Download Spin–spin coupling tensors as determined by experiment and

Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
www.elsevier.com/locate/pnmrs
Spin –spin coupling tensors as determined by experiment
and computational chemistry
Juha Vaaraa,1, Jukka Jokisaarib,*, Roderick E. Wasylishenc,2, David L. Brycec,3
a
b
Department of Chemistry, P.O. Box 55 (A. I. Virtasen aukio 1), FIN-00014 University of Helsinki, Helsinki, Finland
NMR Research Group, Department of Physical Sciences, P.O. Box 3000, FIN-90014 University of Oulu, Oulu, Finland
c
Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada T6G 2G2
Accepted 2 September 2002
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1. Scope of the review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2. NMR spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3. Symmetry aspects and tensorial properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4. Nonrelativistic theory of the spin– spin coupling tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. High field approximation in NMR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. NMR in isotropic media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Liquid crystal NMR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2. Liquid crystal solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3. J tensor contribution to Dexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4. Vibration and deformation effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5. Limitations in the quantitative determination of J tensors. . . . . . . . . . . . . . . . . . . . . . . .
2.3.6. Qualitative determination of J aniso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.7. Results derived from LCNMR experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. Solid-State NMR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2. Solid-State NMR determination of J tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3. Results from single crystal studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4. Results from studies of stationary powders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.5. Results from spinning powder samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
234
234
234
237
237
240
240
241
242
242
242
244
246
247
248
250
254
254
255
256
258
260
* Corresponding author. Tel.: þ 358-8-553-1308; 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).
1
Tel.: þ358-9-191-50181; fax: þ358-9-191-50169.
2
Tel.: þ1-780-492-4336; fax: þ 1-780-492-8231.
3
Also at: Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada, B3H 4J3.
0079-6565/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 0 7 9 - 6 5 6 5 ( 0 2 ) 0 0 0 5 0 - X
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J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
2.5. High-resolution molecular beam spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6. NMR relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Quantum chemical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1. Correlated ab initio methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Density-functional theory methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Basis set requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4. Effects of nuclear motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5. Relativistic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6. Solvation and intermolecular forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7. Couplings for large systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8. Quantum chemical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.1. Symmetric components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.2. Antisymmetric components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
267
271
272
272
274
276
277
279
281
282
283
283
295
296
298
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
1 X
g I ·ð1 2 sM Þ ·B0
HNMR ¼ 2
2p M M M
X
þ
IM ·ðD0MN þ JMN Þ·IN :
ð1Þ
M,N
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) 233–304
IPPP
Nomenclature
B3LYP
CAS(SCF)
CC
CCSD
CCSDPPA
CCSDT
CCSD(T)
CDFT
CI
CISD
CLOPPA
CP
CPMAS
DFT
DHF
DNA
DSO
DZP
EFG
EOM
EOM-CC
FC
FCI
FOPPA
FPT
FWHH
GGA
GIAO
HFA
three-parameter Becke – Lee –
Yang– Parr
complete active space (self-consistent field)
coupled cluster
coupled cluster singles and
doubles
coupled cluster singles and
doubles polarization propagator
approximation
coupled cluster singles, doubles
and triples
coupled cluster singles, doubles
and non-iterative triples
current density-functional theory
configuration interaction
configuration interaction singles
and doubles
contributions from localized
orbitals within the polarization
propagator approximation
cross polarization
cross polarization magic-angle
spinning
density-functional theory
Dirac – Hartree – Fock
deoxyribonucleic acid
diamagnetic nuclear spin-electron orbit
double-zeta plus polarization
electric field gradient
equations-of-motion
equation-of-motion
coupled
cluster
Fermi contact
full configuration interaction
first-order polarization propagator approximation
finite perturbation theory
full width at half height
generalized gradient approximation
gauge-including atomic orbital
high-field approximation
LC
LCNMR
LDA
LR
MAS
MBER
MBMR
MCLR
MCSCF
MO
MP2
MQMAS
NMR
NOE
NQR
PAS
PES
PPA
PSO
QCISD(T)
RAS(SCF)
REX
RHF
RPA
SCF
SD
SOPPA
SOPPA(CCSD)
235
inner projections of the polarization propagator
liquid crystal
liquid crystal nuclear magnetic
resonance
local density approximation
linear response
magic-angle spinning
molecular beam electric resonance
molecular beam magnetic resonance
multiconfiguration self-consistent field linear response
multiconfiguration self-consistent field
molecular orbital
second-order Møller – Plesset
perturbation theory
multiple quantum magic-angle
spinning
nuclear magnetic resonance
nuclear Overhauser enhancement
nuclear quadrupole resonance
principal axis system
potential energy surface
polarization propagator approximation
paramagnetic nuclear spin-electron orbit
quadratic configuration interaction singles, doubles and noniterative triples
restricted active space (self-consistent field)
relativistic extended Hückel
restricted Hartree –Fock
random phase approximation
self-consistent field
spin dipole
second-order polarization propagator approximation
second-order polarization propagator approximation with
coupled cluster singles and
doubles amplitudes
236
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
SOS
SOS-CI
SOS-DFPT
SS
TEPS
sum-over-states
sum-over-states configuration
interaction
sum-over-states density-functional perturbation theory
solid state
tetraethyldiphosphine disulfide
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
HNMR ðB0 ; {IM }Þ
¼ kHðB0 ; E ; {ri }; {si }; {RM }; {IM }ÞlE;{ri };{si };{RM } ;
ð3Þ
where the effects of external electric fields E , 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
X
EmM ·mM þ
M
þ
X
M
mM ·EmM ;B0 ·B0 þ
X
1
B ·E
·B
2 0 B0 ;B0 0
TLC
TMPS
TZP
VAS
ZORA
thermotropic liquid crystal
tetramethyldiphosphine
disulfide
triple-zeta plus polarization
variable angle spinning
zeroth-order regular approximation
the hyperfine coupling tensor of the nucleus M. Both
properties vanish for a closed-shell system. EB0 ;B0 is
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 I
has 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 hyperfine 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
›2 EðmM ; B0 Þ sM;et ¼ det þ
:
ð5Þ
›mM;e ›B0;t m ¼0;B ¼0
M
mM ·EmM ;mN ·mN þ · · ·;
M,N
sM corresponds to the modification, caused by the
presence of the electron cloud, of the Zeeman
interaction of bare nuclei with B0 ;
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 EB0 and EmM are related to the
permanent magnetic moment of the molecule and
0
HZ ¼ 2
1X
1 X
mM ·B0 ¼ 2
g I ·B ;
h M
2p M M M 0
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).
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
237
The terms bilinear in IM and IN correspond to the
spin – spin coupling tensor,
The general symmetric and antisymmetric Cartesian
components of J are
JMN;et ¼ 2D0MN;et
S
JMN;
et ¼
þ
"
›2 EðmM ; mN Þ gM gN
:
2p
›mM;e ›mN;t mM ¼0;mN ¼0
ð7Þ
D0MN ¼ 2
" m0
3R R 2 1R2MN
gM gN MN MN
:
2p 4p
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 symmetric 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 ¼
2p 1
J
" gM gN MN;et
ð9Þ
to remove the parametric dependence on the magnetogyric 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
ðJMN;et þ JMN;te Þ 2 JMN det
ð12Þ
1
2
ðJMN;et 2 JMN;te Þ;
ð13Þ
and
A
JMN;
et ¼
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
P dipole – dipole interaction of
bare nuclei, HDD ¼ M,N IM ·D0MN ·IN ; where
1
2
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]:
S
S
JMN;
et ¼ JNM;et ;
A
A
JMN;
et ¼ 2JNM;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].
In general, J is described by a 3 £ 3 matrix,
expressable as a sum of zeroth-, first-, and secondrank tensors,
1.4. Nonrelativistic theory of the spin – spin
coupling tensor
JMN ¼ JMN 1 þ JAMN þ JSMN :
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
ð10Þ
The rank-0 contribution, J1, corresponds to the
isotropic spin –spin coupling constant,
JMN ¼
1
3
TrJMN ¼
1
3
ðJMN;xx þ JMN;yy þ JMN;zz Þ:
ð11Þ
238
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 structure 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 =ð4pe 0 "cÞ: The assumptions underlying 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 Oða2 Þ
quantum mechanical operators that contribute
through second-order expressions, and one Oða4 Þ
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 depending 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,
X 1ðriM ·riN Þ 2 riN riM
1
ð2Þ
HDSOðMNÞ
¼ a4 gM gN IM ·
·IN
3 3
2
riM
riN
i
X liM
;
3
i riM
IM £ riM
;
3
riM
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,
X
4p 2
ð1Þ
HFCðMÞ
¼
a ge gM
dðriM Þsi ·IM
ð18Þ
3
i
and
ð1Þ
HSDðMÞ
¼
2
X 3riM riM 2 1riM
1 2
a ge gM
si ·
·IM ;
5
2
riM
i
ð19Þ
respectively. Here, ge is the free electron gvalue for which the latest standard value is
2.0023193043737(82) [35] and dðriM Þ 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
1 ›2 EðIM;e ; IN;t Þ JMN;et ¼
2p ›IM;e ›IN;t I ¼I ¼0
M;e
N;t
ð15Þ
ð16Þ
The diamagnetic coupling is obtained from the
bilinear operator of Eq. (15) as a ground state
expectation value
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
4
AM ðri Þ ¼ a2 gM
SD
SD=FC
þ JMN;
et þ JMN;et :
ð20Þ
and
ð1Þ
¼ a2 gM IM ·
HPSOðMÞ
energy, including the contributions to the momentum
from the vector potential
In the a.u. system, the numerical values of the following
constants are equal to unity: "; e, me ; and 4pe 0 : Then, the speed of
light in vacuum, c ¼ 1=a and the permeability of vacuum m0 ¼
4pa2 ; in a.u.
¼
DSO
JMN;
et ¼
DSO
JMN;
et
þ
PSO
JMN;
et
þ
FC
JMN;
et
1 4
a gM gN k0l
4p
X det ðriM ·riN Þ 2 riN;e riM;t
£
l0l:
3 3
riM
riN
i
ð21Þ
The DSO terms generally contribute to the trace as well
as symmetric and antisymmetric parts of J. Typically,
J DSO is either numerically small in comparison with the
other contributions to the coupling constant, or
occasionally (in couplings involving hydrogen, particularly JHH ) largely cancelled by the PSO term to be
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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.
ð1Þ
For these, two first-order perturbations HM
¼ hM;e IM;e
ð1Þ 0
0
and HN ¼ hN;t IN;t give rise to a general term
JMN;et
contributes to all parts of the J tensor listed in Eq.
(10). J PSO is often the second-most important
contribution to the coupling constant after J FC : For
example, in systems where the valence s-electrons of
a certain atom contribute little to bonding, the J PSO
contribution to coupling involving this atom may
exceed J FC : This happens particularly in couplings to
the 19F nucleus as well as other halogen nuclei.
X k0lðhM;e IM;e þ h0N;t IN;t ÞlnlknlðhM;e IM;e þ h0N;t IN;t Þl0l 1
›2
¼
2p ›IM;e ›IN;t n–0
E0 2 En
I
¼
1 X k0lhM;e lnlknlh0N;t l0l þ k0lh0N;t lnlknlhM;e l0l
1
Rh ; h0 S :
;
2p n–0
2p M;e N;t 0
E 0 2 En
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 lnS l 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 Oða6 Þ;
however.
Like the DSO term, the PSO term
**
++
X liM;e X liN;t
1 4
PSO
JMN;et ¼
a gM gN
;
;
ð23Þ
3
3
2p
riM
i
i riN
0
239
M;e ¼IN;t ¼0
ð22Þ
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
ðt ¼ x; y; zÞ; corresponding to the
operators hPSOðNÞ
t
Cartesian components of the liM vector operator
in Eq. (16). The situation thus resembles that
encountered in the calculation of s, where firstorder wave functions with respect to the components 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 gaugeincluding atomic orbital (GIAO) ansatz in the
calculation of the spin –spin coupling tensors [31].
Fig. 1. Schematic illustration of the second-order processes
contributing to the nuclear spin–spin coupling interaction.
240
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
The Fermi contact term
!2
1 4p
FC
a4 g2e gM gN det
JMN;et ¼
2p 3
++
**
X
X
£
dðriM Þsi;e ; dðriN Þsi;e
i
i
ð24Þ
0
is isotropic, JFC ¼ 1J FC ; 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 dðrN Þsz ; to obtain J FC : This term poses
otherwise heavy computational requirements, however, 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
SD
JMN;
et
1 a4 2
gg g
¼
2p 4 e M N
2
X
X 3riM;n riM;e 2 den riM
£
si;n ;
5
riM
n¼x;y;z
i
S
0
DSO
PSO
SD
FC
JMN ¼ JMN
þ JMN
þ JMN
þ JMN
S
DSO;S
PSO;S
SD;S
SD=FC
JMN;
et ¼ JMN;et þ JMN;et þ JMN;et þ JMN;et
R
2
X 3riN;n riN;t 2 dtn riN
si;n
5
riN
i
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
SD=FC
nuclei. Interestingly, the separation of JMN
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 ;
ð27Þ
A
DSO;A
PSO;A
SD;A
JMN;
et ¼ JMN;et þ JMN;et þ JMN;et :
ð25Þ
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
1 4p a4 2
SD=FC
g g g
JMN;
et ¼
2p"*3* 2 e M N
++
2
X
X 3riN;e riN;t 2 det riN
dðriM Þsi;e ;
si;e
5
riN
i
i
0
**
++ #
2
X
X 3riM;t riM;e 2 det riM
si;t ; dðriN Þsi;t
þ
5
riM
i
i
0
ð26Þ
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 ; z0 Þ : B0 ¼ B0
z^ 0 : 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,
X
ENMR ¼ EZ þ
kmM lHs lmM l
þ
X
M,N
M
kmM mN lHD0 þ HJ lmM mN l;
ð28Þ
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
where the lmM l and lmM mN l 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
HFA
HNMR
¼2
B0 X
g I 0 þ Hs þ HD0 þ HJ ;
2p M M M;z
ð29Þ
and for the time average,
kTz0 z0 l ; T þ T aniso ¼
set ¼ 12 ð3 cos ue z0 cosutz0 2 det Þ:
The assumption
B X
Hs ¼ 0
g s 0 0I 0;
2p M M M;z z M;z
kset Tet l ¼ Set kTet l
and for the coupling interactions
X 0
1
HD0 ¼
DMN;z0 z0 IM;z0 IN;z0 2 ðIMþ IN2 þ IM2 INþ Þ
4
M,N
ð31Þ
X
1
HJ ¼
JMN;z0 z0 IM;z0 IN;z0 2 ðIMþ IN2 þ IM2 INþ Þ
4
M,N
X
3
þ
J ðI I þ I I Þ;
ð32Þ
4 M,N MN Mþ N2 M2 Nþ
where the tracelessness of D0 has been used. Eqs. (31)
and (32) involve the ladder operators for nuclear
spins,
IM2 ¼ IM;x0 2 iIM;y0 :
IMþ ¼ IM;x0 þ iIM;y0
et
where ue a is the angle between the e and a axes. It
then follows that
X
Tz0 z0 ¼ cos ue z0 cos utz0 Tet
et
¼
1X
3
e
Tee þ
2X1
3
et
2
ð3 cos ue z0 cos utz0 2 det ÞTet
ð35Þ
ð36Þ
ð37Þ
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 kset l ¼ 12 k3 cos ue z0 cos utz0 2 det l:
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,
T¼
ð33Þ
The spectral observables in the HFA correspond to the
time average of the components of the NMR tensors,
T ¼ s; D0 , and J, along the direction of the external
magnetic field, kTz0 z0 l: Transformation of T between
any two sets of Cartesian axes, ðe ; t; nÞ and ða; b;cÞ;
may be accomplished using
X
Tab ¼ cos ue a cos utb Tet ;
ð34Þ
1X
2X
kTee l þ
ks T l;
3 e
3 et et et
where e and t denote any of the molecule-fixed
coordinates ðx; y;zÞ; and
where the operator for the shielding interaction is
ð30Þ
241
1
3
TrT ¼
1
3
ðkTxx l þ kTyy l þ kTzz lÞ
ð40Þ
and
T aniso ¼
2
2X
S : kTl ¼
S kT l;
3
3 et et et
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 timeaveraged 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
242
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
spin Hamiltonian takes the form
X
B0 X
gM ð1 2 sM ÞIM;z0 þ
JMN IM ·IN :
2p M
M,N
iso
HNMR
¼2
ð42Þ
The spectral observables are reduced to their isotropic
parts, Eq. (40), the shielding constant
sM ¼
1
3
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 intermolecular dipole – dipole interactions vanish. On the other
hand, intramolecular dipolar interactions, the anisotropic 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]
B0 X
g ð1 2 sM 2 saniso
M ÞIM;z0
2p M M
X
X 1 aniso
þ
JMN IM ·IN þ
DMN þ JMN
2
M,N
M,N
LC
HNMR
¼2
ð3IM;z0 IN;z0 2 IM ·IN Þ;
ð44Þ
where
DMN ¼
1
2
0
aniso
DMN
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
*
+
m0 "gM gN
1
RDD ¼
8p2
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 molecules 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. However, in order to obtain reliable, solvent-independent 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 characterization 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
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
243
cases uniaxial, meaning that there exists cylindrical
symmetry around n, and that the directions n and
2 n 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]
1 X
Me ¼
x B;
ð47Þ
m0 t et t
where m0 is the permeability in vacuo and the xet are
the components of the diamagnetic (volume) susceptibility (magnetizability) tensor, xd ; which is diagonal
in the uniaxial phase.The energy density, rB ; due to
the magnetization can be represented as
ð B0
rB ¼ 2
M·dB
0
¼2
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 of
13
C6-benzene oriented in a liquid-crystalline solution.
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
5
In most applications described in this context, the LCs consist of
elongated molecules. Such LCs are called calamitic.
B20
2
xd þ Dxd P2 ðcos uBn Þ ;
3
2m0
ð48Þ
where xd ¼ 13 Trxd is the isotropic diamagnetic
susceptibility, Dxd is the anisotropy of the
susceptibility tensor, and P2 ðcos uBn Þ ¼ 12 ð3
cos2 uBn 21Þ is the second-order Legendre polynomial (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. 3. Molecular orientational order in a nematic phase. Reprinted
with permission from Ref. [55]. Copyright (1998) Wiley–VCH.
244
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
that xd , 0 always):
1. when Dxd . 0; rB reaches a minimum with
P2 ðcos uBn Þ 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 P2 ðcos
uBn Þ 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 anisotropic coupling,
Dexp
MN ¼ DMN þ
1 aniso
2 JMN ;
ð49Þ
includes a contribution from the J tensor. From Eq.
(41), this contribution can be represented as
aniso
JMN
¼
X
2
P2 ðcos uBn Þ SD
et JMN;et ;
3
et
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 laboratoryfixed axis system is ðx0 ; y0 ; z0 Þ where B0 ¼ ð0; 0; B0 Þ: The moleculefixed axis system is ðx; y; zÞ:
In the most general case, Eq. (50) can be written as
nh
i
aniso
JMN
¼ 23 JMN;zz 2 12 ðJMN;xx þ JMN;yy Þ SD
zz
D
þ 12 ðJMN;xx 2 JMN;yy ÞðSD
xx 2 Syy Þ
ð50Þ
where JMN;et is the component of J in the
molecule-fixed coordinate system. SD
et in turn is
the component of the Saupe orientational order
tensor (with respect to n) [44] as defined in
Eq. (39). The factor P2 ðcos uBn Þ 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 orientational 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.
S
SD
þ2JMN;xy
xy
þ
S
2JMN;xz
SD
xz
þ
P2 ðcos uBn Þ:
S
2JMN;yz
SD
yz
ð51Þ
The factor
DJMN ¼ JMN;zz 2
1
2
ðJMN;xx þ JMN;yy Þ
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
h
i
aniso
D
D
1
JMN
¼ 23 DJMN SD
zz þ 2 ðJMN;xx 2 JMN;yy ÞðSxx 2 Syy Þ
£ P2 ðcos uBn Þ
ð53Þ
and for molecules with at least a 3-fold symmetry axis
to an even simpler form
aniso
¼
JMN
2
3
DJMN SD
zz P2 ðcos uBn Þ:
ð54Þ
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
245
Table 1
Independent second-rank orientational order parameters for molecules of various symmetry in uniaxial liquid crystal phases (From
Ref. [57])
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]
Molecular
point group
Saupe
tensor
component
Wigner matrix
element
Szz
kD20;0 l
Sxx 2 Syy
3 1=2
Syz
i 38 1=2 ðkD20;2 l 2 kD20;22 lÞ
3 1=2
Cxy
20
Sxz
2 38 1=2 ðkD20;1 l þ kD20;21 lÞ
3 1=2
Cxz
20
Syz
i 38 1=2 ðkD20;1 l þ kD20;21 lÞ
3 1=2
Cyz
20
C1, Ci
Cs, C2, C2h
C2v, D2, D2h
C3, S6
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
Number of
independent
order
parameters
5
3
2
1
Order parameters
Szz, Sxx 2 Syy, Sxy, Sxz, Syz
Szz, Sxx 2 Syy, Sxy
Szz, Sxx 2 Syy
Szz
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
n
aniso 1
2
JMN
¼ 3 Dc JMN SD
zz ½ð3cos ucz 21Þ
þhc ðcos2 uaz 2cos2 ubz Þ
h
D
2
2
1
þðSD
2S
Þ
xx
yy cos ucx 2cos ucy þ 3 hc
io
ðcos2 uax 2cos2 uay 2cos2 ubx þcos2 uby Þ
P2 ðcosuBn Þ
ð55Þ
and
aniso 1
2
D
D
¼ 3 Dc JMN ½SD
JMN
zz ð3cos ucz 21ÞþðSxx 2Syy Þ
ðcos2 ucx 2cos2 ucy ÞP2 ðcosuBn Þ;
ð56Þ
respectively. In Eqs. (55) and (56),
Dc JMN ¼JMN;cc 2 12 ðJMN;bb þJMN;aa Þ
2
ðkD20;2 l þ kD20;22 lÞ
Snyder
motional
constant
1 1=2
5
3 1=2
5
C3z2 2r2
Cx2 2y2
is the J tensor anisotropy in its PAS, and
3 JMN;aa 2JMN;bb
hc ¼
2
Dc JMN
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 independent order parameters, S33 and S11 2 S22 : Consequently, 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
2
DMN ¼ 2 12 FMN ½SD
33 ð3 cos uMN;3 2 1Þ
D
2
2
þ ðSD
11 2 S22 Þðcos uMN;1 2 cos uMN;2 Þ
ð57Þ
P2 ðcos uBn Þ
ð59Þ
246
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
and
aniso
¼
JMN
1
D3 JMN SD
33 þ 2 ðJMN;11 2 JMN;22 Þ
i
D
ðSD
11 2 S22 Þ P2 ðcos uBn Þ:
2
3
factor is omitted. Similarly,
h
In Eq. (59), the factor F is defined as
*
+
m0 "gM gN
1
FMN ¼
¼ RDD
MN :
8p2
R3MN
aniso
S
¼2SD
JMN
33 ½3D3 JMN þ h ðJMN;11 2JMN;22 ÞP2 ðcosuBn Þ:
ð61Þ
One should note that uMN;a ða ¼ 1; 2;3Þ is now the angle
between the RMN vector and the a axis of PAS(S ),
D3 JMN 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 J aniso possess
similar dependence on the orientational order parameters they generally do not vanish under the same
conditions; the zero condition for the former is
3 cos2 uMN;3 2 1
S11 2 S22
¼2 2
;
S33
cos uMN;1 2 cos2 uMN;2
ð62Þ
i.e. determined by the molecular geometry alone,
whereas J aniso vanishes when
S11 2 S22
22D3 JMN
¼
:
S33
JMN;11 2 JMN;22
ð65Þ
ð60Þ
ð63Þ
with S33 – 0: Thus, the experimentally determined
Dexp ¼ D þ ð1=2ÞJ aniso 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 biomacromolecules partially oriented in dilute LC solutions,
Eq. (59) is generally written in the form [61]
h
2
DMN ¼ 2 12 FMN SD
33 ð3 cos a 2 1Þ
i
ð64Þ
þ 32 hS3 sin2 a cos 2b P2 ðcosuBn Þ;
D
D
where hS3 ¼ ðSD
is the asymmetry and
11 2 S22 Þ=D3 S
D
D
D
D
D3 S ¼ S33 2 1=2ðS11 þ S22 Þ ¼ ð3=2ÞSD
33 the anisotropy
of the S tensor, and the angles a ¼ 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
Thus, in the situation where DMN ¼0;
"
2 3 cos2 a 2 1
aniso
D
JMN ¼ 2S33 3D3 JMN 2
3 sin2 acos2b
#
ðJMN;11 2 JMN;22 Þ P2 ðcosuBn Þ
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
Sýkora 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
23
separation ksMN R23
MN l ¼ SMN kRMN l; Eq. (38), is not
strictly valid. Here, sMN ¼ P2 ðcosuMN;B Þ is the
component of set ; Eq. (37), along RMN : Then,
the dipole – dipole coupling must be represented as
*
+
m0 "gM gN sMN
P2 ðcosuBn Þ:
ð67Þ
DMN ¼ 2
8p 2
R3MN
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
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
and deformation contributions to the dipole – dipole
couplings. For small-amplitude motion, the various
contributions to the dipolar coupling can be
separated as follows:
ah
h
d
DMN ¼ Deq
MN þ DMN þ DMN þ DMN :
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
eq
ah
h
d
Dexp
MN ¼ DMN þ DMN þ DMN þ DMN þ
1 aniso
2 JMN :
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 reference [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 deformations 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 J aniso ; 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 underdetermined. Occasionally this can be overcome by
carrying out experiments in several LC solvents
247
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 J aniso 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 parameters 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 internuclear 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=2ÞJ aniso (as compared to D ) is usually valid for X1H
(X ¼ 1H, 13C, 14N, 15N, 19F, etc.) couplings.
Uncertainty in the determination of J aniso 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 l2Dexp
IS þ JIS l: 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 J aniso as well. In principle, JIS
can be determined in LC phases by performing,
e.g. variable angle spinning (VAS) experiments [73].
The anisotropic contribution, J aniso ; for molecules
with high symmetry depends exclusively on the
248
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 3
Various contributions (in Hz) to D exp in benzene
Couplinga
D eq
Dh
Dd
D calc
D exp
3
2701.16
2134.94
287.65
22108.96
2269.33
268.43
246.20
2248.79
247.88
231.10
9.78
0.98
0.40
158.55
5.08
0.48
0.18
1.83
0.03
20.05
21.99
20.27
20.14
214.23
20.58
20.08
20.01
20.44
0.05
0.08
2693.37
2134.23
287.39
21964.64
2264.83
268.03
246.02
2247.40
247.80
231.07
2693.368(7)
2134.220(9)
287.417(12)
21964.637(14)
2264.790(9)
268.131(9)
246.022(12)
2248.217(21)
247.569(20)
231.613(32)
Coupling ratioc
Experimental valueb
Theoretical value
DoHH =Dm
HH
DoHH =DpHH
m
DHH =DpHH
DoCC =Dm
CC
DoCC =DpCC
p
Dm
CC =DCC
5.166(5)
7.932(5)
1.535(6)
5.218(4)
7.852(11)
1.505(2)
5.1962
8.0000
1.5396
5.1962
8.0000
1.5396
DHH
DHH
5
DHH
1
DCH
2
DCH
3
DCH
4
DCH
1
DCC
2
DCC
3
DCC
4
b
aniso
The difference between D calc and D exp for the CC couplings is due to ð1=2ÞJCC
[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.
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 symmetric 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
Syy Þ=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 J aniso
Molecular symmetry can constrain the ratios of D
couplings. This can be used to reveal whether Dexp
includes a significant contribution from ð1=2ÞJ aniso
exp
or not. If the examination of the ratio Dexp
MN =DOP
reveals a deviation from the corresponding ratio of
direct dipolar couplings (D ¼ Dexp 2 ð1=2ÞJ aniso ¼
Deq þ Dah þ Dh þ Dd ; see Eq. (69)), the couplings
(or at least some of them) may be affected by
ð1=2ÞJ aniso :
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)
Dexp
gM gN kR23
MN l
MN
exp ¼
gO gP kR23
DOP
OP l
"
!#
21
21
1
kR23
kR23
MN l
OP l
DJMN
£ 12
2 DJOP
3
F 0MN
F 0OP
ð72Þ
where F 0MN ¼ FMN kR23
MN l: The expression in the square
brackets equals 1, i.e. the ratio of the D couplings
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 4
Liquid crystals used in studies of J tensors by their code name and
composition
Code name
Composition
EBBA
HAB
Phase 4
N-( p-ethoxybenzylidine)-p-n-butylaniline
p,p0 -di-n-heptylazoxybenzene
Eutectic mixture of p-methoxy-p0 -nbutylazoxybenzenes
Mixture of Phase 4 and p-ethoxy-p0 -nbutylazoxybenzenes
Mixture of phenylcyclohexanes,
biphenylcyclohexane and phenylcyclohexane
esters
Mixture of azoxy compounds and a biphenyl ester
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%)
Mixture of 4-n-trans,trans-bicyclohexyl-40
-carbonitrile (36%), 4-n-propyl-trans,transbicyclohexyl-40 -carbonitrile (34%), 4-n-heptyl
-trans,trans-bicyclohexyl-40 -carbonitrile (30%)
Mixture of alkylphenylcyclohexanes,
alkylcyclohexanebiphenyls, and
bicyclohexanebiphenyls
Mixture of alkylbicyclohexanes and
alkyltricyclohexanes
Phase 5
Phase 1221
ZLI 997
ZLI 1132
ZLI 1167
ZLI 1982
ZLI 2806
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 =
23
DJOP ¼ ðgM gN =gO gP ÞkR23
MN l=kROP l: For the cases
M;N;O;P all equal to either 1H or 13C, Eq. (72)
reduces to the form
Dexp
kR23
MN l
MN
exp ¼
DOP
kR23
OP l
1
23
23
1 2 0 ðDJMN kRMN l 2 DJOP kROP lÞ ; ð73Þ
3F
where F 0 ¼ F 0MN ¼ F 0OP and the condition (2) above
23
becomes DJMN =DJOP ¼ kR23
OP l=kRMN l:
It follows from pthe
ffiffi hexagonal symmetry of
benzene that Rm ¼ 3Ro 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
249
pffiffi
isotopic species, Do : Dm : Dp ¼ 1 : 3=9 : 1=8 < 1 :
0:1925 : 0:1250: It has been found that the 1H1H
couplings in benzene indeed fulfill these ratios
aniso
and thus the JHH
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 provided 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]:
8
2
1
3 <
g 3
exp
DJCC ¼
pCC DCC 2 42 H
Szz :
gC
2 1
3
1 2
D
2 pCH 2 Dexp
2
J
S
CH zz
CH
3
323 9
2
=
1
gH 3 exp 2 3 5
2
pHH DHH
;
gC
ð74Þ
where the anisotropy of JHH is assumed to be
negligible, and pMN ¼ 1 þ DhMN =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 experimental coupling, i.e. for D ¼ Dexp 2 ð1=2ÞJ aniso :
Eq. (74) can be approximated by a linear equation
DJCC ¼ AD2 JCH þ B
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], (D2 JCH ¼ 28:2 Hz,
DJCC ¼ 47:5 Hz), is relatively close to the abovementioned straight line. Consequently, this finding
can be regarded as the first experimental evidence
of the anisotropy of a 13C1H spin – spin coupling
250
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 1 pCH :
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
Dexp
gC 1 2 13 DJSeC R3SeC =F 0SeC
SeC
exp ¼ 8
gSe 1 2 83 DJSeSe R3SeC =F 0SeSe
DSeSe
ð76Þ
(a series expansion cannot be applied here because DJ
may be relatively large) where the numerical values of
F 0SeC and F 0SeSe are 5780:3 and 4398:6 Å3 Hz, respectively. 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 Dexp
SeSe
coupling cannot be determined from the experimental
spectra without knowing DJSeSe : The coupling Dexp
SeC
can be deduced to be negative on the basis of good
agreement between experimental and calculated
nuclear shielding tensor anisotropies, DsC and DsSe ;
Dexp
SeC 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 Hückel
(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 yield þ330 and þ1330 Hz for
DJSeC and DJSeSe ; respectively [8]. It is evident that the
aniso
anisotropic contribution ð1=2ÞJSeSe
¼ ð1=3ÞDJSeSe Szz
exp
dominates in DSeSe 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 J aniso 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 symmetry (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
aniso
contribution ð1=2ÞJXH
to Dexp
XH can generally be
neglected, and consequently the determination of
molecular structure and orientational order parameters 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
Dexp
XC couplings are used to determine orientational
order parameters of LC molecules [82 –85] and
biomacromolecules dissolved in dilute liquidcrystalline 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 Dexp
XC or DXC : As
pointed out above, D and ð1=2ÞJ aniso do not vanish
simultaneously, and therefore in certain circumstances 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 the
77
Se13C and 77Se77Se coupling tensors in carbon
diselenide [8] and the 199Hg13C coupling tensor in
dimethyl mercury [81]. Earlier on, results for
111,113
Cd13C [86], 29Si13C and 119Sn13C [87] and
77
31
Se P [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 :
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
251
Table 5
Indirect spin–spin coupling tensors as determined experimentally by LCNMR
Coupling and molecule
a
Results and comments
Reference
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.
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.
[72]
13 1
C H coupling
Ethyne C2H2
13 13
C C coupling
Acetonitrile CH3CN
Benzene C6H6
Ethane C2H6
Ethene C2H4
DJCC values of 115 ^ 24 and 112 ^ 25 Hz were determined for acetonitrile dissolved in
[75]
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.
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.
[71]
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
kDJCC l ¼ 30 ^ 33 Hz:
For the JCC, see above.
[70]
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:
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:
1
JCC: 55.811 ^ 0.004…55.98 ^ 0.01 Hz
2
JCC: 22.519 ^ 0.009… 2 2.434 ^ 0.007 Hz
3
JCC: 10.090 ^ 0.006…10.12 ^ 0.02 Hz.
DJCC was determined separately in five LC solvents (ZLI 1167, ZLI 1982 and three mixtures
[72]
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.
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.
Due to the D2h point group symmetry of ethene, two order parameters, Szz and Sxx 2 Syy, are
[72]
aniso
; is given by
needed to describe its orientation. Therefore, the anisotropic contribution, JCC
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):
kDJCC l ¼ 11 Hz; and kJCC;xx 2 JCC;yy l 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.
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)
252
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 5 (continued)
a
Coupling and molecule
Results and comments
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.
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/ethyne
[72]
A similar procedure as described above in determination of DJCC was applied to the
determination of D2 J15 NC The analysis resulted in D2 J15 NC ¼ 218 ^ 7 Hz:
A value of 2.9 ^ 0.2 Hz was measured for D2 J15 NC in CDCl3.
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 Dexp
NC (altogether 10 couplings and four parameters) led to the following results:
kDJC14 N l ¼ 8:7 ^ 1:7 Hz; and kDJ14 NC l 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.
JC14 N ¼ 7:63 ^ 0:04 Hz and J14 NC ¼ 6:30 ^ 0:09 Hz when determined in a CDCl3 solution.
[71]
14,15
N13C coupling
Acetonitrile CH3CN
Methylisocyanide CH3NC
19 13
F C coupling
Difluoromethane CH2F2
Fluoromethane CH3F
p-Difluorobenzene
p-C6H4F2
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
aniso
; in principle allow the determination of
consequently the anisotropic contributions, JFC
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.
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 þ 689.8 ^ 62.5 Hz in the individual LCs.
Performing a joint fit to 44 couplings corrected for both harmonic vibrations and
aniso
; leads to
deformation effects, and taking into account the deformation contribution to JFC
kDJFC l ¼ 404 ^ 31 Hz:
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 kDJFC l 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 kDJFC l ¼ 350 Hz:
JFC ¼ 2161.30 ^ 0.04 Hz in the isotropic state of the LC.
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 Dexp
CH of methane is vanishingly small). Only
aniso
harmonic vibrations were considered. Due to the C2v symmetry of the molecule, the n JFC
n
n
n
depend on both D JFC and JFC,xx 2 JFC,yy, see Eq. (53).
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 Jyy
3
JFC
8 ^ 9 Hz
25 ^ 10 Hz
4
JFC
111 ^ 17 Hz
2130 ^ 18 Hz
The z axis of the molecule-fixed frame lies along the CF bond.
[75]
[76]
[77]
[78]
[69]
[78]
[79]
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
253
Table 5 (continued)
Coupling and molecule
29
Si13C coupling
Methylsilane CH3SiH3
Se13C coupling
Carbon diselenide CSe2
a
Results and comments
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.
The experiments did not yield enough independent information to resolve the two tensorial
properties (see above), and therefore the ratio DnJFC/(nJFC,xx 2 nJFC,yy) was fixed to the
value given by the ab initio calculation.
The results are shown below:
Coupling tensor
DJ
Jxx 2 Jyy
1
JFC
400 ^ 90
13 ^ 3
2
JFC
239 ^ 2
220.5 ^ 1.1
3
JFC
17.6 ^ 0.2
13.7 ^ 0.1
4
JFC
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.
[60]
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]
exp
The ratio of the experimental anisotropic couplings, lDexp
SeC =DSeSe l 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.
JSeC ¼ 2226.59 ^ 0.36 Hz in CDCl3 solution and 2226 ^ 6 Hz in the isotropic state of
the ZLI 1132 LC.
[8]
77
199
Hg13C coupling
Dimethylmercury (CH3)2Hg The DJHgC ranges from 655 ^ 56 to 864 ^ 15 Hz when determined in four LCs (ZLI 1167,
[81]
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.
JHgC varies from 690.3 to 693.8 Hz in the isotropic state of the LCs used.
19 19
F F coupling
In this particular case, the deformation contribution to 1 Dexp
[78]
Trifluoromethane CHF3
CH 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.
The resulting D2JFF is 2200 Hz.
2
JFF could not be determined experimentally because of the chemical equivalence of the 19F
nuclei.
For details, see 19F13C coupling/p-difluorobenzene
p-Difluorobenzene
p-C6H4F2
Coupling tensor
DJ
Jxx 2 Jyy
5
JFF
230 ^ 15 Hz
236 ^ 15 Hz
[79]
236.5 ^ 0.5 Hz
238.4 ^ 0.5 Hz
[60]
5
JFF ¼ 17.445 Hz in the isotropic phase of the ZLI 997 (32.1 wt%)/ZLI 1982 (67.9 wt%) LC
mixture.
(continued on next page)
254
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 5 (continued)
Coupling and molecule
Se77Se coupling
Carbon diselenide CSe2
a
Results and comments
Reference
For the details, see 77Se13C coupling/carbon diselenide and text.
Depending upon the sign of the ratio of the D exp, DJSeSe is either 2654 or þ 1212 Hz. Both
aniso
contributions that are much larger than
of these are so large that they lead to the ð1=2ÞJSeSe
.
J
the purely dipolar contribution DSeSe in Dexp
SeSe was not determined experimentally
SeSe
because of the magnetic equivalence of the 77Se nuclei
[8]
77
a
For the code names and composition of LC solvents, see Table 4.
2.4. Solid-State NMR spectroscopy
2.4.1. Introduction
In the case of crystalline solids, the constituent
molecules are oriented in certain directions determined 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 corresponding 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 Tz0 z0 as in isotropic
media and LCs. The Hamiltonian corresponding to
spin-1=2 nuclei in the solid state thus takes the
SS
HFA
¼ HNMR
:
form specified in Eqs. (29) – (32), HNMR
The solid-state NMR observables vary depending 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 throughout 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
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.
goniometer whose rotation axis, z00 in the goniometerfixed frame ðx00 ; y00 ; z00 Þ; 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
Tz0 z0 ¼ c0 þ c1 cosw þ s1 sin w þ c2 cos 2w þ s2 sin 2w;
ð77Þ
where w is the turn angle of the goniometer and
c0 ¼
1
2
ðTx00 x00 þ Ty00 y00 Þsin2 u þ Tz00 z00 cos2 u;
c1 ¼ TxS00 z00 sin 2u;
c2 ¼
1
2
s1 ¼ TyS00 z00 sin 2u;
ðTx00 x00 2 Ty00 y00 Þsin2 u;
ð78Þ
s2 ¼ TxS00 y00 sin2 u:
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 TzS0 z0 ¼ sin2 u cos2 wTaa þ sin2 u sin2 wTbb
þ cos2 uTcc ;
ð79Þ
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
255
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 solidstate literature it is customary to define the experimentally measured dipolar coupling constant as
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 ; z00 Þ and the external magnetic field B0 : The laboratoryand molecule-fixed frames are as in Fig. 4.
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, Dc T; 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
Reff ¼ RDD 2
DJ
3
ð80Þ
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 rovibrational effects for diatomic molecules in the gas
phase, given the availability of high-quality experimental data. Similarly, more complicated corrections 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
256
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
that we report in the solid-state context are defined
as Dc J ¼ Jk 2 J’ ; or in the more general case,
Dc J ¼ Jcc 2 ðJaa þ Jbb Þ=2: Some literature uses a
so-called reduced anisotropy dJ ¼ ð2=3ÞDc J:
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 experimental 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 particularly 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 1 JPP : A 31P NMR study of a single
crystal of the related compound, tetramethyldiphosphine 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 DJ199 Hg31 P ; on
the order of 4 –5 kHz, provide conclusive evidence for
non-Fermi contact coupling mechanisms. It is possible 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
207
Pb19F coupling
PbF2 (cubic)
Results and comments
Reference
DcJ ¼ 8130 ^ 300 Hz (preferred) or 210 ^ 300 Hz.
J ¼ ^(2150 ^ 50) Hz (negative sign preferred).
[95]
DcJ ¼ 462 Hz (preferred) or 10362 Hz.
J is axially symmetric within experimental error.
DcJ # 450 Hz
[92]
DcJ ¼ 1220 ^ 75 or 2600 ^ 75 Hz
J ¼ ^(225 ^ 10) Hz
[97]
J ¼ ^(170 ^ 40) Hz
B(pseudodipolar) ¼ 2300 ^ 70 or 2990 ^ 70 Hz
[98]
DcJ ¼ 5404 ^ 150 Hz (site 1) and 5385 ^ 150 Hz (site 2).
J ¼ 8199 ^ 25 Hz
J is axially symmetric within experimental error, with the unique
component coincident with the unique component of D0
DcJ ¼ 4000 ^ 500 Hz, J ¼ 5550 Hz
J is axially symmetric within experimental error, with the unique
component coincident with the unique component of D0 .
[99]
31 31
P P coupling
Tetraethyldiphosphine disulfide
Tetramethyldiphosphine disulfide
113,115
[96]
31
In P coupling
InP
115
In31P coupling
InP
199
Hg31P coupling
HgPCy3(NO3)2 (Cy ¼ cyclohexyl)
Hg(PPh3)2(NO3)2
[100]
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 orientations, 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 direct
115 31
In P 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 J115 In31 P :
Tomaselli et al., also carried out a study of
J113;115 In31 P 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 115 In nuclei. The value of
J113;115 In31 P ; ^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;115 In31 P 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 crosspolarization 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;115 In31 P 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 convincing results. Furthermore, it is one of the most reliable
determinations of DJ for the case where lReff l is not
larger than lRDD l:
The problems that arise in the interpretation of
spin – spin coupling tensors in situations where the D0
257
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 þ 1 sideband matching condition. The fit to these
data corresponds to lJ113 In31 P l 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.
and J tensors are non-coincident have been discussed
for MAS powdered samples (vide infra). For experiments 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
258
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 RDD
19 F13 C by about 3% for motional
averaging, a value of DJ19 F13 C 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, nonzero 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)phosphaethyne 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-butylphenyl)-1,1-bis(trimethylsilyl)phosphaethene, an
assumption of negligible anisotropy in J results in
an NMR-derived bond length of 1.72 Å, which is only
3% longer than the X-ray value, 1.665 Å [105].
Motional averaging is known to account for differences 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; 2 1380 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 DJ199 Hg31 P are known
from 31P NMR spectroscopy of stationary powder
samples [113 –115]. For the series [HgPR3(NO3)2]2
(see Table 7), J199 Hg31 P ranges from 8008 to 10566 Hz,
and DJ199 Hg31 P 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.
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
259
Table 7
Indirect nuclear spin–spin coupling tensors determined from NMR spectroscopy of stationary powder samples (results in Hz)
Coupling and molecule
31 13
P C coupling
2,4,6-t-Bu3C6H2PyC(SiMe3)2
2,4,6-t-Bu3C6H2PyC(SiMe3)2
2,4,6-t-Bu3C6H2CP
199
Hg13C coupling
K2Hg(CN)4
31 31
P P coupling
1,2-Bis(2,4,6-tri-t-butylphenyl)diphosphene
Tetraphenyldiphosphine
115 31
In P coupling
Br3InP(4-(CH3O)C6H4)3
195 31
Pt P coupling
trans-Pt(PPh3)2Cl2
trans-Pt(PCy3)2Cl2
trans-Pt(PEt3)2Cl2
cis-Pt(PPh3)2Cl2
cis-Pt(Et3)2Cl2
Cl2Pt(PPh2CH2PPh2)
Cl2Pt(PPh2CH2CH2PPh2)
Cl2Pt(PPh2CH2CH2CH2PPh2)
199
Hg31P coupling
(EtO)2P(O)Hg(OOCCH3)
(EtO)2P(O)HgI
(EtO)2P(O)Hg(SCN)
[HgPR3(NO3)2]2
R ¼ phenyl
m-tolyl
p-tolyl
mesityl
p-MeOPh
cyclohexyl
[HgP(o-tolyl)3(NO3)2]2
Results and comments
Reference
DcJ ¼ 777 or 15 117, J ¼ 91. See text for discussion.
DcJ ¼ 1008 or 10 638, J ¼ 90. Coupling to aryl 13C. See text for
discussion.
DcJ ¼ 1233 or 19 821, J ¼ 59. See text for discussion.
[105]
[105]
DcJ ¼ 950 ^ 60, J ¼ 1540 ^ 2
[106]
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.
DcJ ¼ 300, J ¼ 2200 ^ 100. See text for discussion.
[107]
DcJ ¼ 1178 ^ 150 (preferred) or 22558 ^ 150
J ¼ 1109 ^ 9
[110]
DcJ ¼ 1865 ^ 250, J ¼ 2624 ^ 25
DcJ ¼ 1602 ^ 250, J ¼ 2420 ^ 25
DcJ ¼ 1536 ^ 250, J ¼ 2392 ^ 25
DcJ ¼ 2184 ^ 600 or 3282 ^ 600 (site 1)
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)
DcJ ¼ 1356 ^ 600 or 4104 ^ 600
J ¼ 3448 ^ 25
DcJ ¼ 2130, J ¼ 3064
DcJ ¼ 1660, J ¼ 3591
DcJ ¼ 840, J ¼ 3354
[111]
[111]
[111]
[111]
DcJ ¼ 2700 ^ 250, J ¼ 13 324 ^ 15
DcJ ¼ 1500 ^ 250, J ¼ 12 623 ^ 15
DcJ ¼ 1600 ^ 250, J ¼ 12 119 ^ 15
[113]
[113]
[113]
[114]
DcJ ¼ 4545 ^ 500, J ¼ 9572 ^ 15
DcJ ¼ 5235 ^ 200, J ¼ 9165 ^ 15
DcJ ¼ 5470 ^ 200, J ¼ 9144 ^ 15
DcJ ¼ 5560 ^ 500, J ¼ 10 468 ^ 15 (site 1)
DcJ ¼ 5560 ^ 500, J ¼ 10 566 ^ 15 (site 2)
DcJ ¼ 4765 ^ 250, J ¼ 9327 ^ 15 (site 1)
DcJ ¼ 3740 ^ 375, J ¼ 9309 ^ 15 (site 2)
DcJ ¼ 5525 ^ 200, J ¼ 8008 ^ 15
DcJ ¼ 5170 ^ 250, J ¼ 9660
In this case, the maximum possible splitting
Dn 2 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
[105]
[109]
[111]
[112]
[115]
[111]. This study has provided several large values of
DJ195 Pt31 P ; 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 transPt(PCy3)2Cl2, RDD is 822 Hz from the Pt–P bond length
of 2.337 Å [111], while Reff is only 35% of this value.
260
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 crystallographically 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, DJ199 Hg31 P
is non-zero for this compound. Reprinted with permission from Ref.
[100]. Copyright (1995) American Chemical Society.
Phosphorus-31 dipolar-chemical shift NMR experiments carried out on a series of metal compounds
featuring cyclic phosphino ligands yielded three
values of DJ195 Pt31 P [112]. For the series Cl2Pt(PPh2CH2PPh2), Cl2Pt(PPh2CH2CH2PPh2), Cl2Pt(PPh2CH2CH2CH2PPh2), DJ195 Pt31 P 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 Reff
195 Pt31 P 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 J199 Hg13 C 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 Å [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 J115 In31 P 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 significantly 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 summarized 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 cos2 b 2 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
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
261
Fig. 8. (a) 81.033 MHz 31P CPMAS NMR spectrum of solid Br3In-P(4-(CH3O)C6H4)3, depicting splittings due to the isotropic J115 In31 P 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
DJ115 In31 P 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.
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: Regardless 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 centreband 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,
B11B, 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 assumptions 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 1 J; 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 interhalogen 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
11
262
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 8
Indirect nuclear spin–spin coupling tensors determined from solid-state NMR of spinning samples
Coupling and molecule
199
Results and comments
Reference
Dc 2 J199 Hg14 N ¼ 69 ^ 15 Hz; 2 J199 Hg14 N ¼ 20:6 ^ 2:0 Hz
Dc 2 J199 Hg15 N ¼ 297 ^ 15 Hz; 2 J199 Hg15 N ¼ 29:0 ^ 2:0 Hz
199
Hg MAS NMR
[106]
Reff ¼ 24020 ^ 350 Hz, 1J ¼ ^1300 ^ 10 Hz
Reff ¼ 23740 ^ 350 Hz, 1J ¼ ^1260 ^ 10 Hz
Reff ¼ 23260 ^ 350 Hz, 1J ¼ ^1530 ^ 10 Hz
119
Sn 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
D2c J ¼ 23150 and 22950 Hz (2 sites), J ¼ ^2275 Hz
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.
[123]
14,15
Hg
N coupling
K2Hg(CN)4
119
Sn19F coupling
Me3SnF
(i-Bu)3SnF
Ph3SnF
Mes3SnF
(n-Bu)3SnF
Mn31P coupling
Mn2(CO)9PPh3
[124]
55
PhCH2C(O)Mn(CO)4(PPh3)
MesCH2C(O)Mn(CO)4(PPh3)
PhCH2C(O)Mn(CO)4[P(C6H11)3]
MesCH2C(O)Mn(CO)4[P(C6H11)3]
PhCH2Mn(CO)4(PPh3)
PhCH2Mn(CO)4[P(tolyl)3]
PhCH2Mn(CO)4[P(PhF)3]
63
Cu31P coupling
Several
Triphenylphosphine-copper(I)
complexes
63,65
Cu31P coupling
[(PBz3)2Cu][CuBr2]
[(PBz3)2Cu][PF6]
77
Se31P coupling
(Me)3PSe
(Ph)3PSe
113
Cd31P coupling
Cd(NO3)2·2PMe2Ph
119
Sn35Cl coupling
(Benzyl)3SnCl
SnCl2(acac)2
Ph3SnCl
DcJ ¼ 1027 or 5400 Hz, J ¼ ^297 Hz
31
P MAS NMR. The two possible values of DcJ arise because the sign of C55QMn
is not known. hQ is assumed to be zero.
DcJ ¼ 678 ^ 42 Hz, J ¼ 216 ^ 4 Hz
DcJ ¼ 589 ^ 24 Hz, J ¼ 233 ^ 2 Hz
DcJ ¼ 639 ^ 41 Hz, J ¼ 220 ^ 2 Hz
DcJ ¼ 495 ^ 10 Hz, J ¼ 232 ^ 2 Hz
DcJ ¼ 412 ^ 13 Hz, J ¼ 202 ^ 2 Hz
DcJ ¼ 508 ^ 22 Hz, J ¼ 196 ^ 3 Hz
DcJ ¼ 538 ^ 30 Hz, J ¼ 204 ^ 1 Hz
31
P MAS NMR.
See text for discussion of these data, in particular the small reported errors.
[125]
DcJ
31
¼ 600 Hz, J ¼ 900–2000 Hz
P MAS NMR. In some cases, the sign of C63QCu and magnitude of hQ are
estimated by the method of Vega [128].
[127]
DcJ ¼ 750 ^ 50 Hz, J ¼ 1535 ^ 10 Hz
DcJ ¼ 720 ^ 50 Hz, J ¼ 1550 ^ 10 Hz
31
P MAS NMR. Interaction tensor orientations are dictated by symmetry to be
coincident. Similarly, the value of hQ for copper is zero by symmetry.
[129]
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
DcJ ¼ 590 ^ 150 Hz, J ¼ 2735 Hz from 31P CPMAS NMR
[130]
DcJ ¼ 22600 or 21101 Hz, J ¼ 2285 Hz
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)
[131]
DcJ ¼ 2438 Hz with ‘substantial possible error’, J ¼ 227 Hz
DcJ ¼ 2740 Hz, J ¼ ^276 Hz
119
Sn MAS NMR. C35QCl is assumed to be negative, and hQ is assumed to be
zero. For SnCl2(acac)2, the magnitude of C35QCl is estimated.
DcJ ¼ 2350 Hz. 119Sn MAS NMR. C35QCl is assumed to be negative, and hQ is
assumed to be zero.
[132]
[126]
[130]
[133]
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
263
Table 8 (continued)
Coupling and molecule
119
Sn55Mn coupling
( p-XC6H4)3SnMn(CO)5
X ¼ CH3
X ¼ H (a,b)
X ¼ H (c)
X ¼ H (d)
X ¼ OCH3
X ¼ F (a)
X ¼ F (b)
X ¼ Cl
X ¼ SCH3
X ¼ SO2CH3
Ph3SnMn(CO)5 (3 sites)
[Mn(CO)5]2SnPh2
207
Pb55Mn coupling
Ph3PbMn(CO)5
[Mn(CO)5]2PbPh2
119
Sn117Sn coupling
(benzyl3Sn)2O
125
Te123Te coupling
(Me4N)2Te2
Results and comments
Reference
[134]
DcJ ¼ 622 ^ 42 Hz, J ¼ 132 ^ 2 Hz
DcJ ¼ 354 ^ 4 Hz, J ¼ 135 ^ 2 Hz
DcJ ¼ 515 ^ 33 Hz, J ¼ 141 ^ 3 Hz
DcJ ¼ 352 ^ 4 Hz, J ¼ 141 ^ 2 Hz
DcJ ¼ 398 ^ 13 Hz, J ¼ 149 ^ 1 Hz
DcJ ¼ 401 ^ 13 Hz, J ¼ 165 ^ 2 Hz
DcJ ¼ 566 ^ 16 Hz, J ¼ 151 ^ 2 Hz
DcJ ¼ 305 ^ 12 Hz, J ¼ 160 ^ 2 Hz
DcJ ¼ 501 ^ 21 Hz, J ¼ 170 ^ 1 Hz
DcJ ¼ 584 ^ 15 Hz, J ¼ 250 ^ 3 Hz
119
Sn MAS NMR
(X ¼ H: four molecules (a–d) in the unit cell; X ¼ F: two (a,b).)
DcJ ¼ 353 ^ 8 Hz, J ¼ 135 ^ 1 Hz
DcJ ¼ 345 ^ 8 Hz, J ¼ 142 ^ 2 Hz
DcJ ¼ 507 ^ 55 Hz, J ¼ 141 ^ 1 Hz
119
Sn MAS NMR
RDD ¼ 2560 ^ 4 Hz, Reff ¼ 223.6 ^ 0.6 Hz, J ¼ 139 ^ 1 Hz
DcJ not reported. 119Sn MAS NMR
Recrystallized from octane:
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
RDD ¼ 293 ^ 3 Hz, Reff ¼ 5.6 ^ 0.4 Hz, J ¼ 228 ^ 1 Hz
DcJ not reported. 207Pb MAS NMR
[135]
[136]
[135]
[136]
D2c J ¼ 1263 ^ 525 Hz (preferred), or 429 ^ 525 Hz
J 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=3ÞDc J; is reported in Ref. [137].
[137]
DcJ ¼ 24270 ^ 800 Hz, J ¼ ^2960 ^ 5 Hz
123
Te 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]
[138]
2
All results rely on the assumption that J is axially symmetric and coincident with the direct dipolar tensor, unless otherwise stated.
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
264
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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,
d¼
3CQ SðS þ 1Þ 2 3m2
½RDD ð3 cos2 bD 2 1
20nS
Sð2S 2 1Þ
þ hQ sin2 bD cos 2aD Þ 2
þ hQ sin2 bJ cos 2aJ Þ:
1
DJð3 cos2 bJ 2 1
3
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 diagonalization is required [149].
Clearly, there are several parameters which must
be determined accurately in order to obtain convincing 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 experiments 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 and
65
Cu, both spin-3/2 nuclei, yielded precise values of
þ 720 ^ 50 Hz and þ 750 ^ 50 Hz for DJ [129].
Shown in Fig. 9 is a demonstration of the sensitivity of
the simulated 31P MAS NMR spectra of [(PBz3)2Cu][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
Q
be coincident. The values of CCu
were obtained via
31
NQR experiments. P CPMAS measurements were
made at three applied field strengths, and the analysis
involved a complete diagonalization of the Hamiltonian, rather than the perturbation approach discussed 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 J207 Pb55 Mn and
J119 Sn55 Mn ; it was recognized that in cases where
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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=63 Cu31 P ¼ 750 Hz: Reprinted with permission from Ref. [129].
Copyright (1998) Academic Press.
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 C55QMn ; 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 C55QMn 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, DJ55 Mn31 P for MesCH2C(O)Mn(CO)4[P(C6H11)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
265
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 Å
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 discussed 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
D2 J199 Hg14 N was found to be þ 69 ^ 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 D2 J
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 DJ77 Se31 P ¼ þ590 ^ 150 Hz from the 31P
CPMAS NMR spectrum of triphenylphosphine selenide [130]. The value determined in this manner for
trimethylphosphine selenide, DJ77 Se31 P ¼ þ640 ^
260 Hz [130], is in excellent agreement with the
266
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
value determined in a nematic LC phase,
þ 680 ^ 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 D2 J119 Sn117 Sn ¼ þ1263 ^ 525 Hz for solid
(benzyl3Sn)2O [137]. This is in contrast to the negligible value which was reported for (cyclohexyl3Sn)2S
[155]. The linear arrangement of the Sn – O – Sn
moiety in (benzyl 3Sn)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 crystallographically equivalent tin atoms (Fig. 10).
By spinning off the magic angle, Reff is reintroduced with a scaling factor of 20:0619: Since the
chemical shift parameters may be determined beforehand 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
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 DJ119 Sn117 Sn may be determined. Reprinted with permission from
Ref. [137]. Copyright (1998) Elsevier.
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 measurements 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 DJ113 Cd31 P 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 different pulse sequences depending on whether the spin
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 spin1/2 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 quadrupolar, 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).
267
There are several versions of ‘molecular beam’
spectroscopy, including molecular beam electric
resonance (MBER), molecular beam magnetic resonance (MBMR), molecular beam maser spectroscopy,
molecular beam absorption spectroscopy, and molecular 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 spectrometer 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 therefore 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
h21 Hhf ¼ VK : QK þ c1 IK ·J þ c2 IF ·J
þ c3 IK ·dT ·IF þ c4 IK ·IF
ð82Þ
If we are interested in the J ¼ 1 rotational level, higherorder terms such as the nuclear magnetic octupole and
nuclear electric hexadecapole interaction are zero. The
268
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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.
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 1 S
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]:
DJMN
;
ð83Þ
c3 ¼ RDD 2
3
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 1 S 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
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 rovibrationally averaged value of c3 (direct); however interpretation of these data in the language of NMR J has been
lacking.
While it is certainly true in many NMR experiments 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 constants, 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).
Müller 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
7
See, for example, footnote b in Table III of Refs. [176,177].
269
several reliable values of DK from the very highresolution hyperfine data which are now available
[143] (Table 10).
In combination with high-level ab initio calculations, 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 opportunity 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 knowledge, almost no information on J (c3 and c4
parameters) has been extracted for polyatomic
molecules. Even for some 1 S 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 this
270
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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
7
LiH
7
LiH
7
LiH
7
LiD
7 19
Li F
143.45(3)
11.346(7)(n ¼ 0, J ¼ 1)
11.03(8)(n ¼ 1, J ¼ 1)
11.329(12)(n ¼ 0, J ¼ 2)
1.7430(70)
11.4292(42) 2 0.2122(86)ðn þ ð1=2ÞÞ þ
0.0039(29)ðn þ ð1=2ÞÞ2
1.1789(78)
1.0710(61)
0.62834(68) 2 0.0050(11)ðn þ ð1=2ÞÞ
3.85(25)
3.7(2)
0.4269(15) 2 0.0042(2)ðn þ ð1=2ÞÞ 2
0.00021(9)[J(J þ 1)]
0.3922(16) 2 0.0029(5)ðn þ ð1=2ÞÞ þ
0.00014(11)[J(J þ 1)]
0.4749(27) 2 0.0065(10)ðn þ ð1=2ÞÞ
0.2606(15) 2 0.0035(5)ðn þ ð1=2ÞÞ
0.035(12)
2.45(37)
0.79681 ^ 0.00036 2 (0.00642 ^
0.00027)ðn þ ð1=2ÞÞ
0.92(12)(n ¼ 0, J ¼ 1)
0.92713(53) 2 0.00917(93)ðn þ ð1=2ÞÞ þ
0.00097(29)ðn þ ð1=2ÞÞ2
0.028(2) 2 0.000(3)ðn þ ð1=2ÞÞ þ
0.0002(7)ðn þ ð1=2ÞÞ2
Not determined
0.3026(50)
2.859(9)
7.15a (n ¼ 0, J ¼ 1)
6.93(26)(n ¼ 1, J ¼ 1)
5.202(146)a
2.62(3)(n ¼ 0, J ¼ 1)
2.62(1)(n ¼ 0, J ¼ 2)
3.50(15)
20.09(10)(n ¼ 0, J ¼ 2)
20.13(10)(n ¼ 0, J ¼ 2)
20.56(50)
20.13(15)
21.55(8)(n ¼ 0, J ¼ 2)b
21.65(5)(n ¼ 0, J ¼ 2)
21.68(8)(n ¼ 0, J ¼ 2)
21.77(5)(n ¼ 0, J ¼ 2)
22.59(2)(n ¼ 0, J ¼ 3)
22.48(10)(n ¼ 0, J ¼ 3)
1.58(5)(n ¼ 0, J ¼ 13)
1.528(18)(n ¼ 0, J ¼ 13)
1.519(18)(n ¼ 0, J ¼ 15)
0.50(2)
0.135(10)(n ¼ 0, J ¼ 1)
0.17(4)(n ¼ 1, J ¼ 1)
0.160(5) (n ¼ 0, J ¼ 2)
0.005(10)
0.1744(21) 2 0.0042(21)ðn þ ð1=2ÞÞ
1987
1975
1975
1975
1975
1992
[176,177]
[183]
[183]
[183]
[183]
[184]
0.0711(89)
0.0604(70)
0.06223(36) þ 0.00041(26)ðn þ ð1=2ÞÞ
0.150(250)
20.2(2)
0.0859(18)
1972
1972
1999
1964
1965
1987
[185]
[185]
[186]
[187]
[188]
[189]
0.078(3)
1987
[189]
0.0578(13)
0.0317(7)
0.009(6)
0.86(40)
0.23766 ^ 0.00032 2 (0.00245 ^
0.00022)ðn þ ð1=2ÞÞ
0.61(10)(n ¼ 0, J ¼ 1)
0.62745(30) 2 0.00903(22)ðn þ ð1=2ÞÞ
1988
1988
1984
1972
2002
[173]
[173]
[190]
[191]
[192]
1967
1999
[174]
[193]
0.060(4) þ 0.002(5)ðn þ ð1=2ÞÞ þ
0.0006(12)ðn þ ð1=2ÞÞ2
0.306(30)
1.0667(65)
0.840(6)
4.86(28)(n ¼ 0, J ¼ 1)
6.47(84) (n ¼ 1, J ¼ 1)
5.73(105)
22.15(3)(n ¼ 0, J ¼ 1)
22.11(1)(n ¼ 0, J ¼ 2)
213.3(7)
21.52(10)
21.54(10)
21.11(50)
21.28(15)
26.39(8)(n ¼ 0, J ¼ 2)
26.35(5)(n ¼ 0, J ¼ 2)
26.91(8)(n ¼ 0, J ¼ 2)
26.84(5)(n ¼ 0, J ¼ 2)
26.57(1)(n ¼ 0, J ¼ 3)
26.67(5)(n ¼ 0, J ¼ 3)
3.66(3)(n ¼ 0, J ¼ 13)
3.708(22)(n ¼ 0, J ¼ 13)
3.701(23)(n ¼ 0, J ¼ 15)
1977
[194]
1976
1985
1977
1995
1995
1995
1972
1972
1964
1969
1969
1969
1969
1970
1970
1970
1970
1970
1970
1980
1999
1999
[181]
[180]
[195]
[178]
[178]
[178]
[196]
[196]
[182]
[197]
[197]
[197]
[197]
[198]
[198]
[198]
[198]
[199]
[199]
[172]
[179]
[179]
7
Li81Br
Li79Br
7 127
Li I
23
Na19F
23
Na19F
23
Na81Br
7
23
Na79Br
39
K19F
K19F
39 35
K Cl
87
Rb19F
85
Rb19F
41
133
Cs19F
Cs19F
133
133
Cs35Cl
23
Na39K
Na2
35 19
Cl F
79 19
Br F
79 19
Br F
127 19
I F
115 19
In F
115 19
In F
205 19
Tl F
203 35
Tl Cl
205 35
Tl Cl
203 37
Tl Cl
205 37
Tl Cl
203 79
Tl Br
205 79
Tl Br
203 81
Tl Br
205 81
Tl Br
203 127
Tl I
205 127
Tl I
127
I2
127
I2
127
I2
23
a
b
Values for which no rovibrational dependence is given are for the n ¼ 0, J ¼ 1 state unless otherwise indicated.
The value for c3 reported in Ref. [178] is of opposite sign due to use of a different convention.
Extensive data are available for TlBr, e.g. for five vibrational states and two rotational states.
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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
LiF
LiBr
LiI
NaBr
KF
CsF
CsCl
Na2
ClF
BrF
IF
InF
TlF
TlCl
TlBr
TlI
I2a
2.89
3.92
5.15
6.65
9.76
10.9
41.8
39.4
127
75.7
171
252
286.4
2202
2224
2361
2474
763
21.22
3.94
18.1
18.4
43.9
23.8
46.5
67.9
25.71
281.8
2206
2257
89.9
173
262
448
664
2785
[183]
[184]
[185]
[186]
[189]
[173]
[193]
[194]
[180]
[195]
[178]
[178]
[196]
[182]
[197]
[198]
[199]
[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.
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 n ¼ 0; J ¼ 15 state. Clearly, it would be of
interest to further investigate the rotational dependence 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
271
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
272
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
in the rotating frame:
! 1
1
1 2
¼
12 x :
T1r
T1r dip
3
3. Quantum chemical methods
ð84Þ
Here, the total rate of relaxation is equal to the
relaxation due the pure dipolar interaction, ð1=T1r Þdip ;
multiplied by a factor involving the ratio of DJ to RDD ;
represented by x. Depending on the value of x, relaxation 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 contributions 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 wellcharacterized and for which a particular spin pair
may be predicted to have a substantial DJ and a small
but accurately-known direct dipolar coupling constant. 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 contributions 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 :
3.1. 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 wellknown 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 experimental 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-Hartree – Fock methods based on the RHF reference state
may be suspect. In practice, multiconfiguration selfconsistent 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
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 ruleof-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] features 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
J FC 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
273
Table 11
Calculated (MCLR) spin – spin coupling constants in C2H2
compared to the experimental results extrapolated to the equilibrium molecular geometry (results in Hz)
Method
3
MCLR [213]
Experimental [214,215]
10.80
10.89
†
†
†
†
JHH
1
JCH
244.27
242.70
2
1
53.08
53.82
184.68
185.04
JCH
JCC
and is unable to provide the anisotropic
properties. A major drawback of FPT is that
numerical instabilities may arise when supplementing 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 propagator-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 investigation 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 N 5 where N is the
number of basis functions.
The second-order polarization propagator
approach (SOPPA) of Oddershede and coworkers [228,229] is another analytic secondorder method roughly at or better than the MP2
level. While SOPPA (scaling as N 5 ) is still
somewhat subject to the triplet instability
problem, its accuracy is very useful for
274
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 N 6 ; while property calculations scale
as N 5 :
† 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 socalled quadratic one, meaning unrelaxed analytic 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 formulation. 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 ðN 6 Þ is currently the most accurate
black-box model for systems where static
electron correlation is of little importance.
Analytic CCSD(T) ðN 7 Þ 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 ðN 8 Þ and the CC3
model of Ref. [236] with approximate triples
ðN 7 Þ:
† 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, requiring 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 linearscaling 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,
N 3 – 4 : The drawback of DFT is that there exists no systematic way of improving the exchange-correlation
functional of electron density Exc ½r that lies at the
heart of DFT, from one calculation to another. The
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
three ‘generations’ of Exc ½r 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 exchangecorrelation 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 functionals, 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
275
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 secondorder 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 agreement with experiment.
† Malkin et al. [256,257] presented a combined FPT
and sum-over-states density-functional perturbation 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 calculation. 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:
PSO
JMN;
et /
occ X
virt
3
X
kfk llM;e =rM
lfa lkfa llN;t =rN3 lfk l
k
a
xc
1k 2 1a 2 DEk!a
ð85Þ
where the orbital energy denominators 1k 2 1a
have been subjected to the ‘Malkin correction’,
xc
DEk!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
276
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
co-workers [260] made an interesting connection
xc
of DEk!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 J FC
towards group 17 elements, particularly for
19
F. 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 applications 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 typically 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 with
19
F 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 approximate self-interaction correction at the LDA level was
not found to lead to a systematic improvement of the
total J, albeit J PSO 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 J SD and J FC : 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 combination 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 contributes to the favourable situation as the error in the
total J is dominated by the second-order
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 perturbations 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 stateof-the-art basis sets for valence properties.
A few systematic studies of the basis set requirements 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 as ðHÞ ¼ 150; 000
whereas in ordinary basis sets max½as ðHÞ 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].
277
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 3 JHH
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 framework [290,291] would be interesting in the context of
J, as well.
Nair and Chandra [292] have used energyoptimized 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 dfunction 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
278
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Fig. 12. Schematic diagram of the various factors affecting the
comparison of experimentally and theoretically determined NMR
parameters. Drawn after Ref. [296].
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 dependence, the prototypical example being the increase of
J FC 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 of
3
JHH 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 hypersurface
X ›J J ¼ Je þ
Q
›Q k e k
k
!
1X
›2 J
þ
Q Q þ ···
ð86Þ
2 kl ›Qk ›Ql e k l
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 representation used for the property tensors [305,306].
The property surface is averaged over the nuclear
motion as
X ›J kJlT ¼ Je þ
kQ lT
›Qk e k
k
!
1X
›2 J
þ
kQ Q lT þ · · ·;
ð87Þ
2 kl ›Qk ›Ql e k l
where the nomenclature kAlT specifies either the
temperature average of A or its average in a particular
rovibrational state, occupied with a certain temperature-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 anharmonic 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 calculation at the thermally averaged ra geometry, where
!
1 X ›2 J
T
kJl < Ja þ
kQ2 lT :
ð88Þ
2 k ›Q2k e k
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 kQk lT
and kQk Ql lT ; 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 available by solving the rovibrational Schrödinger
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].
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 automation, 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. Calculations 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
279
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 D1 JFC 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
D1 J (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 Pyykkö
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 Pyykkö [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
LiH
7
LiF
19 7
KF
39
Na2
23
ClF
35
a
Li1H
F Li
K19F
Na23Na
Cl19F
J
DJ
J
DJ
J
DJ
J
DJ
J
DJ
re value
Rovib. state
Rovib. average
Exp.a
152.47
212.39
193.10
177.43
76.59
109.22
1243.6
229.88
832.24
2805.68
n ¼ 0, J ¼ 1
151
213
199.0
176.9
78.2
109.5
1245
230
829
2800
135(10)
257(21)
172.3(32)
173.2(28)
57.8(13)
125.7(51)
1067(7)
248(15)
840(6)
2907(27)
n ¼ 0, J ¼ 0
n ¼ 0, J ¼ 0
n¼0
n ¼ 0, J ¼ 1
Experimental microwave spectroscopic results for the specified rovibrational states. For references, see Ref. [143].
280
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
electronic structure can be categorized into scalar
relativistic and spin – orbit effects. The rough consequence of the former is a contraction of the atomic sand 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 spinHamiltonian 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, completely new terms or combinations of non-relativistically 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 nonrelativistic limits [328 – 330]. For J, the leading
relativistic correction terms are Oða6 Þ; 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 Hückel (REXNMR) [7,74] is
a semi-empirical method based on the relativistic
parametrization (obtained by Dirac –Fock atomic
calculations) of the extended Hückel 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 perturbation theory [37,38,333]. A second-order correction 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
J FC and neglects J SD : 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 framework. 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 properties of J [144 – 146]. Autschbach and Ziegler [267]
found the spatial origin of the relativistic increase of
J FC 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
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 pseudopotentials. 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 intermolecular interactions, repulsion, dispersion, electrostatic,
and induction forces, as well as hydrogen bonding
effects [349]. There are both indirect effects manifested 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 characterized 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 bonding is not within reach in a pure reaction field
model.
† The supermolecule method has been applied,
where parts of the immediate molecular surroundings 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-
281
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 shortrange intermolecular interactions, and can be
combined with the reaction field model for longrange 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 –formamide 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
FC
effects on 1 JNLi
in LiNH2, LiN(CH3)2, and (LiNH2)2
due to explicit solvation by one to four water
molecules. Ref. [364] used IPPP-CLOPPA to investigate the effect of the electric field due to a solvent
water molecule on 1 JCH in CH4 and HCN. The 1 JCH 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 interaction in He2. Ref. [378] gives an estimate of
282
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
the corresponding FC contribution in Xe2. Ref. [376]
considers JNN over the hydrogen bond in a methyleneimine 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 presentday 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 throughbond 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 coupling 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 calculated in Ref. [256]. The dependence of 1 JCa Cb and
1
JCa Ha on the backbone conformation of a model
dipeptide [383,384], the dependence of 3 JCC on the
conformation of an open chain natural product
fragment [385], the HH and CH couplings in Me a1
D -xylopyranoside [386], JCH in cyclohexane-related
1
systems [387], JFH in (HF)n clusters [388], internucleotide JNH and JNN between DNA base pairs
[368], and 1 JHD in Os(II)-dihydrogen complexes [346]
have also been investigated. In most of these
applications, J SD has been neglected.
An earlier investigation of JHD in Os(II)-dihydrogen complexes was carried out at the SCF and MP2
levels in Ref. [344].
Refs. [389 – 391] report a hybrid DFT (B3LYP)
level study of n JCC and n JCH in 2-deoxy-b-D ribofuranose and related systems using a FPT
calculation of J FC : 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 n JHH
in terpenes. The results are hampered by serious
methodological deficiencies as discussed before. In
Ref. [216], the uncoupled SCF method was used for
FC
FC
FC
JHH
; JCH
; and JCC
in a number of light main-group
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
molecules. In this context, FPT calculations for J FC 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 1 JCC 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].
FC
A Hartree – Fock level study of 1 JNLi
in model
systems exhibiting the LiN-bond was carried out in
FC
Ref. [363]. 1 JCH
in five- and six-membered heterocyclic 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 characterized by inadequate electron correlation treatment
and/or, by today’s standards, modest basis sets.
Following that, the more recent papers with up-todate methods are commented upon.
Early work using Hartree – Fock-level methods.
Lazzeretti et al. carried out a coupled Hartree – Fock
þ
2
study of PH2
2 , PH3, and PH4 [40] as well as NH2 ,
þ
2
NH4 , and BH4 [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
283
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 ) contributions to the SD/FC cross-term have been presented
separately. For 1 J; 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 AlH2
4 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 1 J and D1 J
with respect to the direction of the vector joining the
heavy nuclei. The SOS-CI method comprises a noniterative 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 calculations 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 2 JHH ; the later
284
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 13
Quantum chemically calculated components of the symmetric part J1 þ J S of the 1H1H spin–spin coupling tensors
System
Bonds
Theorya
Values
Reference
BH2
4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2h
2i
2
2
2
2
2
2
2
2
2
2
3
3
3
3k
3l
3m
3k
3l
3n
SCF/(s )
SOPPA(CCSD)/(s )
RAS/HIV
RAS/HIII
SCF/(s )
SCF/(s)
RAS/HIII
RAS/HIII
SOPPA(CCSD)/(s )
CAS/(s )
CAS þ SOe/HIVu4
CAS/HIVu4
SOPPA/(s )
SOPPA/(s )
MP2/(s )
RAS/HIII
RAS/HIII
RAS/HIII
FOPPA/(s )
SOPPA(CCSD)/aug-cc-pVTZ(m )
FOPPA/(s )
RAS/HIII
RAS/HIII
SCF/(s )
SCF/(s )
SCF/(s )
CAS þ SOe/HIVu4
CAS/HIVu4
MP2/6-31G(**)
CAS þ SOe/HIVu4
CAS/HIVu4
CAS þ SOe/HIVu3
CAS/HIVu3
SOPPA(CCSD)/(s )
RAS/HIV
ZORA DFT(GGA)/Slater
RAS/HIV
RAS/HIV
RAS/HIII
RAS/HIII
RAS/HIII
RAS/HIII
Jaa ¼ 212.10, Jbb ¼ 219.96, Jcc ¼ 2 20.67
Jaa ¼ 28.25, Jbb ¼ 29.48, Jcc ¼ 225.63
DJ ¼ 5.3, Jxx 2 Jyy ¼ 15.3b, Jaa ¼ 4.5, Jbb ¼ 6.8, Jcc ¼ 2 8.5
DJ ¼ 28.3, Jaa ¼ 28.0, Jbb ¼ 28.8, Jcc ¼ 225.5
Jaa ¼ 8.96, Jbb ¼ 215.21, Jcc ¼ 234.89
Jaa ¼ 25.13, Jbb ¼ 221.83, Jcc ¼ 235.75
DcJ ¼ 218.97, J ¼ 222.91c,d
DcJ ¼ 217.65, J ¼ 219.05c,d
Jaa ¼ 1.21, Jbb ¼ 6.38, Jcc ¼ 233.33
Jaa ¼ 0.56, Jbb ¼ 5.48, Jcc ¼ 234.84
Jaa ¼ 0.75, Jbb ¼ 5.25, Jcc ¼ 234.54
Jaa ¼ 0.73, Jbb ¼ 5.19, Jcc ¼ 234.60
Jaa ¼ 1.16, Jbb ¼ 5.66, Jcc ¼ 234.22
Jaa ¼ 21.14, Jbb ¼ 1.69, Jcc ¼ 235.93
Jaa ¼ 20.93, Jbb ¼ 25.12, Jcc ¼ 248.93
Jaa ¼ 11.2, Jbb ¼ 211.8, Jcc ¼ 11.9
DJ ¼ 210.53, Jaa ¼ 25.37, Jbb ¼ 26.58, Jcc ¼ 222.61f
DJ ¼ 6.06, Jxx 2 Jyy ¼ 16.65g, Jaa ¼ 3.35, Jbb ¼ 5.61, Jcc ¼ 211.04
Jaa ¼ 25.55, Jbb ¼ 27.19, Jcc ¼ 27.96
Jaa ¼ 0.01, Jbb ¼ 2.43, Jcc ¼ 5.34
Jaa ¼ 0.42, Jbb ¼ 21.71, Jcc ¼ 25.48
DJ ¼ 27.33, Jaa ¼ 29.50, Jbb ¼ 2 9.73, Jcc ¼ 2 26.49
DJ ¼ 21.96, Jaa ¼ 0.15, Jbb ¼ 3.69, Jcc ¼ 3.72
Jaa ¼ 5.92, Jbb ¼ 28.38, Jcc ¼ 29.42j
DJ ¼ 4.26, Jaa ¼ 27.79, Jbb ¼ 223.52, Jcc ¼ 224.56
Jaa ¼ 5.04, Jbb ¼ 26.09, Jcc ¼ 26.93
Jaa ¼ 26.52, Jbb ¼ 215.73, Jcc ¼ 224.67
Jaa ¼ 26.70, Jbb ¼ 215.80, Jcc ¼ 224.85
Jaa ¼ 214.92, Jbb ¼ 221.07, Jcc ¼ 2 26.18
Jaa ¼ 213.38, Jbb ¼ 214.14, Jcc ¼ 2 24.81
Jaa ¼ 213.74, Jbb ¼ 214.91, Jcc ¼ 2 25.67
Jaa ¼ 213.26, Jbb ¼ 217.02, Jcc ¼ 2 23.03
Jaa ¼ 214.05, Jbb ¼ 218.59, Jcc ¼ 2 24.70
DJ ¼ 3.20, J ¼ 11.31
DJ ¼ 3.4, J ¼ 10.8
DJ ¼ 12, J ¼ 10
DJ ¼ 4.0, Jxx 2 Jyy ¼ 21.2b, Jaa ¼ 8.4, Jbb ¼ 9.7, Jcc ¼ 13.1
DJ ¼ 5.0, Jxx 2 Jyy ¼ 20.8b, Jaa ¼ 14.4, Jbb ¼ 15.8, Jcc ¼ 20.9
DJ ¼ 2.2, J ¼ 7.2
Jaa ¼ 0.7, Jbb ¼ 2 0.8, Jcc ¼ 2.4
Jaa ¼ 10.5, Jbb ¼ 10.6, Jcc ¼ 14.2
DJ ¼ 1.23, J ¼ 3.80
[41]
[321]
[72]
[72]
[41]
[41]
[324]
[324]
[318]
[318]
[38]
[38]
[280,318]
[228]
[227]
[404,405]
[78]
[78]
[406]
[321]
[406]
[80]
[80]
[40]
[40]
[40]
[38]
[38]
[227]
[38]
[38]
[38]
[38]
[214,215]
[72]
[144]
[72]
[72]
[72]
[404,405]
[404,405]
[80]
CH4
C2H4
C2H6
NH2
2
NHþ
4
CH3CN
CH3NC
H2O
H2O
H2O
H2O
H2O
H2O
H2O
HCONH2
CH3F
CH2F2
AlH2
4
SiH4
SiH4
CH3SiH3
CH3SiH3
PH2
2
PH3
PHþ
4
H2S
H2S
H2S
H2Se
H2Se
H2Te
H2Te
C2H2
C2H2
C2H2
C2H4
C2H4
C2H6
HCONH2
HCONH2
CH3SiH3
Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.),
or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), 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
(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.
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 14
Quantum chemically calculated components of the symmetric part J1 þ J S of the
a
285
13 1
C H spin–spin coupling tensors
System
Bonds
Theory
Values
Reference
CH4
CH4
C2H2
C2H2
C2H2
C2H4
1
1
1
1
1
1
SOPPA(CCSD)/(s )
MP2/(s )
SOPPA(CCSD)/(s )
RAS/HIV
ZORA DFT(GGA)/Slater
RAS/HIV
[321]
[227]
[214,215]
[72]
[144]
[72]
C2H6
C6H6
HCN
CH3CN
CH3NC
HCONH2
CH3F
CH2F2
1
1
1
1
1
1
1
1
RAS/HIII
CAS/(s )
RAS/HIV
RAS/HIII
RAS/HIII
RAS/HIII
RAS/HIII
RAS/HIII
CHF3
CH3SiH3
C2H2
C2H2
C2H2
C2H4
C2H6
C6H6
HNC
CH3CN
HCONH2
HCONH2
CH3SiH3
C6H6
CH3NC
C6H6
1
1
2
2
2
2
2
2
2
2
2f
2g
2
3
3
4
RAS/HIII
RAS/HIII
SOPPA(CCSD)/(s )
RAS/HIV
ZORA DFT(GGA)/Slater
RAS/HIV
RAS/HIII
CAS/(s )
RAS/HIV
RAS/HIII
RAS/HIII
RAS/HIII
RAS/HIII
CAS/(s )
RAS/HIII
CAS/(s )
DJ ¼ 225.6, J ¼ 123.8b
DJ ¼ 64.44, J ¼ 130.63b
DJ ¼ 263.41, J ¼ 254.95
DJ ¼ 262.4, J ¼ 232.1
DJ ¼ 238.7, J ¼ 262
DJ ¼ 2.6, Jxx 2 Jyy ¼ 228.8c, Jaa ¼ 123.9, Jbb ¼ 158.0,
Jcc ¼ 161.2
DJ ¼ 6.0, Jaa ¼ 102.7, Jbb ¼ 128.3, Jcc ¼ 128.5
DJ ¼ 28.0, Jaa ¼ 144.0, Jbb ¼ 190.8, Jcc ¼ 195.4
DJ ¼ 263.34, J ¼ 249.27
DcJ ¼ 228.82, J ¼ 142.43d
DcJ ¼ 224.70, J ¼ 143.50d
Jaa ¼ 161.5, Jbb ¼ 192.9, Jcc ¼ 195.2
DJ ¼ 6.10, Jaa ¼ 122.00, Jbb ¼ 149.84, Jcc ¼ 152.62
DJ ¼ 27.02, Jxx 2 Jyy ¼ 212.26e, Jcc ¼ 186.00, Jbb ¼ 184.15,
Jaa ¼ 156.88
DJ ¼ 231.19, J ¼ 236.79
DJ ¼ 7.10, Jaa ¼ 96.73, Jbb ¼ 124.85, Jcc ¼ 125.63
DJ ¼ 31.12, J ¼ 51.73
DJ ¼ 28.2, J ¼ 50.1
DJ ¼ 39.3, J ¼ 52.3
DJ ¼ 5.2, Jxx 2 Jyy ¼ 6.0c, Jaa ¼ 21.1, Jbb ¼ 23.0, Jcc ¼ 28.1
DJ ¼ 21.8, Jaa ¼ 22.5, Jbb ¼ 26.6, Jcc ¼ 27.0
DJ ¼ 29.2, Jaa ¼ 21.7, Jbb ¼ 27.1, Jcc ¼ 213.5
DJ ¼ 33.34, J ¼ 16.44
DcJ ¼ 5.12, J ¼ 215.46d
Jaa ¼ 0.4, Jbb ¼ 2.0, Jcc ¼ 6.0
Jaa ¼ 22.2, Jbb ¼ 24.1, Jcc ¼ 26.3
DJ ¼ 0.41, Jaa ¼ 2.79, Jbb ¼ 3.58, Jcc ¼ 3.84
DJ ¼ 3.3, Jaa ¼ 8.6, Jbb ¼ 12.5, Jcc ¼ 13.9
DcJ ¼ 5.21, J ¼ 2.63d
DJ ¼ 26.9, Jaa ¼ 0.6, Jbb ¼ 25.3, Jcc ¼ 29.2
[72]
[70]
[324]
[324]
[324]
[404,405]
[78]
[78]
[78]
[80]
[214,215]
[72]
[144]
[72]
[72]
[70]
[324]
[324]
[404,405]
[404,405]
[80]
[70]
[324]
[70]
Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.),
or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), 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.
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 1 JOH [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 1 J 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
286
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 15
Quantum chemically calculated components of the symmetric part J1 þ J S of the
15
N1H spin–spin coupling tensors
System
Bonds
Theorya
Valuesb
Reference
NH2
2
NH3
NHþ
4
HNC
N2H4
BH2NH2
BH3NH3
HCONH2
HCONH2
PH2NH2
HCN
CH3NC
HCONH2
CH3CN
1
1
1
1
1
1
1
1f
1g
1
2
2
2
3
SCF/(s )
MP2/(s )
SCF/(s )
RAS/HIV
SOS-CI/(s )
SOS-CI/(s )
SOS-CI/(s )
RAS/HIII
RAS/HIII
SOS-CI/(s )
RAS/HIV
RAS/HIII
RAS/HIII
RAS/HIII
Jaa ¼ 22.26, Jbb ¼ 46.39, Jcc ¼ 68.41
DJ ¼ 15.70, J ¼ 260.07
DJ ¼ 220.12, J ¼ 68.96c
DJ ¼ 36.39, J ¼ 2112.61
DJ ¼ 21.82, J ¼ 285.36d,e
DJ ¼ 20.28, J ¼ 298.44d
DJ ¼ 21.52, J ¼ 288.48d
Jaa ¼ 274.8, Jbb ¼ 2100.4, Jcc ¼ 2103.5
Jaa ¼ 274.6, Jbb ¼ 299.6, Jcc ¼ 2102.7
DJ ¼ 25.65, J ¼ 2103.82d,h
DJ ¼ 219.51, J ¼ 26.44
DcJ ¼ 21.91, J ¼ 4.46i
Jaa ¼ 215.6, Jbb ¼ 216.0, Jcc ¼ 217.4
DcJ ¼ 24.30, J ¼ 22.03i
[41]
[227]
[41]
[324]
[325]
[325]
[325]
[404,405]
[404,405]
[325]
[324]
[324]
[404,405]
[324]
Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.),
or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), 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
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 J S of the
19 1
F H spin–spin coupling tensors
System
Bonds
Theorya
Values
Reference
HF
HF
HF
HF
HF
HF
CH3F
CH2F2
1
1
1
1
1
1
2
2
DHF/cc-pVTZ(u )
SCF/cc-pVTZ(u )
CAS/cc-pV5Z
CAS þ SOb/HIVu3
CAS/HIVu3
MP2/(s )
RAS/HIII
RAS/HIII
[335]
[335]
[143]
[38]
[38]
[227]
[78]
[78]
CHF3
p-C6H4F2
p-C6H4F2
2
3
4
RAS/HIII
RAS/HII(m )
RAS/HII(m )
DJ ¼ 160.19, J ¼ 610.42
DJ ¼ 158.68, J ¼ 612.23
DJ ¼ 115.98, J ¼ 476.09
DJ ¼ 127.3, J ¼ 534.7
DJ ¼ 126.7, J ¼ 534.8
DJ ¼ 2715.92, J ¼ 570.01
DJ ¼ 256.73, Jaa ¼ 8.59, Jbb ¼ 37.00, Jcc ¼ 100.76
DJ ¼ 23.51, Jxx 2 Jyy ¼ 44.71c, Jaa ¼ 22.87, Jbb ¼ 45.38,
Jcc ¼ 87.38
DJ ¼ 40.57, Jaa ¼ 60.17, Jbb ¼ 70.67, Jcc ¼ 107.11
DJ ¼ 16.0, Jxx 2 Jyy ¼ 15.1d, Jaa ¼ 2.6, Jbb ¼ 211.8, Jcc ¼ 12.4
DJ ¼ 21.4, Jxx 2 Jyy ¼ 24.6d, Jaa ¼ 2.7, Jbb ¼ 8.2, Jcc ¼ 9.6
[78]
[60]
[60]
Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.),
or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), 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
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.
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
287
Table 17
Quantum chemically calculated components of the symmetric parts K1 þ K S of the X1H spin–spin coupling tensors other than those listed in
Tables 13 –16
a
System
X
Bonds
Theory
Values
LiH
BH2
4
B2H4
B2H6
B2H6
BH2NH2
BH3NH3
BH2PH2
BH3PH3
H2O
H2O
H2O
H2O
H2O
H2O
H2O
HCONH2
HCONH2
HCONH2
AlH2
4
SiH4
SiH4
SiH4
CH3SiH3
CH3SiH3
PH2
2
PH3
PH3
PHþ
4
P2H4
BH2PH2
BH3PH3
PH2NH2
H2S
H2S
H2S
HCl
HCl
HCl
HCl
HCl
HCl
H2Se
H2Se
HBr
HBr
HBr
HBr
H2Te
H2Te
7
Li
B
11
B
11 f
B
11 g
B
11
B
11
B
11
B
11
B
17
O
17
O
17
O
17
O
17
O
17
O
17
O
17
O
17
O
17
O
27
Al
29
Si
29
Si
29
Si
29
Si
29
Si
31
P
31
P
31
P
31
P
31
P
31
P
31
P
31
P
33
S
33
S
33
S
35
Cl
35
Cl
35
Cl
35
Cl
35
Cl
35
Cl
77
Se
77
Se
79
Br
79
Br
79
Br
79
Br
125
Te
125
Te
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
3j
3k
1
1
1
1
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
CAS/cc-pV5Z
SCF/(s )
SOS-CI/(s )
SOS-CI/(s )
SOS-CI/(s )
SOS-CI/(s )
SOS-CI/(s )
SOS-CI/(s )
SOS-CI/(s )
SOPPA(CCSD)/(s )
CAS/(s )
CAS þ SOh/HIVu4
CAS/HIVu4
SOPPA/(s )
SOPPA/(s )
MP2/(s )
RAS/HIII
RAS/HIII
RAS/HIII
FOPPA/(s )
SOPPA(CCSD)/aug-cc-pVTZ(m )
MP2/6-31G(**)
FOPPA/(s )
RAS/HIII
RAS/HIII
SCF/(s )
MP2/6-31G(**)
SCF/(s )
SCF/(s )
SOS-CI/(s )
SOS-CI/(s )
SOS-CI/(s )
SOS-CI/(s )
CAS þ SOh/HIVu4
CAS/HIVu4
MP2/6-31G(**)
DHF/cc-pVTZ(u)
SCF/cc-pVTZ(u)
CAS/aug-cc-pVQZ
CAS þ SOh/HIVu3
CAS/HIVu3
MP2/6-31G(**)
CAS þ SOh/HIVu4
CAS/HIVu4
DHF/(s )
SCF/(s )
CAS þ SOh/HIVu3
CAS/HIVu3
CAS þ SOh/HIVu3
CAS/HIVu3
DK ¼ 22.8, K ¼ 32.3b
[143]
DK ¼ 210.35, K ¼ 24.28c
[41]
DK ¼ 0.40, K ¼ 40.08d,e
[325]
DK ¼ 20.59, K ¼ 47.12d
[325]
DK ¼ 0.20, K ¼ 18.06d
[325]
DK ¼ 0.09, K ¼ 40.83d
[325]
DK ¼ 1.00, K ¼ 38.52d
[325]
DK ¼ 20.49, K ¼ 47.72d
[325]
DK ¼ 1.27, K ¼ 41.21d
[325]
Kaa ¼ 45.50, Kbb ¼ 48.37, Kcc ¼ 56.33
[318]
Kaa ¼ 47.38, Kbb ¼ 49.94, Kcc ¼ 57.24
[318]
Kaa ¼ 46.47, Kbb ¼ 50.14, Kcc ¼ 57.29
[38]
Kaa ¼ 46.45, Kbb ¼ 50.27, Kcc ¼ 57.34
[38]
Kaa ¼ 45.19, Kbb ¼ 49.49, Kcc ¼ 57.09
[280,318]
Kaa ¼ 39.58, Kbb ¼ 47.03, Kcc ¼ 54.33
[228]
DK ¼ 10.15, Kxx 2 Kyy ¼ 51.29, K ¼ 45.87i
[227]
Kaa ¼ 1.6, Kbb ¼ 2.6, Kcc ¼ 7.2
[404,405]
Kaa ¼ 20.4, Kbb ¼ 20.8, Kcc ¼ 3.0
[404,405]
Kaa ¼ 20.4, Kbb ¼ 21.0, Kcc ¼ 21.6
[404,405]
DK ¼ 2.00, K ¼ 46.82c
[406]
DK ¼ 9.53, K ¼ 80.42c
[321]
DK ¼ 53.61, K ¼ 78.47c
[227]
DK ¼ 11.72, K ¼ 98.95c
[406]
DK ¼ 22.43, Kaa ¼ 73.90, Kbb ¼ 74.10, Kcc ¼ 81.72
[80]
DK ¼ 21.07, Kaa ¼ 22.33, Kbb ¼ 24.78, Kcc ¼ 25.05
[80]
Kaa ¼ 12.93, Kbb ¼ 14.95, Kcc ¼ 27.65l
[40]
DK ¼ 221.34, K ¼ 41.41
[227]
DK ¼ 29.33, Kaa ¼ 33.49, Kbb ¼ 38.22, Kcc ¼ 60.50
[40]
DK ¼ 9.84, K ¼ 135.96c
[40]
DK ¼ 216.33, K ¼ 32.01d,e
[325]
DK ¼ 24.99, K ¼ 64.24d
[325]
DK ¼ 28.36, K ¼ 82.96d
[325]
DK ¼ 29.84, K ¼ 39.80d,m
[325]
Kaa ¼ 22.80, Kbb ¼ 30.39, Kcc ¼ 66.24
[38]
Kaa ¼ 23.74, Kbb ¼ 30.57, Kcc ¼ 66.19
[38]
DK ¼ 19.62, Kxx 2 Kyy ¼ 285.75, K ¼ 40.98i
[227]
DK ¼ 71.40, K ¼ 26.27
[335]
DK ¼ 70.84, K ¼ 27.46
[335]
DK ¼ 51.9, K ¼ 50.0
[143]
DK ¼ 54.68, K ¼ 37.62
[38]
DK ¼ 54.51, K ¼ 37.66
[38]
DK ¼ 256.63, K ¼ 20.75
[227]
Kaa ¼ 24.46, Kbb ¼ 17.05, Kcc ¼ 121.69
[38]
Kaa ¼ 6.21, Kbb ¼ 19.28, Kcc ¼ 121.30
[38]
DK ¼ 216.29, K ¼ 215.82
[335]
DK ¼ 206.10, K ¼ 4.81
[335]
DK ¼ 140.05, K ¼ 34.07
[38]
DK ¼ 138.43, K ¼ 34.69
[38]
Kaa ¼ 228.98, Kbb ¼ 18.56, Kcc ¼ 191.72
[38]
Kaa ¼ 8.44, Kbb ¼ 23.22, Kcc ¼ 190.97
[38]
(continued on next page)
11
Reference
288
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 17 (continued)
a
System
X
Bonds
Theory
Values
Reference
HI
HI
HI
HI
PbH4
127
1
1
1
1
1
DHF/(s )
SCF/(s )
CAS þ SOh/HIVu3
CAS/HIVu3
ZORA DFT(LDA)/Slater
DK ¼ 369.82, K ¼ 2113.20
DK ¼ 340.17, K ¼ 212.97
DK ¼ 216.57, K ¼ 40.22
DK ¼ 213.78, K ¼ 41.01
DK ¼ 672, K ¼ 1121c
[335]
[335]
[38]
[38]
[144]
I
I
127
I
127
I
207
Pb
127
Either the principal values of the symmetrized tensor (Kii, i ¼ a; b; c with lKaa l # lKbb l # lKcc l), specified Cartesian components (xx, xy,
etc.), or the anisotropy with respect to a certain axis (e.g. Dz K ¼ Kzz 2 1=2ðKxx þ Kyy Þ), 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.
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 D1 JCC 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 contribution 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 properties. 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 properties 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
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 18
Quantum chemically calculated components of the symmetric part J1 þ J S of the
a
289
13 13
C C spin–spin coupling tensors
System
Bonds
Theory
Values
Reference
C2H2
C2H2
C2H2
C2H2
C2H4
C2H4
C2H4
C2H6
C2H6
H2CyCyCH2
HCxC–CxCH
HCxC–CxCH
C6H6
CH3CN
H2CyCyNH
CH2CO
OCyCyCO
C6H6
CH3NC
C6H6
1
1
1
1
1
1
1
1
1
1
1d
1e
1
1
1
1
1
2
2
3
SOPPA(CCSD)/(s )
RAS/HIV
EOM/6-31G**
ZORA DFT(GGA)/Slater
RAS/HIV
EOM/6-31G**
ZORA DFT(GGA)/Slater
RAS/HIII
ZORA DFT(GGA)/Slater
EOM/6-31G**
EOM/6-31G**
EOM/6-31G**
RAS/HII
RAS/HIII
EOM/6-31G**
EOM/6-31G**
EOM/6-31G**
RAS/HII
RAS/HIII
RAS/HII
DJ ¼ 49.55, J ¼ 190.00
DJ ¼ 47.5, J ¼ 181.2
DJ ¼ 24.32, J ¼ 216.99
DJ ¼ 72.1, J ¼ 186.6
DJ ¼ 26.5, Jxx 2 Jyy ¼ 244.3b, Jaa ¼ 39.2, Jbb ¼ 83.6, Jcc ¼ 87.9
DJ ¼ 1.29, J ¼ 82.37c
DJ ¼ 38.8, J ¼ 59.2c
DJ ¼ 32.1, J ¼ 38.8
DJ ¼ 34.0, J ¼ 23.8
DJ ¼ 25.39, J ¼ 109.64c
DJ ¼ 24.62, J ¼ 225.87
DJ ¼ 3.61, J ¼ 157.92
DJ ¼ 11.0, Jaa ¼ 44.9, Jbb ¼ 78.2, Jcc ¼ 89.5
DJ ¼ 36.57, J ¼ 71.97c
DJ ¼ 11.19, J ¼ 111.21c
DJ ¼ 212.74, J ¼ 112.43c
DJ ¼ 6.36, J ¼ 221.07
DJ ¼ 212.7, Jaa ¼ 20.6, Jbb ¼ 20.8, Jcc ¼ 213.5
DJ ¼ 11.64, J ¼ 25.23c
DJ ¼ 12.8, Jaa ¼ 13.3, Jbb ¼ 16.4, Jcc ¼ 27.6
[214,215]
[72]
[407]
[144]
[72]
[407]
[144]
[72]
[144]
[407]
[407]
[407]
[70]
[324]
[407]
[407]
[407]
[70]
[324]
[70]
Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.),
or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), 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
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 J S of the
15
N13C spin– spin coupling tensors
System
Bonds
Theorya
Valuesb
Reference
HCN
HNC
CH3CN
CH3NC
CH3NC
CH3CN
H2CyCyNH
CH2N2
HCNO
HCONH2
1
1
1
1c
1d
2
1
1
1
1
RAS/HIV
RAS/HIV
RAS/HIII
RAS/HIII
RAS/HIII
RAS/HIII
EOM/6-31G**
EOM/6-31G**
EOM/6-31G**
RAS/HIII
DJ ¼ 254.64, J ¼ 219.83
DJ ¼ 250.49, J ¼ 210.47
DJ ¼ 250.70, J ¼ 221.55
DJ ¼ 247.06, J ¼ 212.57
DJ ¼ 217.04, J ¼ 219.26
DJ ¼ 27.66, J ¼ 2.82
DJ ¼ 25.89, J ¼ 231.87e
DJ ¼ 26.92, J ¼ 228.54e
DJ ¼ 29.16, J ¼ 259.16
Jaa ¼ 26.6, Jbb ¼ 214.9, Jcc ¼ 232.4
[324]
[324]
[324]
[324]
[324]
[324]
[407]
[407]
[407]
[404,405]
Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.),
or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), 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
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.
290
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 20
Quantum chemically calculated components of the symmetric part J1 þ J S of the
a
19 13
F C spin–spin coupling tensors
System
Bonds
Theory
Values
Reference
CH3F
CH2F2
1
1
RAS/HIII
RAS/HIII
[78]
[78]
CHF3
p-C6H4F2
1
1
RAS/HIII
RAS/HII(m )
p-C6H4F2
2
RAS/HII(m )
p-C6H4F2
p-C6H4F2
3
4
RAS/HII(m )
RAS/HII(m )
DJ ¼ 207.84, J ¼ 2156.56
DJ ¼ 10.39, Jxx 2 Jyy ¼ 2280.33b, Jaa ¼ 236.62,
Jbb ¼ 2261.19, Jcc ¼ 2364.34
DJ ¼ 2173.34, Jaa ¼ 25.47, Jbb ¼ 2333.49, Jcc ¼ 2387.25
DJ ¼ 368.8, Jxx 2 Jyy ¼ 11.5c, Jaa ¼ 61.1, Jbb ¼ 2301.9,
Jcc ¼ 2313.4
DJ ¼ 236.9, Jxx 2 Jyy ¼ 219.4c, Jaa ¼ 215.8, Jbb ¼ 64.5,
Jcc ¼ 78.8
DJ ¼ 37.5, Jxx 2 Jyy ¼ 29.1c, Jaa ¼ 2.2, Jbb ¼ 223.5, Jcc ¼ 31.9
DJ ¼ 219.2, Jxx 2 Jyy ¼ 234.0c, Jaa ¼ 23.2, Jbb ¼ 25.4,
Jcc ¼ 30.8
[78]
[60]
[60]
[60]
[60]
Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.),
or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), 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
See footnote g in Table 13.
c
See footnote d in Table 16.
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 properties of certain type of couplings between different
molecules containing similar structural units.
Kaski et al. [70] investigated the n JCH
ðn ¼ 1; 2; 3; 4Þ and n JCC ðn ¼ 1; 2; 3Þ coupling tensors
in benzene using CASSCF and RASSCF LR calculations. 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, convergence of the calculated results towards the experimental 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 D1 JCC : For the n JCC 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 21
Quantum chemically calculated components of the symmetric parts K1 þ K S of the X13C spin–spin coupling tensors other than those listed in
Tables 14 and 18–20
System
X
Bonds
Theorya
Values
Reference
HCONH2
CH2CO
OCyCyCO
CH3SiH3
17
1
1
1
1
RAS/HIII
EOM/6-31G**
EOM/6-31G**
RAS/HIII
Kaa ¼ 237.3, Kbb ¼ 42.2, Kcc ¼ 2157
DK ¼ 101.1, K ¼ 276.88b
DK ¼ 124.1, K ¼ 291.28
DK ¼ 98.76, K ¼ 100.7
[404,405]
[407]
[407]
[80]
O
O
17
O
19
Si
17
Either the principal values of the symmetrized tensor (Kii, i ¼ a; b; c with lKaa l # lKbb l # lKcc l), specified Cartesian components (xx, xy,
etc.), or the anisotropy with respect to a certain axis (e.g. Dz K ¼ Kzz 2 1=2ðKxx þ Kyy Þ), 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
See footnote d in Table 13.
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
291
Table 22
Quantum chemically calculated components of the symmetric parts K1 þ K S of the NX spin–spin coupling tensors other than those listed in
Tables 15 and 19
System
X
Bonds
Theorya
Values
Reference
BH2NH2
BH3NH3
CH2N2
N2H4
HCNO
HCONH2
PH2NH2
11
1
1
1
1
1
2
1
SOS-CI/(s )
SOS-CI(s )
EOM/6-31G**
SOS-CI(s )
EOM/6-31G**
RAS/HIII
SOS-CI(s )
DK ¼ 30.98, K ¼ 101.4b,c
DK ¼ 20.7, K ¼ 18.2b,c
DK ¼ 104.8, K ¼ 39.1b
DK ¼ 91.33, K ¼ 4.4b,c,d
DK ¼ 124.4, K ¼ 2171.9
Kaa ¼ 30.6, Kbb ¼ 239.1, Kcc ¼ 41.6
DK ¼ 179.3, K ¼ 230.42b,c,e
[325]
[325]
[407]
[325]
[407]
[404,405]
[325]
B
B
15
N
15
N
17
O
17
O
31
P
11
Either the principal values of the symmetrized tensor (Kii, i ¼ a; b; c with lKaa l # lKbb l # lKcc l), specified Cartesian components (xx, xy,
etc.), or the anisotropy with respect to a certain axis (e.g. Dz K ¼ Kzz 2 1=2ðKxx þ Kyy Þ), 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
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.
the ð1=2ÞJ aniso contribution to Dexp
CC 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 2 JCH 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 (HCONH 2 ) 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 J aniso gives
a negligible contribution to Dexp :
The prototypical hydrocarbon series ethane, ethene, 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 1 JCC 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
D1 JCC (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 D1 JCC evolve from the SD/
FC dominance in the sp3 carbons to the large PSO
term of the sp1 case.
292
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 23
Quantum chemically calculated components of the symmetric parts K1 þ K S of the 19FX spin– spin coupling tensors other than those listed in
Tables 16 and 20
System
X
Bonds
Theorya
Values
Reference
LiF
BF
OF2
ClF3
CH2F2
7
Li
B
17
O
19
F
19
F
1
1
1
2c
2
CAS/cc-pV5Z
CAS/cc-pV5Z
RAS/cc-pCVQZ
RAS/cc-pVQZ
RAS/HIII
[143]
[143]
[27]
[27]
[78]
CHF3
p-C6H4F2
19
2
5
RAS/HIII
RAS/HII(m )
NaF
AlF
ClF
ClF
ClF3
ClF3
KF
BrF
IF
TlF
23
1
1
1
1
1g
1h
1
1
1
1
CAS(s )
CAS/aug-cc-pVQZ
CAS/aug-cc-pVQZ
ZORA DFT(GGA)/Slater
RAS/cc-pVQZ
RAS/cc-pVQZ
CAS/(s )
ZORA DFT(GGA)/Slater
ZORA DFT(GGA)/Slater
ZORA DFT(GGA)/Slater
DK ¼ 240.25, K ¼ 45.28b
DK ¼ 129.4, K ¼ 261.3
Kaa ¼ 84.1, Kbb ¼ 286.7, Kcc ¼ 607
Kaa ¼ 6.4, Kbb ¼ 24.4, Kcc ¼ 83.1
DK ¼ 224.64, Kxx 2 Kyy ¼ 213.11d, Kaa ¼ 16.11, Kbb ¼ 34.19,
Kcc ¼ 47.29
DK ¼ 221.81, Kaa ¼ 214.07, Kbb ¼ 17.82, Kcc ¼ 39.21
DK ¼ 23.40, Kxx 2 Kyy ¼ 23.58e, Kaa ¼ 20.23, Kbb ¼ 1.37,
Kcc ¼ 4.95
DK ¼ 165.1, K ¼ 64.8
DK ¼ 188.4, K ¼ 2213
DK ¼ 2721, K ¼ 747f
DK ¼ 2982, K ¼ 872
Kaa ¼ 211.7, Kbb ¼ 298.3, Kcc ¼ 638
Kaa ¼ 83, Kbb ¼ 2167, Kcc ¼ 528
DK ¼ 207.3, K ¼ 148.0b
DK ¼ 22123, K ¼ 1886
DK ¼ 22955, K ¼ 2241
DK ¼ 2324, K ¼ 22034
11
F
F
19
Na
Al
35
Cl
35
Cl
35
Cl
35
Cl
39
K
79
Br
127
I
205
Tl
27
[78]
[60]
[143]
[143]
[143]
[144]
[27]
[27]
[143]
[144]
[144]
[144]
Either the principal values of the symmetrized tensor (Kii, i ¼ a; b; c with lKaa l # lKbb l # lKcc l), specified Cartesian components (xx, xy,
etc.), or the anisotropy with respect to a certain axis (e.g. Dz K ¼ Kzz 2 1=2ðKxx þ Kyy Þ), 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 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.
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 semiquantitative, with particularly 1 JSiC and D1 JSiC overand underestimated, respectively. This may be partially due to neglecting correlation of the Si semicore
aniso
would
orbitals [240,257]. The neglect of 1 JSiC
correspond to only a 1% error in the CSi bond length.
Ref. [60] investigated the couplings to 19F in pC6H4F2 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 correlation 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 1 JFC 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
J aniso to the experimentally observable long-range
3;4 exp
DFC and 5 Dexp
FF couplings was estimated to exceed
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
293
Table 24
Quantum chemically calculated components of the symmetric parts K1 þ K S of the XY spin–spin coupling tensors other than those listed in
Tables 13 –23
System
XY
Bonds
Theorya
Values
Reference
B2H4
B2H6
BH2PH2
BH3PH3
Na2
KNa
P2H4
(CH3)3PSe
TlCl
TlBr
TlI
11 11
1
1
1
1
1
1
1
1
1
1
1
SOS-CI(s )
SOS-CI(s )
SOS-CI(s )
SOS-CI(s )
CAS/Partridge
CAS/Partridge
SOS-CI(s )
DFT(GGA)(s )
ZORA DFT(GGA)/Slater
ZORA DFT(GGA)/Slater
ZORA DFT(GGA)/Slater
DK ¼ 19.92, K ¼ 72.19b
DK ¼ 2.47, K ¼ 22.95b
DK ¼ 79.88, K ¼ 77.36b
DK ¼ 50.22, K ¼ 34.81b
DK ¼ 235.5, K ¼ 1480c
DK ¼ 273.1, K ¼ 3230
DK ¼ 111.92, K ¼ 263.33b,d
Kaa ¼ 2375, Kbb ¼ 21127, Kcc ¼ 21137
DK ¼ 2971, K ¼ 22185
DK ¼ 5926, K ¼ 23153
DK ¼ 8911, K ¼ 23818
[325]
[325]
[325]
[325]
[143]
[143]
[325]
[130]
[144]
[144]
[144]
B B
B B
11 31
B P
11 31
B P
23
Na23Na
23
Na39K
31 31
P P
31 77
P Se
35 205
Cl Tl
79 205
Br Tl
127 205
I Tl
11 11
Either the principal values of the symmetrized tensor (Kii, i ¼ a; b; c with lKaa l # lKbb l # lKcc l), specified Cartesian components (xx, xy,
etc.), or the anisotropy with respect to a certain axis (e.g. Dz K ¼ Kzz 2 1=2ðKxx þ Kyy Þ), 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
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.
3%. This situation results from (1) the aromatic
system being able to convey the components of longrange 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 computational 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 JSD
FF should not be neglected.
The question of the value, even the order of
magnitude, of D1 JFC 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 for
1
JFC and 2 JFF in the systems studied.
Bryce and Wasylishen compared CASSCF calculations 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 agreement 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 K FC and
DK SD=FC follow the opposite trend. While the FC
contribution dominates in most (but not all, notably in
294
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
ClF) coupling constants, the dominance of DK SD=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
JFF ðClF3 Þ and JFO ðOF2 Þ with their respective experimental counterparts was very successful, however
a discrepancy was observed for the weighted-average
JClF ðClF3 Þ: Besides the usual possible explanations
(neglect of relativistic effects, rovibrational corrections and solvent modelling), the flexibility of the ccpVQZ basis set used for ClF3 may be inadequate in
the core region.
Other methods. In Ref. [130], 1 JSeP 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, 2 656 Hz (exp.) vs.
2 820 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 SOPPA(CCSD) calculations using good, augmented
basis sets for the prototypical CH4 and SiH4 molecules
[321]. While both the 1 JXH and 2 JHH 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 D1 K 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 J FC
dominates the isotropic couplings, DJ SD=FC is the
largest contribution only in JCH : The PSO mechanism is very important for both D2 JCC and D2 JCH ; and
DSO in D3 JHH : 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 nonrelativistic 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. þ 1% 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 nonrelativistic and spin – orbit corrected CASSCF levels,
using basis sets close to convergence [38]. The 1 KXH
were modified by the SO-corrections towards
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 2 JHH ; 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 D1 KXH increases due to the SO interaction, while
that of 1 KXH decreases.
Autschbach and Ziegler applied their two-component relativistic ZORA DFT method in Ref.
[144]. The agreement of 1 KPbH 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 1 K and
D1 K in the XF (X ¼ Cl, Br, I) series, with K PSO 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 relativistic effects increase both K PSO and lDK PSO l;
the spin – orbit interaction seems to partially cancel
this effect. There is, similarly, a substantial effect of
the scalar relativity on the (small) K FC 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 1 KCC are qualitatively correct in
295
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 1 J and
2
J 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 2 JHH 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 2 JHH : For 2 JHH ; the
antisymmetric components are noted to be very small.
The value of 1 JAOH is even less than the value of
2 A
JHH based on SOPPA, CASSCF LR, and SOPPA(CCSD) calculations [228,318]. The corresponding 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.
296
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
Table 25
Quantum chemically calculated absolute values of the components of the antisymmetric part J A of spin–spin coupling tensors
System
Coupling
Bonds
Theorya
lValuel
Reference
BH2
4
CH4
NH2
2
NHþ
4
H2O
H2O
H2O
H2O
AlH2
4
SiH4
SiH4
PH2
2
PH3
PHþ
4
NH2
2
H2O
H2O
H2O
H2O
PH2
2
PH3
OF2
ClF3
ClF3
H1H
H1H
1 1
HH
1 1
HH
1 1
HH
1 1
HH
1 1
HH
1 1
HH
1 1
HH
1 1
HH
1 1
HH
1 1
HH
1 1
HH
1 1
HH
15 1
NH
17 1
OH
17 1
OH
17 1
OH
17 1
OH
31 1
PH
31 1
PH
17 19
O F
19 19
F F
35 19
Cl F
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
2e
1f
SCF/(s )
SOPPA(CCSD)/(s )
SCF/(s )
SCF/(s )
SOPPA(CCSD)/(s )
CAS/(s )
SOPPA/(s )
SOPPA/(s )
FOPPA/(s )
SOPPA(CCSD)/aug-cc-pVTZ
FOPPA/(s )
SCF/(s )
SCF/(s )
SCF/(s )
SCF/(s )
SOPPA(CCSD)/(s )
CAS/(s )
SOPPA/(s )
SOPPA/(s )
SCF/(s )
SCF/(s )
RAS/cc-pCVQZ
RAS/cc-pCVQZ
RAS/cc-pVQZ
lJxzl ¼ 1.66b
lJxzl ¼ 1.31b
lJxyl ¼ 4.77c
lJxzl ¼ 5.40b
lJxyl ¼ 1.62c
lJxyl ¼ 1.74c
lJxyl ¼ 1.71c
lJxyl ¼ 19.48c
lJxzl ¼ 1.59b
lJxzl ¼ 3.66b
lJxzl ¼ 1.75b
lJxyl ¼ 2.33c
lJxzl ¼ 0.33b, lJyzl ¼ 2.52d
lJxzl ¼ 2.17b
lJxyl ¼ 0.62b
lJxyl ¼ 0.21b
lJxyl ¼ 0.09b
lJxyl ¼ 0.19b
lJxyl ¼ 0.44b
lJxyl ¼ 2.35b
lJxyl ¼ 1.04b
lJyzl ¼ 109b
lJxyl ¼ 292b
lJxyl ¼ 150b
[41]
[321]
[41]
[41]
[318]
[318]
[280,318]
[228]
[406]
[321]
[406]
[40]
[40]
[40]
[41]
[318]
[318]
[280,318]
[228]
[40]
[40]
[27]
[27]
[27]
a
b
c
d
e
f
1
1
Results in Hz.
See footnote a in Table 13.
Component in the local symmetry plane.
With the molecule in the xy plane and the C2 axis along the y direction.
Symmetry plane is xy.
See footnote c in Table 23.
See footnote h in Table 23.
4. Conclusions
An effort has been made to summarize recent
progress in understanding indirect spin – spin coupling, 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 contributions 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-independent molecular structures and orientational order
parameters, and second, the separation of the often
minute indirect anisotropic contribution, ð1=2ÞJ aniso ;
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=2ÞJ aniso is
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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, fluoromethanes, methyl cyanides and methylsilane, as well
as fluorobenzene. In particular, the methodological
progress is exemplified in the classic question of the
value of D1 JFC in monofluoromethane, where modern
LCNMR techniques have narrowed the range of
experimental results to 350 – 400 Hz from the unrealistic 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=2ÞJ aniso
may even dominate Dexp and, hence, introduce large
uncertainty in the structural and orientational order
parameters.
The amount of data obtained from NMR measurements on solid samples has increased enormously;
however, there are still problems associated with
correcting measured effective dipolar coupling tensors 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,
297
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 Cederberg and co-workers where the vibrational dependence of spin – spin coupling constants is measured to
a precision of better than 1 Hz. For example, in the
case of CsF, J133 Cs19 F ¼ 0:62745ð30Þ 2 0:00903ð22Þ
£ ðn þ 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 associated with the experimental values are larger than for
the isotropic part. Continued efforts in ab initio
calculations of J are very well motivated. Systematically improving methods provide reliable benchmarks 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 promising; however, further benchmarking studies are
still necessary. The available DFT exchange-correlation functionals have not been parametrized for
298
J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304
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 significance 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, intermolecular 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,
Renée Siegel, Takahiro Ueda, and Se-Woung Oh. JV
is on leave from the NMR Research Group, Department 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, Yrjö, and Kalle Väisälä 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, Dalhousie University, the Izaak Walton Killam Trust, and the
Walter C. Sumner Foundation for postgraduate
scholarships.
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