RAS/HIV DJ ¼ 33.34, J ¼ 16.44 [324]

CH3CN 2 RAS/HIII DcJ ¼ 5.12, J ¼ 215.46d [324]

HCONH2 2f RAS/HIII Jaa ¼ 0.4, Jbb ¼ 2.0, Jcc ¼ 6.0 [404,405]

HCONH2 2g RAS/HIII Jaa ¼ 22.2, Jbb ¼ 24.1, Jcc ¼ 26.3 [404,405]

CH3SiH3 2 RAS/HIII DJ ¼ 0.41, Jaa ¼ 2.79, Jbb ¼ 3.58, Jcc ¼ 3.84 [80]


CH3NC 3 RAS/HIII DcJ ¼ 5.21, J ¼ 2.63d [324]


Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaal # lJbbl # lJccl), specified Cartesian components (xx, xy, etc.),

or the anisotropy with respect to a certain axis (e.g. DzJ ¼ Jzz 2 1=2ðJxx þ JyyÞ), is indicated. Unless otherwise noted, in systems possessing a

unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz.a See footnote a in Table 13.b Anisotropy along the bond in question.c See footnote b in Table 13.d See footnotes c and d in Table 13.e See footnote g in Table 13.f Coupling to trans-hydrogen with respect to the oxygen atom.g Coupling to cis-hydrogen with respect to the oxygen atom.


Table 15

Quantum chemically calculated components of the symmetric part J1 þ JS of the 15N1H spin–spin coupling tensors

System Bonds Theorya Valuesb Reference

NH22 1 SCF/(s ) Jaa ¼ 22.26, Jbb ¼ 46.39, Jcc ¼ 68.41 [41]

NH3 1 MP2/(s ) DJ ¼ 15.70, J ¼ 260.07 [227]

NH4þ 1 SCF/(s ) DJ ¼ 220.12, J ¼ 68.96c [41]

HNC 1 RAS/HIV DJ ¼ 36.39, J ¼ 2112.61 [324]

N2H4 1 SOS-CI/(s ) DJ ¼ 21.82, J ¼ 285.36d,e [325]

BH2NH2 1 SOS-CI/(s ) DJ ¼ 20.28, J ¼ 298.44d [325]

BH3NH3 1 SOS-CI/(s ) DJ ¼ 21.52, J ¼ 288.48d [325]

HCONH2 1f RAS/HIII Jaa ¼ 274.8, Jbb ¼ 2100.4, Jcc ¼ 2103.5 [404,405]

HCONH2 1g RAS/HIII Jaa ¼ 274.6, Jbb ¼ 299.6, Jcc ¼ 2102.7 [404,405]

PH2NH2 1 SOS-CI/(s ) DJ ¼ 25.65, J ¼ 2103.82d,h [325]

HCN 2 RAS/HIV DJ ¼ 219.51, J ¼ 26.44 [324]

CH3NC 2 RAS/HIII DcJ ¼ 21.91, J ¼ 4.46i [324]


CH3CN 3 RAS/HIII DcJ ¼ 24.30, J ¼ 22.03i [324]



unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz.a See footnote a in Table 13.b Results converted for 15N when necessary.c See footnote b in Table 14.d Anisotropy with respect to the axis joining the heavy nuclei of the system. See also footnote d in Table 13.e Values at the dihedral angle value corresponding to the equilibrium geometry.f See footnote f in Table 14.g See footnote g in Table 14.h Planar configuration, dihedral angle 908 [325].i See footnotes c and d in Table 13.

Table 16

Quantum chemically calculated components of the symmetric parts J1 þ JS of the 19F1H spin–spin coupling tensors


HF 1 DHF/cc-pVTZ(u ) DJ ¼ 160.19, J ¼ 610.42 [335]

HF 1 SCF/cc-pVTZ(u ) DJ ¼ 158.68, J ¼ 612.23 [335]

HF 1 CAS/cc-pV5Z DJ ¼ 115.98, J ¼ 476.09 [143]

HF 1 CAS þ SOb/HIVu3 DJ ¼ 127.3, J ¼ 534.7 [38]

HF 1 CAS/HIVu3 DJ ¼ 126.7, J ¼ 534.8 [38]

HF 1 MP2/(s ) DJ ¼ 2715.92, J ¼ 570.01 [227]

CH3F 2 RAS/HIII DJ ¼ 256.73, Jaa ¼ 8.59, Jbb ¼ 37.00, Jcc ¼ 100.76 [78]

CH2F2 2 RAS/HIII DJ ¼ 23.51, Jxx 2 Jyy ¼ 44.71c, Jaa ¼ 22.87, Jbb ¼ 45.38,

Jcc ¼ 87.38

[78]

CHF3 2 RAS/HIII DJ ¼ 40.57, Jaa ¼ 60.17, Jbb ¼ 70.67, Jcc ¼ 107.11 [78]

p-C6H4F2 3 RAS/HII(m ) DJ ¼ 16.0, Jxx 2 Jyy ¼ 15.1d, Jaa ¼ 2.6, Jbb ¼ 211.8, Jcc ¼ 12.4 [60]

p-C6H4F2 4 RAS/HII(m ) DJ ¼ 21.4, Jxx 2 Jyy ¼ 24.6d, Jaa ¼ 2.7, Jbb ¼ 8.2, Jcc ¼ 9.6 [60]



unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz.a See footnote a in Table 13.b See footnote e in Table 13.c See footnote g in Table 13.d Anisotropy with respect to the z direction of the FF internuclear axis, with the molecule in the xz plane.


Table 17

Quantum chemically calculated components of the symmetric parts K1 þ KS of the X1H spin–spin coupling tensors other than those listed in

Tables 13–16

System X Bonds Theorya

Values Reference

LiH 7Li 1 CAS/cc-pV5Z DK ¼ 22.8, K ¼ 32.3b [143]

BH42 11B 1 SCF/(s ) DK ¼ 210.35, K ¼ 24.28c [41]

B2H411B 1 SOS-CI/(s ) DK ¼ 0.40, K ¼ 40.08d,e [325]

B2H611Bf 1 SOS-CI/(s ) DK ¼ 20.59, K ¼ 47.12d [325]

B2H611Bg 1 SOS-CI/(s ) DK ¼ 0.20, K ¼ 18.06d [325]

BH2NH211B 1 SOS-CI/(s ) DK ¼ 0.09, K ¼ 40.83d [325]

BH3NH311B 1 SOS-CI/(s ) DK ¼ 1.00, K ¼ 38.52d [325]

BH2PH211B 1 SOS-CI/(s ) DK ¼ 20.49, K ¼ 47.72d [325]

BH3PH311B 1 SOS-CI/(s ) DK ¼ 1.27, K ¼ 41.21d [325]

H2O 17O 1 SOPPA(CCSD)/(s ) Kaa ¼ 45.50, Kbb ¼ 48.37, Kcc ¼ 56.33 [318]

H2O 17O 1 CAS/(s ) Kaa ¼ 47.38, Kbb ¼ 49.94, Kcc ¼ 57.24 [318]

H2O 17O 1 CAS þ SOh/HIVu4 Kaa ¼ 46.47, Kbb ¼ 50.14, Kcc ¼ 57.29 [38]

H2O 17O 1 CAS/HIVu4 Kaa ¼ 46.45, Kbb ¼ 50.27, Kcc ¼ 57.34 [38]

H2O 17O 1 SOPPA/(s ) Kaa ¼ 45.19, Kbb ¼ 49.49, Kcc ¼ 57.09 [280,318]

H2O 17O 1 SOPPA/(s ) Kaa ¼ 39.58, Kbb ¼ 47.03, Kcc ¼ 54.33 [228]

H2O 17O 1 MP2/(s ) DK ¼ 10.15, Kxx 2 Kyy ¼ 51.29, K ¼ 45.87i [227]

HCONH217O 2 RAS/HIII Kaa ¼ 1.6, Kbb ¼ 2.6, Kcc ¼ 7.2 [404,405]

HCONH217O 3j RAS/HIII Kaa ¼ 20.4, Kbb ¼ 20.8, Kcc ¼ 3.0 [404,405]

HCONH217O 3k RAS/HIII Kaa ¼ 20.4, Kbb ¼ 21.0, Kcc ¼ 21.6 [404,405]

AlH42 27Al 1 FOPPA/(s ) DK ¼ 2.00, K ¼ 46.82c [406]

SiH429Si 1 SOPPA(CCSD)/aug-cc-pVTZ(m ) DK ¼ 9.53, K ¼ 80.42c [321]

SiH429Si 1 MP2/6-31G(**) DK ¼ 53.61, K ¼ 78.47c [227]

SiH429Si 1 FOPPA/(s ) DK ¼ 11.72, K ¼ 98.95c [406]

CH3SiH329Si 1 RAS/HIII DK ¼ 22.43, Kaa ¼ 73.90, Kbb ¼ 74.10, Kcc ¼ 81.72 [80]

CH3SiH329Si 2 RAS/HIII DK ¼ 21.07, Kaa ¼ 22.33, Kbb ¼ 24.78, Kcc ¼ 25.05 [80]

PH22 31P 1 SCF/(s ) Kaa ¼ 12.93, Kbb ¼ 14.95, Kcc ¼ 27.65l [40]

PH331P 1 MP2/6-31G(**) DK ¼ 221.34, K ¼ 41.41 [227]

PH331P 1 SCF/(s ) DK ¼ 29.33, Kaa ¼ 33.49, Kbb ¼ 38.22, Kcc ¼ 60.50 [40]

PH4þ 31P 1 SCF/(s ) DK ¼ 9.84, K ¼ 135.96c [40]

P2H431P 1 SOS-CI/(s ) DK ¼ 216.33, K ¼ 32.01d,e [325]

BH2PH231P 1 SOS-CI/(s ) DK ¼ 24.99, K ¼ 64.24d [325]

BH3PH331P 1 SOS-CI/(s ) DK ¼ 28.36, K ¼ 82.96d [325]

PH2NH231P 1 SOS-CI/(s ) DK ¼ 29.84, K ¼ 39.80d,m [325]

H2S 33S 1 CAS þ SOh/HIVu4 Kaa ¼ 22.80, Kbb ¼ 30.39, Kcc ¼ 66.24 [38]

H2S 33S 1 CAS/HIVu4 Kaa ¼ 23.74, Kbb ¼ 30.57, Kcc ¼ 66.19 [38]

H2S 33S 1 MP2/6-31G(**) DK ¼ 19.62, Kxx 2 Kyy ¼ 285.75, K ¼ 40.98i [227]

HCl 35Cl 1 DHF/cc-pVTZ(u) DK ¼ 71.40, K ¼ 26.27 [335]

HCl 35Cl 1 SCF/cc-pVTZ(u) DK ¼ 70.84, K ¼ 27.46 [335]

HCl 35Cl 1 CAS/aug-cc-pVQZ DK ¼ 51.9, K ¼ 50.0 [143]

HCl 35Cl 1 CAS þ SOh/HIVu3 DK ¼ 54.68, K ¼ 37.62 [38]

HCl 35Cl 1 CAS/HIVu3 DK ¼ 54.51, K ¼ 37.66 [38]

HCl 35Cl 1 MP2/6-31G(**) DK ¼ 256.63, K ¼ 20.75 [227]

H2Se 77Se 1 CAS þ SOh/HIVu4 Kaa ¼ 24.46, Kbb ¼ 17.05, Kcc ¼ 121.69 [38]

H2Se 77Se 1 CAS/HIVu4 Kaa ¼ 6.21, Kbb ¼ 19.28, Kcc ¼ 121.30 [38]

HBr 79Br 1 DHF/(s ) DK ¼ 216.29, K ¼ 215.82 [335]

HBr 79Br 1 SCF/(s ) DK ¼ 206.10, K ¼ 4.81 [335]

HBr 79Br 1 CAS þ SOh/HIVu3 DK ¼ 140.05, K ¼ 34.07 [38]

HBr 79Br 1 CAS/HIVu3 DK ¼ 138.43, K ¼ 34.69 [38]

H2Te 125Te 1 CAS þ SOh/HIVu3 Kaa ¼ 228.98, Kbb ¼ 18.56, Kcc ¼ 191.72 [38]

H2Te 125Te 1 CAS/HIVu3 Kaa ¼ 8.44, Kbb ¼ 23.22, Kcc ¼ 190.97 [38]



single-bonds are rather good. The method was limited

to the one-particle/one hole excitation level, hence

besides basis set limitations, further correlation

contributions are to be expected. The calculated DJ

may be compared with later theoretical work for

ethene and ethyne (see below) that seems to have

settled at qualitatively different total D1JCC values,

mainly due to the quite different magnitudes of the

SD/FC contribution as compared to the EOM work.

As in the case of isotropic J, the small basis sets and

modest correlation treatment used in this study [407]

of DJ seem to be the main reasons for the differences.

The fact that the DSO coupling anisotropies are very

different from what the current calculations are able to

provide [72], is harder to understand as this contri-

bution is not affected very much by correlation or

basis set effects.

Fukui et al. investigated the simple first- and

second-row hydrides CH4, SiH4, NH3, PH3, H2O,

H2S, HF, and HCl using FPT MP2 calculations and

modest Pople-type basis sets [227]. While the

agreement with the experimental isotropic JXH is

reasonable in the first-row hydrides, the results for all

JHH as well as JXH in the second row hydrides are

disappointing. In addition to the fact that both the basis

sets used as well as the MP2 correlation treatment

leave lots of room for improvement, there seems to be

something wrong in the calculated anisotropic proper-

ties. Later calculations (cited below) systematically

disagree with the results of Fukui et al. [227] in the

order of magnitude and even the sign of the individual

Cartesian components of the tensors.

MCSCF studies. Barszczewicz et al. carried out

one of the first theoretical investigations of the

tensorial properties of J that can be considered

modern in terms of adequate treatment of the electron

correlation problem and large one-electron basis sets

[324]. The HCN, HNC, CH3CN and CH3NC systems

were investigated at the RASSCF LR level using

moderately large active spaces and the HIII and HIV

basis sets. The J values were in semi-quantitative

agreement with the experimental results, giving

confidence also to the calculated anisotropic proper-

ties for which the experimental data set is much more

sparse. The remaining errors in J for these systems

may be caused by solvent effects to a large extent. It

should be noted in this context that the calculated DJ

for non-axial couplings in CH3CN and CH3NC were

Table 17 (continued)

System X Bonds Theorya

Values Reference

HI 127I 1 DHF/(s ) DK ¼ 369.82, K ¼ 2113.20 [335]

HI 127I 1 SCF/(s ) DK ¼ 340.17, K ¼ 212.97 [335]

HI 127I 1 CAS þ SOh/HIVu3 DK ¼ 216.57, K ¼ 40.22 [38]

HI 127I 1 CAS/HIVu3 DK ¼ 213.78, K ¼ 41.01 [38]

PbH4207Pb 1 ZORA DFT(LDA)/Slater DK ¼ 672, K ¼ 1121c [144]

Either the principal values of the symmetrized tensor (Kii, i ¼ a; b; c with lKaal # lKbbl # lKccl), specified Cartesian components (xx, xy,

etc.), or the anisotropy with respect to a certain axis (e.g. DzK ¼ Kzz 2 1=2ðKxx þ KyyÞ), is indicated. Unless otherwise noted, in systems

possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Reduced couplings are given in units of

1019 T2 J21.a See footnote a in Table 13.b Corrections carried out for the n ¼ 0, J ¼ 1 rovibrational state.c See footnote b in Table 14.d See footnote d in Table 15.e See footnote e in Table 15.f Coupling to terminal hydrogen.g Coupling to bridging hydrogen.h See footnote e in Table 13.i Anisotropy with respect to the z direction along the C2 molecular symmetry axis. The y direction is perpendicular to the plane of the

molecule. See also footnote d in Table 13.j See footnote f in Table 14.k See footnote g in Table 14.l See footnote j in Table 13.

m See footnote h in Table 15.


Table 18

Quantum chemically calculated components of the symmetric part J1 þ JS of the 13C13C spin–spin coupling tensors



C2H2 1 RAS/HIV DJ ¼ 47.5, J ¼ 181.2 [72]

C2H2 1 EOM/6-31G** DJ ¼ 24.32, J ¼ 216.99 [407]


C2H4 1 RAS/HIV DJ ¼ 26.5, Jxx 2 Jyy ¼ 244.3b, Jaa ¼ 39.2, Jbb ¼ 83.6, Jcc ¼ 87.9 [72]

C2H4 1 EOM/6-31G** DJ ¼ 1.29, J ¼ 82.37c [407]

C2H4 1 ZORA DFT(GGA)/Slater DJ ¼ 38.8, J ¼ 59.2c [144]

C2H6 1 RAS/HIII DJ ¼ 32.1, J ¼ 38.8 [72]


H2CyCyCH2 1 EOM/6-31G** DJ ¼ 25.39, J ¼ 109.64c [407]

HCxC–CxCH 1d EOM/6-31G** DJ ¼ 24.62, J ¼ 225.87 [407]

HCxC–CxCH 1e EOM/6-31G** DJ ¼ 3.61, J ¼ 157.92 [407]

C6H6 1 RAS/HII DJ ¼ 11.0, Jaa ¼ 44.9, Jbb ¼ 78.2, Jcc ¼ 89.5 [70]

CH3CN 1 RAS/HIII DJ ¼ 36.57, J ¼ 71.97c [324]

H2CyCyNH 1 EOM/6-31G** DJ ¼ 11.19, J ¼ 111.21c [407]

CH2CO 1 EOM/6-31G** DJ ¼ 212.74, J ¼ 112.43c [407]

OCyCyCO 1 EOM/6-31G** DJ ¼ 6.36, J ¼ 221.07 [407]


CH3NC 2 RAS/HIII DJ ¼ 11.64, J ¼ 25.23c [324]




unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz.a See footnote a in Table 13.b See footnote b in Table 13.c See footnote d in Table 13.d Over the triple bond.e Over the single bond.

Table 19

Quantum chemically calculated components of the symmetric part J1 þ JS of the 15N13C spin–spin coupling tensors

System Bonds Theorya Valuesb Reference

HCN 1 RAS/HIV DJ ¼ 254.64, J ¼ 219.83 [324]

HNC 1 RAS/HIV DJ ¼ 250.49, J ¼ 210.47 [324]

CH3CN 1 RAS/HIII DJ ¼ 250.70, J ¼ 221.55 [324]

CH3NC 1c RAS/HIII DJ ¼ 247.06, J ¼ 212.57 [324]

CH3NC 1d RAS/HIII DJ ¼ 217.04, J ¼ 219.26 [324]

CH3CN 2 RAS/HIII DJ ¼ 27.66, J ¼ 2.82 [324]

H2CyCyNH 1 EOM/6-31G** DJ ¼ 25.89, J ¼ 231.87e [407]

CH2N2 1 EOM/6-31G** DJ ¼ 26.92, J ¼ 228.54e [407]

HCNO 1 EOM/6-31G** DJ ¼ 29.16, J ¼ 259.16 [407]




unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz.a See footnote a in Table 13.b Results converted for 15N when necessary.c See footnote d in Table 18.d See footnote e in Table 18.e See footnote d in Table 13.


reported [324] in their PAS(J ). Thus, they should not

be directly compared to experimental data obtained by

LCNMR, as the latter refer to the frame used to

represent the orientation tensor. The calculations of

Ref. [324] nicely imply transferability of the proper-

ties of certain type of couplings between different

molecules containing similar structural units.

Kaski et al. [70] investigated the nJCH

ðn ¼ 1; 2; 3; 4Þ and nJCC ðn ¼ 1; 2; 3Þ coupling tensors

in benzene using CASSCF and RASSCF LR calcu-

lations. While the size of the molecule prohibited

reaching definite convergence of results as a function

of the size of the basis set (the standard HII and

modified triple-zeta sets were used) and particularly

the length of the determinantal expansion, conver-

gence of the calculated results towards the exper-

imental data could be established. In particular, the

FC and SD/FC terms in the tensor were seen to

dramatically decrease upon improving the correlation

treatment. The PSO term was finally left as the

dominant contributor to D1JCC: For the nJCC that

constituted the main objective, the experimental sign

patterns of both J and DJ were reproduced.

The magnitudes of most of the calculated parameters

are somewhat overestimated. Together with

the experimental findings, the results indicate that

Table 20

Quantum chemically calculated components of the symmetric part J1 þ JS of the 19F13C spin–spin coupling tensors


CH3F 1 RAS/HIII DJ ¼ 207.84, J ¼ 2156.56 [78]

CH2F2 1 RAS/HIII DJ ¼ 10.39, Jxx 2 Jyy ¼ 2280.33b, Jaa ¼ 236.62,

Jbb ¼ 2261.19, Jcc ¼ 2364.34

[78]

CHF3 1 RAS/HIII DJ ¼ 2173.34, Jaa ¼ 25.47, Jbb ¼ 2333.49, Jcc ¼ 2387.25 [78]

p-C6H4F2 1 RAS/HII(m ) DJ ¼ 368.8, Jxx 2 Jyy ¼ 11.5c, Jaa ¼ 61.1, Jbb ¼ 2301.9,

Jcc ¼ 2313.4

[60]


Jcc ¼ 78.8

[60]

p-C6H4F2 3 RAS/HII(m ) DJ ¼ 37.5, Jxx 2 Jyy ¼ 29.1c, Jaa ¼ 2.2, Jbb ¼ 223.5, Jcc ¼ 31.9 [60]


Jcc ¼ 30.8

[60]



unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz.a See footnote a in Table 13.b See footnote g in Table 13.c See footnote d in Table 16.

Table 21

Quantum chemically calculated components of the symmetric parts K1 þ KS of the X13C spin–spin coupling tensors other than those listed in

Tables 14 and 18–20

System X Bonds Theorya Values Reference

HCONH217O 1 RAS/HIII Kaa ¼ 237.3, Kbb ¼ 42.2, Kcc ¼ 2157 [404,405]

CH2CO 17O 1 EOM/6-31G** DK ¼ 101.1, K ¼ 276.88b [407]

OCyCyCO 17O 1 EOM/6-31G** DK ¼ 124.1, K ¼ 291.28 [407]

CH3SiH319Si 1 RAS/HIII DK ¼ 98.76, K ¼ 100.7 [80]




1019 T2 J21.a See footnote a in Table 13.b See footnote d in Table 13.


the ð1=2ÞJaniso contribution to DexpCC in aromatic systems

is in the range of 2% or less. The JCH tensors were

only calculated using a modest CASSCF wave

function, with the calculated 2JCH having the wrong

sign as compared to experiment and the other

parameters overestimated.

RASSCF LR calculations were reported for all the

coupling tensors in formamide (HCONH2) in

Refs. [404,405]. This is a biologically relevant model

molecule and displays a rich variety of NMR

observables. The active spaces were large; however,

the largest one was given only a single-reference wave

function treatment (due to computational limitations at

the time) with single and double excitations into

the virtual orbitals. Consequently, there may still be

room for improvement in the correlation treatment.

The HIII basis set was used, hence additional error

limits of a few % must be allowed due to the lack of

tight functions. The results for J compare well with

experiment apart from couplings to 17O for which

experimental results are not available. Experiments

[404,405] for the anisotropic observables could only

verify the qualitative features of the calculated data

due to the low experimental order parameters and

hence large uncertainty. It is likely that most of the

calculated anisotropic couplings are reliable, judging

also from the convergence of the results in the sequence

of improved wave functions. A possible exception is

formed by the 17O couplings. Using the experimental S

tensor obtained in the work, the calculated Janiso gives

a negligible contribution to Dexp:

The prototypical hydrocarbon series ethane, eth-

ene, and ethyne was studied at the RASSCF LR level

using large active spaces and the HIV (HIII for

ethane) basis set [72]. The goal was to investigate the

properties of 1JCC as a function of the hybridization of

the coupled carbons. Judging by the generally well

calculated isotropic J for all the couplings, the

anisotropic properties should also be of high quality.

Indeed, a qualitative agreement of the theoretical

D1JCC (and JCC;xx 2 JCC;yy for C2H4) with the results

of LCNMR experiments [72] was found. Both the

theoretical and experimental results point out,

together with the previous study on benzene [70],

that the tensorial properties of JCC may be neglected

in comparison with the direct coupling regardless of

the hybridization. The anisotropy along the CC bond

displays a minimum for the sp2-hybridized ethene,

despite the monotonically decreasing JCC from ethyne

to ethene and ethane. A SOPPA(CCSD) or a full

CCSD calculation could be used to verify this. The

different contributions to D1JCC evolve from the SD/

FC dominance in the sp3 carbons to the large PSO

term of the sp1 case.

Table 22

Quantum chemically calculated components of the symmetric parts K1 þ KS of the NX spin–spin coupling tensors other than those listed in

Tables 15 and 19


BH2NH211B 1 SOS-CI/(s ) DK ¼ 30.98, K ¼ 101.4b,c [325]

BH3NH311B 1 SOS-CI(s ) DK ¼ 20.7, K ¼ 18.2b,c [325]

CH2N215N 1 EOM/6-31G** DK ¼ 104.8, K ¼ 39.1b [407]

N2H415N 1 SOS-CI(s ) DK ¼ 91.33, K ¼ 4.4b,c,d [325]

HCNO 17O 1 EOM/6-31G** DK ¼ 124.4, K ¼ 2171.9 [407]

HCONH217O 2 RAS/HIII Kaa ¼ 30.6, Kbb ¼ 239.1, Kcc ¼ 41.6 [404,405]

PH2NH231P 1 SOS-CI(s ) DK ¼ 179.3, K ¼ 230.42b,c,e [325]




1019 T2 J21.a See footnote a in Table 13.b See footnote d in Table 13.c See footnote d in Table 15.d See footnote e in Table 15.e See footnote h in Table 15.


Kaski et al. studied CH3SiH3 using RASSCF LR

with a large active space [80]. The basis was the

standard HIII set, thus a priori restricting the accuracy

somewhat. The agreement with experiment is semi-

quantitative, with particularly 1JSiC and D1JSiC over-

and underestimated, respectively. This may be par-

tially due to neglecting correlation of the Si semicore

orbitals [240,257]. The neglect of 1JanisoSiC would

correspond to only a 1% error in the CSi bond length.

Ref. [60] investigated the couplings to 19F in p-

C6H4F2 using the RASSCF LR method. For systems

of this size, compromises in the basis set—a HII set

supplemented with tight s-primitives—and the corre-

lation treatment had to be made. The results generally

show a qualitative agreement, of signs and orders of

magnitude, as well as evolution of results when

improving the wave function, with the observed J.

The same applies for the anisotropic properties from

the LCNMR experiment, although the analysis of the

experimental data was not completely independent of

the calculation. The experimental 1JFC coupling

tensor would likely be particularly difficult to

reproduce theoretically. A calculation featuring a

more efficient electron correlation treatment, as well

as estimates of intermolecular and rovibrational

effects would be interesting. The contribution of

Janiso to the experimentally observable long-range3;4D

expFC and 5D

expFF couplings was estimated to exceed

Table 23

Quantum chemically calculated components of the symmetric parts K1 þ KS of the 19FX spin–spin coupling tensors other than those listed in

Tables 16 and 20


LiF 7Li 1 CAS/cc-pV5Z DK ¼ 240.25, K ¼ 45.28b [143]

BF 11B 1 CAS/cc-pV5Z DK ¼ 129.4, K ¼ 261.3 [143]

OF217O 1 RAS/cc-pCVQZ Kaa ¼ 84.1, Kbb ¼ 286.7, Kcc ¼ 607 [27]

ClF319F 2c RAS/cc-pVQZ Kaa ¼ 6.4, Kbb ¼ 24.4, Kcc ¼ 83.1 [27]

CH2F219F 2 RAS/HIII DK ¼ 224.64, Kxx 2 Kyy ¼ 213.11d, Kaa ¼ 16.11, Kbb ¼ 34.19,

Kcc ¼ 47.29

[78]

CHF319F 2 RAS/HIII DK ¼ 221.81, Kaa ¼ 214.07, Kbb ¼ 17.82, Kcc ¼ 39.21 [78]

p-C6H4F219F 5 RAS/HII(m ) DK ¼ 23.40, Kxx 2 Kyy ¼ 23.58e, Kaa ¼ 20.23, Kbb ¼ 1.37,

Kcc ¼ 4.95

[60]

NaF 23Na 1 CAS(s ) DK ¼ 165.1, K ¼ 64.8 [143]

AlF 27Al 1 CAS/aug-cc-pVQZ DK ¼ 188.4, K ¼ 2213 [143]

ClF 35Cl 1 CAS/aug-cc-pVQZ DK ¼ 2721, K ¼ 747f [143]

ClF 35Cl 1 ZORA DFT(GGA)/Slater DK ¼ 2982, K ¼ 872 [144]

ClF335Cl 1g RAS/cc-pVQZ Kaa ¼ 211.7, Kbb ¼ 298.3, Kcc ¼ 638 [27]

ClF335Cl 1h RAS/cc-pVQZ Kaa ¼ 83, Kbb ¼ 2167, Kcc ¼ 528 [27]

KF 39K 1 CAS/(s ) DK ¼ 207.3, K ¼ 148.0b [143]

BrF 79Br 1 ZORA DFT(GGA)/Slater DK ¼ 22123, K ¼ 1886 [144]

IF 127I 1 ZORA DFT(GGA)/Slater DK ¼ 22955, K ¼ 2241 [144]

TlF 205Tl 1 ZORA DFT(GGA)/Slater DK ¼ 2324, K ¼ 22034 [144]




1019 T2 J21.a See footnote a in Table 13.b Corrections carried out for the n ¼ 0, J ¼ 0 rovibrational state.c Coupling between equatorial and axial fluorine atoms.d See footnote g in Table 13.e See footnote d in Table 16.f Corrections carried out for the n ¼ 0, J ¼ 1 rovibrational state.g Coupling to the equatorial fluorine atom.h Coupling to the axial fluorine atom.


3%. This situation results from (1) the aromatic

system being able to convey the components of long-

range J and (2) the small value, due to the R23

dependence on the internuclear distance, of the

corresponding D.

Lantto et al. studied fluorine-substituted methanes

CH42nFn ðn ¼ 1; 2; 3Þ at the RASSCF level, using

large active spaces and HIII basis sets [78]. While

there are both correlation and basis set deficiency

errors remaining (the basis particularly lacking tight

functions and hence expected to be converged up to

ca. 5%), the quality of the tensorial properties of JHH;

JCH; JFC; and JFF is expected to be at least

semi-quantitative. The corresponding J values are

very satisfactory, contrary to prior calculations at the

DFT level [257,259]. The work, together with Ref.

[60] for p-C6H4F2, provides the first reliable compu-

tational estimates for the couplings to 19F in the

literature. In these systems, the need to calculate all of

the contributions to the tensors is particularly clear.

Especially JSDFF should not be neglected.

The question of the value, even the order of

magnitude, of D1JFC in CH3F has attracted a lot of

attention in the past (see Ref. [78] for some of the

references). It seems that the current theoretical value

of 208 Hz [78], has settled the issue. The contribution

of J to Dexp was found to be in the 1–1.5% range for1JFC and 2JFF in the systems studied.

Bryce and Wasylishen compared CASSCF calcu-

lations using medium-size active spaces and mostly

correlation-consistent basis sets, to molecular beam

spectroscopic data for light diatomic molecules

containing elements ranging from the alkali metals

to halogens [143]. This comparison is particularly

fruitful as the experimental J and DJ are practically

free from environmental effects. A qualitative agree-

ment with experiment was reached, and further

improvement may be sought both from larger active

spaces and basis sets that contain tight functions.

Contributions from the different coupling mechanisms

were reported for both K and DK; giving rise to

interesting preliminary trends for the two quantities

across the Periodic Table. Briefly, the magnitude of

the total K and DK; as well as the PSO and SD

contributions to DK; increase from left to right along a

given period in the Table. The magnitudes of KFC and

DKSD=FC follow the opposite trend. While the FC

contribution dominates in most (but not all, notably in

Table 24

Quantum chemically calculated components of the symmetric parts K1 þ KS of the XY spin–spin coupling tensors other than those listed in

Tables 13–23

System XY Bonds Theorya Values Reference

B2H411B11B 1 SOS-CI(s ) DK ¼ 19.92, K ¼ 72.19b [325]

B2H611B11B 1 SOS-CI(s ) DK ¼ 2.47, K ¼ 22.95b [325]

BH2PH211B31P 1 SOS-CI(s ) DK ¼ 79.88, K ¼ 77.36b [325]

BH3PH311B31P 1 SOS-CI(s ) DK ¼ 50.22, K ¼ 34.81b [325]

Na223Na23Na 1 CAS/Partridge DK ¼ 235.5, K ¼ 1480c [143]

KNa 23Na39K 1 CAS/Partridge DK ¼ 273.1, K ¼ 3230 [143]

P2H431P31P 1 SOS-CI(s ) DK ¼ 111.92, K ¼ 263.33b,d [325]

(CH3)3PSe 31P77Se 1 DFT(GGA)(s ) Kaa ¼ 2375, Kbb ¼ 21127, Kcc ¼ 21137 [130]

TlCl 35Cl205Tl 1 ZORA DFT(GGA)/Slater DK ¼ 2971, K ¼ 22185 [144]

TlBr 79Br205Tl 1 ZORA DFT(GGA)/Slater DK ¼ 5926, K ¼ 23153 [144]

TlI 127I205Tl 1 ZORA DFT(GGA)/Slater DK ¼ 8911, K ¼ 23818 [144]




1019 T2 J21.a See footnote a in Table 13.b See footnote d in Table 13 and footnote d in Table 15.c Corrections carried out for the n ¼ 0 vibrational state.d See footnote e in Table 15.


ClF) coupling constants, the dominance of DKSD=FC is

far less clear. The results underline the general

necessity to calculate all five coupling terms unless

substantial experience has been gathered for the type

of coupling and system under study.

Further interesting trends were depicted in Ref.

[143] concerning the change of K and DK down the

columns (groups) of the Periodic Table. Based on

calculations for the lighter systems and experimental

data for the heavier ones, the magnitudes of K and DK

were found to increase along the series XF (X ¼ B,

Al, In, Tl), TlX (X ¼ F, Cl, Br, I), and XF (X ¼ Cl,

Br, I). This is accompanied by a dramatic increase in

both DK=K and the indirect contribution to Dexp:

Bryce and Wasylishen carried out RASSCF

calculations using medium-size active spaces on

ClF3 and OF2 [27]. Comparison of the calculated

JFFðClF3Þ and JFOðOF2Þ with their respective exper-

imental counterparts was very successful, however

a discrepancy was observed for the weighted-average

JClFðClF3Þ: Besides the usual possible explanations

(neglect of relativistic effects, rovibrational correc-

tions and solvent modelling), the flexibility of the cc-

pVQZ basis set used for ClF3 may be inadequate in

the core region.

Other methods. In Ref. [130], 1JSeP in (CH3)3PSe

was investigated using the DFT/GGA method with the

FPT/SOS-DFPT ansatz, thus omitting the JSD term.

A fair agreement with the experimental solid-state

coupling constant was obtained, 2656 Hz (exp.) vs.

2820 Hz (calc.). The calculated principal values

were similarly overestimated as compared to the

experimental ranges given in the paper. While

separation of the possible model construction and

methodological errors (particularly the lack of

relativity) is difficult in this case, this level of

agreement is certainly useful already. This work

appears to be the first application of DFT to the

anisotropic properties of J.

The water molecule has been studied using the

SOPPA(CCSD) method and a large basis set [318].

The results of a small CASSCF calculation using the

same basis were quoted for comparison, as well as

SOPPA results originating from Ref. [280].

Although the latter were at a slightly different

molecular geometry, it appears that SOPPA is in this

case a better approximation to the apparently very

accurate SOPPA(CCSD) numbers than the basic

CASSCF wave function used [318]. Full tensors

were reported.

Sauer et al. reported high-accuracy SOP-

PA(CCSD) calculations using good, augmented

basis sets for the prototypical CH4 and SiH4 molecules

[321]. While both the 1JXH and 2JHH are dominated by

the FC mechanism, the anisotropic properties of the

couplings obtain relevant contributions from SD/FC,

PSO, and DSO mechanisms. It is noteworthy that

the D1K parameters have opposite signs in the two

systems. The K values are, after the rovibrational

treatment, in very good agreement with experiment.

Hence, the anisotropic properties are also most likely

reliable.

Wigglesworth et al. carried out SOPPA(CCSD)

calculations using large basis sets for all of the

coupling tensors in C2H2 [214,215]. While JFC

dominates the isotropic couplings, DJSD=FC is the

largest contribution only in JCH: The PSO mechan-

ism is very important for both D2JCC and D2JCH; and

DSO in D3JHH: The calculated J values are in good

agreement with experimental estimates for the

equilibrium geometry (see Table 11). The values

of DJ agree well with the earlier calculations of

Kaski et al. [72].

Relativistic effects. Visscher et al. compared non-

relativistic and fully relativistic (four-component)

SCF results for the HX (X ¼ F, Cl, Br, I) series of

molecules [335]. Relativistically optimized basis sets

for Br and I were used, as well as uncontracted

cc-pVTZ sets for the other elements. While the

uncorrelated method is as such inadequate for J

couplings, the results are indicative of the importance

of relativistic effects on the couplings to a heavy atom.

The conclusion is that relativity affects J significantly,

particularly for HBr and HI, whereas the effects on DJ

are smaller. There are opposite changes in the absolute

magnitude of DJ and J, increase and decrease,

respectively. The relative anisotropy DJ=J increases

for the lighter members of the series, but HBr and HI

feature a sign change in J. The effect of switching

from a point-like nuclear model to a Gaussian

distribution is ca. þ1% for JIH:

Vaara et al. investigated the H2X (X ¼ O, S, Se,

Te) and HX (X ¼ F, Cl, Br, I) systems at both non-

relativistic and spin–orbit corrected CASSCF levels,

using basis sets close to convergence [38]. The 1KXH

were modified by the SO-corrections towards


the experimental values (particularly for the H2X

series), but the correction is not large enough to

reproduce the experimental trend for the heavier

systems. Comparison with the findings of Refs. [335,

336] underlines the importance of the scalar

relativistic effects for couplings involving heavy

atoms. For 2JHH; the SO correction is probably

adequate, and quantitatively accurate values should

be sought by improving the electron correlation

treatment beyond the CASSCF level. The magnitude

of D1KXH increases due to the SO interaction, while

that of 1KXH decreases.

Autschbach and Ziegler applied their two-com-

ponent relativistic ZORA DFT method in Ref.

[144]. The agreement of 1KPbH with experiment at

the LDA level is excellent for PbH4, implying also a

good calculated value for the corresponding DK: In

the other systems studied in this work, GGA was

found to be superior to LDA. Qualitative agreement

was obtained with experiment for both the 1K and

D1K in the XF (X ¼ Cl, Br, I) series, with KPSO and

the total value overestimated for ClF as compared

to the MCSCF results of Ref. [143]. In the case of

IF, the authors demonstrated the effects of relativity

on the individual contributions. While scalar relati-

vistic effects increase both KPSO and lDKPSOl;the spin–orbit interaction seems to partially cancel

this effect. There is, similarly, a substantial effect of

the scalar relativity on the (small) KFC contribution,

and a very small opposing spin–orbit effect. These

conclusions cannot be generalized to other systems,

however, as exemplified by the TlX (X ¼ F, Cl, Br,

I) series. There, the inclusion of scalar relativity in

the model worsens the agreement of the calculated

result with the experimental data, and the large

spin–orbit contributions restore the qualitative

agreement. For these systems, the choice of either

LDA or GGA is irrelevant in comparison with the

effect of relativity. The total K and DK under- and

overestimate, respectively, their experimental

counterparts.

While the numbers calculated by the ZORA DFT

method are not in fully satisfactory agreement with

the experiment, the method has reached a useful

level of accuracy for systems that have previously

been beyond the reach of meaningful modelling. In

the prototypical hydrocarbon series C2H2, C2H4, and

C2H6, the values of 1KCC are qualitatively correct in

Ref. [267] but further removed from the experiment

than the non-relativistic RASSCF data of Ref. [72].

The calculated DFT DKCC as well as KCC decrease

monotonically in the series, in contrast to DKCC at

the MCSCF level [72], which has a minimum for

ethene.

3.8.2. Antisymmetric components

Table 25 displays the results for components of JA:

In general, only very few reports of JA exist,

although the antisymmetric components are available

from practically all of the programs in current use. For

consistency, the procedure of diagonalizing the J1 þ

JS part and expressing the components of JA in the

PAS(J ) frame should be adopted.

Few papers report the full tensors from which the

antisymmetric components can be extracted. Refs.

[40,41,406] reported SCF level calculations for 1J and2J in simple first- and second-row hydrides. As these

numbers do not contain any electron correlation

contribution, they should be used with caution.

More reliable SOPPA(CCSD) calculations were

carried out in Ref. [321] for the couplings in CH4

and SiH4. SOPPA, SOPPA(CCSD), and CASSCF LR

were compared for 2JHH using good basis sets for

water in Ref. [318]. In 1984, a SOPPA calculation

[228] was performed for H2O, but the antisymmetric

component is overestimated for 2JHH: For 2JHH; the

antisymmetric components are noted to be very small.

The value of 1JAOH is even less than the value of

2JAHH based on SOPPA, CASSCF LR, and SOP-

PA(CCSD) calculations [228,318]. The correspond-

ing terms for BH [41], NH [41], and PH couplings

[40], albeit from uncorrelated calculations, are only

slightly larger.

Whereas the antisymmetric components of the

couplings involving proton seem to be negligibly

small, the ab initio RASSCF work reported in Ref.

[27] on couplings possessing Cs local symmetry in

ClF3 and OF2 demonstrated similar order of

magnitude of the antisymmetric components to

the corresponding J. The same comments as before,

concerning the data on J1 þ JS in these systems,

apply here as well. The antisymmetry seems to

increase rapidly as heavier elements are involved.

The semi-empirical REXNMR results of Ref. [3]

(not tabulated) for H2Te2 point to the same

conclusion.


4. Conclusions

An effort has been made to summarize recent

progress in understanding indirect spin–spin coup-

ling, J; tensors. From an experimental point of view,

considerable progress has been realized in the three

main methods used in their characterization. The

importance of NMR of solute molecules in liquid

crystal solvents (LCNMR) emerges in cases where the

contribution from the J tensor to the experimental

anisotropic coupling Dexp is relatively small. Key

issues associated with this method are the contri-

butions of the vibrational motions and of the

correlation of the vibrational and reorientational

motions (the deformation effects) to the dipolar

couplings. After extensive theoretical studies in the

1980s, these contributions can be treated for systems

with small amplitude motions using existing computer

programs and available force fields. Proper treatment

of these contributions is important for two reasons:

first, the determination of accurate, solvent-indepen-

dent molecular structures and orientational order

parameters, and second, the separation of the often

minute indirect anisotropic contribution, ð1=2ÞJaniso;

from the corresponding Dexp: The method is restricted

by the fact that the determination of the complete

structure and orientation tensor requires a large

number of Dexp couplings in which ð1=2ÞJaniso is

Table 25

Quantum chemically calculated absolute values of the components of the antisymmetric part JA of spin–spin coupling tensors

System Coupling Bonds Theorya lValuel Reference

BH42 1H1H 2 SCF/(s ) lJxzl ¼ 1.66b [41]

CH41H1H 2 SOPPA(CCSD)/(s ) lJxzl ¼ 1.31b [321]

NH22 1H1H 2 SCF/(s ) lJxyl ¼ 4.77c [41]

NH4þ 1H1H 2 SCF/(s ) lJxzl ¼ 5.40b [41]

H2O 1H1H 2 SOPPA(CCSD)/(s ) lJxyl ¼ 1.62c [318]

H2O 1H1H 2 CAS/(s ) lJxyl ¼ 1.74c [318]

H2O 1H1H 2 SOPPA/(s ) lJxyl ¼ 1.71c [280,318]

H2O 1H1H 2 SOPPA/(s ) lJxyl ¼ 19.48c [228]

AlH42 1H1H 2 FOPPA/(s ) lJxzl ¼ 1.59b [406]

SiH41H1H 2 SOPPA(CCSD)/aug-cc-pVTZ lJxzl ¼ 3.66b [321]

SiH41H1H 2 FOPPA/(s ) lJxzl ¼ 1.75b [406]

PH22 1H1H 2 SCF/(s ) lJxyl ¼ 2.33c [40]

PH31H1H 2 SCF/(s ) lJxzl ¼ 0.33b, lJyzl ¼ 2.52d [40]

PH4þ 1H1H 2 SCF/(s ) lJxzl ¼ 2.17b [40]

NH22 15N1H 1 SCF/(s ) lJxyl ¼ 0.62b [41]

H2O 17O1H 1 SOPPA(CCSD)/(s ) lJxyl ¼ 0.21b [318]

H2O 17O1H 1 CAS/(s ) lJxyl ¼ 0.09b [318]

H2O 17O1H 1 SOPPA/(s ) lJxyl ¼ 0.19b [280,318]

H2O 17O1H 1 SOPPA/(s ) lJxyl ¼ 0.44b [228]

PH22 31P1H 1 SCF/(s ) lJxyl ¼ 2.35b [40]

PH331P1H 1 SCF/(s ) lJxyl ¼ 1.04b [40]

OF217O19F 1 RAS/cc-pCVQZ lJyzl ¼ 109b [27]

ClF319F19F 2e RAS/cc-pCVQZ lJxyl ¼ 292b [27]

ClF335Cl19F 1f RAS/cc-pVQZ lJxyl ¼ 150b [27]

Results in Hz.a See footnote a in Table 13.b Component in the local symmetry plane.c With the molecule in the xy plane and the C2 axis along the y direction.d Symmetry plane is xy.e See footnote c in Table 23.f See footnote h in Table 23.


safely negligible. Usually this means that couplings

involving at least one proton have to be used. On the

other hand, an increasing number of couplings result

in more difficult analyses of the spectra. Therefore,

LCNMR experiments performed in highly ordered LC

environments are restricted to spin systems with high

symmetry and at least a few hydrogen atoms.

As shown earlier, reliable data have been derived

by LCNMR, e.g. for the CC, NC, FC, SiC, and FF

spin–spin coupling tensors in hydrocarbons, fluoro-

methanes, methyl cyanides and methylsilane, as well

as fluorobenzene. In particular, the methodological

progress is exemplified in the classic question of the

value of D1JFC in monofluoromethane, where modern

LCNMR techniques have narrowed the range of

experimental results to 350–400 Hz from the unrea-

listic early results that are scattered over literally

thousands of Hz. JCC; JNC; and JFC are not only

valuable from the electronic structure calculation

point of view, but also because the corresponding Dexp

are increasingly used in studies of protein structure

and orientation in dilute liquid-crystalline solutions,

as well as in studies of the orientational behaviour of

liquid crystal molecules. In certain cases, ð1=2ÞJaniso

may even dominate Dexp and, hence, introduce large

uncertainty in the structural and orientational order

parameters.

The amount of data obtained from NMR measure-

ments on solid samples has increased enormously;

however, there are still problems associated with

correcting measured effective dipolar coupling ten-

sors for motional averaging. Although molecular

motion in solids is highly restricted, vibrations and

librations will lead to some averaging of the dipolar

interaction (typically 1–5%). Quantitative corrections

of the measured effective dipolar coupling constants,

Reff ; for such motion are difficult if not impossible.

Often researchers have failed to consider how

molecular motion might influence the anisotropic

spin–spin coupling constant data they report. There is

clearly a need for further single-crystal NMR data on

systems where the Reff are significantly different in

magnitude than the direct dipolar coupling constants,

RDD: Finally, one advantage of NMR investigations of

solids is that spin-pairs that involve quadrupolar

nuclei can be examined because quadrupolar nuclei

often have relatively long nuclear relaxation times in

the solid state compared to solution. In fact,

the presence of a quadrupolar nucleus can be critical

in characterizing J-tensors. In such systems, it is very

important to carry out measurements at more than one

applied magnetic field strength. Also, it is important to

recognize that the most reliable data will generally

result from systems where symmetry demands that the

electric field gradient tensor at the quadrupolar

nucleus is axially symmetric.

The availability of high-resolution molecular beam

data is very important as it provides highly accurate

and precise spin–spin coupling data on isolated

diatomics which serve as most suitable experimental

benchmarks for testing computational methods.

Particularly significant is the recent work of Ceder-

berg and co-workers where the vibrational depen-

dence of spin–spin coupling constants is measured to

a precision of better than 1 Hz. For example, in the

case of CsF, J133Cs19F ¼ 0:62745ð30Þ2 0:00903ð22Þ

£ ðnþ ð1=2ÞÞ kHz; with one standard deviation of

uncertainty estimates in the last two digits shown in

parentheses.

The development of quantum chemical methods,

their efficient implementation, and the rapid increase

of computer resources have revolutionized theoretical

calculations of J. For small molecules consisting of

light elements, the present ab initio methods are

approaching quantitative agreement with experiment.

Comparison of the experimental coupling constants

with the most accurate calculations still leaves room

for improvement in the latter. Regarding the rank-2

part of J, it is more difficult to assess the accuracy of

the theoretical calculations because the errors associ-

ated with the experimental values are larger than for

the isotropic part. Continued efforts in ab initio

calculations of J are very well motivated. System-

atically improving methods provide reliable bench-

marks for more approximate approaches.

Coupled cluster methods beyond CCSD are likely

to constitute one of the main directions where

progress can be expected. Parallelization and linear

scaling techniques would increase the range of

systems accessible to ab initio quantum chemical

methods.

For medium-size systems, the recent analytical

DFT implementations for J calculations are prom-

ising; however, further benchmarking studies are

still necessary. The available DFT exchange-corre-

lation functionals have not been parametrized for


hyperfine properties or transition metal systems.

Problems in the DFT performance are apparent

already in couplings to fluorine, as discussed in this

review. The long-term goal is to develop more

systematic functionals with less or no need for

empirical parametrization, as well as a better

understanding of the quantitative role of the current

dependence of the exchange-correlation functionals.

The anisotropic properties of J increase in signifi-

cance in systems containing heavier elements. There,

one has to resort to comparison with experimental

data when judging the accuracy of the practical

(ZORA) DFT method that both includes relativity

and is available for J. Correlated relativistic ab initio

methods for J at four- and two-component levels

would indeed be very desirable. The reliability of

DFT is nevertheless at the present time sufficient to

make qualitative conclusions of chemical trends and

to be of substantial assistance in steering the

direction of experimental work.

The roles of rovibrational averaging, intermolecu-

lar and solvation effects, as well as configurational

sampling in more complex systems, remain relatively

unexplored in the context of J tensors.

Acknowledgements

JV and JJ would like to thank Jaakko Kaski, Perttu

Lantto, Juhani Lounila, Kenneth Ruud, and Olav

Vahtras for research cooperation, and Henrik

Konschin for discussions (JV). REW and DLB

thank Prof. James Cederberg for rubidium fluoride

molecular beam data in advance of publication, and

the members of the solid-state NMR group of

the University of Alberta for valuable comments:

Kirk Feindel, Guy Bernard, Michelle Forgeron,

Kristopher Ooms, Kristopher Harris, Myrlene Gee,

Renee Siegel, Takahiro Ueda, and Se-Woung Oh. JV

is on leave from the NMR Research Group, Depart-

ment of Physical Sciences, University of Oulu,

Finland, and has been supported by The Academy

of Finland (grant 48578), the Magnus Ehrnrooth Fund

of the Finnish Society of Sciences and Letters, and the

Vilho, Yrjo, and Kalle Vaisala Foundation of the

Finnish Academy of Science and Letters. JJ is grateful

to The Academy of Finland for financial support

(grant 43979). The computational resources were

partially provided by the Center for Scientific

Computing, Espoo, Finland. REW thanks the Natural

Sciences and Engineering Research Council

(NSERC) of Canada for funding. REW holds a

Canada Research Chair in physical chemistry at the

University of Alberta. DLB thanks NSERC, Dalhou-

sie University, the Izaak Walton Killam Trust, and the

Walter C. Sumner Foundation for postgraduate

scholarships.

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Spin–spin coupling tensors as determined by experiment and computational chemistry

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