1338
NMR line shapes from AB spin systems
in solids — The role of antisymmetric spin–spin
coupling
Kristopher J. Harris, David L. Bryce, and Roderick E. Wasylishen
Abstract: NMR parameters such as indirect nuclear spin–spin coupling (J), nuclear magnetic shielding (s), direct dipolar
coupling (D), and electric field gradient (V) are properly described by second-rank tensors. Each may be decomposed into
isotropic, symmetric, and antisymmetric components; the number of these three components which may be nonzero is a distinguishing attribute of each interaction tensor. The rank-1 antisymmetric portion of J (Janti) holds the distinction of remaining the only nonzero part of these fundamental NMR interaction tensors which has never been observed experimentally.
Accordingly, effects from Janti are usually ignored, but it is important to consider when this is valid. An experimental strategy for observing Janti in powdered samples of tightly coupled homonuclear spin pairs, based on ideas originally presented
by Andrew and Farnell (Mol. Phys. 1968, 15, 157), is described. The theory of Andrew and Farnell is extended to powder
samples, and methods for analyzing NMR spectra from powdered samples are presented. It is found that, in certain rare
cases, Janti has the potential to affect the NMR line shapes from AB spin systems, but that even in these systems, the most
intense features of the spectra are not affected and may be analyzed independently of Janti. Furthermore, Janti will only have
an observable effect on the NMR spectra when its magnitude is comparable with that of Jiso and with the difference in
chemical shifts (in Hz) between the two sites. Finally, the first experimental attempts to measure Janti are reported, and experimental proof that no elements of Janti(119Sn,119Sn) in hexa(p-tolyl)ditin are larger than 2900 Hz is given. The benefits of
modern double-quantum filtering NMR pulse sequences in isolating effects from Janti are also illustrated.
Key words: J coupling, solid-state NMR spectroscopy, indirect nuclear spin–spin coupling,
relaxation, antisymmetric tensor.
119Sn
NMR, symmetry, NMR
Résumé : Les interactions de RMN, tel le couplage spin-spin nucléaire indirect (J), le blindage magnétique nucléaire (s), le
couplage dipolaire direct (D) et le gradient du champ électrique (V) peuvent être décrits correctement par des tenseurs de
deuxième rang. Chacune peut être décomposée en composants isotropes, symétriques et antisymétriques; le nombre de ces
trois composants qui peut être différente de zéro est un attribut caractéristique de chaque tenseur d’interaction. La portion antisymétrique de rang 1 de J (Janti) se distingue par le fait qu’il s’agit de la seule partie de ces tenseurs fondamentaux d’interaction RMN dont la valeur est différente de zéro qui n’a pas encore été observée expérimentalement. Il en découle qu’on
ignore généralement les effets attribuables à Janti, mais il est important de considérer quand cela est valide. On décrit une stratégie expérimentale basée sur les idées proposées originalement par Andrew et Farnell (Mol. Phys. 1968, 15, 157) pour observer Janti dans des échantillons en poudre de paires de spin homonucléaires fortement couplés. La théorie de Andrew et
Farnell a été étendue aux échantillons en poudre et on présente des méthodes qui permettent d’analyser les spectres RMN des
échantillons en poudre. On a trouvé que dans certains rares cas, le couplage Janti a le pouvoir d’affecter les formes des raies
RMN des systèmes de spin AB, mais que même avec ces systèmes, les caractéristiques les plus intenses des spectres ne sont
pas affectées et elles peuvent être analysées indépendamment de Janti. De plus, Janti n’aura un effet observable sur les spectres
RMN que lorsque son amplitude sera comparable à celle de Jiso et à la différence dans les déplacements chimiques (en Hz)
entre les deux sites. Finalement, on rapporte les premiers efforts experimentaux à mesurer Janti, et on démontre experimentallement qu’il n’y a pas d’éléments de Janti(119Sn, 119Sn) plus grand que 2900 Hz dans l’hexa(p-tolyl)di-étain. On illustre aussi
les bénéfices des séquences modernes d’impulsions RMN avec filtrage à double quantum pour isoler les effets de Janti.
Mots-clés : couplage J, spectroscopie RMN à l’état solide, couplage spin-spin nucléaire indirect, RMN du
RMN, tenseur antisymmétrique.
119Sn,
relaxation
[Traduit par la Rédaction]
Received 14 March 2009. Accepted 5 May 2009. Published on the NRC Research Press Web site at canjchem.nrc.ca on 16 September
2009.
This article is part of a Special Issue dedicated to Professor T. Ziegler. This article is dedicated to Professor Tom Ziegler in recognition
of his many important contributions to science and in particular his central role in the development of methods to calculate NMR
parameters.
K.J. Harris, D.L. Bryce,1 and R.E. Wasylishen.2 Department of Chemistry, University of Alberta, Edmonton, AB T6G 2G2, Canada.
1Present
address: Department of Chemistry, University of Ottawa, Ottawa, ON K1N 6N5, Canada.
author (e-mail: [email protected]).
2Corresponding
Can. J. Chem. 87: 1338–1351 (2009)
doi:10.1139/V09-089
Published by NRC Research Press
Harris et al.
Introduction
One of the most useful aspects of NMR spectroscopy for
relating spectroscopic observables to molecular structure and
dynamics is the inherent orientation dependence of the fundamental NMR interactions.1 This orientation dependence is
routinely taken advantage of in solid-state NMR spectroscopy to determine, for example, internuclear coupling tensors, magnetic shielding tensors, and electric field gradient
tensors. Over the last decade, this orientation dependence
has also been exploited in solution NMR spectroscopy by
employing dilute liquid crystalline solvents which introduce
a small degree of molecular alignment, thereby preventing
complete isotropic averaging of direct dipolar coupling tensors and magnetic shielding tensors.2–7 Clearly, an improved
understanding of the tensor nature of NMR interactions is
desirable.
The most important fundamental NMR parameters in diamagnetic molecules include the electric field gradient (V)
and nuclear magnetic shielding (s), as well as the direct dipolar (D), and indirect nuclear spin–spin (J) coupling, and
are properly represented by second-rank tensors.8 The J tensor may be written in Cartesian form as follows, with up to
nine elements9
2
3
J xx J xy J xz
½1
J ¼ 4 J yx J yy J yz 5
J zx J zy J zz
In general, any second-rank tensor may be decomposed
into three contributions. The rank-0 contribution is independent of orientation, and is given by (for J)
½2
1
1
Jiso ¼ TrðJÞ ¼ ðJxx þ Jyy þ Jzz Þ
3
3
This term is also commonly known as the ‘‘scalar’’ coupling
constant in the NMR spectroscopy literature of both solids
and solutions. The isotropic portions of the V and D tensors
are zero, while the isotropic part of s is simply the nuclear
magnetic shielding constant, siso. The shielding tensor and
the isotropic shielding constant are directly related to the
experimentally observable chemical shift tensor (d) and isotropic chemical shift (diso). Thus, the nonzero isotropic portions of NMR interaction tensors (Jiso, diso) are routinely
measured.
The rank-2 symmetric part of J is given by
½3
1
to PAS
Jsym ¼ ðJ þ Jt Þ Jiso 1
! Jsym;PAS
2
2
3
J11 Jiso
0
0
5
¼4
0
0
J22 Jiso
0
0
J33 Jiso
where Jt is the transpose of J, and ‘‘symmetric’’ refers to
the fact that Jijsym ¼ Jjisym in any axis system. As shown in
eq. [3], when Jsym is represented in its principal axis system
(PAS) there are no off-diagonal elements and the diagonal
elements are referred to as the principal components. Note
that, although there are three principal components, their
sum is zero and Jsym,PAS therefore contains only two independent elements. However, it should be noted that Jsym is
only fully described if its eigenvectors, i.e., the vectors that
1339
form its PAS, are known in addition to the two independent
elements of Jsym,PAS. Because the PAS can be specified via
three Euler angles that relate it to some reference frame in
the molecule, Jsym is generally described as containing five
independent parameters. By definition, the principal components are ordered |J33 – Jiso| ‡ |J11 – Jiso| ‡ |J22 – Jiso|.10,11
The V, D, and s tensors can all have nonzero rank-2 symmetric contributions, but of course this depends on symmetry for V and s. For example, the nuclear quadrupolar
coupling constant (CQ) and asymmetry parameter (hQ) are
characteristic of Vsym; the direct dipolar coupling constant
used to extract internuclear distances from NMR measurements is characteristic of Dsym; anisotropy and asymmetry
parameters describing nuclear magnetic shielding and chemical shift tensors are representations of ssym. Thus, experimental measurements of the symmetric parts of V, D, and s
(or d), are routinely made. Measurements of the principal
components of Jsym are not nearly as straightforward;11,12
nevertheless, much effort has been directed at making such
measurements in the gas phase, in liquid crystal solutions,
and in the solid state so that a body of reliable data now exists.13,14 Quantum chemical calculation of J tensors is an
important tool in the interpretation of experimental results,15–25 and advances in this area are reviewed regularly.13,26–28
The rank-1 antisymmetric contribution to the NMR interaction tensors is zero for V and D, but not for s or J. For
the J tensor, this contribution is given by10,11
½4
1
Janti ¼ ðJ Jt Þ
22
3
0
Jxy Jyx Jxz Jzx
14
Jyx Jxy
0
Jyz Jzy 5
¼
2 J J
J
J
0
zx
xz
zy
yz
2
3
anti
anti
Jzx
0
Jyx
6 anti
anti 7
0
Jzy
4 Jyx
5
anti
anti
Jzx Jzy
0
This component of J is designated antisymmetric because of
the obvious form of the matrix in eq. [4]. While Jsym can be
diagonalized through a change of basis, Janti remains in the
form shown in eq. [4] under any choice of coordinate system. Because there is no obvious reference frame for Janti,
the simplest method is to report Janti in the PAS of Jsym;
however, it should be noted that symmetry may require one
or more elements of Janti to be zero in a particular coordinate
system, in which case the tensor will still appear to have 6
entries (three unique) in other coordinate systems (compare
with Jsym versus Jsym,PAS). In summary, the rank-0 portion
of J contains one independent parameter, the rank-2 symmetric portion contains five, and the rank-1 antisymmetric
portion contains three. Any J tensor can therefore be represented in an arbitrary coordinate system using the six independent elements of Jiso + Jsym + Janti, and three Euler
angles relating the coordinate system to the PAS of Jsym.
It has been shown that the antisymmetric components of s
affect the spectrum in second order only, causing small peak
shifts and (or) subtle pattern changes.10,29,30 While direct effects on the spectra would generally be very small and thus
difficult to detect, the components of santi can contribute to
nuclear spin relaxation, but only two reports of their influPublished by NRC Research Press
1340
Can. J. Chem. Vol. 87, 2009
ence on relaxation have been published.31,32 Wi and Frydman have shown that cross-correlations with the quadrupolar
Hamiltonian can make direct detection of santi more tenable,
and have reported the antisymmetric components of the 59Co
chemical shift tensor in cobalt(III) tris(acetylacetonate), referenced to 1M K3[Co(CN)6](aq), as danti
zx = 1000 ± 500 ppm,
anti
33
danti
zy = –1000 ± 500 ppm, dxy = 1000 ± 1000 ppm. Nuclear
spin relaxation measurements, similar to those used for santi,
could provide a method to detect the antisymmetric part of
J,34,35 while another experimental strategy may arise from
the symmetry-based pulse-sequence design methodology developed by Levitt and co-workers.36
The rank-1 antisymmetric part of the J tensor remains as
the only contribution to the four fundamental NMR interactions which has yet to be measured experimentally. In 1968,
Andrew and Farnell tackled this problem by considering a
pair of spin-1/2 nuclei in a single crystal, taking into account
the full second-rank tensor nature of the chemical shift, the
direct dipolar coupling tensor, and the indirect nuclear spin–
spin coupling tensor.37 Under rapid MAS, the chemical shift
tensors are averaged to their isotropic values for each of the
two nuclei involved; the direct dipolar coupling tensor is
averaged to its isotropic value (zero); the isotropic J-coupling
constant manifests itself in a fashion analogous to the situation observed in solution, and the anisotropic rank-2
part of the J tensor is averaged to zero. The most interesting aspect of their analysis, however, is with regards to the
antisymmetric rank-1 part of J, which is averaged to zero
in solution, but not under MAS conditions.37,38 The theory
laid out by Andrew and Farnell alone does not provide
much insight into what types of chemical systems may actually lead to an experimental observation of Janti, and a
discussion of such considerations is presented here. Additionally, the theory is extended to include averaging over
the orientations of a micro-crystalline powdered sample,
and methods for analyzing the resultant line shapes are
presented. In the present paper, the discussion describes
one strategy for the measurement of Janti and details some
initial experimental attempts at its characterization.
Theory
The theory for describing NMR spectra from a pair of
non-equivalent spin-1/2 nuclei in solution or in the solid
state without effects from Janti is well established.39–41
The full Hamiltonian operator for a homonuclear pair of
spin-1/2 nuclei in a stationary sample is
½5
b ¼ ð2pÞ1 gbIð1 sI ÞB0 ð2pÞ1 g b
Sð1 ss ÞB0
h1 H
b
þI Jb
S þ bI D b
S
After breaking the interaction tensors into their isotropic,
symmetric, and antisymmetric components, and dropping
nonsecular42 terms, the effects of MAS in averaging orientation dependencies can be included. It is well known that all
terms in the secular Hamiltonian containing Jsym and D
average to zero, and also that the secular part of Î sI B0
is averaged to sIÎZ when B0 is along the laboratory Z axis;37
this last result uses sI as notation for the isotropic shielding
(i.e., 1/3 the trace of sI). Therefore, the secular Hamiltonian
under MAS can be defined as
½6
MAS
b0
h1 H
¼ nL ½ð1 s I ÞbI Z þ ð1 s S Þb
SZ
1
S Z þ Jiso ðbI þb
S þ bI b
SþÞ
þ JisobI Zb
2
S
þ bI Janti;MAS b
where nL =(2p)–1gB0, and Janti,MAS is the MAS average of
Janti, and an infinitely fast MAS rate is assumed. Line-shape
simulations for solid samples generally neglect the term
containing Janti,MAS; however, Andrew and Farnell have presented a theory including effects from the antisymmetric
portion of J on solid-state NMR spectra from a pair of
spin-1/2 nuclei in a single crystal undergoing MAS.37 We
first review Andrew and Farnell’s results for NMR spectra
from a single crystal, and then extend the theory to include
the orientational distribution of a powder sample. Methods
for analyzing the resultant powder pattern in terms of the
components of Janti are then discussed.
The Antisymmetric J-coupling Hamiltonian
Returning to a stationary single crystal in the lab frame,
the only secular terms in the antisymmetric J-coupling Hamanti;L
iltonian between spins I and S are the two involving JYX
:
b
b anti ¼ bI Janti;L S
h1 H
anti;L b b
1 b anti
SXÞ
h H 0 ¼ JYX ðI X S Y bI Yb
½7
1 anti;L b b
SþÞ
¼ i JYX ðI þ S bI b
2
b anti
contains only the flip-flop operator involving
Because H
0
spins I and S, it has a negligible effect on the energy levels
unless the spins I and S have nearly equal chemically
shifted Zeeman interactions; accordingly, Janti only affects
NMR spectra from AB spin systems. The only step remaining before we can predict spectra using the Hamiltonian in
anti;L
.
eq. [6] is determination of the MAS average of JYX
Under MAS, the orientation of each crystallite will become time-dependent, so it is convenient to consider the tensor in the lab frame, Janti,L, as being derived from its
representation in the crystallite axis system depicted in
Fig. 1a, Janti,C, using direction cosines:43
2
anti;C 3
ðaYx aXy aYy aXx ÞJyx
anti;L
anti;C 5
½8
JYX
¼ 4 þðaYx aXz aYz aXx ÞJzx
anti;C
þðaYy aXz aYz aXy ÞJzy
In eq. [8], each aUv is the cosine of the angle between labframe axis U and crystallite axis v. If some external motion
causes the molecular orientation to change, each direction
cosine will be time-dependent. After including expressions
for the time dependence of each aUv, the average value of
anti;L
JYX
under external motion can be derived (see ref. 44 for
an instructive example where U is the laboratory Z axis and
the external motion is MAS). Under isotropic tumbling, the
anti;L
is zero, but not for a solid undergoing
time average of JYX
MAS.37 It is convenient to use the notation A(Q) for the
anti;L
under MAS, because it acts as an efaverage value of JYX
fective antisymmetric coupling constant which differs for
each orientation, Q, of the crystallite.37 The effect of Janti is
therefore to add the following term to the secular Hamiltonian for an AB spin system under MAS:
Published by NRC Research Press
Harris et al.
1341
Fig. 1. (a) Coordinate system and conventions used in the expressions for the A(Q) distribution. The lab frame is defined by the axes
X, Y, Z; the axis of sample rotation is rrot, which is at the ‘‘magic
angle’’ of cos–1(1/H3) relative to the Z axis; and the crystallite axis
system in which J is represented is described with the vectors x, y, z.
Angles given as xa are those between axis a of the crystallite frame
and rrot. (b) Normalized projections of the crystallite axes x, y, z on
the plane perpendicular to rrot shown in (a) are labelled px, py, pz.
Rotation of these projected vectors by –908 about rrot are annotated
as the corresponding primed vectors px0 , py0 , pz0 , and angles given as
3ab’ are those between pa and pb0 (3yx’ is shown as an example).
that the value of A(Q) for each crystallite is independent of
time. Furthermore, the rotational independence of the expression requires that any crystallites whose initial orientations are related to each other by rotation around the
sample’s spinning axis are characterized by the same value
of A(Q). However, any crystallites whose orientations do
not become superimposed by the motion of the rotor will
exhibit a different value of A(Q), and a powder will therefore be characterized by a distribution of these effective
coupling constants.
It is interesting to contrast the effects of MAS on Janti with
its effect on the other anisotropic terms in the secular Hamiltonian. By defining ssym analogously to the definition of Jsym
in eq. [3], the only orientation dependence, discounting
anti;L
JYX
, is contained in terms involving ssym, Jsym, and D. For
these more familiar terms, the secular Hamiltonian’s dependence on crystallite orientation is encoded in direction cosines
between the applied magnetic field and the principal component axes of the involved tensor, i.e., terms of the form aZv.37
Because the average squared value under MAS (or in solution) for this form of direction cosine, ha2Zv i, is 1/3 for any
crystallite orientation, the average of each secular Hamiltonian term involving ssym, Jsym, and D is zero.37,44 This
averaging explains why these three terms need not be included in the Hamiltonian, and do not affect spectra of samples undergoing MAS (at a sufficiently rapid rate). All of the
orientation-dependent terms in the secular Hamiltonian other
b anti
depend on the angles between each crystallite axis
than H
0
anti
b 0 instead depends on the
and the lab-frame Z axis, while H
angles between each crystallite axis system and the lab-frame
X and Y axes. Because the direction cosines describing the
anti;L
are of a different form, the effect of MAS on
value of JYX
anti
J differs from its effect on ssym, Jsym, and D; accordingly,
anti;L
the average value of JYX
under MAS is not zero, but a constant, A(Q), that differs for each nondegenerate crystallite
orientation.
½9
anti;MAS
b0
h1 H
1
S bI b
SþÞ
¼ i AðQÞðbI þb
2
Andrew and Farnell have provided an expression for the
effective antisymmetric coupling constant for a crystallite
undergoing MAS, see eq. [39] of reference 37. Application
of a few algebraic manipulations produces an analogous,
but slightly simpler, form for A(Q):
2
anti;C 3
ðsin xy sin xx cos 3yx0 ÞJyx
pffiffiffi
anti;C 5
½10
AðQÞ ¼ 1= 34 þð sin xz sin xx cos 3zx0 ÞJzx
anti;C
þð sin xz sin xy cos 3zy0 ÞJzy
Equation [10] makes use of the coordinate system and definitions shown in Fig. 1. As described in the previous section, the components of Janti,C are those in the crystallite
reference frame shown in Fig. 1, which could be, for example, the PAS of Jsym. While the depiction in Fig. 1 is for the
crystallite axis system at time zero, all angles in eq. [10] are
constant during the motion of the rotor, reflecting the fact
The A(Q) distribution
To predict NMR spectra using the above Hamiltonian operator it is necessary to determine the shape of the A(Q) distribution, and this was investigated here using numerical
powder averaging. To investigate the behaviour of A(Q) independent of the magnitude of the Janti,C components, it is
convenient to scale eq. [10] by the largest component of
anti
is the largest elJanti,C. Considering first the case when Jzy
ement,
2
3
ðsin xy sin xx cos 3yx0 Þcyx
pffiffiffi
anti;C
½11
AðQÞ ¼ Jzy ð1= 3Þ4 þðsin xz sin xx cos 3zx0 Þczx 5
þðsin xz sin xy cos 3zy0 Þ
anti;C
¼ Jzy
fzy ðQ; cyx ; czx Þ
where fzy(Q; cyx, czx) is a function of the crystallite orientation, which is again denoted by Q. The function fzy(Q; cyx,
czx) involves only the ratio of elements in Janti,C through the
anti;C
; note that while the theoretical
parameters cij ¼ Jijanti;C =Jzy
range for these parameters is –1 £ cij £ 1, all results were
found to be independent of the sign of the cij constants. See
the Experimental section for specific details on the numeriPublished by NRC Research Press
1342
Fig. 2. (a) Numerically generated plots of fzy(Q; cyx, czx) versus fraction of crystallites displaying that value; (i) The smallest range of the
fzy distribution, ±1/H3, is found with the parameters cyx = czx = 0;
(ii) The largest range of the fzy distribution, ±1, occurs when cyx =
czx = 1; (iii) Example of the range found for the fzy distribution
with intermediate values of the cij constants, in this plot, cyx = 0
and czx = 0.8. (b) Plot of f 2zy ðQ; cyx ; czx Þ, with parameters cyx = czx =
1, the same as in (a) (ii).
Can. J. Chem. Vol. 87, 2009
distribution is displayed in Fig. 2b, where the increasing probability towards zero is apparent. Interestingly, there is a strik2
ing similarity between fzy
ðQ; cyx ; czx Þ and the powder pattern
for an isolated spin-1/2 nucleus with an axially symmetric
magnetic shielding tensor. From the shape of the distribution
in Fig. 2b, and the range of Amax noted above, it may be seen
that the most probable value of A2 is zero, and the observed
values of A2(Q) will cover the range between 0 and A2max ,
anti;C
anti;C
anti;C
¼ Jzx
¼ Jzy
) and
where A2max is between 1 (when Jyx
anti
1/3 (when two components of J are zero) of the largest
squared component of Janti,C.
NMR spectra from a single crystal
From the above considerations, the secular Hamiltonian
for one crystallite containing an ‘‘isolated’’ AB spin pair
under fast MAS averaging must incorporate the usual terms
involving isotropic nuclear magnetic shieldings for both nuclei and the isotropic J-coupling, and, additionally, must include the antisymmetric J-coupling term defined above
½12
cal calculations. As shown in Fig. 2a, all values of fzy beanti;C
is the only
tween ±1/H3 are equally probable when Jzy
component of Janti,C, while all values in the range ±1 are poanti;C
anti;C
anti;C
¼ Jzx
¼ Jzy
. For all other
pulated equally when Jyx
values of the cij constants, the upper and lower limits of
fzy were found to be intermediate between the extremes
of ±1/H3 and ±1 mentioned above, and all values of the distribution were also found equally probable. There appears to
be no obvious connection between the specific limits of fzy
and the cij constants, except in the two cases of cyx = czx = 0
anti;C
or 1. Results obtained under the assumption that either Jyx
anti;C
or Jzx
are the largest component of Janti,C are entirely analogous. In summary, a powdered sample is populated equally
by crystallites displaying all values of A(Q) between –Amax
and +Amax, where Amax is between 1 and 1/H3 of the largest
component of Janti,C.
As will be shown below, values for the transition frequen2
ðQ; cyx ; czx Þ
cies depend on A2(Q) rather than on A(Q). The fzy
MAS
b0
h1 H
¼ nL ½ð1 s I ÞbI Z þ ð1 s S Þb
SZ
1
b
b
SþÞ
þ JisobI Z S Z þ Jiso ðbI þ S þ bI b
2
1
þ i AðQÞðbI þb
S bI b
SþÞ
2
To predict the NMR spectrum resulting from this Hamiltonian, one must first find the energy levels. For spin-1/2
nuclei, taking the Zeeman states as basis functions, the nonzero elements of the Hamiltonian matrix are the diagonal
ones,
2
3
D
E
1
1
b MAS
aajh1 H
jaa ¼ nL 41 ðsI þ sS Þ5 þ Jiso
0
2
4
2
3
D
E
1
1
b MAS
abjh1 H
jab ¼ þnL 4 ðsI sS Þ5 Jiso
0
2
4
2
3
½13
D
E
1
1
b MAS
bajh1 H
jba ¼ nL 4 ðsI sS Þ5 Jiso
0
2
4
2
3
D
E
1
1
b MAS
bbjh1 H
jbb ¼ þnL 41 ðsI þ sS Þ5 þ Jiso
0
2
4
and two off-diagonal entries which contain contributions
from Jiso and A(Q):
D
E 1
b MAS
bajh1 H
jab
¼ ½Jiso iAðQÞ
0
2
½14
D
E
1
b MAS
jba ¼ ½Jiso þ iAðQÞ
abjh1 H
0
2
b MAS
, and their
Both jaai and jbbi are eigenfunctions of H
0
eigenvalues are simply given by the first and last expressions of eq. [13]. The two remaining eigenfunctions can be
obtained in the usual way by diagonalizing the 2 2 submatrix containing the four remaining nonzero matrix elements.41 Resulting from this diagonalization is the set of
energy levels and wave functions given in Table 1, where
Published by NRC Research Press
Harris et al.
1343
Table 1. Energy levels and wave functions for an AB spin system under MAS, including the
effects of Janti.
Wave functiona
j1 ¼ jaai
j2 ¼ Q1 f½Jiso þ iAðQÞjabi þ ½C DS jbaig
j3 ¼ Q1 f½C DS jabi ½Jiso iAðQÞjbaig
j4 ¼ jbbi
Energy levela
E1 ¼ nL 1 12 ðs I þ s S Þ þ 14 Jiso
E2 ¼ 14 Jiso þ 12 C
E3 ¼ 14 Jiso 12 C
E4 ¼ þnL 1 12 ðs I þ s S Þ þ 14 Jiso
mTa
+1
0
0
–1
Note: See ref. 37 for analogous expressions.
a
See text for definitions of constants.
Table 2. Transition frequencies and intensities for an AB spectrum,
incorporating isotropic chemical shifts and the full J tensor.
Transition
v3,4
v1,2
v2,4
v1,3
Frequency
nL 1 12 ðs I þ s S Þ þ 12 C þ 12 Jiso
nL 1 12 ðs I þ s S Þ þ 12 C 12 Jiso
nL 1 12 ðs I þ s S Þ 12 C þ 12 Jiso
nL 1 12 ðs I þ s S Þ 12 C 12 Jiso
Relative intensitya
1 JCiso
1 þ JCiso
1 þ JCiso
1 JCiso
a
See text for definitions of constants.
the following definitions have been used to simplify the expressions:
½15
½16
DS ¼ nL ðsI sS Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 þ A2 ðQÞ þ D2
C ¼ Jiso
s
½17
Q¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2CðC DS Þ
Here, DS is the difference between the magnetic shieldings
of the two sites, measured in Hz. All results reduce to match
those of the classic expressions for an AB spin system in the
limit of A2(Q) = 0.
The allowed transitions are those where mT, the eigenvalue of the operator for the z-component of total spin anguS Z Þ changes by ±1,41 and values of mT
lar momentum ðbI Z þ b
for each wave function are reported in Table 1. Frequencies
for each allowed transition are given in Table 2. Relative intensities for each transition may be determined from the abb ¼ bI þ b
solute squares of the matrix elements of F
S
41
between the involved wave functions, and are also reported
in Table 2. The spectrum expected from a single crystal
undergoing MAS is presented in Fig. 3a, and is analogous
to the textbook example of an AB spectrum in solution
(where Janti has no effect); however, for a solid sample under
MAS, the splitting between the two central lines is increased
when A(Q) is nonzero. It is interesting to note that when
A(Q) is of substantial magnitude compared with Jiso, the relative intensities of the outer two transitions increase relative
to the case when A(Q) is negligible (for the same ratio of Jiso
to the chemical shift difference between the two sites).
NMR spectra from powdered samples
Because each crystallite of a powder has a different value
of A(Q), each will produce a different four-line NMR
spectrum. The separation between the two inner peaks is
C – |Jiso|, and because C is a function of A2, the inner peaks
will form powder patterns. The outer two peaks will also
have a line shape because, in every crystallite, each is separated by Jiso from the respective inner peak. The smallest
separation between the inner two peaks occurs for crystallites with A2(Q) = 0, designated as C = C0 in Fig. 3b, and this
will also be the most intense portion of the line shape as it is
the most probable value of A2(Q). Crystallites with increasing values of A2(Q) will produce NMR spectra with an increasing separation between the inner peaks, but with
decreasing intensity because of the smaller probability of
A2(Q); see Fig. 3b. The two outer peaks of the spectrum of
each crystallite will mimic that of the inner two, producing
the total NMR spectrum given in Fig. 3b.
The form of the spectrum in Fig. 3b illustrates the information that can be derived from it. The spacings between the
most intense parts of the four line shapes are independent of
the antisymmetric components of J, so these can be analyzed
according to the methods used for AB NMR spectra of solution samples.41,45,46 Because the intensity of the outer two
peaks is generally too small to measure, it is usually necessary to obtain NMR spectra at two applied magnetic field
strengths (or from two isotopomers) to separate the value of
Jiso from that of DS. The largest splitting observed between
the inner two transitions of Fig. 3b occurs for crystallites
where C = Cmax, from which eq. [16] can be solved for Amax.
And, as noted above, the absolute value of the largest
pffiffiffi of the
anti,C
is between 1Amax and 3Amax . Inthree components of J
terestingly, it is not necessary to fit the entire line shape, because all of the information comes from the breadth of the
peak. Analysis of the NMR line shape from an AB spin system under MAS therefore provides the values of Jiso and DS,
and a measure of the largest component of Janti,C.
It is worth noting that symmetry requirements on the
number of components in Janti,C, see below, may make it
constants
and thereby repossible to fix the values of the cij p
ffiffiffi
duce the noted range of 1Amax to 3Amax that experiments
can place on the largest component of Janti,C. We also note
Published by NRC Research Press
1344
Fig. 3. (a) General four-line NMR spectrum for a homonuclear AB
spin pair in a single crystal undergoing fast MAS, taking into account isotropic chemical shifts, isotropic J coupling, and antisymmetric J coupling. Adapted from reference 37. Note that for a tightly
coupled spin pair, the two outer lines will be much less intense than
the two central lines. (b) NMR spectrum corresponding to part (a),
but for a powdered sample; C0 is the value of C when A2 = 0, and
Cmax is the value of C when A2 ¼ A2max . Specific data used for this
anti
anti
anti
example are DS = Jiso = 50 Hz, Jzy
= 100 Hz, and Jyx
¼ Jzx
¼ 0.
The zero frequency in both spectra is nL[1–1/2(sI + sS)].
that the above results are independent of the choice used for
the crystallite axis system. The PAS of Jsym can be used, but
any crystallite axis system related by a set of three Euler angles could be chosen; accordingly, the experiment would
provide an upper limit on the largest component of Janti
transformed to the new coordinate system. As the conclusions must remain valid under a change of the crystallite
pffiffiffi coordinate system, the final result is an upper limit of 3Amax
on the largest component of Janti as represented in any coordinate system. No orientation information regarding Janti is
produced by the experiment.
Geometric interpretation of the antisymmetric
component
The above discussion is focussed on deriving the effect of
Janti on the NMR spectrum. It is possible to gain some insight into the character of the antisymmetric component by
using some simple geometric arguments. Consider a system
Can. J. Chem. Vol. 87, 2009
Fig. 4. Depiction of the geometry used in the classical mechanics
example. In the coordinate system shown, the example angular momentum vectors are shifted to the same origin; (a) ~
S along the
x axis, and ~
I along the y axis; (b) the vectors are interchanged, with
~
I now along the x axis and ~
S along the y axis.
in classical mechanics whose energy is governed by the
analogous equation Eclassical ¼ ~
I Jtrial ~
S, where
2
3
0 Jxy 0
½18
Jtrial ¼ 4 Jyx 0 0 5
0
0 0
In this example, only the Jxy and Jyx elements (the only
terms that affect the quantum energy levels) are nonzero. In
the example orientations shown in Fig. 4, the classical energy, Eclassical = JxyIxSy + JyxIySx, is unaltered when the orientations of the two vectors are interchanged if Jtrial is
symmetric, but is reversed if Jtrial is antisymmetric. In general, an indirect coupling tensor will have both symmetric
and antisymmetric components; the symmetric portion encodes the contribution to the energy that is the same when
the two angular momentum vectors are interchanged, and
the antisymmetric component represents a contribution to
the energy that is equal and opposite. Linear combination of
the two components allows the equation to represent any difference in the energetics when the orientation of the two angular momentum vectors is interchanged. A similar
geometric description of Janti has been given by Robert and
Weisenfeld.38
This geometric picture provides an interesting way to interpret Janti; however, it is important to consider which part
of the picture is quantum mechanically relevant. For there to
be an antisymmetric component in J, there does have to be a
difference in energy when the directions of the nuclear spins
are interchanged in the fashion shown in Fig. 4. However,
one does not measure Janti by placing the spins in the two
orientations shown in the Figure; instead, the difference in
the total energy (Hamiltonian operator) terms describing
these orientations causes an extra flip-flop type operator to
be active. Because it is represented by such an operator,
Janti affects the energy levels of a tightly coupled AB spin
system by mixing the jabi and jbai levels.
Selection of appropriate spin systems and
compounds
In this section, we seek to establish some practical requirements for the observation of spectral features induced
by Janti. There are several requirements which must be satisPublished by NRC Research Press
Harris et al.
fied to ensure the feasibility of measuring Janti. Even if all
the requirements are satisfied, there is no guarantee that Janti
will be of significant magnitude to be measured unambiguously. The requirements may be divided into those which
concern the nuclear spin interactions of the nucleus to be observed, and those that describe the structural and symmetry
properties of the molecule to be studied. The following discussion aims to demonstrate that after consideration of all requirements, only a small fraction of molecules are likely to
provide an observation of Janti. The discussion is focussed
on powdered samples; however, measurement of Janti from a
single crystal has many of the same general requirements.
The first requirement is that a chemical system must be
found for which the two nuclei are strongly J coupled and for
which their isotropic chemical shift difference is as small as
possible. This requirement may be understood by considering
the Hamiltonian, eq. [12], where it may be seen that the only
occurrence of A is as the coefficient of a flip-flop type operator. And, in the usual fashion, a flip-flop operator only affects
the energy levels when the two involved spins have Larmor
frequencies which don’t differ by much more than the magnitude of its coefficient. This behaviour is also encoded in the
energy-level expressions through the value of C, eq. [16],
where it may be seen that the influence of A on the spectrum
increases as the magnitude of A increases relative to the
chemical shift difference. Because of the necessity of a small
difference in chemical shifts, a homonuclear spin pair will be
required. A pair of magnetically equivalent nuclei, which
constitute an A2 spin pair and give rise to identical chemical
shifts, is not a suitable spin pair since J coupling between
these nuclei will not manifest itself in the NMR spectrum.
Hence, what is required is an isolated pair of bonded nuclei
which are nearly (but not exactly) crystallographically equivalent, so they will have very similar isotropic chemical shifts,
but are likely to have a large indirect nuclear spin–spin coupling interaction. Since the chemical shift difference is
smaller in Hz at lower applied magnetic field strengths, in
some cases the observation of Janti will only be possible at
low applied magnetic field strengths. Measurement of Janti is
therefore one of the unusual cases where it is advisable to use
the lowest possible external magnetic field strength.
The second requirement within the present strategy is that
one must have a chemical system for which there is an effectively isolated pair of spin-1/2 nuclei. Quadrupolar nuclei
are not feasible candidates due to the typically dominant
quadrupolar interaction; however, it is possible that another
method for measuring antisymmetric J coupling may involve quadrupolar nuclei as the quadrupolar interaction was
used by Wi and Frydman to estimate the antisymmetric part
of the chemical shift tensor.33 Isotopes such as 1H, 19F, and
possibly 203/205Tl will likely not be suitable due to strong homonuclear dipolar couplings preventing them from forming
isolated spin pairs. Lastly, because of the fact that spin pairs
must be observed, the spectral intensity will scale with natural abundance squared, and as such, natural abundance will
play an important role in selecting a nucleus (unless isotopic
labelling is available).
Although Abragam states that the components of Janti are
expected to be negligible relative to the symmetric part of
J,47,48 multiconfigurational self-consistent field calculations
on ClF3 and OF2 have indicated that the magnitudes of the
1345
Table 3. Number of independent components of
Janti for each point group, assuming the two nuclei
are not exchanged by a symmetry operation.38,53
Point group symmetry
about the two
coupled nuclei
C1
Cs, Cn (n > 2)
All others
Number of unique
antisymmetric
components
3
1
0
components of the Janti tensor are comparable to those of
the principal components of the Jsym tensor in these compounds.49 Thus, a spin pair with a large value of Jiso favours
the likelihood of there being an antisymmetric component of
substantial magnitude. This requirement rules out certain
spin pairs including 13C and 15N because in a magnetic field
of 4.7 T, e.g., the value of 1Jiso(13C,13C) would have to be
well in excess of 50 Hz for a spin pair with a chemical shift
difference of 1 ppm. Such a situation is not likely given that
one-bond carbon–carbon coupling constants are typically on
the order of 50 Hz.50 Because Jiso values generally become
larger moving down any group in the periodic table, the
heavier elements present the best chance of possessing observable Janti components.
Practically, one must be concerned with the chemical shift
anisotropy of the nuclei in the AB pair. Heavier nuclei such
as 199Hg and 207Pb tend to have substantial CS tensor
spans.51,52 When carrying out a MAS NMR experiment on a
powdered sample, this fact introduces the additional complication that the spectral intensity is distributed over a large
spectral width in the form of spinning sidebands. Thus, not
only would a large span reduce the effective signal-to-noise
ratio of the NMR spectrum, but it would also complicate the
appearance and interpretation of the spectrum. Hence it is
preferable to find a system for which the nuclei have relatively small CS tensor spans. This requirement is not critical, especially if a single-crystal NMR experiment is to be
performed. However, this discussion is focussed on measurements made using powdered samples.
The molecule under study must be of sufficiently low
symmetry about the bond axis of the coupled nuclei to ensure that the antisymmetric component of J is not forced to
be zero by symmetry. The number of symmetry-allowed
components in Janti under various point groups is presented
in Table 3 (see also references 38 and 53).
While the anisotropy of the symmetric part of the s tensor
may pose a challenge in the acquisition of high-quality
NMR spectra of the AB spin pair, we do not expect the antisymmetric component of s to hamper or influence the measurement of Janti by this method. The effect of santi on NMR
line shapes is generally expected to be very small.10,29,30
However, in some cases it is possible for the inclusion of
extra terms in the Hamiltonian to magnify effects from the
typically nonsecular terms involving santi, e.g., a strong
quadrupolar interaction is known to have such an effect.33
Inspection of the wave functions in Table 1 shows that this
b anti;MAS , as terms in the Hamiltonian
is not the case with H
containing santi remain nonsecular. Accordingly, we do not
expect the predicted effects of Janti to be obscured by the
presence of santi terms.
Published by NRC Research Press
1346
In the present work, the nuclei deemed most suitable for a
definitive measurement of Janti are 29Si, 31P, 77Se, 111/113Cd,
115/117/119Sn, 123/125Te, and 129Xe. There is also a possibility
that some heavier nuclei could provide valuable results, particularly if single crystal spectra are investigated using MAS.
Can. J. Chem. Vol. 87, 2009
Fig. 5. Structure of hexa(p-tolyl)ditin. (a) View along the tin–tin
bond; (b) side view. Hydrogen atoms are omitted for clarity.
Data collection and analysis
Now that the most promising candidates for the observation of homonuclear antisymmetric J coupling have been
narrowed down, suitable compounds must be identified. Tin
seems to be the most promising element due to the high receptivities of the 119Sn and 117Sn isotopes, in addition to
moderate chemical shift anisotropies and well-developed
synthetic chemistry. Tellurium is another promising element,
and some tellurium systems are also discussed.
Tellurium-125 solid-state NMR spectroscopy of diaryl
ditellurium compounds
Tellurium-125 CP MAS NMR spectra were acquired from
p,p’-dimethoxydiphenyl ditelluride, I; p,p’-ditolyl ditelluride,
II; 1-naphthyl ditelluride, III; and diphenyl ditelluride, IV.
One-bond 125Te–125Te coupling constants are generally of
significant magnitude, e.g., 1Jiso(125Te, 125Te) in (Me4N)2Te2
is ±3568 Hz in the solid state,54 making these promising
samples for investigations of Janti. However, measurement
of antisymmetric J coupling in the present case is hampered
in several ways. First, there is a large tellurium chemical
shift anisotropy55 which distributes the powder pattern
across ~1200 ppm. Second, the chemical shift difference between the tellurium sites, e.g., 74.4 ppm or 4698 Hz at 4.7 T
for II, is of the same order of magnitude as the expected
value of Jiso, thereby reducing the ‘‘AB’’ character of the
spin system. The situation is even less promising in the
case of I, for which there are two molecules in the asymmetric unit.56 This distributes the desired 125Te NMR signal
over twice as many sites, thereby reducing the signal-tonoise ratio. In summary, 125Te NMR spectra of these compounds were of such poor quality that the resonances from
the AB spin pairs could not be resolved at 4.7 T (spectra
not shown), and there was therefore no opportunity for detecting whether the components of the AB spectrum possessed a line shape.
Tin-119 solid-state NMR spectroscopy of hexa(ptolyl)ditin
Many ditin compounds of the type R3Sn–SnR3 possess
magnetically equivalent tin atoms which are related by an
inversion centre. Others have C3v symmetry, which by symmetry requires all elements of Janti to be zero.38,53 One candidate that satisfies the stringent requirements is hexa(ptolyl)ditin (V), which has two crystallographically nonequivalent tin atoms57 (Fig. 5). Solution 119Sn NMR in
CDCl3 has provided a value of diso = –141.9 ppm for the
then chemically equivalent tin atoms, and a value of
1J (119Sn,119Sn) of 4570 Hz.57
iso
The 119Sn CP MAS NMR spectrum of solid powdered V
acquired at 7.05 T is presented in Fig. 6a; 119Sn and 117Sn
CP MAS spectra were also obtained at 4.70 T (not shown).
The form of the spectrum, which appears more complicated
than the four-line ideal spectrum presented in Fig. 3, is due
to the fact that the 119Sn isotope is not 100% abundant. We
also note that nearly all of the spectral intensity is concentrated in the centrebands, as 119Sn CP NMR spectra of a stationary sample of V indicate a CS tensor span for tin of less
than 200 ppm. Values for the NMR parameters derived from
these spectra are presented in Table 4. The isotropic chemical shifts of the two Sn sites in V are easily determined
as –142.5 and –154.2 ppm from the peaks of the uncoupled
119Sn nuclei, providing a value for D of 1310 ± 20 Hz. In
S
Fig. 6a, the splittings (i) and (ii) between the outer two pairs
of lines reflect the couplings of sites 1 and 2 to 117Sn nuclei.
These two couplings are found to be the same (as expected),
and the spectra yield a final value of 4175 ± 20 Hz for
1J (119Sn,117Sn) in V. Because isotope effects are expected
iso
to be insubstantial here,58 a value for 1Jiso(119Sn,119Sn) of
Published by NRC Research Press
Harris et al.
Fig. 6. (a) Tin-119 VACP MAS spectrum of hexa(p-tolyl)ditin obtained at 7.05 T. The two dominant peaks result from the uncoupled
nuclei, and are therefore far more intense than peaks from the
coupled isotopomers. The splitting (i) is equal to 1Jiso(119Sn, 117Sn)
for site 1 and the splitting (ii) is 1Jiso(119Sn, 117Sn) for site 2. The
splitting corresponding to C0 – |Jiso(119Sn, 119Sn)| is also marked in
the figure. (b) Tin-119 POST-C7 spectrum of hexa(p-tolyl)ditin also
obtained at 7.05 T, where the double-quantum pulse sequence filters
out all but homonuclear spin pairs, leaving only the central part of
the AB pattern. The edges of the line shape provide a maximum
value for Cmax – |Jiso(119Sn, 119Sn)|, and therefore of the largest
component of Janti.
4368 Hz may be obtained through multiplication by the ratio
of the magnetogyric ratios, g(119Sn)/ g(117Sn).9 The splitting
C0 – |Jiso|, see Fig. 6a, is 140 ± 50 Hz, which agrees within
error with the splitting of 192 ± 7 Hz expected from the
above values of Jiso and DS. Note that for molecules containing two 119Sn nuclei, the two outer transitions are not observed in the 119Sn NMR spectrum shown in Fig. 6a.
Because the integrated intensity of each of the transitions is
approximately 2.2% of each of the inner transitions, it would
be extremely difficult to observe them. It would be possible
to observe effects of Janti solely from the inner transitions;
however, spectral crowding by peaks from the other isotopomers prevents observation of the entire line shape and
could be responsible for obscuring the effects of Janti.
In an effort to determine some line shape for the inner two
peaks, a two-dimensional J-resolved 119Sn NMR spectrum
was obtained at 4.70 T.59,60 In the J-resolved experiment,
heteronuclear dipolar interactions as well as anisotropic
chemical shifts are refocused at the top of the echo, thereby
1347
Table 4. Tin-119 NMR parameters determined for
hexa(p-tolyl)ditin.
Parameter
diso(site 1)
diso(site 2)
Jiso(119Sn,119Sn)
Janti(119Sn,119Sn) elements
Value
–142.5±0.2 ppm
–154.2±0.2 ppm
4368±20 Hz
<2900 Hz
Fig. 7. 2D J-resolved 119Sn CP MAS NMR spectrum of
hexa(p-tolyl)ditin obtained at 4.70 T. The isotropic chemical shifts
of the uncoupled nuclei in sites 1 and 2 are indicated. The splitting
C0 – |Jiso(119Sn,119Sn)| is easily resolved in the second dimension.
providing better resolution in the indirect (F1) dimension.59
Furthermore, because the peaks from the AB pattern are separated along the F1 dimension, this experiment presents the
opportunity to observe some anisotropic line shape that
could be obscured in the one-dimensional spectrum. Shown
in Fig. 7 is a 2D J-resolved 119Sn CP MAS NMR spectrum
of hexa(p-tolyl)ditin. The projection of this spectrum onto
the isotropic dimension corresponds to the one-dimensional
CP MAS spectrum. In the indirect J-resolved dimension, the
splitting C0 – |Jiso(119Sn,119Sn)| is much better resolved than
it is in the isotropic dimension, and therefore provides a
more accurate measurement. The precision of the measurement in the indirect dimension is limited by the number of
points which define the spectrum. In the spectrum shown in
Fig. 7, 16 points define the 200 Hz spectral window in the
indirect dimension, thereby providing one data point every
12.5 Hz. The splitting C0 – |Jiso(119Sn,119Sn)| is 85.0 ±
Published by NRC Research Press
1348
12.5 Hz, which compares favourably with the splitting of
87 ± 5 Hz expected from Jiso and the DS = 875 ± 20 Hz
measured from the 1D spectrum. Unfortunately, when the
signal is spread out into two dimensions, the signal-to-noise
ratio of the inner two peaks of the AB spectrum is reduced
such that any line shape present was not detected.
Another method of decongesting the spectral overlap seen
in Fig. 6a is to use a double-quantum filter, DQF, to remove
any peaks that do not arise from a homonuclear spin pair.
The spectrum shown in Fig. 6b is an example of this method
where we have used the POST-C7 experiment which leaves
only peaks from 119Sn nuclei that are relatively strongly
dipolar coupled to one another.61,62 The spectrum in Fig. 6b
therefore corresponds to that of Fig. 6a, except that only the
inner two peaks from the AB spin system remain. Before
providing a concrete evaluation, we note that Amax only
produces a significant effect on the spectrum when its
magnitude is comparable to both Jiso and DS. For example,
the breadth of the each of the inner peaks of Fig. 3b,
Cmax – C0, would only be 11 Hz if Amax were 450 Hz (10%
of Jiso). Comparing the DQF spectrum to the theoretical
form of Fig. 3b shows that the edges of the peaks in the
DQF spectrum provide an upper limit to Cmax, and therefore
an upper limit on the largest element of Janti. The breadth of
the AB doublet is 500 Hz, which translates into a value of
1700 Hz for Amax, and Amax is from 1 to 1/H3 of the largest
element of Janti. The DQF spectrum therefore provides an
upper limit of 2900 Hz for the largest element of Janti in
compound V.
In cases where a peak shoulder is directly observed, an
exact value of Amax can be extracted; however, only the
peak width can be determined from the presented spectrum
of hexa(p-tolyl)ditin, and therefore only an upper limit on
Amax can be reported. It should also be noted that line broadening from other sources acts to increase the upper limit
provided by this analysis. For example, the sample calculation presented in the preceding paragraph shows that even if
Amax were zero in this compound, line broadening causing
an apparent increase in Cmax – C0 of only 11 Hz would lead
to an upper limit of 450 Hz for Amax. Because of interference from line-broadening effects, the actual value of Janti
elements may be anywhere between 0 and the upper limit
of 2900 Hz. Despite the limitations of analyzing a spectrum
for which the theoretically expected shoulder is not observed, it is interesting that we can place a definite upper
limit on Janti elements.
Can. J. Chem. Vol. 87, 2009
that the most probable frequency, i.e., the highest part of
each peak, is independent of Janti. Therefore, one need not
worry about Janti introducing errors in determinations of the
other NMR observables.
While the above theory shows that it is possible to analyze spectra independently of Janti, the discussion also
presents a strategy for its measurement. The fundamental
reason for measuring the antisymmetric parts of NMR interaction tensors, and indeed for measuring the anisotropic
symmetric parts, is to make use of the fact that up to nine
independent elements are available for a given interaction
tensor (see eq. [1]). Routine use is made of the isotropic
portions of the chemical shift and J-coupling tensors; the
availability of up to eight more parameters to describe a
given interaction signifies the opportunity to provide much
more complete descriptions of the nuclear environment.
The potential existence of sizable antisymmetric contributions to J also has other important implications. For example, the different mechanisms which contribute to J
coupling are listed in Table 5, where it may be seen that
each mechanism possesses different symmetry properties.63
Because Janti contains no contribution from the Fermi-contact (FC) coupling mechanism, measurement of a nonzero
antisymmetric coupling constant would represent some of
the most convincing experimental evidence for non-FC contributions to J. Furthermore, it is noted that in principle the
antisymmetric part of J contributes to nuclear spin relaxation although its effect has never been observed experimentally.34,35 While the experimental strategy described herein is
applicable to tightly coupled homonuclear spin pairs, antisymmetric J coupling is a general phenomenon which will
in principle affect nuclear spin relaxation even in situations
where it cannot be measured directly. That heteronuclear J
couplings may contain important antisymmetric contributions has been demonstrated by high-level quantum chemical calculations on selected systems.49
Table 5. Relationships between the mechanisms which
contribute to J and the symmetry properties of J.63
Mechanism
DSO
PSO
FC
SD
FCSD
Isotropic
ß
Symmetric
ß
Antisymmetric
ß
ß
Experimental and theoretical implications
Given that quantum chemical calculations have shown
that the magnitudes of Janti elements have the potential to
be comparable to those of Jsym,49 it is important to consider
their effect on NMR spectra. Effects from Janti have generally been ignored in spectral analysis, despite the fact that
Andrew and Farnell have predicted measureable effects on
AB spectra from single crystals spinning at the magic angle.37 Aside from interest in Janti itself, it is important to
consider whether the presence of large elements in Janti
could lead to errors when analyzing AB spectra of powders
undergoing MAS. While an observable difference in peak
position is predicted for each crystallite, it is found here
Conclusions
The present work has described an experimental strategy
for measuring the antisymmetric part of indirect nuclear
spin–spin coupling tensors for tightly coupled (AB) homonuclear spin pairs in solid samples undergoing MAS. Considerations for identifying appropriate spin systems and
chemical compounds have been developed, and we also
note that modern quantum-chemical calculations would
likely be helpful in selecting suitable candidate molecules.
Analysis of spectra from samples undergoing MAS, which
averages all interactions other than Janti to their isotropic
values, appears to be an ideal strategy. Janti would affect
Published by NRC Research Press
Harris et al.
spectra from stationary samples, but it would be difficult to
separate Janti from the orientationally dependent Jsym and D
interactions (even with single crystal data). The effect of
Janti components on NMR line shapes from powdered samples undergoing MAS is derived, and the method of spectral
analysis discussed. The main finding is that NMR peak
shapes from AB spin systems will be affected, but that the
most intense portion of the spectra may be analyzed independent of Janti. It is also found that Janti must be comparable in magnitude to Jiso and the difference in chemical shifts
between the two sites to affect the NMR spectra. In particular, the method of analysis allows one to determine a measure of the largest element of Janti. Using this approach, the
119Sn NMR spectrum of hexa(p-tolyl)ditin was analyzed to
show that all elements of Janti in this compound must be
smaller than 2900 Hz.
Experimental details
p,p’-Dimethoxydiphenyl ditelluride, I, (dark red powder),
p,p’-ditolyl ditelluride, II, (orange powder), 1-naphthyl ditelluride, III, (light orange powder), and diphenyl ditelluride,
IV, (orange powder) were obtained from Sigma-Aldrich.
Tellurium-125 one-dimensional VACP MAS NMR spectra
of compounds I-IV were acquired on a Chemagnetics CMX
Infinity 200 spectrometer operating at a frequency of
63.16 MHz. MAS rates of at least 10 kHz were found to be
essential to resolve spinning sidebands satisfactorily; a double-resonance 4 mm MAS probe was used for all experiments. Telluric acid, Te(OH)6, was used as a crosspolarization setup sample.64 Typical experimental parameters were: 10–20 s recycle delay, 2.50 ms p/2 pulse,
10.0 ms contact time, 76.8 ms acquisition time, 4k–8k scans
recorded. High-power proton decoupling was used for all
experiments. For compounds I, II, and IV, the temperature
around the sample was kept below room temperature, typically & 10 8C, to prevent melting of the samples.
Hexa-(p-tolyl)ditin, V, (white powder) was synthesized following a procedure adapted from references 65 and 66. The
compound was recrystallized from benzene, which apparently
affords the type-B polymorph.57 Tin-119 VACP MAS NMR
spectra of powdered samples of hexa(p-tolyl)ditin were acquired at a frequency of 74.63 MHz using a Chemagnetics
CMX Infinity 200 spectrometer, or at 112.01 MHz with a
Bruker Avance 300 spectrometer. Tin-117 VACP MAS
NMR spectra were acquired at 71.31 MHz using a Chemagnetics CMX Infinity 200 spectrometer. A 4 mm DR MAS
probe was used for all experiments. Tetracyclohexyltin was
used as a setup sample and secondary chemical shift reference, diso = –97.35 ppm, with respect to a solution of SnMe4
at 0 ppm.67,68 All 119Sn and 117Sn CP/MAS experiments used
a 40 s recycle delay, 5–6 ms contact times, and either 2.40 ms
(at 4.70 T) or 4.00 ms p/2 pulses. High-power proton decoupling (gB1/2p > 60 kHz) was used for all experiments. The
POST-C7 pulse sequence61,62 was used to provide a doublequantum filtered 119Sn spectrum; an MAS rate of 8.929 kHz
and an experimentally optimized 119Sn B1 field near the theoretical maximum of 62.5 kHz was applied. The DQ conversion efficiency of the POST-C7 sequence was found to be
approximately half the theoretical value of 47% for the
895.96 ms DQ excitation block used.69 Heteronuclear decou-
1349
pling was applied only during the acquisition period of the
POST-C7 experiment.70 Two-dimensional J-resolved 119Sn
MAS NMR spectra59,60 were acquired using similar parameters, with 16 points acquired in the indirect dimension, a spectral window of 200 Hz in the indirect dimension, and 128
scans per increment.
Numerical calculations of the A(Q) distribution and its effect on NMR spectra were performed with purpose-built C
code. The powder distribution was generated by calculating
the value of fzy(Q;cyx,czx), see eq. [11], for a large set of
crystallite orientations;71 orientations were generated in the
rotor frame, to take advantage of the above-noted rotational
symmetry of fzy(Q;cyx,czx) about the axis of sample rotation.
Specifically, crystallites were initially aligned with z along
rrot, then rotated about rrot through the angle a (0 < a < 2p),
and then about the initial y axis through the angle b (0 < b <
p). A total of 832039 crystallite orientations were used,
where the a and b Euler angles were sampled according to
the Zaremba–Conroy–Wolfsberg, ZCW, distribution.72–74
Each xa and 3ab angle in eq. [11] was then calculated
using standard linear algebra, and used to create plots of
2
ðQ; cyx ; czx Þ, or the NMR spectral frequenfzy(Q;cyx,czx), fzy
cies.
Acknowledgments
We thank all members of the solid-state NMR group at
the University of Alberta for helpful comments. We are
grateful to Professor Cynthia Jameson for advice. K.J.H.
thanks the province of Alberta and the University of Alberta
for funding. We thank the Natural Sciences and Engineering
Research Council (NSERC) of Canada for research grants.
R.E.W. thanks the University of Alberta for financial support and also thanks the Canada Research Chairs Program
for funding a Canada Research Chair in Physical Chemistry
held by R.E.W. at the University of Alberta.
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