Cartesian and SphericalTensors in NMRHamiltoniansPASCAL P. MAN1,2
1Sorbonne Universit�es, UPMC Univ Paris 06, FR 2482, Institut des mat�eriaux de Paris-Centre, College de France,11 place Marcelin Berthelot, 75231 Paris Cedex 05, France2CNRS, FR 2482, Institut des mat�eriaux de Paris-Centre, College de France, 11 place Marcelin Berthelot, 75231Paris Cedex 05, France
ABSTRACT: NMR Hamiltonians are anisotropic due to their orientation dependence
with respect to the strong, static magnetic field. They are best represented by product of
two rank-2 tensors: one is the space-part tensor T and the other is the spin-part tensor
A. We reformulated the dot product of Cartesian tensors and the dyadic product of
spherical tensors in NMR Hamiltonian as the double contraction of these two tensors. As
the double contract has two definitions (double inner product and double outer product
of two rank-2 tensors), there are two sets of spherical tensor components in terms of Car-
tesian tensor components for any rank-2 tensor, two Cartesian Hamiltonians, and two
spherical Hamiltonians. We succeeded in determining the spherical components of ten-
sors A and T that verify the Cartesian Hamiltonian defined by the double inner product
of two rank-2 tensors and the spherical Hamiltonian defined by the double outer product
of two rank-2 tensors. In particular, we established the spherical components of space-
part tensor T in terms of Cartesian tensor components provided by Cook and De Lucia.
Throughout the article, Wigner active rotation matrix is used to illustrate the active rota-
tion of spherical vector, spherical harmonics, and spherical tensor as well as the passive
rotation via their rotational invariants. VC 2014 Wiley Periodicals, Inc. Concepts Magn
Reson Part A 42A: 197–244, 2013.
KEY WORDS: rotational invariant; Wigner active rotation matrix; spherical tensor; dou-
ble contraction; dual basis
I. INTRODUCTION
NMR Hamiltonians are anisotropic due to their orienta-
tion dependence with respect to the strong, static mag-
netic field B0. They are best represented by product of
two rank-2 tensors: one denoted by T is the space-part
tensor and the other by A is the spin-part tensor. Fur-
thermore, tensor A5V� U is formed either with two
spin operators (V5I and U5S) or with a spin operator
and a magnetic field (V5I and U5B0), whereas T is a
Received 2 June 2013; revised 30 August 2013;
accepted 3 December 2013
Correspondence to: P.P. Man; E-mail: [email protected]
Concepts in Magnetic Resonance Part A, Vol. 42A(6) 197–244 (2013)
Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cmr.21289
� 2014 Wiley Periodicals, Inc.
197
pure rank-2 tensor. These two rank-2 tensors are
expressed either with Cartesian components or with
spherical components. In 3D Euclidean space, Carte-
sian rank-2 tensor relates a physical vector to another
physical vector. The typical example in NMR (1) is the
chemical shift tensor r, which relates the chemical shift
field BCS to B0. Each component of BCS depends on
the three components of B0. This directional depend-
ency information is provided by r. However, Cartesian
tensors are not suitable for rotation studies of coordi-
nate systems in NMR. Spherical tensors are more suita-
ble for these coordinate transformations but they
involve more mathematical concepts.
The passage from Cartesian to spherical tensors
requires relations expressing spherical components in
terms of Cartesian components. Unfortunately, rela-
tions deduced by Cook and De Lucia (2) and reported
by Mehring (3) are not well explained due to some
missing mathematical definitions, although they are
extensively used by the NMR community. We provide
a general procedure for the deduction of these relations
in this article.
In Section II, we gather useful definitions, opera-
tions, and symbols involved in the manipulations of
Cartesian and spherical tensors. Change of coordinate
systems involves change of vector and tensor
components.
We introduce the dual basis in Section III that pro-
vides a straightforward method of calculating vector
components even in a nonorthogonal Cartesian coordi-
nate system. We also introduce the important concept
of covariant and contravariant components using a
common example about a position vector in a nonor-
thogonal 2D Cartesian coordinate system (4–6),
because covariant spherical components of rank-2 ten-
sors are implicitly used in NMR literature. Although
distinction between covariant and contravariant Carte-
sian components is not necessary in orthogonal Carte-
sian coordinate system with normed basis, it is
compulsory to distinguish these two types of compo-
nents for spherical tensor because these components
are complex numbers. In other words, covariant spheri-
cal components are complex conjugates of contravar-
iant spherical components.
In Section IV, we provide an introduction of Carte-
sian tensor, in particular the contragredient and cogre-
dient transformations of tensor components as well as
the decomposition of rank-2 tensor, which is related to
the reducible property of tensor.
As the transformation of tensor components under
rotation of coordinate system is simpler when spherical
tensors instead of Cartesian tensors are used, we first
introduce in Section V, the active rotation operators
about a single axis and about three rotated axes using
Euler angles applied to a space function. The descrip-
tion of active rotation may be confusing because two
approaches are available in the literature. We may use
one or two coordinate systems. With one coordinate
system, we rotate the space function itself. In two coor-
dinate system approach, we orientate the coordinate
system attached to the space function with respect to a
fixed coordinate system, the two systems coincide ini-
tially. We also express the active rotation operator
about axes of the fixed coordinate system.
In Section VI, we extend our study of active rotation
to spherical harmonics before that of spherical tensor
components. Spherical harmonics have arguments as
space function. When two-coordinate system approach
is used, the arguments allow us to localize the spherical
harmonics in the right coordinate system. Furthermore,
spherical harmonics of order l have 2l 1 1 components
as spherical tensor of rank r has 2r 1 1 components.
But spherical tensor components are not associated
with arguments, which may lead difficulty for localiz-
ing these components in the right coordinate system.
Understanding the transformation law of spherical har-
monics under rotation of coordinate system will facili-
tate that of spherical tensor.
In Section VII, we discuss spherical tensor. As the
construction of rank-2 tensor is from coupling two
rank-1 tensors or vectors, we first describe in detail var-
ious conventions for defining spherical rank-1 tensor.
Then we present three procedures for obtaining spheri-
cal components of rank-2 tensor in terms of Cartesian
components using the double inner product (DIP) and
double outer product (DOP) of two rank-2 tensors. As a
result, each of the two tensors A and T has two sets of
spherical components. Active rotation and rotational
invariance of spherical tensor are discussed using
spherical tensor as a geometry entity.
In Section VIII, NMR Hamiltonian is reformulated
using the DIP and DOP of two rank-2 tensors. As a
result, there are also two expressions for Cartesian
Hamiltonian and two others for spherical Hamiltonian.
Rotational invariance of NMR Hamiltonian is also
introduced.
Finally, in Section IX we associate the pair of space-
and spin-part tensors with NMR Cartesian Hamiltonian
determined by DIP and spherical Hamiltonian deter-
mined by DOP. This step is facilitated by our Wolfram
Mathematica-5 notebooks.
II. USEFUL DEFINITIONS, OPERATIONS,AND SYMBOLS
We collect definitions, operations, and relations most
frequently used in rank-2 tensor calculations.
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Throughout the article, we do not use Einstein summa-
tion rule because two types of indices are involved.
The first concerns with Cartesian rank-2 tensor compo-
nents Tij with indices i and j taking x, y, and z. In Ein-
stein summation, indices i and j vary from 1 to 3. The
second concerns with spherical rank-2 tensor compo-
nents Tkq with index k varying from 0 to 2 and index qfrom 2k to 1k.
Cartesian Tensor
Several symbols are available for vector and tensor
operators, where the dot is contraction operator. How-
ever, they are not universally accepted conventions.
1. Dot product (or scalar product or inner product)of two vectors a and b, represented with �ð Þ, iscommutative (a � b5b � a).
2. Dyadic product (or outer product, or direct prod-uct, or tensor product) of two vectors is repre-sented with �ð Þ. It forms a matrix rather than ascalar as in the dot product of two vectors. Theorder of vectors in a dyad is important:b� c 6¼ c� b. A dyadic is a linear combinationof dyads (7). A dyad b� c has physical meaningonly when operating on a vector.
3. Dyadic dot product (or dyadic inner product orsimple contraction) between two tensors orbetween a tensor and a vector is also representedwith �ð Þ. A dyad a� b is a rank-2 tensor, whichlinearly transforms a vector c into a vector withthe direction a (8–11):
a� bð Þ � c5a b � cð Þ: (1)
In other words, the dyadic dot product of dyada� b with vector c produces a new vector a mu-ltiplied by a scalar b � c. This operation is also c-alled simple contraction, because initially a rank-2 tensor and a vector are involved, but the resultis a vector. We also have
a � b� cð Þ5 a � bð Þc: (2)
The dyadic dot product between two dyads is defin-ed by (7,9–11)
a� bð Þ � c� dð Þ5 b � cð Þ a� dð Þ; (3)
that between two basis tensors (or unit dyads) by
ei � ej
� �� em � en
� �5 ej � em
� �ei � en
� �5djm ei � en
� �; i; j;m; n5x; y; zð Þ;
(4)
where djm is the Kronecker delta symbol and ei(i 5x, y, z) are orthonormal Cartesian basis vectors. Thevectors in Eq. (3) or basis vectors in Eq. (4) that ar-e beside each other are the ones which are “dotted”together. The result is a scalar multiplied by a dyadas shown in Eq. (3) or by a basis tensor as shown
in Eq. (4), that is, a rank-2 tensor. The order isimportant: a � b� cð Þ 6¼ b� cð Þ � a and a� bð Þ�c� dð Þ 6¼ c� dð Þ � a� bð Þ. Equation (4) is involved
for example in the dyadic dot product between tworank-2 tensors F and G:
F �G5X
i;j5x;y;z
fij ei � ej
� � !�
Xm;n5x;y;z
gmn em � en
� � !
5X
i;j;m;n5x;y;z
fijgmn ei � ej
� �� em � en
� �5
Xi;j;m;n5x;y;z
fijgmn ej � em
� �ei � en
� �5
Xi;j;m;n5x;y;z
fijgmndjm ei � en
� �5
Xi;j;n5x;y;z
fijgjn ei � en
� �:
(5)
Notice that a contraction of the inner indices j andm occurs during the dyadic dot product of tworank-2 tensors as in the multiplication of two matri-ces. When two indices of a tensor are equated and asummation is performed over this repeated index,the process is called contraction.
4. Double contraction of two rank-2 tensors is a sca-lar. Therefore, this operator is commutative. Thedouble contraction is defined such that it operatesbetween tensors of at least rank 2. The doublecontraction of two rank-2 tensors A and B occursin two ways:
A : B5X
i;j
AijBij5Tr ATB� �
5Tr ABT� �
; (6)
A::B5A :BT5X
i;j
AijBji5Tr ABf g: (7)
Equations (6) and (7) define the DIP represented with
:ð Þ and the DOP represented with ::ð Þ of two rank-2
tensors, respectively. Similarly, two definitions also
exist for the double contraction of two dyads.
The DIP of two dyads a� b and c� d is defined by
(7–13)
a� bð Þ: c� dð Þ5 a � cð Þ b � dð Þ; (8)
and that of two basis tensors ei � ej and em � en by
(7,11,12)
ei � ej
� �: em � en
� �5 ei � em
� �ej � en
� �5dimdjn;
ði; j;m; n5x; y; zÞ:(9)
The DOP of two dyads a� b and c� d is defined
by (8,10,13–17)
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a� bð Þ:: c� dð Þ5 a � dð Þ b � cð Þ; (10)
and that of two basis tensors by (15,16)
ei � ej
� �:: em � en
� �5 ei � en
� �ej � em
� �5dindjm;
ði; j;m; n5x; y; zÞ:(11)
The latter has some similarity with Eq. (4).
Spherical Tensor
Throughout the article, subscripts are used for indexing
covariant spherical components. Contravariant spheri-
cal components are replaced by the complex conjugates
of covariant spherical components.
The dyadic product between two spherical tensors
Vk1 and Uk2 of ranks k1 and k2 is defined by (18–22)
Vk1 � Uk2
� rs
5Xq1;q2
Vk1q1Uk2q2hk1q1k2q2jrsi; (12)
where the numerical factors hk1q1k2q2jrsi are called
Clebsch-Gordan coefficients (23–27). In spherical ten-
sor algebra, the dot product of two spherical tensors Vk
and Uk of same rank k is defined conventionally by
(20,27–30)
Vk � Uk5Xk
q52k
21ð ÞqVkqUk2q; (13)
and not by Eq. (12) with k1 5 k2 5 k and r 5 s 5 0.
The two forms differ only by a constant factor (31–36):
Vk � Uk5 21ð Þkffiffiffiffiffiffiffiffiffiffiffiffi2k11p
Vk � Uk
� 00: (14)
Combining the above two relations yields
(19,27,33,37)
Vk � Uk
� 00
51ffiffiffiffiffiffiffiffiffiffiffiffi
2k11p
Xk
q52k
21ð Þk2qVkqUk2q: (15)
The omission of the factor 2k11ð Þ21=2in Eq. (15)
can be viewed as a scaling. To remind us this omission,
a new symbol 8ð Þ for the dyadic product of two spheri-cal tensors is introduced (37):
Vk8Uk
� 00
5 21ð ÞkXk
q52k
21ð Þ2qVkqUk2q: (16)
Equations (13) and (16) differ in sign if k is odd
and are equivalent if k is even. In particular, for rank-1
tensors (k 5 1), the two spherical tensors V1 and U1
are described by their covariant spherical
components in terms of Cartesian components
(3,12,21,25,27,28,30,32,35,38–45):
V111521ffiffiffi2p ðVx1iVyÞ
V105Vz
V12151ffiffiffi2p ðVx2iVyÞ
;
U111521ffiffiffi2p ðUx1iUyÞ
U105Uz
U12151ffiffiffi2p ðUx2iUyÞ
:
8>>>>>>><>>>>>>>:
8>>>>>>><>>>>>>>:
(17)
Equation (16) becomes (46)
V18U1
� 00
5 V121U1112V10U101V111U121
� �52 VxUx1VyUy1VzUz
� �;
(18)
and Eq. (13) becomes (27)
V1 � U15X11
q521
21ð ÞqV1qU12q52V121U111
1V10U102V111U1215VxUx1VyUy1VzUz:
(19)
As Eq. (19) has the usual meaning of dot product of
two vectors V1 and U1, it is preferred to Eq. (18),
which has a negative factor. There is agreement
between Cartesian and spherical rank-1 tensors.
III. DUAL BASIS
We introduce the dual basis associated with a normed
basis of nonorthogonal Cartesian coordinate system in
Euclidean plane for simplicity. This dual basis provides
a straightforward method of calculating vector compo-
nents. A Euclidean space is a real vector space fur-
nished by the dot product. With nonorthogonal axes,
the distinction between contravariant and covariant
components of a vector is more easy to establish
(6,47,48). By convention, contravariant components
are written with upper index notation, whereas covari-
ant components are denoted by lowered indices.
In this section, we follow standard approach for con-
travariant and covariant notations that use integer num-
bers in superscript and subscript for Cartesian vector
components and basis vectors. Two presentations of
dual basis are used, a concrete one in geometry and an
abstract one in linear algebra.
In Geometry
Figure 1 shows a vector OP in the nonorthogonal Car-
tesian coordinate system X1;X2f g with normed basis
vectors e1; e2
� �of Euclidean plane, that is,
jje1jj5jje2jj51; e1 � e2 6¼ 0: (20)
The contravariant components of OP are gathered
in a column matrix ð x1 x2 ÞT by convention, where
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the superscript T stands for matrix transposition. The
jth contravariant component consists of the projection
of OP onto the jth axis parallel to the other axis. The
vector OP expressed with its contravariant components
in basis e1; e2
� �is,
OP5x1e11x2e2: (21)
Conversely, the covariant components of OP are
gathered in a row matrix x1 x2ð Þ by convention. The
jth covariant component consists of the projection of
OP onto the jth axis perpendicular to that axis. The
meaning of covariant components of OP is not
obvious, because they do not allow us to express OP in
basis e1; e2
� �simply.
We cannot determine the contravariant components
xi of OP by multiplying the latter in Eq. (21) with ej
because the basis vectors e1; e2
� �are not orthogonal.
We have to introduce the dual basis e1; e2� �
associated
with basis e1; e2
� �. These basis vectors are defined by
the dot product (5,10)
ei � ej5dij; (22)
where dij is the Kronecker delta symbol
dij5
1; if i5j
0; if i 6¼ j:
((23)
Therefore, e1 and e2 are perpendicular to each other,
as are e2 and e1. The dual basis vectors defined in Eq.
(22) allow us to determine the contravariant compo-
nents xi of OP by multiplying the latter in Eq. (21)
with ej (47), that is,
OP � e15 x1e11x2e2
� �� e15x1; (24)
OP � e25 x1e11x2e2
� �� e25x2: (25)
First, we determine the dual basis vectors e1; e2� �
.
In Fig. 2, elements of dual coordinate system are col-
ored in red. Equation (22) allows us to draw the dual
coordinate system N1;N2f g. The norm of the jth dual
basis vector ej is defined by the dot product
ej � ej5jjejjjjjejjjcos u51: (26)
This means that cos u � 0. In other words, each vec-
tor of the basis makes an acute angle with the vector of
the other basis whose index has the same value (4). As
basis vectors e1; e2
� �are normed, we deduce the norm
of each dual basis vector:
jjejjj5 1
cos u: (27)
The projection of e2 onto X2 parallel to N1 is e2 and
that of e1 onto X1 parallel to N2 is e1. The dual coordi-
nate system and its basis have been defined. We have to
clarify the meaning of covariant components xi of OP.
Second, we determine the relations between the
components of OP in these two coordinate systems. In
the dual coordinate system, OP has its own contravar-
iant n1; n2� �
and covariant n1; n2
� �components. Figure
2 shows that
jjOAjjcos u5jjOBjj or n1jje1jjcos u5x1jje1jj: (28)
Taking into account Eqs. (20) and (27), we deduce
that
n15x1: (29)
Figure 1 Contravariant (x1, x2) and covariant (x1, x2)
components of vector OP in Euclidean plane with nonor-
thogonal Cartesian system {X1, X2} but normed basis
vectors {e1, e2}.
Figure 2 Contravariant (n1, n2) and covariant (n1, n2)
components of vector OP in the dual basis {e1, e2} of
basis {e1, e2} in Euclidean plane. The dual coordinate
system {N1, N2} is colored in red.
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If we apply the intercept theorem (49), also known
as Thales’ theorem in elementary geometry, to the tri-
angle OAB in Fig. 2, we also get Eq. (29). Similarly,
jjODjjcos u5jjOCjj or x1jje1jjcos u5n1jje1jj: (30)
Taking into account Eqs. (20) and (27), we deduce
that
n15x1cos 2u: (31)
It is easy to check that we also have
n25x2
n25x2cos 2u:
((32)
With normed basis vectors e1; e2
� �, the covariant
components n1; n2
� �of OP in dual basis e1; e2
� �are
those of the contravariant components x1; x2� �
of OP
in basis e1; e2
� �multiplied by cos 2u. In contrast, the
contravariant components n1; n2� �
of OP in the dual
basis e1; e2� �
are identical to the covariant components
ðx1; x2Þ of OP in basis e1; e2
� �:
OP5n1e11n2e25x1e11x2e2: (33)
The covariant components xi of OP allow us to
express the latter in the dual basis.
Finally, we provide the meanings of contravariant
and covariant components of a vector. Figure 3 is a
simplified version of the two coordinate systems where
the contravariant and covariant components of OP are
shown. With normed basis vectors e1; e2
� �, the covari-
ant components of OP appear in the axes of the two
coordinate systems X1;X2f g and N1;N2f g. If we
increase the length of a basis vector ej, we must
decrease the contravariant component xj of OP in order
to keep the latter unchanged. In other words, contravar-
iant component xj transforms in the opposite way to
basis vector ej. As a result, the dual basis vector ej
decreases in length due to Eq. (26), therefore, the
covariant component xj of OP in the dual coordinate
system must increase in order to keep OP unchanged.
In other words, covariant components xj in the dual
coordinate system transforms in the same way as basis
vector ej. If the norms of basis vectors e1; e2
� �are dif-
ferent from 1, the covariant components xj of OP are
defined only in the dual coordinate system N1;N2f g(4,5). Basis fe1; e2g is also called covariant basis and
the dual basis fe1; e2g called contravariant basis.
In orthogonal Cartesian coordinate system with
normed basis, the latter and its dual basis are the
same (u 5 0 in Eq. (26)), and covariant and contra-
variant components of a vector are identical. As a
result, vectors and tensors are written usually with
subscripts.
In Linear Algebra
Consider a finite dimensional vector space K, its dual
space K� is the vector space of all linear transforma-tions from K to the real numbers R. If K5RN is a
Euclidean space, a basis b5 e1; :::; eN
� �of K is associ-
ated with its dual basis (or cobasis) b�5 e1; :::; eN� �
of
K� verifying
ei � ej5dij; i; j51; :::;Nð Þ: (34)
This means that a dual basis vector ei verifies two
properties: it is normal to all basis vectors ej with index
j 6¼ i, and the dot product of ei with its dual ei of the
same index is unit.
A vector V of K is expressed in b basis as
V5v1e11 � � �1vNeN; (35)
where the coefficients vi are called the contravariantcomponents of V in K. Every dual vector (or covector)
W� of K� is expressed in basis b� as
W�5w1e11 � � �1wNeN; (36)
where the coefficients wi are called the covariant com-ponents of W� in K�.
By convention, V is represented by a column matrix
v1 : : vN� �T
and W� by a row matrix
w1 : : wNð Þ. Therefore, V and W� in Eqs. (35)
and (36) can be written in matrix forms:
V5 e1 . . . eNð Þv1
�
vN
0BB@
1CCA;W�5 w1 . . . wNð Þ
e1
�
eN
0BB@
1CCA:(37)
In other words, basis vectors ei and covariant com-
ponents wi are gathered in row matrices, whereas dual
Figure 3 Contravariant (x1, x2) and covariant (x1, x2)
components of vector OP in Euclidean plane with normed
but nonorthogonal basis vectors {e1, e2} and non-normed
dual basis vectors {e1, e
2}.
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basis vectors ei and contravariant components vi are
gathered in column matrices.
The contravariant components vi of V and the
covariant components wi of W� can be deduced from
ei � V5ei � e1 . . . eNð Þ
v1
�
vN
0BBBB@
1CCCCA5vi;
(38)
W� � ei5 w1 . . . wNð Þe1
�
eN
0BB@
1CCA � ei5wi: (39)
In practice, V �W, the contravariant components
vi of V and its covariant components vi can be deduced
from
vi5ei � V; vi5V� � ei: (40)
Notice the positions of V and V� relative to the sym-
bol of dot product are different: V is on the right-hand
side, whereas V� is on the left-hand side. This is an
important point when spherical basis is involved.
IV. CARTESIAN TENSOR
Cartesian rank-2 tensors are ubiquitous in science and
engineering. Several approaches are available for defin-
ing rank-2 tensors. 1) They generalize scalars and vec-
tors; their components can be determined in any
Cartesian coordinate system. 2) They may act as linear
transformations operating on vectors. 3) They can be
represented by 3 3 3 matrices.
Cartesian tensor has simple transformation law
under rotation of coordinate system. But it is reducible
with respect to this rotation. We use the presentation of
Weissbluth (27) for the decomposition of rank-2 tensor
into a sum of irreducible tensors. This presentation dis-
tinguishes the general case where the coordinate system
may be nonorthogonal and the practical case where the
coordinate system is orthogonal. The general case is
based on linear transformation, whereas the practical
case is founded upon orthogonal transformation, in par-
ticular the orthogonality condition. The definition of
contragredient and cogredient transformations is also
provided. We still use integer numbers for differentiat-
ing vector components and basis vectors in subsections
about transformations.
Linear Transformation
Consider two vector spaces Rm and Rn with dimensions
m and n over the real numbers R. Let fv1; . . . ; vmg be a
basis with covariant basis vectors for Rm and
w1; . . . ;wn
� �a basis with covariant basis vectors for
Rn. A vector V in basis v1; . . . ; vm
� �of Rm is deter-
mined by its contravariant components a1; . . . ; am� �
such that
V5Xm
i51
aivi: (41)
We define f as a linear transformation from Rm to Rn
such that
f Vð Þ5fXm
i51
aivi
!5Xm
i51
aif vi
� �: (42)
Some common linear transformations are rotation,
reflection, axis-scaling, projection, and shearing. Linear
transformations are characterized by the property that
addition and scalar multiplication are preserved. The
linear transformation f is completely determined by its
action on a basis f v1
� �; � � � ; f vm
� �� �. The linear trans-
formation of each basis vector f vi
� �of Rm can be
expressed as
f vi
� �5Xn
j51
wjrji: (43)
We used to represent the values rji in an n 3 mmatrix R, called transformation matrix. In matrix form,
Eq. (43) becomes
f v1
� �� � � f vm
� �� �5 w1 � � � wnð ÞR: (44)
The linear transformation f is entirely determined by
the matrix R. By extension, we sometimes identify the
two quantities
R5 f v1
� �::: f vm
� �� �: (45)
In other words, the columns of the transformation
matrix R are the values of the linear transformation f of
basis v1; � � � ; vm
� �. Combining Eqs. (42) and (43) yields
f Vð Þ5Xm
i51
Xn
j51
wjrjiai: (46)
If we place the m elements ai of V in a column
matrix A, the matrix multiplication RA is a column
matrix A0 with n elements representing the coordinates
of f Vð Þ in basis w1; . . . ;wn
� �:
A05RA; a0j5Xm
i51
rjiai: (47)
Contragredient and CogredientTransformations
A vector V in three-dimensional space with a basis b5
v1; v2; v3
� �may be written as
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 203
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V5 v1 v2 v3ð Þa1
a2
a3
0BB@
1CCA; (48)
where a1, a2, a3 are the contravariant components
of V in the basis b. Basis vectors are gathered in a
row matrix and vector components in a column
matrix.
With another basis b05 v01; v02; v03
� �that is related
with the basis b by a linear transformation represented
by a 3 3 3 matrix R,
v01 v02 v03ð Þ5 v1 v2 v3ð ÞR
5 v1 v2 v3ð Þ
r11 r12 r13
r21 r22 r23
r31 r32 r33
0BBB@
1CCCA; (49)
the vector V is defined by
V5 v01 v02 v03ð Þ
a01
a02
a03
0BB@
1CCA: (50)
The vector V is viewed in two bases, but it is not
affected by the transformation of basis b to basis b0 by
the matrix R. The vector V is an invariant against a
basis transformation. This transformation is also a
change of description. We have
v01 v02 v03ð Þ
a01
a02
a03
0BB@
1CCA5 v1 v2 v3ð Þ
a1
a2
a3
0BB@
1CCA:(51)
Introducing Eq. (49) in Eq. (51) yields
R
a01
a02
a03
0BB@
1CCA5
a1
a2
a3
0BB@
1CCA: (52)
Multiplying the two members of Eq. (52) by R21,
the inverse of R, yields
a01
a02
a03
0BB@
1CCA5R21
a1
a2
a3
0BB@
1CCA: (53)
We have at our disposal two relationships: Eq. (49)
for basis vectors and Eq. (53) for vector components.
To facilitate the comparison of these two relationships
of transformation, we transpose the matrices in Eq.
(49) so that 3 3 3 matrices and column matrices occur
in the same order. That is,
v01
v02
v03
0BB@
1CCA5RT
v1
v2
v3
0BB@
1CCA: (54)
We rename the matrix RT by the matrix D, that is,
RT5D (55)
and
R215 DT� �21
: (56)
As a result, Eqs. (53) and (54) become
a01
a02
a03
0BB@
1CCA5 DT
� �21
a1
a2
a3
0BB@
1CCA; (57)
and
v01
v02
v03
0BB@
1CCA5D
v1
v2
v3
0BB@
1CCA: (58)
The transformation of vector components in Eq.
(57) is called contragredient to that of basis vectors in
Eq. (58) (18,50). Conversely, if we have a transforma-
tion of vector components defined by
c01
c02
c03
0BB@
1CCA5D
c1
c2
c3
0BB@
1CCA; (59)
this transformation of vector components is called cog-redient to that of basis vectors in Eq. (58) (18,50).
The following definitions are provided by
McWeeny (51):
1. Any set of entities T which, when the basis ischanged, must be replaced by a new set according
to T! T05 DT� �21
T, is a tensorial set trans-
forming cogrediently to the vector components
a1, a2, a3 (Eq. (57)), but contragrediently to thebasis vectors (Eq. (58)). It is a contravariant set.
2. Any set of entities T which, when the basis ischanged, must be replaced by a new set accordingto T! T05DT, is a tensorial set transformingcogrediently to the basis vectors v1, v2, v3. It is acovariant set.
204 MAN
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The invariance of a vector V against a change of
coordinate system is expressed by
V5 v01 v02 v03ð Þ
a01
a02
a03
0BBB@
1CCCA5 v1 v2 v3ð Þ
3DT DT� �21
a1
a2
a3
0BBB@
1CCCA5 v1 v2 v3ð Þ
a1
a2
a3
0BBB@
1CCCA;
(60)
where Eqs. (57) and (58) are used. In other words, the
invariance of a vector is expressed by the product of
covariant basis vectors with contravariant components
of the vector in the same basis.
The adjoint of matrix D is determined as
D15 DT� ��
5 D�� �T
: (61)
If D is also unitary, its adjoint verifies
D15D21: (62)
As a result,
D215 DT� ��
5 D�� �T
; (63)
DT� �21
5 D1� ��� �21
5 D21� ��� �21
5D�: (64)
Equation (60) becomes
V5 v01 v02 v03ð Þ
a01
a02
a03
0BBB@
1CCCA5 v1 v2 v3ð Þ
3 D�� �21
D�
a1
a2
a3
0BBB@
1CCCA5 v1 v2 v3ð Þ
a1
a2
a3
0BBB@
1CCCA:
(65)
If the bases are orthonormal then the matrix D is
orthogonal. That is, D215DT (see following subsec-
tion). Equation (60) becomes
V5 v01 v02 v03ð Þ
a01
a02
a03
0BBB@
1CCCA5 v1 v2 v3ð Þ
3D21D
a1
a2
a3
0BBB@
1CCCA5 v1 v2 v3ð Þ
a1
a2
a3
0BBB@
1CCCA:
(66)
Orthogonal Transformation
An orthogonal transformation is a linear transformation
f from a vector space RN to itself, in which is defined
the dot product. Orthogonal transformation preserves
lengths of vectors and angles between them. It is either
a rotation or a reflection (an improper rotation). In finite
dimension spaces, the square matrix R with real entities
is called orthogonal matrix with determinant equal to
61. Its columns are mutually orthogonal vectors with
unit norm, likewise for its rows. If R is a rotation
matrix (52), its determinant is 11; reflection matrix has
determinant 21.
With orthogonal Cartesian coordinate systems, we
do not need to distinguish contravariant and covariant
components of Cartesian tensors and vectors. Consider
two (unprimed and primed) orthogonal Cartesian coor-
dinate systems. If a vector x in the unprimed coordinate
system has components xi, the transformed vector x0 in
the primed coordinate system has components x0i. The
orthogonal matrix R satisfies the condition
x0 � x05 Rxð Þ � Rxð Þ5x � x: (67)
In other words, the norm of the vector in the
unprimed coordinate system,XN
i51xixi, and that in
the primed coordinate system,XN
i51x0ix0i, are pre-
served (53,54):
XN
i51
x0ix0i5XN
i51
xixi: (68)
Applying Eq. (47) twice yields
XN
i51
x0ix0i5XN
i51
XN
m51
rimxm
XN
n51
rinxn
!
5XN
m51
XN
n51
xmxn
XN
i51
rimrin:
(69)
This results the orthogonality condition (53,54) rep-
resented by
XN
i51
rimrin5dmn; m; n51; . . . ;Nð Þ: (70)
This summation is the dot product of mth column
with nth column of the same matrix R. The orthogonal-
ity condition is a consequence of requiring that the
length of a vector remains invariant under rotation of
the coordinate system.
An alternative form of the orthogonality condition is
that the inverse of an orthogonal matrix is equal to its
transpose matrix:
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 205
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RT5R21: (71)
Therefore, Eq. (69) can be rearranged as
XN
i51
x0ix0i5XN
i51
XN
m51
xm RT� �
mi
XN
n51
rinxn
!
5XN
i51
XN
m51
xm R21� �
mi
XN
n51
rinxn
!5XN
m51
xmxm:
(72)
For the dot product of two vectors x and y (50), we
have
XN
i51
x0iy0i5XN
i51
XN
m51
ym R21� �
mi
XN
n51
rinxn
!
5XN
i51
XN
m51
R21� �Tn o
imym
XN
n51
rinxn
!5XN
i51
xiyi:
(73)
The transformed vectors x0 and y0 in the primed
coordinate system are related to those in the unprimed
coordinate system as
x05Rx; (74)
y05 R21� �T
y: (75)
When coordinate system other than Cartesian,
orthogonal, and real is used, the orthogonality condition
(Eq. (70)) is not applicable. However, the dot product
defined by Eq. (73) is still applicable. The two vectors
are also said to be contragredient to each other (50).
The invariance of the dot product is defined by com-
bining two vectors, which are contragredient to each
other. Conversely, the invariance of a vector is defined
by the product of covariant basis vectors with contra-
variant vector components in the same basis. More gen-
erally, the invariance of a tensor is defined by the
product of covariant basis tensors with contravariant
tensor components in the same basis tensors.
Active and Passive Rotations with Eulerangles of a Position Vector
In our previous study of active and passive rotations
with Euler angles in NMR (52), we focused on the rota-
tion of unit basis vectors and components of position
vector in a 3D Euclidean space with Cartesian coordi-
nate systems.
In passive rotation of a position vector from its
principal-axis system (PAS) to its observation-axis
system (OBS), we shown that the unit basis vectors
ðXPAS ;YPAS ;ZPAS Þ and the position vector compo-
nents ðxPAS ; yPAS ; zPAS Þ transform as
xOBS
yOBS
zOBS
0BB@
1CCA5Protated Z3Y2Z1ðg;b;aÞ
XPAS
YPAS
ZPAS
0BB@
1CCA; (76)
xOBS
yOBS
zOBS
0BB@
1CCA5Protated Z3Y2Z1ðg;b;aÞ
xPAS
yPAS
zPAS
0BB@
1CCA; (77)
where the passive rotation matrix is
Protated Z3Y2Z1ðg;b;aÞ5Pfixed Z1Y2Z3ðg;b;aÞ
5
CaCbCg2SaSg SaCbCg1CaSg 2SbCg
2CaCbSg2SaCg 2SaCbSg1CaCg SbSg
CaSb SaSb Cb
0BB@
1CCA:
(78)
The Euler angles being defined with rotated axes,
the first rotation angle is a if the passive rotations occur
about rotated axes. In contrast, the first rotation angle is
g if the passive rotations occur about fixed axes.
The presentation of active rotation of a position vec-
tor involved two coordinate systems: ðO;XPAS ;YPAS ;ZPASÞ and ðO; xOBS ; yOBS ; zOBSÞ. Initially, these two
coordinate systems coincide. The position vector is
attached to ðO; xOBS ; yOBS ; zOBS Þ. The active rotation
of the position vector is described by the rotation of its
attached coordinate system ðO; xOBS ; yOBS ; zOBS Þ. In
active rotation of a position vector from its initial posi-
tion ðx1; y1; z1Þ in PAS and OBS to its final position ðx;y; zÞ in PAS, it is shown that the unit basis vectors and
the position vector components transform as
xOBS yOBS zOBSð Þ5 XPAS YPAS ZPASð Þ3Arotated Z1Y2Z3ða;b; gÞ;
(79)
x
y
z
0BB@
1CCA5Arotated Z1Y2Z3ða;b; gÞ
x1
y1
z1
0BB@
1CCA; (80)
where the active rotation matrix is
Arotated Z1Y2Z3ða;b; gÞ5Afixed Z3Y2Z1ða;b; gÞ
5
CaCbCg2SaSg 2CaCbSg2SaCg CaSb
SaCbCg1CaSg 2SaCbSg1CaCg SaSb
2SbCg SbSg Cb
0BB@
1CCA:
(81)
The Euler angles being defined with rotated axes,
the first rotation angle is a if the active rotations occur
about rotated axes. In contrast, the first rotation angle is
206 MAN
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g if the active rotations occur about fixed axes. As the
rotation in orthonormal bases is an orthogonal transfor-
mation, active rotation matrix (Eq. (81)) is the trans-
pose of passive rotation matrix (Eq. (78)).
The rotational invariance of a position vector men-
tioned in Eq. (66) is verified by the product:
xOBS yOBS zOBSð Þ
xOBS
yOBS
zOBS
0BBBB@
1CCCCA
5 XPAS YPAS ZPASð Þ Protated Z3Y2Z1ðg;b;aÞf gT
3Protated Z3Y2Z1ðg;b;aÞ
xPAS
yPAS
zPAS
0BBBB@
1CCCCA
5 XPAS YPAS ZPASð Þ
xPAS
yPAS
zPAS
0BBBB@
1CCCCA:
(82)
In passive rotation, the column matrix of unit basis
vectors (Eq. (76)) and that of position vector compo-
nents (Eq. (77)) are premultiplied by the passive rota-
tion matrix (Eq. (78)). These two transformations are
cogredient. In active rotation, the row matrix of unit
basis vectors (Eq. (79)) is postmultiplied by the active
rotation matrix, whereas the column matrix of initial
position vector components in PAS (Eq. (80)) is pre-
multiplied by the active rotation matrix (Eq. (81)). We
will find the same properties for the rotation of a gen-
eral ket vector in the angular momentum space, the
spherical harmonics, and spherical tensors. In these
cases, covariant basis vectors and contravariant vector
components are required.
Cartesian Rank-2 Tensor as a Matrix
Tensors are invariant concept independent of coordi-
nate systems as vectors. Scalars and vectors are actually
special cases of tensors. A scalar is a quantity that has
only magnitude. A vector is a quantity that has magni-
tude and direction. A vector is completely defined by
its projections on a Cartesian coordinate system with
orthonormal basis. Similarly, a rank-2 tensor is com-
pletely specified by its nine components on nine basis
tensors, dyadic products of the three orthonormal basis
vectors. Tensors are extremely useful for describing
anisotropic properties in materials.
Consider the conductivity law
J5rE: (83)
The vector E is the applied electric field and the
vector J the electric current. If the media is anisotropic,
r becomes the conductivity tensor that can be repre-
sented by a 3 3 3 square matrix. Assuming linear rela-
tionship, Eq. (83) becomes (54)
Jj5X
i5x;y;z
rjiEi; j5x; y; zð Þ (84)
in an unprimed coordinate system. This relation differs
with that defined in Eq. (47), in which the orthogonal
matrix R transforms the components of a vector to
those of the same vector in another coordinate system.
In contrast, the tensor r transforms a vector E to
another vector J in the same coordinate system. In
other words, the orthogonal matrix R does not repre-
sent a tensor.
If we express J and E in a primed Cartesian coordi-
nate system, in which physical symbols carry a prime,
Eq. (84) becomes
J0j5X
i5x;y;z
r0jiE0i; j5x; y; zð Þ: (85)
The change of coordinate systems is associated with
an orthogonal matrix R, that is,
J0j5X
i5x;y;z
rjiJi;E0i5X
h5x;y;z
rihEh: (86)
So, Eq. (85) becomesXi5x;y;z
r0jiE0i5X
n5x;y;z
rjnJn5X
n5x;y;z
rjn
Xm5x;y;z
rnmEm
5X
n5x;y;z
rjn
Xm5x;y;z
rnm
Xi5x;y;z
rimE0i
5X
i;m;n5x;y;z
rjnrnmrimE0i:
(87)
We have applied Eq. (71) in the above calculation.
Therefore, the transformation law for r is
r0ji5X
m;n5x;y;z
rjnrimrnm: (88)
In matrix form
r05RrRT: (89)
The conductivity tensor r with two indices for its
components is a Cartesian rank-2 tensor. The tensor rtransforms with two copies of the orthogonal matrix R,
one on each index. Conversely, any matrix that
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 207
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
transforms under orthogonal transformation as in Eq.
(88) is a Cartesian rank-2 tensor.
A quantity, which under rotations of coordinate sys-
tem transforms like
T0a1a2:::an5X
b1b2:::bn5x;y;z
ranbn � � � ra2b2ra1b1Tb1b2:::bn;
(90)
where T0 and T have n indices, is a Cartesian rank-ntensor. As a result, the number of copies of R increases
linearly with the tensor rank. In general, a Cartesian
rank-n tensor has 3n components where the number 3
denotes the dimension of the 3D space.
Cartesian Rank-2 Tensor as a Dyadic
The above subsection about Cartesian rank-2 tensor
does not explicitly involve basis vectors, because they
are hidden by vectors and matrices. Now, we introduce
them to determine tensor components by projecting the
tensor onto basis vectors. This means a tensor becomes
a geometrical entity represented in terms of its compo-
nents and a set of unit vectors (50).
Consider a rank-2 tensor T in a Cartesian coordinate
system with orthonormal basis fex; ey; ezg. This tensor
can be regarded to some extent as a linear transforma-tion (or linear operator) which transforms a vector
V5X
i5x;y;z
eivi (91)
into another vector W5T Vð Þ. Other notations are pos-
sible: W5T � V or W5TV as Eq. (83). From the line-
arity of T,
W5X
j5x;y;z
T ej
� �vj: (92)
T transforms basis fex; ey; ezg into another basis
fTðexÞ;TðeyÞ;T ez
� �g.
The component vj of V is defined by the projection
of V on the basis vector ej or as the dot product
vj5ej � V: (93)
Equations. (93) and (1) allow us to express W in Eq.
(92) as
W5X
j5x;y;z
T ej
� �ej � V� �
5X
j5x;y;z
T ej
� �� ej
!� V:
(94)
Therefore,
T5X
j5x;y;z
T ej
� �� ej: (95)
We can express TðejÞ in basis fex; ey; ezg as in Eq.
(43):
T ej
� �5X
i5x;y;z
eiTij: (96)
As a result,
T5X
i;j5x;y;z
Tij ei � ej
� �: (97)
The nine scalars Tij are the Cartesian components of
rank-2 tensor T along the basis tensors ei � ej. A vector
is expressed with the Cartesian basis vectors
fex; ey; ezg. A Cartesian rank-2 tensor is expressed
with the basis tensors composed of the nine unit dyads
ei � ej. The values of the nine components Tij depend
on the coordinate system.
Now we are interested in the projection of a tensor
T onto the Cartesian basis fex; ey; ezg to determine its
component Tij. We first determine the dot product of T
with a basis vector en using Eq. (1):
T � en5X
i;j5x;y;z
Tij ei � ej
� � !� en5
Xi;j5x;y;z
Tijei ej � en
� �
5X
i;j5x;y;z
djneiTij5X
i5x;y;z
eiTin:
(98)
The projection of T on basis vector en is the vector
Tn with components Tin i5x; y; zð Þ. Then we evaluate
the projection of this new vector on basis vector em or
their dot product (7):
em � Tn5 em � T � en5 em �X
i5x;y;z
eiTin
5X
i5x;y;z
em � ei
� �Tin5
Xi5x;y;z
dmiTin5Tmn:(99)
The projection em � T � en of T onto two Cartesian
basis vectors yields the Cartesian tensor component
Tmn.
Equation (97) should also allow us to determine
Cartesian tensor components along the basis tensor em
�en using the DIP of two rank-2 tensors:
T : em � en
� �5X
i;j5x;y;z
Tij ei � ej
� �: em � en
� �; (100)
or the DOP of two rank-2 tensors:
T:: em � en
� �5X
i;j5x;y;z
Tij ei � ej
� �:: em � en
� �: (101)
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We get Tmn with Eq. (100) or its transpose Tnm with
Eq. (101). We will explore this situation for spherical
tensor components.
Decomposition of Cartesian Rank-2 Tensor
Cartesian tensors have simple transformation law under
rotation (see Eq. (88) for rank-2 tensor), but they are
reducible with respect to rotation. We use the presenta-
tion of Weissbluth (27) for the decomposition of Carte-
sian rank-2 tensor into a sum of irreducible tensors.
Any rank-2 tensor T can be written as a sum
(27,39,55,56)
T5Tð1Þ1TðsÞ; (102)
Tð1Þ being an antisymmetric rank-1 tensor with three
independent components and TðsÞ a symmetric rank-2
tensor with six independent components:
Tð1Þij 5
1
2Tij2Tji
� �52T
ð1Þji ; (103)
TðsÞij 5
1
2Tij1Tji
� �5T
ðsÞji : (104)
In matrix forms:
Tð1Þ5
01
2ðTxy2TyxÞ
1
2ðTxz2TzxÞ
1
2ðTyx2TxyÞ 0
1
2ðTyz2TzyÞ
1
2ðTzx2TxzÞ
1
2ðTzy2TyzÞ 0
0BBBBBB@
1CCCCCCA;
(105)
and
TðsÞ5
Txx
1
2ðTxy1TyxÞ
1
2ðTxz1TzxÞ
1
2ðTyx1TxyÞ Tyy
1
2ðTyz1TzyÞ
1
2ðTzx1TxzÞ
1
2ðTzy1TyzÞ Tzz
0BBBBBB@
1CCCCCCA:
(106)
As shown in Eq. (88), the Cartesian components of
rank-2 tensor T are mixed by the orthogonal matrix R.
But its symmetric part transforms as
T0ð ÞðsÞij 51
2T0ð Þij1 T0ð Þji
� �5
1
2
Xm;n5x;y;z
rimrjnTmn
11
2
Xm;n5x;y;z
rjnrimTnm
5X
m;n5x;y;z
rimrjn
1
2Tmn1Tnm
� �5
Xm;n5x;y;z
rimrjnTðsÞmn;
(107)
where we have applied the transformation matrix R as
in Eq. (47). Therefore, the components of the symmet-
ric part mix only with the components of the symmetric
part. Similarly,
T0ð Þð1Þij 51
2T0ð Þij2 T0ð Þji
� �5
1
2
Xm;n5x;y;z
rimrjnTmn21
2
Xm;n5x;y;z
rjnrimTnm
5X
m;n5x;y;z
rimrjn
1
2Tmn2Tnm
� �5
Xm;n5x;y;z
rimrjnTð1Þmn :
(108)
The components of the antisymmetric part mix only
with the components of the antisymmetric part. In sum-
mary, any Cartesian rank-2 tensor T can be written as
the sum of two parts Tð1Þ and TðsÞ, which do not mix
under the transformation matrix R. So far, what we
have described is valid for any linear transformation. It
has been established without invoking the orthogonality
condition (Eq. (70)).
Now we impose the condition that both the
unprimed and the primed coordinate systems are ortho-normal. We can go a step further by decomposing TðsÞ
using the orthogonality condition:
TðsÞ5Tð0Þ1Tð2Þ: (109)
Tð0Þ is a rank-0 tensor or scalar and Tð2Þ a traceless
symmetric rank-2 tensor with five independent
components:
Tð0Þij 5
1
3Tr Tf gdij; (110)
Tð2Þij 5
1
2Tij1Tji
� �2
1
3Tr Tf gdij5T
ð2Þji : (111)
In matrix forms (57),
Tð0Þ5
1
3Tr Tf g 0 0
01
3Tr Tf g 0
0 01
3Tr Tf g
0BBBBBBB@
1CCCCCCCA; (112)
Tð2Þ5
Txx21
3Tr Tf g 1
2Txy1Tyx
� � 1
2Txz1Tzx
� �1
2Tyx1Txy
� �Tyy2
1
3Tr Tf g 1
2Tyz1Tzy
� �1
2Tzx1Txz
� � 1
2Tzy1Tyz
� �Tzz2
1
3Tr Tf g
0BBBBBBB@
1CCCCCCCA:
(113)
First, consider the trace:
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 209
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Tr T0f g5X
i5x;y;z
T0ð Þii5X
i;m;n5x;y;z
rimrinTmn: (114)
But in view of the orthogonality condition, we have
Tr T0f g5X
i;m;n5x;y;z
rimrinTmn
5X
m;n5x;y;z
dmnTmn5X
m5x;y;z
Tmm5Tr Tf g:(115)
Therefore, Tr Tf g is an invariant under orthogonal
transformation of coordinate system. The symmetric
part Tð2Þ transforms as
T0ð Þð2Þij 51
2T0ð Þij1 T0ð Þji
� �2
1
3Tr T0f gdij
51
2
Xm;n5x;y;z
rimrjnTmn11
2
Xm;n5x;y;z
rjnrimTnm
21
3Tr
Xm;n5x;y;z
rimrjnT
( )dij
5X
m;n5x;y;z
rimrjn
1
2Tmn1Tnm
� �� �2
1
3Tr
Xm;n5x;y;z
rimrinT
( ):
(116)
Taking into account the orthogonality condition, we
have
T0ð Þð2Þij 5X
m;n5x;y;z
rimrjn
1
2Tmn1Tnm
� �2
1
3Tr Tf gdmn
� �
5X
m;n5x;y;z
rimrjnTð2Þmn :
(117)
The components of the traceless symmetric rank-2
tensor Tð2Þ mix only with the components of Tð2Þ. A
specific example (54) is furnished by the symmetricelectric quadrupole tensor
Qij5
ð3ij2r2dij
� �q x; y; zð Þdxdydz; ði; j5x; y; zÞ:
(118)
The 2r2dij term represents a subtraction of the sca-
lar trace. Therefore, the electric quadrupole tensor is a
symmetric tensor with zero trace.
In summary, in linear transformation condition, we
have the decomposition
T5Tð1Þ1TðsÞ; (119)
in orthogonal transformation condition, we have the
decomposition
T5Tð0Þ1Tð1Þ1Tð2Þ: (120)
So the set of rank-2 tensors can be decomposed into
three irreducible subsets with respect to the orthogonal
transformation. The scalar subset contains the trace of
the tensor. The vector subset of dimension three con-
tains the antisymmetric components of the tensor. The
quadrupole subset of dimension five contains traceless
symmetric components of the tensor.
Cartesian rank-2 tensor is reducible because there
are some linear combinations of its components that
transform into each other under rotation. The decompo-
sition of Cartesian rank-2 tensor in Eq. (120) is tedious.
As we need to express NMR interactions from a coor-
dinate system to another, it is preferable to use spheri-
cal tensors rather than Cartesian tensors.
V. ACTIVE ROTATION OPERATOR
As angular momentum has a close relationship with
rotation, we first recall some properties of angular
momentum operator. Then, we deduce the expression
of the active rotation operator with the simplest case
about the rotation of a function of space coordinates
that we call space function in short. In fact, it is not as
simple as it might be, because two presentations of
active rotation are available in the literature:
1. The rotation of the function occurs in a singlecoordinate system, called fixed coordinatesystem.
2. The description of the rotation involves two coor-dinate systems, the first one is a fixed coordinatesystem as in case 1, whereas the second coordi-nate system is attached to the function and rotateswith it. The two coordinate systems coincidebefore rotation. In fact, we describe the rotationof the coordinate system attached to the functioninstead of that of the function. Finally, among thevarious rotation parameterizations available (58–60), we choose the popular Euler angles for defin-ing the active rotation operator about rotated axes.As this operator is also involved in the rotation ofspin operators, it is customary to deduce this rota-tion operator about axes of the fixed coordinatesystem for avoiding the anomalous commutationrelations of spin operators (31).
Angular Momentum
The angular momentum K of a massive, spinless parti-
cle with momentum vector p located at distance r of
the origin of a Cartesian coordinate system is defined
by the cross product (or vector product)
K5r3p; (121)
whose components are
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Kx5yp z2zp y;Ky5zp x2xp z;Kz5xp y2yp x:
(122)
In quantum mechanics, p is defined by an operator
in differential form 2i�hr. The operators x, y, and z do
not commutate with the operators px, py, and pz:
½x; px�5½y; py�5½z; pz�5i�h: (123)
The angular momentum operator L and the angular
momentum K are related by (61)
K5�hL: (124)
The components of the angular momentum operator
L are defined as (60)
Lx52i yo
oz2z
o
oy
�; Ly52i z
o
ox2x
o
oz
�;
Lz52i xo
oy2y
o
ox
�:
(125)
The following three operators are important:
L15Lx1iLy; L25Lx2iL; L25L2x1L2
y1L2z : (126)
A common eigenket jl;mi of the angular momen-
tum operator squared L2 and Lz verifies
Lzjl;mi5mjl;mi; L2jl;mi5lðl11Þjl;mi: (127)
L2 commutes with the components of L:
L2; Lx
� 5 L2; Ly
h i5 L2; Lz
� 50: (128)
As Lz commutes with the angular momentum
squared L2, we focus our attention to a given angular
momentum l, having 2l 1 1 orthonormal eigenket vec-
tors jl;mi. The later is characterized by a closure
relation:
X1l
m52l
jl;mihl;mj51: (129)
Here is a useful formula for the manipulation of
angular momentum operator components:
exp ðAÞexp ðBÞexp ð2AÞ5exp ðAÞX11k50
Bk
k!
!exp ð2AÞ
5X11k50
1
k!exp ðAÞBkexp ð2AÞ
5X11k50
1
k!exp ðAÞBexp ð2AÞexp ðAÞBexp ð2AÞexp ðAÞBexp ð2AÞ:::|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
k times
5X11k50
1
k!exp ðAÞBexp ð2AÞf gk
5exp exp ðAÞBexp ð2AÞf g:
(130)
Active Rotation of Space Function
The study of active rotation may involve one or two
coordinate systems. When one coordinate system is
used (61), it corresponds to a space-fixed coordinate sys-
tem ðO; x; y; zÞ (31). When two coordinate systems are
used (28), one corresponds to a space-fixed coordinate
system ðO; x; y; zÞ and the other is attached to the space
function, rotates with it, and is called the rotated coordi-
nate system ðO; x0; y0; z0Þ, body-fixed coordinate system,
or molecule-fixed coordinate system (31). The value of a
space function at the position vector r cannot depend on
the coordinates of r in the space-fixed coordinate sys-
tem, but should depend only on the coordinates of r in
the body-fixed coordinate system. We use the same sym-
bol RAða; zÞ for active rotation of angle a about z
applied to coordinate system (52) and active rotation
operator of angle a about z applied to space function.
In One Coordinate System. Consider the active
rotation (21,31,32,35,61–65) of angle a about z of the
space function Wðr1Þ. A new space function W0AðrÞ in
the same coordinate system ðO; x; y; zÞ is generated
(Fig. 4). By definition of a space function, we find at
the position vector r the value of the space function we
found earlier at r1:
W0AðrÞ5Wðr1Þ: (131)
The two position vectors r1 and r are related by (52)
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r5AzðaÞr1: (132)
Conversely, the position vector r1 is related to the
active rotation of angle 2a about z of r, which is
equivalent to a passive rotation of angle a about z of r:
r15Azð2aÞr5PzðaÞr5
cos a sin a 0
2sin a cos a 0
0 0 1
0BB@
1CCAr:
(133)
Despite we perform an active rotation RAð2a; zÞ of
r, a passive rotation matrix PzðaÞ is used. This means
we must not rely on the rotation matrix to determine
the nature (active or passive) of a rotation. Equation
(131) becomes
W0AðrÞ5Wðr1Þ5W PzðaÞrð Þ: (134)
An active rotation of a space function is equivalent
to a passive rotation of its argument (a position vector).
The components ðx1; y1; z1Þ of r1 and those ðx; y; zÞ of
r are related by
x15x cos a1y sin a;
y152x sin a1y cosa;
z15z:
(135)
If the angle a is small, then sin a 5 e and cos a 5 1,
x15x1ye;
y15y2xe;
z15z:
(136)
Equation (131) becomes
W0Aðx; y; zÞ5Wðx1ye; y2xe; zÞ: (137)
If W is differentiable, then
W0Aðx; y; zÞ5Wðx; y; zÞ1e 2xo
oy1y
o
ox
�Wðx; y; zÞ
5 11e 2xo
oy1y
o
ox
�� �Wðx; y; zÞ:
(138)
Using the definition of angular momentum operator
defined in Eq. (125)
W0Aðx; y; zÞ5 12ieLzð ÞWðx; y; zÞ: (139)
A rotation of angle a about z may be described by a
series of N rotations of angle e5 aN about z. If we con-
sider the limit condition when N ! 11 or e! 0 for
constant a, we establish the active rotation operator RA
ða; zÞ of angle a about z applied to a space function:
limN!11
12iLza
N
� �N5exp ð2iaLzÞ: (140)
We denote (54,61,62,65)
RAða; zÞ5exp ð2iaLzÞ: (141)
Therefore (21,63,66), including the definition RAða;zÞWðrÞ in Eq. (134) yields
W0AðrÞ51
RAða; zÞWðrÞ52
Wðr1Þ53
W PzðaÞrð Þ:(142)
Two interpretations are possible for the action of RA
ða; zÞ in Eq. (142) (62). The first equality denoted by 1
describes an active rotation of the space function from
W to W0A, their arguments r being identical. By defini-
tion, the action of an operator RAða; zÞ upon the space
function WðrÞ must give us a new function W0AðrÞ of
the same argument (63). We may write:
W0AðrÞ51
RAða; zÞW½ �ðrÞ52
Wðr1Þ53
W PzðaÞrð Þ:(143)
In contrast, the other two equalities denoted by 2
and 3 describe the action of RAða; zÞ as a passive rota-
tion of the argument, the space function W being identi-
cal. In quantum mechanics, an active rotation operator
RAða; zÞ is a unitary transformation, that is,
RAða; zÞ½ �15 RAða; zÞ½ �-1: (144)
Therefore,
RAða; zÞWðrÞ5hrjRAða; zÞjWi5h½RAða; zÞ�1rjWi
5h½RAða; zÞ�-1rjWi5W
�½RAða; zÞ�-1
r�
5W�
RPða; zÞr�:
(145)
In Two Coordinate Systems. Rose (28) did not use
figures to describe the rotation of a space function. In
Figure 4 Active rotation of angle a about z axis of a
space function shown in a single coordinate system
(O,x,y,z). w(r1) is the initial space function before rotation
and w0A(r) is the rotated space function.
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fact, he used two coordinate systems and investigated
the active rotation of a space function. Our space-fixed
coordinate system ðO; x; y; zÞ in Fig. 5 is called the
original coordinate system by Rose, and the rotated
coordinate system is ðO; x0; y0; z0Þ in Fig. 5(b). The
rotated coordinate system is attached to the space
function.
Initially, the space function before rotation is Wðr1Þin the original coordinate system [Fig. 5(a)]. Then, we
rotate the coordinate system attached to the space func-
tion. The later is described by Wðr0Þ in its attached
coordinate system ðO; x0; y0; z0Þ where the function
remains the same but its argument becomes r0 [Fig.
5(b)]. The angle a describes the rotation of the attached
coordinate system in Fig. 5(b), whereas it describes the
rotation of the space function in Fig. 5(a) and Fig. 4
when one coordinate system is used. The rotated space
function in Fig. 5(b) is viewed in two coordinate sys-
tems, therefore,
W0AðrÞ5Wðr0Þ: (146)
Equation (146) from Rose differs with Eq. (131) by
the arguments of the space function W. The argument
of W in Eq. (146) is the final position vector r0 in the
rotated coordinate system ðO; x0; y0; z0Þ, whereas in Eq.
(131), it is the initial position vector r1 in the space-
fixed coordinate system ðO; x; y; zÞ or initial coordinate
system (25). In its attached coordinate system
ðO; x0; y0; z0Þ, the two vectors r1 and r0 have the same
coordinate components. Therefore, Eq. (146) from two
coordinate systems and Eq. (131) from one coordinate
system become identical when position vectors are
replaced by their components.
The position vector r0 is identical to r in Fig. 5(b).
This position vector is viewed from two coordinate sys-
tems. This means (52)
r05PzðaÞr: (147)
Reporting Eq. (147) in Eq. (146) yields
W0AðrÞ5Wðr0Þ5W PzðaÞrð Þ: (148)
In expanded form for infinitesimal rotation angle e,
we have
x05x1ye;
y05y2xe;
z05z:
(149)
Formally, Eqs. (149) and (148) from Rose are simi-
lar to Eqs. (136) and (134), which means Rose also per-
formed an active rotation of a space function. As a
result, the active rotation operator is defined by Eq.
(141) as shown Rose. Including the definition RAða; zÞWðrÞ in Eq. (148) yields
W0AðrÞ51
RAða; zÞWðrÞ52
Wðr0Þ53
W PzðaÞrð Þ:(150)
The active rotations of a space function described
with one and two coordinate systems are equivalent.
This is supported by the same series of equalities in
Eqs. (142) and (150). Van de Wiele (67), Messiah (21),
Steinborn and Ruedenberg (68), Brink and Satchler
(35), and Devanathan (40) also use two coordinate sys-
tems denoted by S and S0.
Euler Angles for Active Rotation of SpaceFunction
About Rotated Axes. The first rotation of angle a
about the vector z [Fig. 6(a)] transforms the coordi-
nate system ðO; x; y; zÞ to a new coordinate system
ðO; x0; y0; z0Þ that is attached to the space function.
The active rotation operator exp ð2iaLzÞ transforms
the space function WðrÞ to W0AðrÞ:
Figure 5 Active rotation of angle a about z axis of a
space function shown in two coordinate systems (O,x,y,z)
and (O,x0,y0,z0). (a) Initial configuration: W(r1) is the ini-
tial space function before rotation as in Fig. 4, (O,x,y,z)
and (O,x0,y0,z0) coincide. (b) Final configuration: w0A(r) is
the rotated space function in the coordinate system
(O,x,y,z) as in Fig. 4 and w(r0) is the rotated space func-
tion in its attached coordinate system (O,x0,y0,z0) after
rotation.
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W0AðrÞ5exp ð2iaLzÞWðrÞ: (151)
The two space functions, expressed in the fixed
coordinate system ðO; x; y; zÞ, have the same argument
of position vector r.
Then, the second rotation of angle b about the vec-
tor y0 [Fig. 6(b)] transforms the coordinate system ðO;x0; y0; z0Þ into a new coordinate frame ðO; x00; y00; z00Þthat is attached to the space function. The active rota-
tion operator exp ð2ibLy0 Þ transforms the space func-
tion W0AðrÞ to W00AðrÞ:
W00AðrÞ5exp ð2ibLy0 ÞW0AðrÞ5exp ð2ibLy0 Þexp ð2iaLzÞWðrÞ:
(152)
The two space functions W00AðrÞ and W0AðrÞ are
expressed in the fixed coordinate system ðO; x; y; zÞ.Finally, the third rotation of angle g about the vector
z00 [Fig. 6(c)] transforms the coordinate frame ðO; x00;y00; z00Þ to a new coordinate frame ðO; x000; y000; z000Þ that is
attached to the space function. The active rotation oper-
ator exp ð2igLz00 Þ transforms the space function W00AðrÞto W000AðrÞ:
W000AðrÞ5exp ð2igLz00 ÞW00AðrÞ
5exp ð2igLz00 Þexp ð2ibLy0 Þexp ð2iaLzÞWðrÞ:(153)
The two space functions W000AðrÞ and W00AðrÞ are
expressed in the fixed coordinate system ðO; x; y; zÞ.In the three rotations, the rotation axis belongs to the
coordinate system before the rotation. We denote the
active rotation operator about rotated axes by (21,61)
RrotA ðg;b;aÞ5exp ð2igLz00 Þexp ð2ibLy0 Þexp ð2iaLzÞ:
(154)
The order of the Euler angles in the arguments of
RrotA ðg;b;aÞ follows that of Euler angles in the three
elementary active rotation operators. The first rotation
angle a is the right-hand side argument of RrotA ðg;b;aÞ.
Therefore, we have the generalization of Eq. (142):
W000AðrÞ51
RrotA ðg;b;aÞWðrÞ5
2Wðr1Þ
53
W Protated Z3Y2Z1ðg;b;aÞrð Þ;(155)
for one coordinate system or
W000AðrÞ51
RrotA ðg;b;aÞWðrÞ5
2Wðr000Þ
53
W Protated Z3Y2Z1ðg;b;aÞrð Þ;(156)
for two coordinate systems where the components of
r000 in ðO; x000; y000; z000Þ are identical to those of r1 in
ðO; x; y; zÞ.
About Fixed Axes. We determine the active rotation
operator about fixed axes from that about rotated axes
defined in Eq. (154). First, we express the angular
momentum operators Ly0 and Lz00 about rotated axes in
terms of angular momentum operators about fixed axes
Ly and Lz. From Fig. 6, we deduce that
Ly05exp ð2iaLzÞLyexp ðiaLzÞ; (157)
Lz005exp ð2ibLy0 ÞLzexp ðibLy0 Þ: (158)
Figure 6 Euler angles defining the rotation of a Carte-
sian coordinate system: (a) rotation of angle a about zaxis of (O,x,y,z); (b) rotation of angle b about y0 axis of
(O,x0,y0,z0); and (c) rotation of angle g about z00 axis of
(O,x00,y00,z00).
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Then, we apply the formula defined in Eq. (130) to
Eqs. (157) and (158) (39,65):
exp ð2ibLy0 Þ5exp ð2iaLzÞexp ð2ibLyÞexp ðiaLzÞ;(159)
exp ð2igLz00 Þ5exp ð2ibLy0 Þexp ð2igLzÞexp ðibLy0 Þ:(160)
We report Eq. (159) in Eq. (160):
exp ð2igLz00 Þ5exp ð2iaLzÞexp ð2ibLyÞ3exp ðiaLzÞexp ð2igLzÞexp ðibLy0 Þ:
(161)
We multiply both members of Eq. (161) by
exp ð2ibLy0 Þ, which are part of Eq. (154):
exp ð2igLz00 Þexp ð2ibLy0 Þ5exp ð2iaLzÞ3exp ð2ibLyÞexp ðiaLzÞexp ð2igLzÞ:
(162)
We multiply both members of Eq. (162) by
exp ð2iaLzÞ, which are part of Eq. (154):
exp ð2igLz00 Þexp ð2ibLy0 Þexp ð2iaLzÞ
5exp ð2iaLzÞexp ð2ibLyÞexp ðiaLzÞ
3exp ð2igLzÞexp ð2iaLzÞ:
(163)
As the rotation operators about the same axis com-
mute, we obtain the active rotation operator about fixed
axes (21,40,61,65):
RfixedA ða;b; gÞ5exp ð2iaLzÞexp ð2ibLyÞ
3exp ð2igLzÞ5RrotA ðg;b;aÞ:
(164)
The order of Euler angles in the arguments of RfixedA
ða;b; gÞ follows that of elementary angular momen-
tum operators about fixed axes. This order is the
reverse of that in RrotA ðg;b;aÞ. We have manipulated
the expression of RrotA ðg;b;aÞ, which depends on ele-
mentary angular momentum operators of successively
rotated coordinate systems, so that the expression
depends on elementary angular momentum operators
of the fixed coordinate system, without taking into
account the space function. Equation (155) for one
coordinate system becomes
W000AðrÞ51
RfixedA ða;b; gÞWðrÞ52 Wðr1Þ5
3
W Pfixed Z1Y2Z3ðg;b;aÞrð Þ;(165)
whereas Eq. (156) for two coordinate systems becomes
W000AðrÞ51
RfixedA ða;b; gÞWðrÞ52 Wðr000Þ53
W Pfixed Z1Y2Z3ðg;b;aÞrð Þ:(166)
In the remainder of the article, the active rotation
operator RfixedA ða;b; gÞ about fixed coordinate axes is
replaced by RAða;b; gÞ for simplicity.
VI. SPHERICAL HARMONICS
We widen our investigation of active rotation using
spherical harmonics and Wigner active rotation matrix.
This is a prerequisite for the study of rotation of spheri-
cal tensors. Spherical harmonics of order-l are a set of
2l 1 1 space functions whose arguments are the polar
angles of position vector in a coordinate system. The
arguments of the spherical harmonics allow us to deter-
mine their associated coordinate system. In contrast,
spherical tensors have no arguments. The missing of
arguments does not allow us to determine the coordi-
nate system in which the tensor components are. We
have to rely on spherical harmonics to clarify this
point.
Wigner Active Rotation Matrix
Thanks to commutation rules in Eq. (128), the
active rotation operator about fixed axes (Eq. (164))
commutes with the square of angular momentum
operator L2, whose eigenstates are the set of eigen-
kets jl;mi. Taking into account the relation of
closure of angular momentum eigenkets jl;mi (Eq.
(129)), we have
RAða;b;gÞjl;mi5X1l
m052l
jl;m0ihl;m0j !
RAða;b; gÞjl;mi:
(167)
The rotated ket RAða;b; gÞjl;mi is a linear combi-
nation of jl;mi with coefficients that are elements of
Wigner active rotation matrix:
Dðl;AÞm0m ða;b; gÞ � hl;m0jRAða;b; gÞjl;mi: (168)
Replacing RAða;b; gÞ by its expression (Eq. (164)),
we have
Dðl;AÞm0m ða;b; gÞ
5hl;m0jexp ð2iaLzÞexp ð2ibLyÞexp ð2igLzÞjl;mi5exp ð2iam0Þhl;m0jexp ð2ibLyÞjl;miexp ð2igmÞ
5exp ð2iam0Þdðl;AÞm0m ðbÞexp ð2igmÞ:(169)
For l 51 (28,40,44,54,67,69), this matrix is
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 215
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Wigner active rotation matrix Dðl;AÞða;b; gÞ has
many properties (35,69). As the active rotation operator
RAða;b; gÞ is unitary, it results that Dðl;AÞða;b; gÞ is
also unitary. That is,
Dðl;AÞða;b; gÞn o1
5 Dðl;AÞða;b; gÞn o21
: (171)
As
Dðl;AÞða;b; gÞn o1
Dðl;AÞða;b; gÞ5EðlÞ; (172)
where EðlÞ is the 2l11 dimensional identity matrix, it
results that (28,32,35)Xm00
Dðl;AÞmm00 ða;b; gÞ D
ðl;AÞm0m00 ða;b; gÞ
n o�5dmm0 ; (173)
Xm00
Dðl;AÞm00m0 ða;b; gÞ
n o�Dðl;AÞm00m ða;b; gÞ5dmm0 : (174)
Wigner Active Rotation of Eigenbra andEigenket Vectors
If a coordinate system ðO; x0; y0; z0Þ is obtained by the
rotation RAða;b; gÞ of a fixed coordinate system
ðO; x; y; zÞ, the eigenket vectors jl;mi0 of Lz0 are given
by rotating the eigenket vectors jl;mi of Lz induced by
the rotation of coordinate system:
jl;mi05RAða;b; gÞjl;mi: (175)
The eigenvalue equations for these eigenket vectors
are (70)
L2jl;mi5lðl11Þjl;mi, Lzjl;mi5mjl;mi,
L2jl;mi05lðl11Þjl;mi0; Lz0 jl;mi05mjl;mi0: (176)
Taking into account the closure relation (Eq. (129)),
Eq. (175) becomes (32,69–71)
RAða;b; gÞjl;mi5X1l
m052l
jl;m0ihl;m0j !
3RAða;b; gÞjl;mi5X1l
m052l
jl;m0iDðl;AÞm0m ða;b; gÞ:
(177)
The eigenket vectors jl;mi0 of Lz0 are specified in
terms of those of Lz. Equation (177) means that the
eigenket vectors transform among themselves with
coefficients that are elements of Wigner active rotation
matrix. Equation (177) is a unitary transformation from
the eigenket basis jl;mi of the complete set of commut-
ing observables L2; Lz
� �to the eigenket basis jl;mi0 of
fL2; Lz0 g. As RAða;b; gÞ commutes with L2, an active
rotation RAða;b; gÞ acting on jl;mi will conserve the
quantum number l. However, the projection m on the
fixed z axis may not be conserved:
LzRAða;b; gÞjl;mi5Lz
X1l
m052l
jl;m0iDðl;AÞm0m ða;b; gÞ
5X1l
m052l
m0Dðl;AÞm0m ða;b; gÞ:
(178)
The result is a linear combination of m.
Similarly, the eigenbra vectors 0hl;mj of Lz0 are
given by rotating the eigenbra vectors hl;mj of Lz
induced by the rotation of coordinate system. As
the operator RAða;b; gÞ is not Hermitian, we
have
0hl;mj5hl;mj RAða;b; gÞf g1: (179)
Taking into account the closure relation (Eq. (129)),
Eq. (179) becomes
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0hl;mj5hl;mj RAða;b;gÞf g15X1l
m052l
hl;m0j Dðl;AÞm0m ða;b;gÞ
n o�:
(180)
In Eqs. (177) and (180), eigenket and eigenbra vec-
tors are in row matrices. The eigenbra vectors trans-
form among themselves with coefficients that are
elements of the complex conjugate of Wigner active
rotation matrix. As eigenket and eigenbra vectors
belong, respectively, to the angular momentum space
and its dual space, the transformation law in Eq. (180)
is called contragredient (18,35,50) to that in Eq. (177).
Wigner Active Rotation of a General KetVector in Angular Momentum Space
A general ket vector jf i in the angular momentum
space is defined as
jf i5X1l
m52l
jl;miðamÞ�; (181)
where we use the complex conjugate of covariant
spherical components instead of contravariant spherical
components am. Applying an active rotation RAða;b; gÞto this ket jf i generates another ket jf i0 in the same
eigenket basis:
jf i05X1l
m052l
jl;m0iða0m0 Þ�: (182)
Another expression of jf i0 is defined with the active
rotation operator:
jf i05RAða;b; gÞjf i5RAða;b; gÞX1l
m52l
jl;miðamÞ�:
(183)
Introducing the closure relation (Eq. (129)) yields
jf i05X1l
m;m052l
jl;m0ihl;m0jRAða;b; gÞjl;miðamÞ�
5X1l
m;m052l
jl;m0iDðl;AÞm0m ða;b; gÞðamÞ�:
(184)
Comparing Eq. (184) with Eq. (182) yields
ða0m0 Þ�5X1l
m52l
Dðl;AÞm0m ða;b; gÞðamÞ
�: (185)
This is a traditional matrix multiplication where the
contravariant spherical components ða0m0 Þ�
and ðamÞ�
of the general ket vector are in column matrices.
In an active rotation of a general ket vector jf i, the
same Wigner active rotation matrix Dðl;AÞða;b; gÞ is
involved in the transformation law of angular momen-
tum eigenkets jl;mi (Eq. (177)) and in that of its con-
travariant spherical components ðamÞ�
(Eq. (185)). But
the row matrix of angular momentum eigenkets is post-
multipled by Dðl;AÞða;b; gÞ, whereas the column
matrix of contravariant spherical components is pre-
multiplied by Dðl;AÞða;b; gÞ.
Wigner Active Rotation of SphericalHarmonics
An application of Wigner active rotation matrix is the
transformation of spherical harmonics induced by an
active rotation of the physical system. This rotation
may occured in the same coordinate system or between
two coordinate systems (68).
In the configuration of one coordinate system, the
active rotation of physical system is described by the
Euler angles ða;b; gÞ. In the configuration of two coor-
dinate systems, the second coordinate system is attached
to the physical system and rotates with it. Before rota-
tion, the two coordinate systems coincide. The relative
orientation of these two coordinate systems after rota-
tion is described by the Euler angles ða;b; gÞ.The spherical harmonics, eigenfunctions of orbital
angular momentum, are the coordinate representation
of the angular momentum eigenket vectors jl;mi with
integer values of l and m. The rotated eigenket vectors
jl;mi0 and the fixed eigenket vectors jl;mi are related
(Eq. (177)):
jl;mi05RAða;b; gÞjl;mi5X1l
m052l
jl;m0iDðl;AÞm0m ða;b; gÞ:
(186)
In One Coordinate System. Consider the first con-
figuration shown in Fig. 7. The initial position vector is
named r1 and the final or rotated one is r in the
fixed coordinate system ðO; x; y; zÞ. The position eigen-
kets jr1i and jri associated with r1 and r are related by
an active rotation operator:
jri5RAða;b; gÞjr1i: (187)
Conversely, we have
jr1i5 RAða;b; gÞf g21jri: (188)
As RAða;b; gÞ is a unitary operator, the dual of Eq.
(188) is
hr1j5hrjRAða;b; gÞ: (189)
We multiply each member of Eq. (186) by hrj:
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 217
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hrjl;mi05hrjRAða;b; gÞjl;mi5X1l
m052l
hrjl;m0iDðl;AÞm0m ða;b; gÞ:
(190)
Taking into account Eq. (189), we have
hr1jl;mi5X1l
m052l
hrjl;m0iDðl;AÞm0m ða;b; gÞ: (191)
The polar angles of the initial position vector r1 and
those of the rotated one r in ðO; x; y; zÞ are u1;/1
� �and u;/ð Þ, respectively. These angles allow us to
rewrite Eq. (191) as
hu1;/1jl;mi5X1l
m052l
hu;/jl;m0iDðl;AÞm0m ða;b; gÞ: (192)
Writing this result in terms of spherical harmonics,
we obtain (64,67)
Yl;mðu1;/1Þ53 X1l
m052l
Yl;m0 ðu;/ÞDðl;AÞm0m ða;b; gÞ: (193)
The spherical harmonics Yl;mðu1;/1Þ whose argu-
ments are the polar angles of the initial position vector
r1 are expressed in terms of Yl;m0 ðu;/Þ whose argu-
ments are the polar angles of the final or rotated posi-
tion vector r. The spherical harmonics in the two
members of Eq. (193) are identical but their arguments
are different. This corresponds to the equality denoted
by 3 in Eq. (142).
By extension, including the definition of the active
rotation of spherical harmonics in Eqs. (190) and (193)
yields
Y0l;mðu;/Þ51
RAða;b; gÞYl;mðu;/Þ52
Yl;mðu1;/1Þ
53 X1l
m052l
Yl;m0 ðu;/ÞDðl;AÞm0m ða;b; gÞ:
(194)
Equation (194) about active rotation of spherical
harmonics and Eq. (165) about that of space function
have similar structure.
In Two Coordinate Systems. Consider the second
configuration (70) that is shown in Fig. 8. The rotated
position vector is named r in the fixed coordinate sys-
tem ðO; x; y; zÞ [Figs. 8(a, b)] and r0 in the rotated coor-
dinate system ðO; x0; y0; z0Þ [Fig. 8(b)] or body-fixed
coordinate system (64). The position eigenkets jri and
jr0i associated with r and r0 are related by a passiveFigure 7 Active rotation of spherical harmonics shown in
a single coordinate system (O,x,y,z): u1 and /1 are the polar
angles of the position vector r1 before rotation whereas u
and / are those of the position vector r after rotation.
Figure 8 Active rotation of spherical harmonics shown
in two coordinate systems (O,x,y,z) and (O,x0,y0,z0) related
by the Euler angles a, b, and g. (a) Initial configuration:
r1 with polar angles u1 and /1 is the initial position vec-
tor before rotation as in Fig. 7, (O,x,y,z) and (O,x0,y0,z0)coincide. (b) Final configuration: r with polar angles u
and / is the rotated position vector in (O,x,y,z) as in Fig.
7, and r0 with polar angles u0 and /0 is the position vector
in its attached coordinate system (O,x0,y0,z0) after rotation.
218 MAN
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rotation operator, representing the passive rotation of
the rotated position vector from ðO; x; y; zÞ to
ðO; x0; y0; z0Þ. This passive rotation operator is the
inverse of RAða;b; gÞ:
jr0i5 RAða;b; gÞf g21jri: (195)
Its dual is
hr0j5hrjRAða;b; gÞ: (196)
We multiply each member of Eq. (186) by hrj:
hrjl;mi05hrjRAða;b; gÞjl;mi5X1l
m052l
hrjl;m0iDðl;AÞm0m ða;b; gÞ:
(197)
Taking into account Eq. (196), we have
hr0jl;mi5X1l
m052l
hrjl;m0iDðl;AÞm0m ða;b; gÞ: (198)
The polar angles of the position vectors r and r0 are
u;/ð Þ in ðO; x; y; zÞ and u0;/0ð Þ in ðO; x0; y0; z0Þ,respectively. These angles allow us to rewrite Eq. (198)
as
hu0;/0jl;mi5X1l
m052l
hu;/jl;m0iDðl;AÞm0m ða;b; gÞ: (199)
Writing this result in terms of spherical harmonics,
we obtain (21,35,54,68,70)
Yl;mðu0;/0Þ53 X1l
m052l
Yl;m0 ðu;/ÞDðl;AÞm0m ða;b; gÞ: (200)
The spherical harmonics Yl;mðu0;/0Þ whose argu-
ments are the polar angles of the position vector r0
in its attached coordinate system are expressed in
terms of Yl;m0 ðu;/Þ whose arguments are the polar
angles of the final or rotated position vector r in
the fixed coordinate system. As the position vector
and its attached coordinate system rotate together,
its polar angles ðu1;/1Þ before rotation in the ini-
tial coordinate system (Fig. 8(a)) are identical to
those ðu0;/0Þ after rotation in the rotated coordinate
system:
ðu0;/0Þ5ðu1;/1Þ: (201)
The spherical harmonics in the two members of Eq.
(200) are identical but their arguments are different.
This corresponds to the equality denoted by 3 in Eq.
(150). If we replace the polar angles ðu0;u0Þ in Eq.
(200) by those defined in Eq. (201), Eqs. (200) and
(193) become identical. Equations (193) and (200) fol-
low the second interpretation of a rotation operator,
which only changes the arguments of spherical harmon-
ics, leaving the spherical harmonics unchanged. The
spherical harmonics transform among themselves with
exactly the same matrix of coefficients as that for 2l 1 1
angular momentum eigenket vectors jl;mi (Eq. (186)).
Including the definition RAða;b; gÞYl;mðu;/Þ in
Eqs. (197) and (200) yields
Y0l;mðu;/Þ51
RAða;b; gÞYl;mðu;/Þ52
Yl;mðu0;/0Þ
53 X1l
m052l
Yl;m0 ðu;/ÞDðl;AÞm0m ða;b; gÞ:
(202)
Equation (202) about active rotation of spherical
harmonics and Eq. (166) about that of space function
have similar structure.
Example 1. Consider a position vector A in a three-
dimensional Cartesian coordinate system ðO; x; y; zÞassociated with an orthonormal basis fex; ey; ezg. The
Cartesian components of A in this basis are ðax; ay; azÞ.We apply an active rotation of Euler angles a5
b50 and g5 p2
about the z axis to A. The Carte-
sian components ða0x; a0y; a0zÞ of the rotated vector
A0 in the same orthonormal basis is given by the
direct Cartesian rotation of A (52,71,72):
a0x
a0y
a0z
0BBB@
1CCCA5
cosp2
2sinp2
0
sinp2
cosp2
0
0 0 1
0BBBBB@
1CCCCCA
ax
ay
az
0BBB@
1CCCA
5
0 21 0
1 0 0
0 0 1
0BBB@
1CCCA
ax
ay
az
0BBB@
1CCCA5
2ay
ax
az
0BBB@
1CCCA:
(203)
There is a second way to solve this problem. We
can relate the Cartesian components of A to the spheri-
cal harmonics of order l 5 1 (Table 1):
Y1;11ðu1;/1Þ52
ffiffiffiffiffiffi3
4p
rax1iay
dffiffiffi2p ; Y1;0ðu1;/1Þ
5
ffiffiffiffiffiffi3
4p
raz
dffiffiffi2p ; Y1;21ðu1;/1Þ5
ffiffiffiffiffiffi3
4p
rax2iay
dffiffiffi2p ;
(204)
where d5jjAjj.It is tempting to use Eq. (193) defined for the config-
uration of one coordinate system. But this relation (Eq.
(193)) expresses the spherical harmonics Y1;mðu1;/1Þwith initial polar angles in terms of spherical harmonics
Y1;m0 ðu;/Þ with final polar angles. In short, the initial
covariant spherical components of A are expressed in
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 219
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terms of rotated covariant spherical components of A0.Therefore, we must express the final spherical harmon-
ics Y1;m0 ðu;/Þ in terms of initial spherical harmonics
Y1;mðu1;/1Þ.We multiply the two members of Eq. (193) by the
complex conjugate of Wigner active rotation matrix:
X1l
m52l
Yl;mðu1;/1Þ Dðl;AÞm00m ða;b; gÞ
n o�
5X1l
m;m052l
Yl;m0 ðu;/ÞDðl;AÞm0m ða;b; gÞ D
ðl;AÞm00m ða;b; gÞ
n o�:
(205)
Taking into account Eq. (173), we have
X1l
m52l
Yl;mðu1;/1Þ Dðl;AÞm00m ða;b; gÞ
n o�
5X1l
m052l
Yl;m0 ðu;/Þdm0m00 :
(206)
Therefore (66–68),
Yl;m00 ðu;/Þ5X1l
m52l
Dðl;AÞm00m ða;b; gÞ
n o�Yl;mðu1;/1Þ:
(207)
Spherical harmonics are in column matrices. Con-
travariant spherical components are provided by the
complex conjugates of Eq. (207):
Yl;m00 ðu;/Þn o�
5X1l
m52l
Dðl;AÞm00m ða;b; gÞ Yl;mðu1;/1Þ
n o�:
(208)
The active rotation RAða;b; gÞ takes a position vec-
tor with the direction ðu1;/1Þ into the direction ðu;/Þ.
The arguments or polar angles of spherical harmonics
allow us to deduce not only the coordinate system in
which is the position vector but also that of spherical
harmonics. Unfortunately, this is not always the case
for spherical tensor components.
In an active rotation of spherical harmonics of
order-l Yl, the same Wigner active rotation matrix
Dðl;AÞða;b; gÞ is involved in the transformation law of
angular momentum eigenkets jl;mi (Eq. (186)) and in
that of its contravariant spherical harmonics fYl;mðu;/Þg
�(Eq. (208)). But the row matrix of angular
momentum eigenkets is postmultipled by
Dðl;AÞða;b; gÞ, whereas the column matrix of contra-
variant spherical harmonics is premultiplied by
Dðl;AÞða;b; gÞ.In our case, Dð1;AÞða;b; gÞ defined in Eq. (170)
becomes
Dð1;AÞ a50;b50; g5p2
� �5
2i 0 0
0 1 0
0 0 1i
0BB@
1CCA: (209)
Replacing covariant spherical harmonics in Eq.
(207) by covariant spherical components of A and A0
yields
2a0x1ia0yffiffiffi
2p
a0z
a0x2ia0yffiffiffi2p
0BBBBBB@
1CCCCCCA5
1i 0 0
0 1 0
0 0 2i
0BB@
1CCA
2ax1iayffiffiffi
2p
az
ax2iayffiffiffi2p
0BBBBBB@
1CCCCCCA:
(210)
It results that
a0x52ay; a0y5ax; a0z5az; (211)
in agreement with the direct Cartesian rotation in Eq.
(203). In this example, one coordinate system is used.
Definition of Spherical Tensor
Two definitions of spherical tensor are found in the lit-
erature. The first one defines the components of the
spherical tensor with a transformation law based on
that of spherical harmonics (28,64) or on that of angu-
lar momentum eigenket vectors (40). The second one is
based on a geometry definition (18,50) where spherical
tensor components are defined in a tensor basis as vec-
tor components are defined in a vector basis. We pres-
ent the first one; the second is reported in the next
section about spherical tensor.
The ð2l11Þ entities, Tlm, for m52l;2l11; . . . ;1l,are said to form the components of a spherical tensor of
Table 1 First (21,27,30–32,34,39,40,54,73–79) FewNormalized Spherical Harmonics Yl;mðh;/Þ and dlYl;m
ðx; y; zÞ with x5d sin h cos /, y5d sin h sin /, and
z5d cos h
l m dlYl;mðx; y; zÞ Yl;mðu;/Þ
0 0ffiffiffiffi1
4p
q ffiffiffiffi1
4p
q1 0
ffiffiffiffi3
4p
qz
ffiffiffiffi3
4p
qcos u
1 617
ffiffiffiffi3
8p
qx6iyð Þ 7
ffiffiffiffi3
8p
qsin ue6i/
2 0ffiffiffiffi5
4p
q ffiffi14
q3z22r2ð Þ
ffiffiffiffi5
4p
q ffiffi14
q3cos 2u21ð Þ
2 617
ffiffiffiffi5
4p
q ffiffi32
qz x6iyð Þ 7
ffiffiffiffi5
4p
q ffiffi32
qcos usin ue6i/
2 62ffiffiffiffi5
4p
q ffiffi38
qx6iyð Þ2
ffiffiffiffi5
4p
q ffiffi38
qsin 2ue62i/
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rank l if they transform under rotations like the spheri-
cal harmonics Yl;m of order l. That is,
TlmðrÞ ! Tlmðr0Þ53 X1l
m052l
Tlm0 ðrÞDðl;AÞm0m ða;b; gÞ:
(212)
Here r0 is the position vector of a physical point in
the rotated coordinate system (the body-fixed coordi-
nate system), while r is the position vector of the same
point in the original (space-fixed) coordinate system
(64,80). This description is that of Eq. (200), corre-
sponding to the configuration of two coordinate sys-
tems. The spherical tensor components in both
members of Eq. (212) are identical, in agreement with
Eq. (202) about spherical harmonics.
Most often, spherical tensor components Tlm, con-
trary to spherical harmonics Yl;m, have no arguments.
Equation (212) without arguments for spherical tensor
components becomes odd. It is the reason why Mueller
(72) introduces the following notation for Eq. (212),
TRlm 5
3 X1l
m052l
Tlm0Dðl;AÞm0m ða;b; gÞ; (213)
where TRlm are the spherical tensor components in its
attached coordinate system after rotation and Tlm0 those
in the space-fixed coordinate system after rotation.
Equation (202) about active rotation of spherical har-
monics becomes
RAða;b; gÞTlm 52
TRlm 5
3 X1l
m052l
Tlm0Dðl;AÞm0m ða;b; gÞ:
(214)
Equality denoted by 1 in Eq. (202) is missing in
Eq. (214).
Brink and Satchler (35) define the same relation as
T0lm5X1l
m052l
Tlm0Dðl;AÞm0m ða;b; gÞ; (215)
ða;b; gÞ being the Euler angles of the rotation that
takes the old, unprimed, axes into the new, primed,
axes. Equation (215) expresses a component T0lmdefined with respect to the new axes in terms of the
components Tlm defined with respect to the old axes.
Without arguments for the spherical tensor compo-
nents, we have some difficulties to recognize the loca-
tion of these tensor components in two coordinate
systems involved in Eqs. (213) and (215). We can
reformulate these two equations using a single coordi-
nate system, that is, the space-fixed coordinate system.
The components TRlm in Eq. (213) are those of the spher-
ical tensor in its attached coordinate system after rota-
tion. These components are also those of the spherical
tensor before rotation in the space-fixed coordinate sys-
tem. In other words, they are the initial components of
the spherical tensor before rotation. Conversely, the
components Tlm0 in Eq. (213) are those of the rotated
spherical tensor in the space-fixed coordinate system.
In other words, they are the final components of the
spherical tensor in the space-fixed coordinate system
after rotation. In short, in a space-fixed coordinate sys-
tem, Eq. (215) expresses the initial components of a
spherical tensor before rotation in terms of the final
components of the spherical tensor after rotation.
In practice, we need to express the final spherical
tensor components Tlm0 in terms of the initial spherical
tensor components TRlm. To this end, we multiply the
two members of Eq. (213) by the complex conjugate of
Wigner active rotation matrix and sum over m:
Xl
m52l
TRlm D
ðl;AÞm00m ða;b; gÞ
� ��
5X1l
mm052l
Tlm0Dðl;AÞm0m ða;b; gÞ D
ðl;AÞm00m ða;b; gÞ
� ��:
(216)
Equation (173) allows us to simplify the above
relation:
Xl
m52l
TRlm D
ðl;AÞm00m ða;b; gÞ
� ��5X1l
m052l
Tlm0dm0m00 : (217)
Finally, we have our desired relation for covariant
spherical components:
Tlm005Xl
m52l
Dðl;AÞm00m ða;b; gÞ
� ��TR
lm; (218)
or for contravariant spherical components:
Tlm00� ��
5Xl
m52l
Dðl;AÞm00m ða;b; gÞ TR
lm
� ��: (219)
VII. SPHERICAL TENSOR
As Cartesian tensors are reducible with respect to the
rotation of coordinate system, they are not suitable in
NMR studies involving rotation transformations. In
contrast, spherical tensors are irreducible, whose com-
ponents transform linearly among themselves under
rotations. That is, they do not contain within them ten-
sors of lower ranks. The disadvantages of spherical ten-
sors are the mathematical manipulations, which
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 221
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demand much more effort than with Cartesian
tensors. In particular there are various phase conven-
tions (38).
In the previous subsection, we have provided the
first definition of spherical rank-r tensor, which has
2r 1 1 components, combines, and transforms like
angular momentum eigenket vectors jl;mi (Eq. (186)),
with r5l and s5m (32,69). The transformation of a
spherical tensor involves Wigner active rotation matrix
D r;Að Þ a;b; gð Þ, which is a (2r 1 1)X(2r 1 1) square
matrix, whose size increases with the rank r of the
spherical tensor. In contrast, r copies of rotation matrix
R are involved in the transformation of Cartesian rank-
r tensors (Eq. (90)). As spherical tensors are related to
the quantum mechanical treatment of angular momen-
tum, their components are indexed by angular momen-
tum r and s rather than by Cartesian coordinates x, y, z.As the components of Cartesian vectors and tensors,
those of spherical tensors can be written with contra-
variant or covariant notations. The angular momentum
algebra of most of the literature is based on covariant
spherical components but without explicit reference to
their covariant nature (45).
Here is the second definition of spherical tensor Tr
of rank r:
Tr5X1r
s52r
Trs
� ��trs (220)
is a geometrical entity (50). It is a generalization of
a vector, where Trs are covariant spherical tensor
components and trs covariant spherical basis tensors.
This definition of spherical tensor is identical to that
of Eq. (30) used by Mueller (72). Covariant spheri-
cal tensor components have no arguments in contrast
to spherical harmonics. This may lead confusion
about the associated coordinate system as we shown
in the previous subsection. Fortunately, the basis
tensors trs allow us to find the corresponding coordi-
nate system.
In the remainder of this article, we use covariant
notation mainly, that is, subscript indices for spherical
components, basis vectors, and basis tensors. Their
contravariant elements are replaced by the complex
conjugates (or duals) of their covariant elements. Our
aim is the determination of spherical rank-2 tensor
components in terms of Cartesian components. As
spherical rank-2 tensor is deduced from the coupling of
two spherical rank-1 tensors, we first detail the determi-
nation of spherical rank-1 tensor components, and then
we present three procedures for determining spherical
rank-2 tensor components. The transformation laws are
illustrated with the active rotation and the rotational
invariance of spherical tensor.
Rank-1 Tensor or Vector Components
A vector A can be expressed in terms of orthonormalbasis vectors along the three Cartesian coordinate axes.
Let these orthonormal Cartesian basis vectors be ex, ey,
and ez. Therefore,
A5Axex1Ayey1Azez; (221)
where Ax, Ay, and Az are the Cartesian components of
this vector. We may also use covariant (25,45) spheri-
cal basis vectors e111; e10; e121
� �defined by
(2,3,25,27,28,30,32,35,40,45,69,81–83)
e111521ffiffiffi2p ex1iey
� �e105ez
e12151ffiffiffi2p ex2iey
� �
8>>>>><>>>>>:
(222)
to describe this vector A. This is a complex basis, so
vector with real components in Cartesian basis may
have complex components in spherical basis. The
matrix representations fMx;My;Mzg of the three Car-
tesian basis vectors fex; ey; ezg are
Mx5
1
0
0
0BB@
1CCA; My5
0
1
0
0BB@
1CCA; Mz5
0
0
1
0BB@
1CCA: (223)
Those of the covariant spherical basis vectors are
(27,69)
M11521ffiffiffi
2p
1
i
0
0BB@
1CCA; M05
0
0
1
0BB@
1CCA;
M2151ffiffiffi2p
1
2i
0
0BB@
1CCA:
(224)
As Cartesian basis vectors are orthonormal, that is,
ei � ej5di;j; i; j5x; y; zð Þ; (225)
there is also an orthogonality relation for the covariant
spherical basis (25,30,40):
e1q
� ��� e1s5dq;s; q; s511; 0;21ð Þ: (226)
The symbol (*) means complex conjugation. These
covariant spherical basis vectors satisfy the identity
(12,27,30,40,81):
222 MAN
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
e1s
� ��5 21ð Þse12s; s511; 0;21ð Þ: (227)
Equations (226) and (227) can be checked using the
matrices in Eq. (224). The three unit vectors e111
� ��,
e121
� ��, and e10
� ��are in fact the dual spherical basis
of e111; e10; e121
� �. They are also called contravariant
spherical basis (25,45). We do not apply contravariant
convention, which involves superscript indices, but we
use an equivalent definition, which involves the com-
plex conjugate of covariant spherical basis vectors or
components.
Introducing Eq. (222) of covariant spherical basis
vectors into Eq. (221) of vector A, the latter has the fol-
lowing expression (27):
A51ffiffiffi2p Ax1iAy
� �e1211Aze101
1ffiffiffi2p 2Ax1iAy
� �e111;
(228)
or (28,30)
A52A111e1211A10e102A121e111
5X
s511;0;21
21ð ÞsA1se12s;(229)
with covariant spherical components
(3,12,21,25,27,28,30,32,35,38–45)
A111521ffiffiffi2p Ax1iAy
� �A105Az
A12151ffiffiffi2p Ax2iAy
� � :
8>>>>><>>>>>:
(230)
These components are sometimes called standardcomponents of a vector (18,21,84). These covariant
spherical components transform under rotation like
the spherical harmonics Y1;m with m 5 21, 0, 11. It
is worth noting that covariant spherical basis vectors
(Eq. (222)) and covariant spherical components (Eq.
(230)) have the same structure. In other words, the
relations between vector components in different
bases are the same as the relations between basis
vectors (25). This agrees with our discussion on the
meanings of contravariant and covariant components
in Section III. However, covariant spherical compo-
nents and covariant spherical basis vectors have
opposite indices s in Eq. (229), because they both
use covariant notations.
Fortunately, we may express A in Eq. (229) with
dual spherical basis vectors defined in Eq. (227) as
(32,81)
A5X
s511;0;21
A1s 21ð Þse12s
� 5A111 e111
� ��1A10 e10
� ��1A121 e121
� ��
5 A111 A10 A121ð Þ
e111
� ��e10
� ��e121
� ��
0BBBBB@
1CCCCCA:
(231)
With vector A expressed in dual spherical basis,
covariant spherical components and complex conju-
gates of covariant spherical basis vectors have the same
index s. The matrix form of A follows the standard con-
vention as shown in Eq. (37). Therefore, the covariant
spherical components of A are given by the dot product
(3,30,32,81)
A1s5A � e1s; s521; 0;11ð Þ; (232)
where A is expressed in the dual spherical basis (Eq.
(231)).
In the dual spherical basis, Eq. (231) shows that
spherical components and basis vectors have the same
index s. It is not the case in the covariant spherical
basis, Eq. (229) shows that they have opposite indices.
However, Eq. (229) can be rewritten in the covariant
spherical basis using covariant spherical components in
Eq. (230) as (55,82)
A5X
s511;0;21
21ð ÞsA1s
� e12s
5 A121
� ��e1211A10e101 A111
� ��e111
5 e111 e10 e121ð Þ
A111
� ��A10
A121
� ��
0BBBB@
1CCCCA:
(233)
As a result, the complex conjugates of covariant
spherical components and covariant spherical basis
vectors also have the same index s. The matrix form of
A follows the standard convention as shown in Eq.
(37). Therefore, the complex conjugate of covariant
spherical components are given by the dot product (82)
A1s
� ��5 e1s
� �� � A; s521; 0;11ð Þ; (234)
where A is expressed in the covariant spherical basis
(Eq. (233)).
It is meaningless to distinguish contravariant and
covariant components in orthonormal Cartesian basis,
but it is not the case when spherical bases are involved.
Any vector A (Eq. (221)) with Cartesian components
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 223
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
can be expanded either in the dual spherical basis
as in Eq. (231) or in the covariant spherical basis
as in Eq. (233). Contravariant spherical components
of A in Eq. (233) are the complex conjugates of
its covariant spherical components in Eq. (231).
This is in agreement with the language of quantum
mechanics (21,73,74) where bra space is the dual
of ket space (32,85), that is, if a vector B in ket
space is
jBi5 k1
� ��ju1i1 k2
� ��ju2i1 k3
� ��ju3i; (235)
whose components are gathered in a column matrix
ð ðk1Þ� ðk2Þ
� ðk3Þ� ÞT. In bra space, B has the fol-
lowing expression
hBj5k1hu1j1k2hu2j1k3hu3j; (236)
whose components are gathered in a row matrix
k1 k2 k3ð Þ. The vector A expressed in dual spheri-
cal basis as in Eq. (231) may be rewritten as
hAj5A111he111j1A10he10j1A121he121j; (237)
and that expressed in the covariant spherical basis as in
Eq. (233) may be rewritten as
jAi5 A111
� ��je111i1 A10
� ��je10i1 A121
� ��je121i:(238)
It is easy to deduce that the covariant spherical com-
ponents A1s and its complex conjugates A1s
� ��are pro-
vided by
A1s5hAje1si; A1s
� ��5he1sjAi s511; 0;21ð Þ:
(239)
Some presentation in the literature expresses the
vector A in the dual spherical basis (86) as in Eq.
(231). In Dirac notation, it is expressed in bra space as
Eq. (237).
The definition of covariant spherical components
(Eq. (230)) is not unique, Landau and Lifshitz (87)
as well as Fano and Racah (18) use another
convention:
A111521ffiffiffi2p i Ax1iAy
� �A105iAz
A12151ffiffiffi2p i Ax2iAy
� � :
8>>>>><>>>>>:
(240)
The latter is also used by Sanctuary (88). In fact,
covariant spherical and Cartesian components are
related by a unitary matrix (89–91):
A111 A10 A121ð Þ
5 Ax Ay Az
� �2
iffiffiffi2p j 0
iffiffiffi2p j
1ffiffiffi2p j 0
1ffiffiffi2p j
0 ij 0
0BBBBBB@
1CCCCCCA:
(241)
Condon and Shortley’s phase convention (85) is
obtained if j is set to 2i in Eq. (241), resulting Eq.
(230). Fano and Racah’s phase convention is obtained
if j51, resulting Eq. (240). Heine (23) sets j5iffiffiffi2p
.
Wigner (55) applies the dual or contravariant (45) com-
ponents of those in Eq. (230), that is,
A111
� ��52
1ffiffiffi2p Ax2iAy
� �A10
� ��5Az
A121
� ��5
1ffiffiffi2p Ax1iAy
� �:
8>>>>>><>>>>>>:
(242)
Rank-2 Tensor Components
We present three procedures for determining spherical
rank-2 tensor components in terms of Cartesian compo-
nents. The first one is a general procedure proposed by
Cook and De Lucia (2), in which we introduce and
explore the two definitions of double contraction of
two rank-2 tensors applied to basis tensors. The second
one is a simplified procedure abundantly developed in
the literature, in which a dyad A5V� U is chosen as a
representative of rank-2 tensor. The third one explores
the properties of spherical harmonics.
General procedure. We construct the nine Cartesian
basis tensors for rank-2 tensor fei � ej; i; j5x; y; zg by
taking the dyadic product of Cartesian basis vectors
fex; ey; ezg. The result of applying the dyadic product
to a pair of vectors is a matrix. The matrix representa-
tions Mij (i, j 5 x, y, z) of these nine basis tensors are
reported in Table 2; they are also provided by Mueller
(72). These matrices verify the relation:
Mi �Mj
� �Mm �Mn
� �5 ej � em
� �Mi �Mn
� �5djm Mi �Mn
� �; i; j;m; n5x; y; zð Þ;
(243)
corresponding to the dyadic dot product between two
basis tensors defined in Eq. [4].
To obtain the covariant spherical basis tensors trs
(r 5 0, 1, 2; s 5 2r, .,1r) for rank-2 tensors, we form
the dyadic product fe1p � e1q; p; q511; 0;21g with
224 MAN
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
the covariant spherical basis e111; e10; e121
� �defined
in Eq. (222), and use Clebsch-Gordan coefficients (23–27) for standard coupling two angular momenta (41,81)
to couple two covariant spherical bases:
trs5X
p;q521;0;11
h1p1qjrsi e1p � e1q
� �: (244)
Various expressions of Clebsch-Gordan coefficients
can be found in the literature (22,30). With explicit val-
ues of Clebsch-Gordan coefficients (2,27,81) we get
t0051ffiffiffi3p e111 � e1212e10 � e101e121 � e111
� �;
(245)
t11151ffiffiffi2p e111 � e102e10 � e111
� �; (246)
t1051ffiffiffi2p e111 � e1212e121 � e111
� �; (247)
t121521ffiffiffi2p e121 � e102e10 � e121
� �; (248)
t2125e111 � e111; (249)
t21151ffiffiffi2p e10 � e1111e111 � e10
� �; (250)
t2051ffiffiffi6p e111 � e12112e10 � e101e121 � e111
� �;
(251)
t22151ffiffiffi2p e10 � e1211e121 � e10
� �; (252)
t2225e121 � e121: (253)
The basis tensors t00 and t2s
s522;21; 0;11;12ð Þ are symmetric, while the basis
tensors t1s s521; 0;11ð Þ are antisymmetric.
Then we replace covariant spherical basis vectors
e111; e10; e121
� �in Eqs. (245) and (253) by Cartesian
basis vectors fex; ey; ezg using Eq. (222), the covariant
spherical basis tensors trs in terms of Cartesian basis
vectors (27) become
t00521ffiffiffi3p ex � ex1ey � ey1ez � ez
� �; (254)
t11151
2ez � ex2ex � ez1i ez � ey2ey � ez
� �n o;
(255)
t1051ffiffiffi2p i ex � ey2ey � ex
� �; (256)
t12151
2ez � ex2ex � ez2i ez � ey2ey � ez
� �n o;
(257)
t21251
2ex � ex2ey � ey1i ex � ey1ey � ex
� �n o;
(258)
t211521
2ex � ez1ez � ex1i ey � ez1ez � ey
� �n o;
(259)
t2051ffiffiffi6p 2ex � ex12ez � ez2ey � ey
� �; (260)
t22151
2ex � ez1ez � ex2i ey � ez1ez � ey
� �n o;
(261)
Table 2 Matrix Representations Mij (i, j 5 x, y, z) ofUnit Dyad Tensors ei � ej from Orthonormal Carte-
sian Basis Vectors ei (i 5 x, y, z) for Cartesian Rank-
2 Tensor
Mxx5Mx �Mx5
1
0
0
0BB@
1CCA� 1 0 0ð Þ5
1 0 0
0 0 0
0 0 0
0BB@
1CCA
Mxy5Mx �My5
1
0
0
0BB@
1CCA� 0 1 0ð Þ5
0 1 0
0 0 0
0 0 0
0BB@
1CCA
Mxz5Mx �Mz5
1
0
0
0BB@
1CCA� 0 0 1ð Þ5
0 0 1
0 0 0
0 0 0
0BB@
1CCA
Myx5My �Mx5
0
1
0
0BB@
1CCA� 1 0 0ð Þ5
0 0 0
1 0 0
0 0 0
0BB@
1CCA
Myy5My �My5
0
1
0
0BB@
1CCA� 0 1 0ð Þ5
0 0 0
0 1 0
0 0 0
0BB@
1CCA
Myz5My �Mz5
0
1
0
0BB@
1CCA� 0 0 1ð Þ5
0 0 0
0 0 1
0 0 0
0BB@
1CCA
Mzx5Mz �Mx5
0
0
1
0BB@
1CCA� 1 0 0ð Þ5
0 0 0
0 0 0
1 0 0
0BB@
1CCA
Mzy5Mz �My5
0
0
1
0BB@
1CCA� 0 1 0ð Þ5
0 0 0
0 0 0
0 1 0
0BB@
1CCA
Mzz5Mz �Mz5
0
0
1
0BB@
1CCA� 0 0 1ð Þ5
0 0 0
0 0 0
0 0 1
0BB@
1CCA
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 225
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t22251
2ex � ex2ey � ey2i ex � ey1ey � ex
� �n o:
(262)
The matrix representations Mrs (r 5 0, 1, 2;
s 5 2r,., 1r) of trs are reported in Table 3. Those pro-
vided by van Kleef (86) are their complex conjugates.
All basis tensors but t00 have a vanishing trace. They
satisfy the identity (12,81)
trs
� ��5 21ð Þr1s
tr2s: (263)
It is easy to check that the DIP of two spherical basis
tensors tkq and trs is represented by the trace of the
product of two matrices as (92)
tkq:trs5Tr Mkq
� ��Mrs
n o5Tr Mkq Mrs
� ��n o; (264)
with k; r50; 1; 2; 2k � q � 1k; 2r � s � 1rð Þ.In the present case, we have
tkq:trs5Tr Mkq
� �T
Mrs
� �5Tr Mkq Mrs
� �Tn o
:
(265)
In contrast, the DOP of two spherical basis tensors
is represented by
tkq::trs5Tr MkqMrs
n o: (266)
Now we determine the covariant spherical compo-
nents Trs (r 5 0, 1, 2; s 5 2r,., 1r) of a rank-2 tensor
T first using the DIP of two rank-2 tensors (12):
Trs5T :trs; (267)
trs being defined in Eqs. (254–262). This relation is
similar to Eq. (232) for the covariant spherical compo-
nent of a vector. Tensor T is defined (2,3) by Cartesian
components Tij (i, j 5 x, y, z) as in Eq. (97). Therefore
(2),
Trs5X
i;j5x;y;z
ei � ej
� �Tij
!:trs; (268)
and (22,27,83,93–95)
T00 1f g521ffiffiffi3p Txx1Tyy1Tzz
� �; (269)
T10 1f g5 1ffiffiffi2p i Txy2Tyx
� �; (270)
T161 1f g5 1
2Tzx2Txz6i Tzy2Tyz
� �n o; (271)
T20 1f g5 1ffiffiffi6p 3Tzz2 Txx1Tyy1Tzz
� �n o; (272)
T261 1f g571
2Txz1Tzx6i Tyz1Tzy
� �n o; (273)
T262 1f g5 1
2Txx2Tyy6i Txy1Tyx
� �n o: (274)
The symbols 1f g or 2f g in tensor component for-
mulas indicate that the DIP or the DOP of two rank-2
tensors are used. It is worth noting that covariant spher-
ical basis tensors (Eqs. (254–262)) and covariant spher-
ical tensor components (Eqs. (269–274)) of rank-2
tensor have the same structure.
Similarly, the DOP of two rank-2 tensors is
Trs5X
i;j5x;y;z
ei � ej
� �Tij
!::trs; (275)
then (2,3,96–103)
T00 2f g521ffiffiffi3p Txx1Tyy1Tzz
� �; (276)
T10 2f g521ffiffiffi2p i Txy2Tyx
� �; (277)
T161 2f g521
2Tzx2Txz6i Tzy2Tyz
� �n o; (278)
Table 3 Matrix Representations Mrs (r 5 0, 1, 2;s 5 2r,., 1r) of Covariant Spherical Basis Tensors trs
for Spherical Rank-2 Tensor
M005 1ffiffi3p
1 0 0
0 1 0
0 0 1
0BB@
1CCA
M105 iffiffi2p
0 1 0
21 0 0
0 0 0
0BB@
1CCA
M1615 12
0 0 21
0 0 7i
1 6i 0
0BB@
1CCA
M205 1ffiffi6p
21 0 0
0 21 0
0 0 2
0BB@
1CCA
M26157 12
0 0 1
0 0 6i
1 6i 0
0BB@
1CCA
M2625 12
1 6i 0
6i 21 0
0 0 0
0BB@
1CCA
226 MAN
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T20 2f g5 1ffiffiffi6p 3Tzz2 Txx1Tyy1Tzz
� �n o; (279)
T261 2f g571
2Txz1Tzx6i Tyz1Tzy
� �n o; (280)
T262 2f g5 1
2Txx2Tyy6i Txy1Tyx
� �n o: (281)
Equations (276–281) are those determined by Cook
and De Lucia (2) and reported by Mehring (3) and Spi-
ess (104). However, they are the complex conjugates of
those determined by Mueller (72) who uses contravar-
iant spherical components, whereas we use covariant
spherical components. Equations (269–274) and (276–
281) differ only in the signs of T1sðs521; 0;11Þ. This
is in agreement with our observation in Cartesian
Rank-2 Tensor as a Dyadic subsection: Cartesian rank-
2 tensor component obtained with the DIP becomes its
transpose when the DOP is applied. For symmetric
Cartesian tensor whose components verifying Tij5Tji,
its spherical tensor components T1sðs521; 0;11Þ are
nulls. Equations (269–274) and (276–281) become
identical. Sometimes, Eqs. (269) and (276) are rede-
fined (104–108) as
T00 1f g5T00 2f g5 1
3Txx1Tyy1Tzz
� �: (282)
We can express tensor T with basis tensors or its
complex conjugates as for vector A in Eqs. (231) and
(233):
T5X2
r50
X1r
s52r
Trs trs
� ��5X2
r50
X1r
s52r
Trs
� ��trs: (283)
Taking into account Eq. (263), we obtain
T5X2
r50
X1r
s52r
Trs 21ð Þr1str2s
�
5X2
r50
X1r
s52r
Tr2s 21ð Þr2strs
� 5X2
r50
X1r
s52r
Tr2s 21ð Þr2s� trs:
(284)
Therefore (12,40),
Trs
� ��5 21ð Þr2sTr2s: (285)
This relation about spherical components has the
same structure as Eq. (263) about basis tensors.
To determine the expressions of spherical compo-
nents of a rank-2 tensor A5V� U from two vectors V
5 Vx;Vy;Vz
� �and U5 Ux;Uy;Uz
� �, we proceed as
above. We define tensor A with Cartesian basis
tensors:
A5ðVxex1Vyey1VzezÞ� ðUxex1Uyey1UzezÞ5
Xi;j5x;y;z
ei � ej
� �ViUj:
(286)
Tensor A defined in Eq. (286) is similar to tensor T
defined in Eq. (97) where Tij is replaced with ViUj.
Therefore, two sets of spherical tensor components are
also available. If the DIP is used, then (104)
A00 1f g5 V� U½ �00521ffiffiffi3p VxUx1VyUy1VzUz
� �52
1ffiffiffi3p V � U;
(287)
A10 1f g5 V� U½ �1051ffiffiffi2p i VxUy2VyUx
� �; (288)
A161 1f g5 V� U½ �16151
2VzUx2VxUz6i VzUy2VyUz
� �n o;
(289)
A20 1f g5 V� U½ �2051ffiffiffi6p 2VxUx12VzUz2VyUy
� �5
1ffiffiffi6p 3VzUz2V � U� �
;
(290)
A261 1f g5 V� U½ �261
571
2VxUz1VzUx6i VyUz1VzUy
� �n o;
(291)
A262 1f g5 V� U½ �262
51
2VxUx2VyUy6i VxUy1VyUx
� �n o:
(292)
If the DOP is used, then
A00 2f g5 V� U½ �00
521ffiffiffi3p VxUx1VyUy1VzUz
� �52
1ffiffiffi3p V � U;
(293)
A10 2f g5 V� U½ �10521ffiffiffi2p i VxUy2VyUx
� �; (294)
A161 2f g5 V� U½ �161
521
2VzUx2VxUz6i VzUy2VyUz
� �n o;
(295)
A20 2f g5 V� U½ �2051ffiffiffi6p 2VxUx12VzUz2VyUy
� �5
1ffiffiffi6p 3VzUz2V � U� �
;
(296)
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 227
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
A261 2f g5 V� U½ �261
571
2VxUz1VzUx6i VyUz1VzUy
� �n o;
(297)
A262 2f g5 V� U½ �262
51
2VxUx2VyUy6i VxUy1VyUx
� �n o:
(298)
As for tensor T, these two sets of covariant spherical
components for tensor A differ only with the signs of
A1s (s 5 21, 0 11). Furthermore, for symmetric tensor,
these two sets become identical. Sometimes, Eqs. (287)
and (293) are redefined as (104)
A00 1f g5A00 2f g5V � U5VxUx1VyUy1VzUz: (299)
Simplified procedure. The common method (73,74)
to determine the expressions of spherical components
of a rank-2 tensor A is to consider the latter as the
dyadic product A5V� U of two vectors V5ðVx;Vy;VzÞ and U5ðUx;Uy;UzÞ. The covariant spherical com-
ponents of these two vectors are defined in Eq. (17).
However, dyads represent only a subset of rank-2 ten-
sors. In general, rank-2 tensors are defined by nine
independent Cartesian components, whereas the dyad
A5V� U depends only on six independent parame-
ters, those of the two vectors V and U.
The covariant spherical components of dyad A
(19,27,31,34,36,37,40,94,104,109–111) are deduced
from the coupling of the two vectors V and U using
Clebsch-Gordan coefficients. In this simplified proce-
dure, the symbol of dyadic product (�) only relates the
two vectors. It does not appear between vector compo-
nents. In contrast, it appears between basis vectors in
the general procedure.
A00 1f g5 V� U½ �0051ffiffiffi3p V111U1212V10U101V121U111
� �52
1ffiffiffi3p VxUx1VyUy1VzUz
� �52
1ffiffiffi3p V � U;
(300)
A111 1f g5 V� U½ �11151ffiffiffi2p V111U102V10U111
� �5
1
2VzUx2VxUz1i VzUy2VyUz
� �n o;
(301)
A10 1f g5 V� U½ �1051ffiffiffi2p V111U1212V121U111
� �5
1ffiffiffi2p i VxUy2VyUx
� �;
(302)
A121 1f g5 V� U½ �121521ffiffiffi2p V121U102V10U121
� �5
1
2VzUx2VxUz2i VzUy2VyUz
� �n o;
(303)
A212 1f g5 V� U½ �2125V111U111
51
2VxUx2VyUy1i VxUy1VyUx
� �n o;
(304)
A211 1f g5 V� U½ �21151ffiffiffi2p V10U1111V111U10
� �52
1
2VxUz1VzUx1i VyUz1VzUy
� �n o;
(305)
A20 1f g5 V� U½ �2051ffiffiffi6p V111U12112V10U101V121U111
� �5
1ffiffiffi6p 2VxUx12VzUz2VyUy
� �5
1ffiffiffi6p 3VzUz2V � U� �
;
(306)
A221 1f g5 V� U½ �22151ffiffiffi2p V10U1211V121U10
� �5
1
2VxUz1VzUx2i VyUz1VzUy
� �n o;
(307)
A222 1f g5 V� U½ �2225V121U121
51
2VxUx2VyUy2i VxUy1VyUx
� �n o:
(308)
Equations (300–308) are identical to Eqs. (287–
292). This is the reason for which the symbol 1f g has
been included in the above relations.
The covariant spherical components Trs 1f g of T
(19,22,27,94,95,112) are obtained in replacing ViUj by
Tij in Eqs. (300–308). In other words, they are those
defined in Eqs. (269–274). Conversely, replacing ViUj
by Tji (3,56) in Eqs. (300–308) yields the covariant
spherical components Trs 2f g defined in Eqs. (276–
281). This approach for determining covariant spherical
components of tensor T is not rigorous, because we
deduce rank-2 tensor properties from those of its subset
of dyads.
This simplified procedure generates one set of
covariant spherical components for A verifying the
DIP but two sets for T. The general procedure, which
involves basis tensors, generates two sets of covariant
spherical components for A and T due to the DIP and
DOP of two rank-2 tensors.
Spherical harmonic procedure. We present two
methods for determining spherical rank-1 tensor
228 MAN
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
components from Cartesian components of antisym-
metric tensor Tð1Þ in Eq. (105).
The first method consists in generating a pseudovec-
tor B defined by (25)
Bi51
2eijkTjk; i; j; k5x; y; zð Þ; (309)
where eijk is the Levi-Civita alternator symbol:
eijk5
11 if even permutations of indices i; j; kð Þ
21 if odd permutations of indices i; j; kð Þ
0 if any two indices are equal :
8>><>>:
(310)
As a result, Tð1Þ is written as (19,44)
Tð1Þ5
01
2Txy2Tyx
� � 1
2Txz2Tzx
� �1
2Tyx2Txy
� �0
1
2Tyz2Tzy
� �1
2Tzx2Txz
� � 1
2Tzy2Tyz
� �0
0BBBBBBB@
1CCCCCCCA
5
0 Bz 2By
2Bz 0 Bx
By 2Bx 0
0BBB@
1CCCA:
(311)
The covariant components of the spherical tensor
associated with Tð1Þ can be deduced (25,34) using Eq.
(230):
T111 pvf g521ffiffiffi2p Bx1iBy
� �52
1
2ffiffiffi2p Tyz2Tzy1i Tzx2Txz
� �n o;
(312)
T10 pvf g5Bz51
2Txy2Tyx
� �; (313)
T121 pvf g5 1ffiffiffi2p Bx2iBy
� �5
1
2ffiffiffi2p Tyz2Tzy2i Tzx2Txz
� �n o:
(314)
The symbol pvf g stands for pseudovector.
The second method for determining the expres-
sions of spherical tensor components is the use of
spherical harmonics. The first few normalized
expressions (21,27,30–32,34,39,40,54,75–79), which
use the Condon–Shortley phase convention
(21,31,36,40,63,73,74,78,79,81)
Yl;m u;/ð Þh i�
5 21ð ÞmYl;2m u;/ð Þ; (315)
are gathered in Table 1. Other expressions of spherical
harmonics are available (113,114), in which Y1;61 and
Y2;61 differ in signs with those in Table 1. We define
the spherical tensor components Trs in replacing the
arguments x, y, and z of spherical harmonics with vec-
tor components Bx, By, and Bz:
Trs5dsYr;s Bx;By;Bz
� �: (316)
For spherical rank-1 tensor (73,74,113,115), we
obtain
dY1;0ðx; y; zÞ5ffiffiffiffiffiffi3
4p
rz ) T1;0 shf g5
ffiffiffiffiffiffi3
4p
rBz; (317)
dY1;61ðx; y; zÞ57
ffiffiffiffiffiffi3
8p
rx6iyð Þ
) T1;61 shf g57
ffiffiffiffiffiffi3
4p
r1ffiffiffi2p Bx6iBy
� �:
(318)
The symbol shf g stands for spherical harmonics.
We drop the common factorffiffiffiffi3
4p
qfor renormalization
because we have replaced the unit vector r5 x; y; zð Þ by
vector B. Finally, we proceed as in Eqs. (312–314).
For spherical rank-2 tensor (73,74,113), we replace
the product mn of arguments in spherical harmonics by
Tmn:
d2Y2;0ðx; y; zÞ5ffiffiffiffiffiffi5
4p
r ffiffiffi1
4
r3z22d2ð Þ
) T20 shf g5ffiffiffiffiffiffi15
8p
r ffiffiffi1
6
r3Tzz2T2� �
;
(319)
d2Y2;61ðx; y; zÞ57
ffiffiffiffiffiffi5
4p
r ffiffiffi3
2
rz x6iyð Þ
) T261 shf g57
ffiffiffiffiffiffi15
8p
r1
2Tzx1Txz6i Tzy1Tyz
� �n o;
(320)
d2Y2;62ðx; y; zÞ5ffiffiffiffiffiffi5
4p
r ffiffiffi3
8
rx6iyð Þ2
) T262 shf g5ffiffiffiffiffiffi15
8p
r1
2Txx2Tyy6i Txy1Tyx
� �n o:
(321)
We drop the common factorffiffiffiffi158p
qfor normalization.
In short, we directly obtain:
T00 shf g5 1
3Tr Tf g; (322)
T10 shf g5T10 pvf g5 1
2Txy2Tyx
� �; (323)
T161 shf g5T161 pvf g571
2ffiffiffi2p Tyz2Tzy6i Tzx2Txz
� �n o;
(324)
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 229
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
T20 shf g5 1ffiffiffi6p 3Tzz2 Txx1Tyy1Tzz
� �n o; (325)
T261 shf g571
2Txz1Tzx6i Tyz1Tzy
� �n o; (326)
T262 shf g5 1
2Txx2Tyy6i Txy1Tyx
� �n o; (327)
without using Clebsch-Gordan coefficients. They sat-
isfy the identity (31,38)
Trs
� ��5 21ð ÞsTr2s; (328)
as that of spherical harmonics (Eq. (315)). But this
identity differs with that defined in Eq. (285) from the
general procedure. In fact, we have also the following
relations:
T1s shf g5T1s pvf g52iffiffiffi2p T1s 1f g5 iffiffiffi
2p T1s 2f g;
(329)
T2s shf g5T2s 1f g5T2s 2f g: (330)
The components of Tð0Þ, Tð1Þ, and Tð2Þ transform
under rotations in the same way as the spherical har-
monics of rank zero, one, and two, respectively. These
spherical components are not used in NMR literature
but by Chandra Shekar and Jerschow (116,117) who
apply
T11151ffiffiffi2p Bx1iBy
� �T105Bz
T121521ffiffiffi2p Bx2iBy
� �
8>>>>><>>>>>:
(331)
instead of Eq. (230). Equation (331) is deduced from
other expressions of spherical harmonics (113,114). As
a result, their expressions for T161 are our T171 shf g.
Active Rotation of Spherical Tensor
An active rotation operator RAða;b; gÞ applied to a
spherical rank-r tensor Tr transforms the latter to
another spherical tensor of the same rank T0r . In the
same covariant spherical basis tensors trs0 , T0r is defined
by
T0r5X1r
s052r
T0rs0� ��
trs0 : (332)
This description of transformation is based on the
configuration of a single set of covariant spherical basis
tensors.
We can also rotate the covariant spherical basis ten-
sors trs attached to Tr. In other words, we use the con-
figuration of two sets of covariant spherical basis
tensors to describe the active rotation of Tr. Using
Wigner active rotation matrix, the basis tensors gath-
ered in row matrices transform as
t0rs5X1r
s052r
trs0Dðr;AÞs0s ða;b; gÞ: (333)
This transformation is identical (72) to that of eigen-
ket vectors jl;mi in Eq. (186). The rotated spherical
tensor T0r in the rotated basis tensors t0rs has the same
covariant spherical components as Tr in the initial
covariant spherical basis tensors trs before rotation.
That is,
T0r5X1r
s52r
Trs
� ��t0rs: (334)
Using Eq. (333), we obtain
T0r5X1r
s52r
Trs
� ��X1r
s052r
trs0Dðr;AÞs0s ða;b; gÞ: (335)
Comparing the latter with Eq. (332) yields
T0rs0� ��
5X1r
s52r
Dðr;AÞs0s ða;b; gÞ Trs
� ��: (336)
The complex conjugate of covariant spherical tensor
components are gathered in column matrices. This rela-
tion is identical to Eq. (219) deduced from the first defi-
nition of spherical tensor, but differs in notations for
tensor components. This relation is identical to that in
Eq. (31) of Mueller (72). Equation (336) about contra-
variant spherical tensor components and Eq. (208)
about contravariant spherical harmonics have the same
structure. They are submitted to the same transforma-
tion law.
In active rotation of a spherical rank-r tensor Tr, the
same Wigner active rotation matrix Dðr;AÞða;b; gÞ is
involved in the transformation law of covariant spheri-
cal basis tensors (Eq. (333)) and in that of its contravar-
iant Trs
� ��spherical tensor components (Eq. [336]).
But the row matrix of basis tensors is postmultipled by
Dðr;AÞða;b; gÞ, whereas the column matrix of tensor
components is premultiplied by Dðr;AÞða;b; gÞ.
Example 2. We have seen that Wigner active rota-
tion matrix Dðl;AÞða;b; gÞ is involved in several trans-
formation laws:
1. that of angular momentum eigenket vectors jl;mias in Eq. (177),
2. that of spherical basis tensors tlm as in Eq. (333),3. that of spherical harmonics fYl;mðu1;/1Þg
�as in
Eq. (208),
230 MAN
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
4. that of spherical tensor components Tlm
� ��as in
Eq. (336).
Consider again the case studied in subsection Exam-
ple 1 concerning the active rotation of Euler angles a5
b50 and g5 p2
about the z axis of a vector A. Spherical
and Cartesian components of A and those of the rotated
vector A0 are related as in Eq. (230):
A111521ffiffiffi2p ax1iay
� �A105az
A12151ffiffiffi2p ax2iay
� � and
A0111521ffiffiffi2p a0x1ia0y
� �A0105a0z
A012151ffiffiffi2p a0x2ia0y
� �:
8>>>>>>>><>>>>>>>>:
8>>>>>>>><>>>>>>>>:
(337)
Application of Eq. (336) yields
2a0x2ia0yffiffiffi
2p
a0z
a0x1ia0yffiffiffi2p
0BBBBBB@
1CCCCCCA5
2i 0 0
0 1 0
0 0 1i
0BB@
1CCA
2ax2iayffiffiffi
2p
az
ax1iayffiffiffi2p
0BBBBBB@
1CCCCCCA:
(338)
The Cartesian components of A0 in term of those of
A are
a0x52ay; a0y5ax; a0z5az; (339)
in agreement with those in Eq. (211). The two methods,
active rotation of spherical harmonics (Eq. (208)) and
that of spherical tensor components (Eq. (336)), pro-
vide the same Cartesian components for the rotated
vector A0.
Rotational Invariance of Spherical Tensor
A spherical rank-r tensor Tr is defined in Eq. (220). It
is a rotational invariant if its expression remains
unchanged under rotation of coordinate system. That is,
we perform a passive rotation of the physical system.
Under rotation of coordinate system, the covariant
spherical basis tensors transform as in Eq. (333). In the
new basis tensors t0rs, Tr has new covariant spherical
components T0rs:
Tr5X1r
n52r
T0rn
� ��t0rn: (340)
with
t0rn5X1r
s52r
trsDðr;AÞsn ða;b; gÞ: (341)
Replacing t0rn in Eq. (340) by that in Eq. (341) yields
Tr5X1r
n52r
T0rn
� ��X1r
s52r
trsDðr;AÞsn ða;b; gÞ
5X1r
s52r
trs
X1r
n52r
Dðr;AÞsn ða;b; gÞ T0rn
� ��:
(342)
As a result, the contravariant spherical components
Trs
� ��of Tr before the passive rotation are expressed
in terms of contravariant spherical components T0rn� ��
of Tr after the passive rotation:
Trs
� ��5X1r
n52r
Dðr;AÞsn ða;b; gÞ T0rn
� ��: (343)
But we want to express T0rn� ��
in terms of Trs
� ��.
To this end, we multiply the two members of Eq. (343)
by Dðr;AÞsn0 ða;b; gÞ
n o�and sum on s:
X1r
s52r
Trs
� ��Dðr;AÞsn0 ða;b; gÞ
n o�5X1r
s52r
X1r
n52r
Dðr;AÞsn0 ða;b; gÞ
n o�Dðr;AÞsn ða;b; gÞ T0rn
� ��:
(344)
Taking into account Eq. (174), Eq. (344) becomes
T0rn0� ��
5X1r
s52r
Trs
� ��Dðr;AÞsn0 ða;b; gÞ
n o�; (345)
which is the transformation law for contravariant spher-
ical components. That of covariant spherical compo-
nents is deduced from Eq. (345) by suppressing the
complex conjugate symbol (*). That is,
T0rn05X1r
s52r
TrsDðr;AÞsn0 ða;b; gÞ: (346)
As the same matrix Dðr;AÞða;b; gÞ is involved in
Eqs. (341) and (346), these two transformations are
cogredient. A spherical rank-r tensor is a rotational
invariant if the transformation of its covariant spherical
basis tensors and that of its covariant spherical compo-
nents are obtained by postmultiplying their row matri-
ces by Dðr;AÞða;b; gÞ.Alternatively, if we consider the contravariant
spherical components Trs
� ��(Eq. (345)) and its covari-
ant spherical basis tensors (Eq. (341)), these two trans-
formations are contragredient. A spherical tensor is a
rotational invariant if the row matrix of covariant
spherical basis tensors and that of contravariant spheri-
cal components are postmultiplied by Dðr;AÞða;b; gÞand fDðr;AÞða;b; gÞg�, respectively.
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 231
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
VIII. TENSORS IN NMR HAMILTONIAN
Interactions within a physical system are almost
described in terms of scalars, as we suppose that the
system is isolated and that the space is isotropic (32). A
primary construction involving spherical tensors is con-
traction to a scalar tensor. Formulas suitable for
describing scalar quantities such as NMR Hamiltonians
are well known.
Until now we have focused on the construction of
rank-2 tensors from two rank-1 tensors or vectors. Now
we are interested in the description of NMR Hamilto-
nians defined with two rank-2 tensors: a spin-part A
and a space-part T tensors. Furthermore, tensor A5V
�U is dyadic product of two vectors V and U. We will
reformulate an NMR Hamiltonian as double contrac-
tion of two Cartesian or two spherical rank-2 tensors.
We also present the rotational invariance of NMR
Hamiltonian.
Cartesian Rank-2 Tensors
To establish the relations between Cartesian and spheri-
cal components of rank-2 tensors A5V� U and T, we
first consider the Cartesian Hamiltonian
(3,103,110,118)
H 1; cf g5kV � T � U
5k Vx Vy Vz
� �Txx Txy Txz
Tyx Tyy Tyz
Tzx Tzy Tzz
0BBBB@
1CCCCA
Ux
Uy
Uz
0BBBB@
1CCCCA
5kðVxTxxUx1VyTyxUx1VzTzxUx
1VxTxyUy1VyTyyUy1VzTzyUy
1VxTxzUz1VyTyzUz1VzTzzUzÞ5khVjTjUi;(347)
where k is a constant specific for an interaction, V and
U are vectors, and T is a rank-2 tensor. We will show
that Hamiltonian H 1; cf g in Eq. (347) is compatible
with the DIP (Eqs. (8) and (9)) of two rank-2 tensors.
The character c in H 1; cf g means Cartesian compo-
nents are involved. The various interactions in terms of
their Hamiltonians are expressed in angular frequency
unit, that is, energy unit divided by �h or x-units.
The Zeeman interaction of a nucleus possessing
gyromagnetic ratio g and spin I with the strong, static
magnetic field B0 is (3,110,119)
HZ52gI � Z � B0; (348)
where Z is the unit matrix.
The chemical shift interaction
(3,39,93,108,110,118–123),
HCS 5gI � r � B0; (349)
describes the magnetic field induced by the electronic
charge distribution; r is the chemical shift tensor. In
general, Tr rf g 6¼ 0. The antisymmetric components of
r contribute to the resonance shift only in second-order
and are usually neglected.
The spin–rotation interaction describes the coupling
of rotational angular momentum S of a molecule with a
nuclear spin I (3,39,111,119),
HSR 5I � Rs � S: (350)
Rs is the spin–rotation tensor.
The quadrupole interaction of a nuclear spin I> 1/2
(3,39,100,101,119,120,124–128),
HQ5eQ
2Ið2I21Þ�h I � C � I; (351)
describes the coupling of the nuclear electric quadru-
pole moment eQ with the electric field gradient at the
site of the nucleus. Tensor C is symmetric and trace-
less. The latter property is due to Laplace’s equation.
The dipolar interaction between two nuclear spins I
and S (3,27,39,118–120) is
HD5l0
4p�hgIgS
r3I � S2
3ðI � rÞðS � rÞr2
� �5
l0
4p�hgIgS
r3I � D � S:
(352)
D is a traceless (Tr Df g50) symmetric rank-2
tensor:
D5
123x2
r22
3xy
r22
3xz
r2
23yx
r212
3y2
r22
3yz
r2
23zx
r22
3zy
r212
3z2
r2
0BBBBBBBBB@
1CCCCCCCCCA: (353)
The indirect spin–spin coupling between two
nuclear spins Ið1Þ and Ið2Þ (3,39,98,110,119,129),
HJ5Ið1Þ � J � Ið2Þ; (354)
has a finite trace.
ESR is widely used in the study of paramagnetic
centers. The coupling of electron spins with their sur-
rounding is formally similar to that of nuclear spins
but of different origin and usually with larger cou-
pling tensor (97,130,131). The electronic Zeeman
interaction is
232 MAN
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
Hg5lBS � g � B0; (355)
where lB is the Bohr magneton and S the electron spin.
The g tensor can be treated formally in the same way
as the NMR chemical shift tensor.
The coupling between electron and nuclear spin
magnetic dipoles (130,132) is
HC5XN
C51
S � AC � IC; (356)
where AC is the hyperfine coupling tensor for center C
and IC the nuclear spin.
In Cartesian coordinate system with orthonormal
basis, a rank-2 tensor T can be decomposed into three
contributions T5Tð0Þ1Tð1Þ1Tð2Þ as shown in Section
IV. As a result, an NMR Hamiltonian also consists of
three contributions H5H01H11H2 with
H05kV � Tð0Þ � U
5k Vx Vy Vz
� �
3
1
3Tr Tf g 0 0
01
3Tr Tf g 0
0 01
3Tr Tf g
0BBBBBBBBB@
1CCCCCCCCCA
Ux
Uy
Uz
0BBBB@
1CCCCA;
(357)
H15kV � Tð1Þ � U
5k Vx Vy Vz
� �
3
01
2Txy2Tyx
� � 1
2Txz2Tzx
� �1
2Tyx2Txy
� �0
1
2Tyz2Tzy
� �1
2Tzx2Txz
� � 1
2Tzy1Tyz
� �0
0BBBBBBBBB@
1CCCCCCCCCA
Ux
Uy
Uz
0BBBB@
1CCCCA:
(358)
H25kV � Tð2Þ � U
5k Vx Vy Vz
� �
3
Txx21
3Tr Tf g 1
2Txy1Tyx
� � 1
2Txz1Tzx
� �1
2Tyx1Txy
� �Tyy2
1
3Tr Tf g 1
2Tyz1Tzy
� �1
2Tzx1Txz
� � 1
2Tzy1Tyz
� �Tzz2
1
3Tr Tf g
0BBBBBBBBB@
1CCCCCCCCCA
Ux
Uy
Uz
0BBBB@
1CCCCA:
(359)
Any NMR Hamiltonian can be written as double
contraction of two Cartesian rank-2 tensors A and T.
First, consider the DIP indicated by the number 1 in
1; cf g of the Hamiltonian:
H 1; cf g5kA : T5kX
i;j5x;y;z
Aij ei � ej
� � !:
Xm;n5x;y;z
Tmn em � en
� � !
5kX
i;j;m;n5x;y;z
AijTmn ei � ej
� �: em � en
� �:
(360)
H 1; cf g is a sum of nine products of Cartesian ten-
sor components (7,12,19,81,92,98,103,116,117,133):
H 1; cf g5kA : T5kX
i;j;m;n5x;y;z
AijTmndimdjn5kX
i;j5x;y;z
AijTij:
(361)
In Eq. (361), the orders of the two indices for Carte-
sian rank-2 tensors A and T are identical. Each compo-
nent of A is multiplied by the corresponding
component of T, and the sum of the nine terms is taken.
This definition is analogous to that of the dot product of
two vectors. This means that we have
H 1; cf g5kX
i;j5x;y;z
AijTij5kTr ATT� �
5kTr ATT� �
;
(362)
where A, AT, T, and TT are matrices (11) in the present
circumstance. DIP is redefined in terms of the trace,
independent of any coordinate system. The norm of a
tensor C is
jjCjj25X
i;j5x;y;z
CijCij5Tr CTC� �
5CT: C: (363)
This norm is always positive. As A is dyadic prod-
uct from V and U, that is,
A5V� U5
VxUx VxUy VxUz
VyUx VyUy VyUz
VzUx VzUy VzUz
0BB@
1CCA; (364)
the Hamiltonian becomes
H 1; cf g5kX
i;j5x;y;z
AijTij
5kðVxUxTxx1VxUyTxy1VxUzTxz
1VyUxTyx1VyUyTyy1VyUzTyz
1VzUxTzx1VzUyTzy1VzUzTzzÞ
5khVjTjUi:
(365)
The above expression of H 1; cf g is identical to that
defined in Eq. (347), because the components as
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 233
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
operators in the spin-part tensor commute with those in
the space-part tensor.
In contrast, the DOP is indicated by the number 2 in
2; cf g of the Hamiltonian:
H 2; cf g5kA::T5kX
i;j;m;n5x;y;z
AijTmndindjm5kX
i;j5x;y;z
AijTji:
(366)
The orders of the two indices for Cartesian rank-2
tensors A and T are reversed (16,19,134). This means
that we have (93)
H 2; cf g5kX
i;j5x;y;z
AijTji5kTr ATf g; (367)
where A and T are matrices (93) in the present
circumstance.
The DIP and DOP generate two expressions for
NMR Hamiltonian (Eqs. (362) and (367)). These two
relations show that the matrix of one of the two tensors
has been transposed. Suppose that it is the space-part
tensor T. The transposition affects only the antisym-
metric rank-1 tensor Tð1Þ in the decomposition of
T5Tð0Þ1Tð1Þ1Tð2Þ. Tensor Tð1Þ becomes 2Tð1Þ and
the decomposition becomes T5Tð0Þ2Tð1Þ1Tð2Þ. This
is in line with our early observation that the passage
from the DIP to the DOP transposes the Cartesian com-
ponents of a rank-2 tensor. As a result, the Hamiltonian
decomposition H5H01H11H2 changes to
H5H02H11H2.
If the two vectors V and U as spin operators com-
mute, which is the case for chemical shift Hamilto-
nian HCS where V5I and U5B0, dipole interaction,
spin–rotation interaction, and indirect spin–spin cou-
pling, then Hamiltonian H 2; cf g in Eq. (367)
becomes
H 2; cf g5kX
i;j5x;y;z
AjiTij
5kðVxUxTxx1VyUxTxy1VzUxTxz
1VxUyTyx1VyUyTyy1VzUyTyz
1VxUzTzx1VyUzTzy1VzUzTzzÞ
5kðUxVxTxx1UxVyTxy1UxVzTxz
1UyVxTyx1UyVyTyy1UyVzTyz
1UzVxTzx1UzVyTzy1UzVzTzzÞ:
(368)
Hamiltonian H 2; cf g in Eq. (368) can be rewritten
as (135,136)
H 2; cf g5k Ux Uy Uz
� � Txx Txy Txz
Tyx Tyy Tyz
Tzx Tzy Tzz
0BBB@
1CCCA
Vx
Vy
Vz
0BBB@
1CCCA
5khUjTjVi:(369)
For quadrupole interaction where V5U5I, the
components of the same spin do not commute. Fortu-
nately, the space-part tensor T5C is symmetric, that is,
Tij5Tji. Hamiltonian H 2; cf g in Eq. (367) becomes
identical to H 1; cf g in Eq. (347):
H 2; cf g5kX
i;j5x;y;z
AjiTij
5kX
i;j5x;y;z
AjiTji5H 1; cf g5kX
i;j5x;y;z
AijTij:(370)
This also means Aij5Aji. As a result, Eq. (369) is
also applicable to quadrupole interaction.
Spherical Rank-2 Tensors
In the literature, NMR Hamiltonian H is expressed as
the dot product (Eq. (13)) of two spherical tensors: a
spin-part tensor A5V� U and a space-part tensor T
(3,97,108,110,118,137). Its expression is
Hf2; sg5kX2
r50
X1r
s52r
ð21ÞsArsTr2s
5kfA00T001A10T102ðA111T1211A121T111Þ1A20T20
2ðA211T2211A221T211Þ1A212T2221A222T212g
:
(371)
Once spherical tensor components are replaced by
Cartesian ones, spherical Hamiltonian Hf2; sg should
be identical to Cartesian Hamiltonian Hf1; cg in Eq.
(347). Hamiltonian Hf2; sg is also the sum of nine
products as Hf1; cg. The symbol f2g in Eq. (371) is
added by anticipation and the character s in Hf2; sgmeans that spherical tensor components are involved.
As with Cartesian Hamiltonian, we can express H as
the double contraction of A and T using covariant
spherical basis tensors tkq defined in Eqs. (245–253)
and (254–262) and covariant spherical tensor
components:
H 1; sf g5kA : T5kX2
k50
X1k
q52k
Akqtkq
!:X2
r50
X1r
s52r
Trstrs
!
5kX2
k50
X1k
q52k
X2
r50
X1r
s52r
AkqTrs tkq:trs
� �;
(372)
234 MAN
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
Hf2; sg5kA::T5kX2
k50
X1k
q52k
Akqtkq
!::X2
r50
X1r
s52r
Trstrs
!
5kX2
k50
X1k
q52k
X2
r50
X1r
s52r
AkqTrsðtkq::trsÞ:
(373)
We provide Wolfram Mathematica-5 notebook
(138) that performs tkq:trs and tkq::trs of two rank-2 ten-
sors using Eqs. (265) and (266). The file can be read
with the free application Wolfram CDF Player for
modern web browsers. Results are gathered in Table 4,
which shows that only t1q:t1s (q, s 5 21, 0, 11) and
t1q::t1s differ in signs. Eq. (372) yields
Hf1; sg5kfA00T002A10T101ðA111T1211A121T111Þ1A20T202ðA211T2211A221T211Þ1A212T2221A222T212g
5kX2
r50
X1r
s52r
ð21Þr2sArsTr2s;
(374)
whereas Eq. (373) yields the Hamiltonian in Eq. (371).
There is a sign factor ð21Þr in Eq. (374) but not in Eq.
(371). These two Hamiltonians differ in the signs of
A1qT12q (q 5 21, 0, 11). This is in line with our previ-
ous observation for NMR Hamiltonian expressed with
Cartesian tensor components: only Hamiltonian H1 in
the decomposition H5H01H11H2 changes sign. In
other words, the usual NMR Hamiltonian Hf2; sg in
Eq. (371) is associated with the usual dot product oftwo spherical tensors defined in Eq. (13):
Hf2; sg5kX2
r50
X1r
s52r
ð21ÞsArsTr2s5kX2
r50
Ar � Tr:
(375)
It is the result of the application of the DOP (19).
In contrast, Hamiltonian Hf1; sg in Eq. (374) that is
not used in NMR literature is associated with the
dyadic product of two spherical tensors defined in Eq.
(16):
Hf1; sg5kX2
r50
X1r
s52r
ð21Þr2sArsTr2s5kX2
r50
½Ar8Tr�00:
(376)
It is the result of the application of the DIP (19).
Rotational Invariance of NMR Hamiltonian
The rotational invariance of a vector (Eqs. (60), (65),
(66), and (233)) is expressed by the product of
covariant spherical basis vectors with contravariant
spherical components of the vector in the same basis.
By extension, the rotational invariance of a tensor
(Eq. (220)) is expressed by the product of covariant
spherical basis tensors with contravariant spherical
components of the tensor in the same basis tensors.
The expression of a tensor remains unchanged if the
contravariant spherical tensor components and the
covariant spherical basis tensors transform contragre-
diently (Eqs. (341) and (345)) when a rotation of
coordinate system occurs. The rotational invariance
of the dot product (Eqs. (72) and (73)) is defined by
combining two vectors, which are contragredient to
each other.
As NMR Hamiltonian is a rank-0 tensor, it is obvi-
ously invariant when a rotation of the coordinate sys-
tem occurs. In fact, it is a sum of rotational invariants.
Consider the elements with a fixed value r of tensor
rank in Hf1; sg defined in Eq. (376):
Hf1; sg5kX3
r50
hrf1; sg (377)
Table 4 Non-Zeroed Integers Without Parentheses Resulted from the DIP tkq:trs (k, r 5 0, 1, 2; q 5 2k,., 1k;s 5 2q,., 1q) and Those Inside Parentheses Resulted from the DOP tkq::trs of Two Covariant Spherical BasisTensors
trs
tkq t00 t10 t111 t121 t20 t212 t211 t221 t222
t00 1(1)
t10 21(1) 0 0
t111 0 0 1(21)
t121 0 1(21) 0
t20 1(1) 0 0 0 0
t212 0 0 0 0 1(1)
t211 0 0 0 21(21) 0
t221 0 0 21(21) 0 0
t222 0 1(1) 0 0 0
Results remain valid for ðtkqÞ�:ðtrsÞ
�and ðtkqÞ
�::ðtrsÞ�.
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 235
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
with
hrf1; sg5X1r
s52r
ð21Þr2sAr2sTrs: (378)
The space-part tensor T is a spherical tensor. The
spin-part tensor A5V� U is a spherical tensor opera-
tor whose components are spin operators. The compo-
nents of its adjoint operator A1 is defined by (139)
ðArsÞ1
5ð21Þr2sAr2s: (379)
Equation (378) is rewritten as
hrf1; sg5X1r
s52r
ðArsÞ1Trs: (380)
The 2r 1 1 spherical tensor operators Arq transform
among themselves as (21,28,35,69,70,139)
A0rq5RAða;b; gÞArqfRAða;b; gÞg1
5X1r
s52r
ArsDðr;AÞsq ða;b; gÞ;
(381)
upon rotation of coordinate system. A spherical tensor
operator is defined with respect to a coordinate system
attached to the physical system, because we use the
Wigner active rotation matrix. Applying the adjoint
operator (1) to both sides of Eq. (381) yields
ðA0rqÞ1
5fRAða;b; gÞArqfRAða;b; gÞg1g1
5X1r
s52r
fArsDðr;AÞsq ða;b; gÞg1
:(382)
Applying the properties of the adjoint operator
yields (139)
ðA0rqÞ1
5RAða;b; gÞðArqÞ1fRAða;b; gÞg1
5X1r
s52r
ðArsÞ1fDðr;AÞsq ða;b; gÞg�:
(383)
Equations (381) and (383) show that a spherical ten-
sor operator A is contragredient to its adjoint operator
A1. The adjoints ðArqÞ1
of the spherical tensor opera-
tor A transform according to Eq. (383), whereas the
complex conjugates of spherical components of the
space-part tensor transform according to Eq. (345).
Writing with same subscripts, the two transformation
laws become (71):
T0rq5X1r
s52r
TrsDðr;AÞsq ða;b; gÞ: (384)
ðA0rqÞ1
5X1r
s52r
ðArsÞ1fDðr;AÞsq ða;b; gÞg�; (385)
The covariant spherical components of space-part
tensor transform in the same way as the covariant
spherical basis tensors (Eq. (341)), they transform cog-rediently. They are postmultiplied by the Wigner active
rotation matrix Dðr;AÞ. In contrast, the adjoints of the
covariant spherical components of spin-part tensor are
postmultiplied by the complex conjugate of Dðr;AÞ.Therefore, ðArqÞ
1and Trq transform contragredi-
ently (140). As Dðr;AÞ verifies Eq. (173), it is obvious
that hrf1; sg is a rotational invariant.
The adjoint of a spherical tensor operator is also
defined as (21,28,31,69)
ðArsÞ1
5ð21Þ2sAr2s: (386)
It is involved in (141)
hrf2; sg5X1r
s52r
ð21ÞsAr2sTrs5X1r
s52r
ðArsÞ1Trs; (387)
part of Hf2; sg. The adjoints of spherical tensor opera-
tors in Eqs. (379) and (386) are proportional. Therefore,
hrf2; sg is also a rotational invariant (28).
IX. DISCUSSION
Two expressions of NMR Hamiltonian are used in the
literature: Cartesian Hamiltonian Hf1; cg5khVjTjUiand spherical Hamiltonian Hf2; sg. However, we have
deduced four expressions of NMR Hamiltonian. Two
for Cartesian Hamiltonian:
1. Hf1; cg5khVjTjUi in Eqs. (347) and (365),2. Hf2; cg5khUjTjVi in Eqs. (368) and (369),
and two for spherical Hamiltonian:
1. Hf1; sg in Eqs. (374) and (376),2. Hf2; sg in Eqs. (371) and (375).
These expressions of NMR Hamiltonian are
obtained, respectively, with the DIP and the DOP of
two rank-2 tensors.
We have also at our disposal two sets of covariant
spherical components for space-part tensor T:
1. Trsf1g in Eqs. (269–274),2. Trsf2g in Eqs. (276–281),
and two sets of covariant spherical components for
spin-part tensor A:
1. Arsf1g in Eqs. (287–292),2. Arsf2g in Eqs. (293–298).
These tensor components are also obtained, respec-
tively, with the DIP and DOP of two rank-2 tensors.
We consider the question, which pairs of spherical
tensor components among four (Trsf1g–Arsf1g,
236 MAN
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
Trsf1g–Arsf2g, Trsf2g–Arsf1g, and Trsf2g–Arsf2g)allow us to relate Hf1; sg or Hf2; sg with Hf1; cg5khVjTjUi or Hf2; cg5khUjTjVi. There are also four
combinations of Hamiltonian expressions: Hf1; cg–Hf1; sg, Hf1; cg–Hf2; sg, Hf2; cg–Hf1; sg, and
Hf2; cg–Hf2; sg.We provide Wolfram Mathematica-5 notebook that
solves this problem (142). The notebook inserts the
expressions of each pair Trsfig–Arsfjg into Hf1; sg and
Hf2; sg and compares the results with Hf1; cg5khVjTjUi and Hf2; cg5khUjTjVi. Table 5 gathers the
results.
Table 5 shows that two pairs, Trsf1g–Arsf2g and
Trsf2g–Arsf1g, connect the two expressions of NMR
Hamiltonian used in the literature. NMR community
uses the second pair because the spin-part tensor com-
ponents Arsf1g, which are also generated by the simpli-
fied procedure, are applicable to other scientific fields.
In contrast, the space-part tensor components Trsf2gare specific to NMR.
Life would be simpler if Cartesian and spherical ten-
sors and Hamiltonians are all defined with the same
definition of double contraction of two rank-2 tensors.
For example, inserting Trsf1g and Arsf1g into Hf1; sgyields Hf1; cg5khVjTjUi. That is, only the DIP is
involved. Similarly, inserting Trsf2g and Arsf2g into H
f2; sg yields Hf2; cg5khUjTjVi; only the DOP is
involved.
To clarify our notations, we choose the chemical
shift interaction HCS of a nuclear spin I with a strong,
static magnetic field B0. With the correspondence
V5I, U5B0, and T5r, this interaction is
HCS f1; cg5gI � r � B0
5gX
i;j5x;y;z
IirijB0j5gX
i;j5x;y;z
Aijrij;(388)
and
A5I� B05
IxB0x IxB0y IxB0z
IyB0x IyB0y IyB0z
IzB0x IzB0y IzB0z
0BB@
1CCA: (389)
In spherical tensor notations, the NMR Hamiltonian
for chemical shift interaction is
HCS f2; sg5gX2
r50
X1r
s52r
ð21ÞsArsrr2s; (390)
where the covariant spherical components of the spin-
part interaction (3,96,110,118) are
A00f1g5½I� B0�00521ffiffiffi3p ðIxB0x1IyB0y1IzB0zÞ;
(391)
A10f1g5½I� B0�1051ffiffiffi2p iðIxB0y2IyB0xÞ; (392)
A161f1g5½I� B0�16151
2fIzB0x2IxB0z6iðIzB0y2IyB0zÞg;
(393)
A20f1g5½I� B0�2051ffiffiffi6p f3IzB0z2ðIxB0x1IyB0y1IzB0zÞg;
(394)
A261f1g5½I� B0�261571
2fIxB0z1IzB0x6iðIyB0z1IzB0yÞg;
(395)
A262f1g5½I� B0�26251
2fIxB0x2IyB0y6iðIxB0y1IyB0xÞg:
(396)
Table 1.3 in Ref. (137) presents the covariant spheri-
cal components Arsf2g5½I� X�rs. As the covariant
spherical components of space-part tensor shown in
Table 1.2 in Ref. (137) agree with our definition Trsf2gin Eqs. (276–281), the associated covariant spherical
components of spin-part tensor should be Arsf1g5½I�X�rs so that the pair Trsf2g–Arsf1g verifies the Carte-
sian Hamiltonian Hf1; cg and the spherical Hamilto-
nian Hf2; sg used in NMR literature. Furthermore, if
we replace X in Table 1.3 of Ref. (137) by
B05ð0; 0;B0Þ, the results disagree with those in Table
1.5 of Ref. (137) for chemical shift interaction. Equa-
tion (419) in Ref. (123) also presents the covariant
spherical components Arsf2g.The components of the space-part tensor r
(3,96,108,118,123,137) are
Table 5 Associations Between Spherical Components of Space-Part Tensor (Trs 1f g or Trs 2f g) and Those ofSpin Part Tensor (Ars 1f g or Ars 2f g) for the Four Combinations of Cartesian (H 1; cf g or H 2; cf g) and Spherical(H 1; sf g or H 2; sf g) NMR Hamiltonians
Spherical Hamiltonian
Cartesian Hamiltonian
H 1; cf g H 2; cf g
H 1; sf g Trs 1f g-Ars 1f g or Trs 2f g-Ars 2f g Trs 1f g-Ars 2f g or Trs 2f g-Ars 1f gH 2; sf g Trs 1f g-Ars 2f g or Trs 2f g-Ars 1f g Trs 1f g-Ars 1f g or Trs 2f g-Ars 2f g
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 237
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
r00f2g521ffiffiffi3p ðrxx1ryy1rzzÞ; (397)
r10f2g521ffiffiffi2p iðrxy2ryxÞ; (398)
r161f2g521
2frzx2rxz6iðrzy2ryzÞg; (399)
r20f2g51ffiffiffi6p f3rzz2ðrxx1ryy1rzzÞg; (400)
r261f2g571
2frxz1rzx6iðryz1rzyÞg; (401)
r262f2g51
2frxx2ryy6iðrxy1ryxÞg: (402)
In the usual conditions, the magnetic field B0
is along z axis, that is, B0x5 B0y5 0 and B0z5 B0.As a result, relations become simpler
(3,108,110,122,137):
A00f1g5½I� B0�00521ffiffiffi3p IzB0; (403)
A10f1g5½I� B0�1050; (404)
A161f1g5½I� B0�161521
2ðIx6iIyÞB052
1
2I6B0;
(405)
A20f1g5½I� B0�2051ffiffiffi6p 2IzB0z; (406)
A261f1g5½I� B0�261571
2ðIx6iIyÞB057
1
2I6B0;
(407)
A262f1g5½I� B0�26250: (408)
Notice that the minus sign of A161f1g is missing in
Ref. (98). Our A161f1g in Eq. (405) and A261f1g in
Eq. (407) are the complex conjugates of those of Muel-
ler (72) who uses contravariant spherical tensor compo-
nents, whereas we use covariant spherical tensor
components.
Table 6, as well as Table 2.4 in Ref. (104),
Table 3.1.2 in Ref. (107), and Table 3.1 in Ref.
(116) show that among the main three interactions
(chemical shift, dipole, and quadrupole interactions)
observed in solid state NMR, only chemical shift
has non-zero spherical components A161 of the
spin-part tensor. As the antisymmetric Hamiltonian
H1 of the chemical shift tensor r is usually
neglected, it means that the three components A10
and A161 are not considered (108,143). This is the
reason why the change of signs in the space-part
tensor components T10 and T161 is seldom men-
tioned in the literature. Tab
le6
Sp
heri
cal
Com
ponents
of
Sp
in-P
art
Tenso
rA
wit
hI 6
5I x
6iI
yand
S6
5S
x6
iSy
from
Mehri
ng
(3)
Spin
-par
tte
nsor
aA
00
A10
A16
1A
20
A26
1A
26
2
Spin
–ro
tati
onb
21 ffiffi 3p
I�S
21
2ffiffi 2p
I 1S 2
2I 2
S 1�
�1 2
I zS6
2I 6
S z
��
1 ffiffi 6p
3I zS
z2I�S
��
71 2
I zS6
1I 6
Sz
��
1 2I 6
S 6J-
coupli
ng
21 ffiffi 3p
Ið1Þ�Ið2Þ
21
2ffiffi 2p
Ið1Þ
1Ið
2Þ
22
Ið1Þ
2Ið
2Þ
1
��
1 2Ið
1Þ
zIð
2Þ
62
Ið1Þ
6Ið
2Þ
z
��
1 ffiffi 6p
3Ið
1Þ
zIð
2Þ
z2
Ið1Þ�Ið2Þ
��
71 2
Ið1Þ
zIð
2Þ
61
Ið1Þ
6Ið
2Þ
z
��
1 2Ið
1Þ
6Ið
2Þ
6
Chem
ical
shif
t2
1 ffiffi 3p
I zB
00
21 2
I 6B
0
ffiffiffi 2 3q I zB
07
1 2I 6
B0
0
Dip
ole
00
01 ffiffi 6p
3I z
Sz2
I�S
��
71 2
I zS
61
I 6S
z
��
1 2I 6
S 6Q
uad
rupole
00
01 ffiffi 6p
3I2 z
2I
I11
ðÞ
��
71 2
I zI 6
1I 6
I z�
�1 2
I 6I 6
aS
pie
ss(1
04)
and
Hae
ber
len
(108
)re
pla
ceth
enum
eric
alfa
ctor
21 ffiffi 3p
by
1in
A00.
Tab
le11
(103
).
238 MAN
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
X. CONCLUSIONS
In this article, we have reformulated the dot product of
Cartesian tensors and the dyadic product of spherical
tensors in NMR Hamiltonian as the double contraction
of two rank-2 tensors A and T. The double contraction
is not explicitly used in NMR literature. Tensors have
to be expressed with basis tensors as vectors with basis
vectors in orthogonal Cartesian coordinate system. As
the double contraction has two definitions, NMR Ham-
iltonian expressed with Cartesian tensor components is
the DIP of A and T, whereas that expressed with spher-
ical tensor components is the DOP of A and T. This
reformulation with double contraction also allows us to
deduce the well-known expressions of covariant spheri-
cal tensor components in terms of Cartesian tensor
components presented by Cook and De Lucia (2) and
reported by Mehring (3). However, these authors did
not mention the implication of double contraction.
This article has also clarified the covariant conven-
tion for NMR spherical tensor components. Unfortu-
nately, most expressions in the literature are written in
covariant notation without mentioning this convention.
The latter is important because spherical tensor compo-
nents are complex numbers. Most transformation laws
about active rotation and rotational invariant through-
out the article involve Wigner active rotation matrix.
We are ready to tackle Wigner passive and active rota-
tion matrices to describe active rotation of spin-part
tensor by an excitation pulse or passive rotation of
space-part tensor under rotation of coordinate system.
REFERENCES
1. Gerstein BC, Dybowski CR. 1985. Transient Tech-
niques in NMR of Solids. Orlando: Academic Press.
2. Cook RL, De Lucia FC. 1971. Application of the
theory of irreducible tensor operators to molecular
hyperfine structure. Am J Phys 39:1433–1454.
3. Mehring M. 1983. Principles of High Resolution
NMR in Solids. Berlin: Springer-Verlag.
4. Borisenko AI, Tarapov IE. 1979. Vector and Tensor
Analysis with Applications. New York: Dover.
Available at: http://bookza.org/. Accessed July 2013.
5. Young EC. 1993. Vector and Tensor Analysis, 2nd
ed. New York: Marcel Dekker. Available at: http://
www.scribd.com/doc/6805061/Mathematics-Vector-
Tensor-Analysis. Accessed February 2013.
6. Nolan PJ. Contravariance, covariance, and spacetime
diagrams. Available at: http://www.farmingdale.edu/
faculty/peter-nolan/pdf/relativity/Ch04Rel.pdf. Accessed
February 2014.
7. Kelly P. 2013. Solid Mechanics Part III: Founda-
tions of Continuum Solid Mechanics. Available at:
http://homepages.engineering.auckland.ac.nz/pkel015/
Solid MechanicsBooks/Part_III/index.html. Accessed
April 2013.
8. Wikipedia. Dyadics. Available at: http://en.wikipe-
dia.org/wiki/Dyadics. Accessed March 2013.
9. Itskov M. 2007. Tensor Algebra and Tensor Analy-
sis for Engineers—With Applications to Continuum
Mechanics. Berlin: Springer-Verlag. Available at:
http://bookza.org/. Accessed July 2013.
10. Lebedev LP, Cloud MJ, Eremeyev VA. 2010. Ten-
sor Analysis with Applications in Mechanics. New
Jersey: World Scientific. Available at: http://fr.scribd.
com/doc/123341313/Tensor-Analysis-with-application-
in-Mechanics. Accessed May 2013.
11. Bauer E. 2010. Introduction to tensor analysis.
Available at: http://www.scribd.com/doc/128605826/
tensor-analysis. Accessed February 2014.
12. Zerilli FJ. 1970. Tensor harmonics in canonical
form for gravitational radiation and other applica-
tions. J Math Phys 11:2203–2208.
13. Mase GE. 1970. Theory and Problems of Continuum
Mechanics. New York: McGraw-Hill. Available at:
http://archive.org/details/SchaumsTheoryAndProblems
OfContinuumMechanics. Accessed May 2013.
14. Higgins BG. 2004. A primer on vectors, basis sets
and tensors. Available at: http://www.ekayasolu-
tions.com/ech140/ECH140ClassNotes/Vector_Tensors.
pdf. Accessed March 2013.
15. Phan-Thien N. 2013. Understanding Viscoelasticity:
An Introduction to Rheology. Berlin: Springer-Verlag.
Available at: http://bookza.org/. Accessed July 2013.
16. Reddy JN. 2008. An Introduction to Continuum
Mechanics. Cambridge: Cambridge University Press.
Available at: http://www.libgen.net/view.php?id5
112567. Accessed July 2013.
17. Coope JAR, Snider RF, McCourt FR. 1965. Irreduc-
ible Cartesian tensors. J Chem Phys 43:2269–2275.
18. Fano U, Racah G. 1959. Irreducible Tensorial Sets.
New York: Academic Press.
19. Boca R. 2012. A Handbook of Magnetochemical
Formulae. London: Elsevier. Available at: http://
books.google.fr/books?id56tbuoOCcgnEC. Accessed
April 2013.
20. Piecuch P. 1985. Note on the multipole expansion
in the spherical tensor form. J Phys A: Math Gen
18:L739–L743.
21. Messiah A. 1999. Quantum Mechanics, two volumes
bound as one. Mineola: Dover. Available at: http://
archive.org/details/QuantumMechanicsVolumeI, http:
//archive.org/details/QuantumMechanicsVolumeIi.
Accessed July 2013.
22. Boca R. 1999. Theoretical Foundations of Molecular
Magnetism. Lausanne: Elsevier. Available at: http://
books.google.fr/books?id5W4Ombeaz1t4C. Accessed
April 2013.
23. Heine V. 1993. Group Theory in Quantum Mechan-
ics: An Introduction to its Present Usage. New
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 239
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
York: Dover. Available at: http://archive.org/details/
GroupTheoryInQuantumMechanics. Accessed July
2013.
24. Clebsch-Gordan coefficients, spherical harmonics,
and d functions. Available at: http://pdg.lbl.gov/
2008/reviews/clebrpp.pdf. Accessed February 2013.
25. Varshalovich DA, Moskalev AN, Khersonskii VK.
1988. Quantum Theory of Angular Momentum: Irre-
ducible Tensors, Spherical Harmonics, Vector Cou-
pling Coefficients, 3nj Symbols. Singapore: World
Scientific.
26. Wikipedia. Table of Clebsch-Gordan coefficients.
Available at: http://en.wikipedia.org/wiki/Table_of_
Clebsch%E2%80%93Gordan_coefficients. Accessed
February 2013.
27. Weissbluth M. 1978. Atoms and Molecules, Student
Edition. New York: Academic Press. Available at:
http://bookza.org/. Accessed July 2013.
28. Rose ME. 1957. Elementary Theory of Angular
Momentum. New York: Wiley. Available at: http://
www.libgen.net/view.php?id5904519. Accessed
July 2013.
29. Racah G. 1942. Theory of complex spectra. II. Phys
Rev 62:438–462.
30. Edmonds AR. 1974. Angular Momentum in Quan-
tum Mechanics. Princeton: Princeton University
Press. Available at: http://books.google.com/book-
s?id50BSOg0oHhZ0C. Accessed July 2013.
31. Zare RN. 1988. Angular Momentum: Understanding
Spatial Aspects in Chemistry and Physics. New
York: Wiley.
32. Thompson WJ. 1994. Angular Momentum: An Illus-
trated Guide to Rotational Symmetries for Physical
Systems. New York: Wiley. Available at: http://
bookza.org/. Accessed July 2013.
33. Stone AJ. 1996. The Theory of Intermolecular
Forces. Oxford: Clarendon Press.
34. Condon EU, Odabasi H. 1980. Atomic Structure.
Cambridge: Cambridge University Press. Available
at: http://bookza.org/. Accessed July 2013.
35. Brink DM, Satchler GR. 1968. Angular Momentum,
2nd ed. Oxford: Clarendon. Available at: http://
archive.org/details/AngularMomentum. Accessed
May 2013.
36. Silver BL. 1976. Irreducible Tensor Methods: An Intro-
duction for Chemists. New York: Academic Press.
37. Marian CM. 2003. Spin-orbit coupling in molecules.
In: Lipkowitz KB, Boyd DB, eds. Reviews in Com-
putational Chemistry, Vol. 17. New York: Wiley-
VCH; pp 99–204. Available at: http://books.google.
fr/books?id5luNYUUfBQb4C. Accessed February
2013.
38. Brouder C, Juhin A, Bordage A, Arrio M-A. 2008.
Site symmetry and crystal symmetry: a spherical
tensor analysis. J Phys: Condens Matter 20:455205.
39. Kimmich R. 1997. NMR: Tomography, Diffusome-
try, Relaxometry. Berlin: Springer-Verlag.
40. Devanathan V. 2002. Angular Momentum Techni-
ques in Quantum Mechanics. New York: Kluwer
Academic. Available at: http://bookza.org/. Accessed
July 2013.
41. Rose ME. 1954. Spherical tensors in physics. Proc
Phys Soc A 67:239–247.
42. Nielsen RD, Robinson BH. 2006. The spherical tensor
formalism applied to relaxation in magnetic resonance.
Concepts Magn Reson Part A 28A:270–290.
43. Rowe DJ, Wood JL. 2010. Fundamentals of Nuclear
Models, Foundational Models. New Jersey: World
Scientific.
44. Tinkham M. 2003. Group Theory and Quantum
Mechanics. Mineola: Dover. Available at: http://books.
google.fr/books?id5r4GIU2wJCAEC. Accessed May
2013.
45. Suhonen J. 2007. From Nucleons to Nucleus: Con-
cepts of Microscopic Nuclear Theory. Berlin:
Springer-Verlag. Available at: http://folk.uio.no/
mhjensen/phy981/suhonen.pdf. Accessed May 2013.
46. Sharp RR. 1990. Nuclear spin relaxation in paramag-
netic solutions. Effects of large zero-field splitting in the
electron spin Hamiltonian. J Chem Phys 93:6921–6928.
47. Covariance and contravariance of vectors. Available
at: http://en.wikipedia.org/wiki/Covariance_and_con-
travariance_of_vectors. Accessed February 2013.
48. Rimrott FPJ, Tabarrok B. 1995. Contravariant com-
ponents and covariant projections in gyrodynamics.
Available at: http://www.uni-magdeburg.de/ifme/
zeitschrift_tm/1995_Heft1/Rimrott_Tabarrok.pdf.
Accessed February 2014.
49. Wikipedia. Intercept theorem. Available at: http://
en.wikipedia.org/wiki/Intercept_theorem. Accessed
April 2013.
50. Fano U, Rau ARP. 1996. Symmetries in Quantum
Physics Symmetries in Quantum Physics. San
Diego: Academic Press. Available at: http://bookza.
org/. Accessed July 2013.
51. McWeeny R. 2002. Symmetry: An Introduction to
Group Theory and its Applications. Mineola: Dover.
Available at: http://seeinside.doverpublications.com/
dover/0486421821. Accessed July 2013.
52. Millot Y, Man PP. 2012. Active and passive rota-
tions with Euler angles in NMR. Concepts Magn
Reson Part A 40:215–252.
53. Goldstein H. 1950. Classical Mechanics. Reading:
Addison-Wesley. Available at: http://www.libgen.
net/view.php?id5610060. Accessed July 2013.
54. Arfken G. 1985. Mathematical Methods for Physi-
cists, 3rd ed. Boston: Academic. Available at: http://
www.libgen.net/view.php?id5911859. Accessed
July 2013.
55. Wigner EP. 1959. Group Theory and Its Application
to Quantum Mechanics of Atomic Spectra, [trans-
lated from the German by J. J. Griffin]. New York:
Academic Press. Available at: http://bookza.org/.
Accessed July 2013.
240 MAN
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
56. Chandrakumar N, Subramanian S. 1987. Modern
Techniques in High-Resolution FT-NMR. New
York: Springer-Verlag.
57. Anet FAL, O’Leary DJ. 1991. The shielding tensor.
Part I: Understanding its symmetry properties. Con-
cepts Magn Reson 3:193–214.
58. Siminovitch DJ. 1997. Rotations in NMR: Part I.
Euler–Rodrigues parameters and quaternions. Con-
cepts Magn Reson 9:149–171.
59. Siminovitch DJ. 1997. Rotations in NMR: Part II.
Applications of the Euler–Rodrigues parameters.
Concepts Magn Reson 9:211–225.
60. Kuprov I. 2013. Spin dynamics, module II, lecture
03. Available at: http://spindynamics.org/Spin-
Dynamics–-Part-II–-Lecture-03.php. Accessed August
2013.
61. Goldman M. 1988. Quantum Description of High-
Resolution NMR in Liquids. Oxford: Clarendon.
Available at: http://bookza.org/. Accessed July 2013.
62. Wolf AA. 1969. Rotation operators. Am J Phys 37:
531–536.
63. Davydov AS. 1965. Quantum Mechanics (translated,
edited and with additions by D. Ter Haar). Oxford:
Pergamon. Available at: http://www.libgen.net/view.
php?id517154. Accessed July 2013.
64. Thankappan VK. 1993. Quantum Mechanics, 2nd
ed. New Delhi: New Age International. Available
at: http://www.scribd.com/doc/98432883/Thankappan-
V-K- quantum-Mechanics. Accessed July 2013.
65. Van de Ven FJM. 1995. Multidimensional NMR in
Liquids: Basics Principles and Experimental Meth-
ods. New York: VCH.
66. Ballentine LE. 1998. Quantum Mechanics, A Mod-
ern Development. Singapore: World Scientific.
Available at: http://www-dft.ts.infn.it/resta/fismat/
ballentine.pdf. Accessed July 2013.
67. Van de Wiele J. 2001. Rotations et moments angu-
laires en m�ecanique quantique. Ann Phys Fr 26:1–
169.
68. Steinborn EO, Ruedenberg K. 1973. Rotation and
translation of regular and irregular solid spherical
harmonics. Adv Quantum Chem 7:1–81.
69. Chaichian M, Hagedorn R. 1998. Symmetries in
Quantum Mechanics, From Angular Momentum to
Supersymmetry. Brewer DF, ed. Bristol: Institute of
Physics Publishing. Available at: http://bookza.org/.
Accessed July 2013.
70. Morrison MA, Parker GA. 1987. A guide to rota-
tions in quantum mechanics. Aust J Phys 40:465–
497.
71. Biedenharn LC, Louck JD. 1981. Angular Momen-
tum in Quantum Physics. Reading: Addison-Wiley.
Available at: http://bookza.org/. Accessed July 2013.
72. Mueller LJ. 2011. Tensors and rotations in NMR.
Concepts Magn Reson Part A 38A:221–235.
73. Sakurai JJ. 1994. Modern Quantum Mechanics,
revised ed. Reading: Addison-Wesley. Available at:
http://www.fisica.net/quantica/Sakurai%20-%20Mod-
ern%20Quantum%20Mechanics.pdf. Accessed May
2013.
74. Sakurai JJ, Napolitano JJ. 2011. Modern Quantum
Mechanics, 2nd ed. Boston: Addison-Wesley. Avail-
able at: http://www.theochem.kth.se/junjiang/Quantum_
book/Sakurai,%20Napolitano%20-%20Modern%20
Quantum%20Mechanics,%202ed,%20Addison-Wesley,
%202011.pdf. Accessed May 2013.
75. Wikipedia. Table of spherical harmonics. Available
at: http://en.wikipedia.org/wiki/Table_of_spherical_
harmonics. Accessed May 2013.
76. Merzbacher E. 1970. Quantum Mechanics, 2nd
edtion. New York: Wiley. Available at: http://www.
scribd.com/doc/30029685/Merzbacher-Quantum-
Mechanics. Accessed May 2013.
77. McIntyre DH. 2012. Quantum Mechanics,
A Paradigms Approach. Boston: Pearson. Available
at: http://www.doc88.com/p-709554971349.html.
Accessed July 2013.
78. Tung W-K. 1985. Group Theory in Physics, An
Introduction to Symmetry Principles, Group Repre-
sentations, and Special Functions in Classical and
Quantum Physics. Singapore: World Scientific Pub-
lishing. Available at: http://www.scribd.com/doc/
76828849/Wu-Ki-Tung-Group-Theory-in-Physics.
Accessed May 2013.
79. Baym G. 1990. Lectures on Quantum Mechanics.
New York: Westview Press. Available at: http://
bookza.org/. Accessed July 2013.
80. Fritzsche S, Inghoff T, Tomaselli M. 2003. Maple
procedures for the coupling of angular momenta.
VII. Extended and accelerated computations. Com-
put Phys Commun 153:424–444.
81. Daumens M, Minnaert P. 1976. Tensor spherical
harmonics and tensor multipoles. I. Euclidean space.
J Math Phys 17:1903–1909.
82. Blum K. 1981. Density Matrix Theory and Applica-
tions. New York: Plenum Press. Available at: http://
bookza.org/. Accessed July 2013.
83. Auzinsh M, Budker D, Rochester S. 2010. Optically
Polarized Atoms: Understanding Light-Atom Inter-
actions. Oxford: Oxford University Press. Available
at: http://www.doc88.com/p-087372978030.html.
Accessed May 2013.
84. Cohen-Tannoudji C, Diu B, Lalo€e F. 1977. Quan-
tum Mechanics, Vol. 2. New York: Wiley. Avail-
able at: http://bookza.org/book/. Accessed August
2013.
85. Condon EU, Shortley GH. 1959. The Theory of
Atomic Spectra. Cambridge: Cambridge University
Press. Available at: http://www.scribd.com/doc/
109167215/Atom-1959-Condon-Shortley-the-Theory-of-
Atomic-Spectra. Accessed May 2013.
86. van Kleef EH. 1995. The Cartesian rank two repre-
sentation of spherical tensors with an application to
Raman scattering. Am J Phys 63:626–633.
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 241
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
87. Landau LD, Lifshitz EM. 1977. Quantum mechan-
ics, non-relativistic theory, Vol. 3. In: Course of
Theoretical Physics, 3rd ed. Oxford: Pergamon.
Available at: http://bookza.org/. Accessed July 2013.
88. Sanctuary BC. 1976. Multipole operators for an
arbitraty number of spins. J Chem Phys 64:4352–
4361.
89. Stone AJ. 1976. Properties of Cartesian-spherical
transformation coefficients. J Phys A: Math Gen 9:
485–497.
90. Stone AJ. 1975. Transformation between cartesian
and spherical tensors. Mol Phys 29:1461–1471.
91. Normand J-M, Raynal J. 1982. Relations between
Cartesian and spherical components of irreducible
Cartesian tensors. J Phys A: Math Gen 15:1437–
1461.
92. de Souza Neto EA, Peric D, Owen DRJ. 2008.
Computational Methods for Plasticity: Theory and
Applications. Chichester: Wiley. Available at: http://
www.scribd.com/doc/123933140/EFM. Accessed
April 2013.
93. Dick B. 1987. Response function theory of time-
resolved CARS and CSRS of rotating molecules in
liquids under general polarization conditions. Chem
Phys 113:131–147.
94. Brown JM, Carrington A. 2003. Rotational Spectros-
copy of Diatomic Molecules. Cambridge: Cam-
bridge University Press. Available at: http://www.
scribd.com/doc/79353332/Rotational-Spectroscopy-of-
Diatomic-Molecules-Brown-J-CUP-2003. Accessed
March 2013.
95. Reich S, Thomsen C, Maultzsch J. 2004. Carbon
Nanotubes: Basic Concepts and Physical Properties.
Weinheim: Wiley-VCH. Available at: http://www.
libgen.net/view.php?id5220623. Accessed July
2013.
96. Bain AD. 2006. Operator formalisms: An overview.
Concepts Magn Reson Part A 28A:369–383.
97. Mehring M, Weberruss VA. 2001. Object-Oriented
Magnetic Resonance: Classes and Objects, Calcula-
tions and Computations. San Diego: Academic.
98. Robert JB, Wiesenfeld L. 1982. Magnetic aniso-
tropic interactions of nuclei in condensed matter.
Phys Rep 86:363–401.
99. Kristensen JH, Bilds�e H, Jakobsen HJ, Nielsen
NC. 1999. Application of Lie algebra to NMR spec-
troscopy. Prog Nucl Magn Reson Spectrosc 34:1–
69.
100. Glaubitz C. 2000. An introduction to MAS NMR
spectroscopy on oriented membrane proteins. Con-
cepts Magn Reson 12:137–151.
101. Bain AD. 2003. Exact calculation, using angular
momentum, of combined Zeeman and quadrupolar
interactions in NMR. Mol Phys 101:3163–3175.
102. Bain AD, Berno B. 2011. Liouvillians in NMR: The
direct method revisited. Prog Nucl Magn Reson
Spectrosc 59:223–244.
103. Grandinetti PJ, Ash JT, Trease NM. 2011. Symme-
try pathways in solid-state NMR. Prog Nucl Magn
Reson Spectrosc 59:121–196.
104. Spiess HW. 1978. Rotation of molecules and
nuclear spin relaxation. In: Diehl P, Fluck E, Kos-
feld R, eds. NMR Basic Principles and Progress.
Vol. 15. Berlin: Springer-Verlag. pp 55–214.
105. Schmidt-Rohr K, Spiess HW. 1994. Multidimen-
sional Solid-State NMR and Polymers. San Diego:
Academic.
106. Hodgkinson P, Emsley L. 2000. Numerical simula-
tion of solid-state NMR experiments. Prog Nucl
Magn Reson Spectrosc 36:201–239.
107. Bl€umich B. 2000. NMR Imaging of Materials.
Oxford: Clarendon. Available at: http://bookza.org/.
Accessed July 2013.
108. Haeberlen U. 1976. High Resolution NMR in
Solids: Selective Averaging. New York:
Academic; Adv Magn Reson. Vol. Suppl. 1. Avail-
able at: http://www.sciencedirect.com/science/book/
9780120255610. Accessed May 2013.
109. Saltsidis P, Brinne B. 1995. Solutions to Problems
in Quantum Mechanics. Available at: http://fr.scribd.
com/doc/7209494/Saltsidis-P1-Brinne-B. Accessed
February 2013.
110. Smith SA, Palke WE, Gerig JT. 1992. The Hamilto-
nians of NMR. Part I. Concepts Magn Reson 4:107–
144.
111. McConnell J. 1987. The Theory of Nuclear Mag-
netic Relaxation in Liquids. Cambridge: Cambridge
University Press.
112. Bonin KD, Kresin VV. 1997. Electric - Dipole
Polarizabilities of Atoms, Molecules and Clusters.
Singapore: World Scientific. Available at: http://
www.libgen.net/view.php?id5596470. Accessed
July 2013.
113. Berne BJ, Pecora R. 2000. Dynamic Light Scatter-
ing: With Applications to Chemistry, Biology, and
Physics. New York: Dover. Available at: http://
www.libgen.net/view.php?id5501743. Accessed
July 2013.
114. Yu PY, Cardona M. 2010. Fundamentals of Semi-
conductors: Physics and Materials Properties. Hei-
delberg: Springer-Verlag. Available at: http://www.
libgen.net/view.php?id5275195. Accessed July
2013.
115. Gamliel D, Levanon H. 1995. Stochastic Processes
in Magnetic Resonance. Singapore: World Scientific.
Available at: http://books.google.fr/books?id5vo2__
zB-YfsC. Accessed May 2013.
116. Chandra Shekar S, Jerschow A. 2010. Tensors in
NMR. In: McDermott AE, Polenova T, eds. Solid-
State NMR of Biopolymers. Chichester: Wiley. pp
39–47. Available at: http://books.google.fr/book-
s?id5atsaceAxLEwC. Accessed April 2013.
117. Chandra Shekar S, Jerschow A. 2009. Tensors in
NMR. In: Harris RK, Wasylishen RE, Duer MJ,
242 MAN
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
eds. NMR Crystallography. Chichester: Wiley.
Available at: http://books.google.fr/books?id5JWH_
3ihp3IsC&redir_esc5y. Accessed May 2013.
118. Smith SA, Palke WE, Gerig JT. 1992. The Hamilto-
nian of NMR. Part II. Concepts Magn Reson 4:181–
204.
119. Kuprov I. 2013. Module I, lecture 06: Spin interac-
tion Hamiltonians, part I. Available at: http://spindy-
namics.org/Spin-Dynamics–-Part-I–-Lecture-06.php.
Southampton: University of Southampton.
Accessed August 2013.
120. Duer MJ. 2004. Introduction to Solid-State NMR
Spectroscopy. Oxford: Blackwell Publishing.
121. Facelli JC, Grant DM. 1989. Molecular structure
and carbon-13 chemical shielding tensors obtained
from nuclear magnetic resonance. In: Eliel EL,
Wilen SH, eds. Topics in Stereochemistry. Vol. 19.
New York: Wiley. pp 1–61.
122. Cavadini S. 2010. Indirect detection of nitrogen-14
in solid-state NMR spectroscopy. Prog Nucl Magn
Reson Spectrosc 56:46–77.
123. Grant DM. 2010. Chemical shift tensors. In: McDer-
mott AE, Polenova T, eds. Solid State NMR Studies
of Biopolymers. Chichester: Wiley. Available at:
http://books.google.com/books?id5atsaceAxLEwC.
Accessed April 2013.
124. Freude D. 2000. Quadrupolar nuclei in solid-state
nuclear magnetic resonance. In: Meyers RA, ed.
Encyclopedia of Analytical Chemistry. Chichester:
Wiley. pp 12188–12224. Available at: http://www.
uni-leipzig.de/energy/publist/freude2000.pdf.
Accessed April 2013.
125. Freude D, Haase J. 1993. Quadrupole effects in
solid-state nuclear magnetic resonance. In: Diehl P,
Fluck E, G€unter H, Kosfeld R, Seelig J, eds. NMR
Basic Principles and Progress. Vol. 29. Berlin:
Springer-Verlag. pp 1–90.
126. Man PP. 2011. Quadrupolar interactions. In: Harris
RK, Wasylishen RE, eds. Encyclopedia of Magnetic
Resonance. Chichester: Wiley. Available at: http://
www.pascal-man.com/book/emr2011.pdf. Accessed
April 2013.
127. Man PP. 2000. Quadrupole couplings in nuclear
magnetic resonance, general. In: Meyers RA, ed.
Encyclopedia of Analytical Chemistry. Chichester:
Wiley. pp 12224–12265. Available at: http://www.
pascal-man.com/navigation/publication-since1983.
shtml#-64. Accessed July 2013.
128. Hajjar R, Millot Y, Man PP. 2010. Phase cycling in
MQMAS sequences for half-integer quadrupole spins.
Prog Nucl Magn Reson Spectrosc 57:306–342.
129. Aucar GA. 2008. Understanding NMR J-couplings
by the theory of polarization propagators. Concepts
Magn Reson Part A 32A:88–116.
130. Murphy DM. 2009. EPR (Electron Paramagnetic
Resonance) spectroscopy of polycrystalline oxide
systems. In: Jackson SD, Hargreaves JSJ, eds. Metal
Oxide Catalysis. Weinheim: Wiley-VCH Verlag.
Available at: http://onlinelibrary.wiley.com/doi/10.
1002/9783527626113.ch1/pdf. Accessed August
2013.
131. Slichter CP. 1990. Principles of Magnetic Reso-
nance, 3rd ed. Berlin: Springer-Verlag. Available at:
http://bookza.org/. Accessed August 2013.
132. Lushington GH. 2004. The effective spin Hamilto-
nian concept from a quantum chemical perspective.
In: Kaupp M, B€uhl M, Malkin VG, eds. Calculation
of NMR and EPR Parameters, Theory and Applica-
tions. Weinheim: Wiley-VCH Verlag. pp 33–42.
Available at: http://www.scribd.com/doc/35558268/
Calculation-of-NMR-and-EPR-Parameters. Accessed
July 2013.
133. Mathews J. 1962. Gravitational multipole radiation.
J Soc Indust Appl Math 10:768–780.
134. Maxum B. 2004. Field Mathematics for Electromag-
netics, Photonics, and Materials Science: A Guide
for the Scientist and Engineer—4th Printing. Bel-
lingham: SPIE Press. Available at: http://books.goo-
gle.fr/books?id53iNyCRy_q8wC. Accessed April
2013.
135. Saito H, Ando I, Naito A. 2006. Solid State NMR
Spectroscopy for Biopolymers: Principles and Appli-
cations. Dordrecht: Springer. Available at: http://
148.206.53.231/tesiuami/S_pdfs/Solid%20State%20
NMR%20Spectroscopy%20for%20Biopolymers.pdf.
Accessed May 2013.
136. Saito H, Ando I, Ramamoorthy A. 2010. Chemical
shift tensor—The heart of NMR: Insights into bio-
logical aspects of proteins. Prog Nucl Magn Reson
Spectrosc 57:181–228.
137. Mehring M. 2010. Internal spin interactions and
rotations in solids. In: McDermott AE, Polenova T,
eds. Solid-State NMR of Biopolymers. Chichester:
Wiley. pp 3–27. Available at: http://books.google.fr/
books?id5atsaceAxLEwC. Accessed May 2013.
138. Man PP. 2013. Double dot product of two spherical
basis tensors. Available at: http://www.pascal-man.
com/tensor-quadrupole-interaction/mathematica/trs
DDPtkq.nb. Accessed on February 2014.
139. Louck JD. 2006. Angular momentum theory. In:
Drake GWF, ed. Springer Handbooks of Atomic,
Molecular, and Optical Physics. New York:
Springer-Verlag. Available at: http://www.libgen.
net/view.php?id517569. Accessed August 2013.
140. Tavan P, Schulten K. 1980. An efficient approach to
CI: General matrix element formulas for spin
coupled particle-hole excitations. J Chem Phys 72:
3547–3576.
141. Jerschow A. 2005. From nuclear structure to the
quadrupolar NMR interaction and high-resolution
spectroscopy. Prog Nucl Magn Reson Spectrosc 46:
63–78.
142. Man PP. 2013. Cartesian and spherical tensors in
NMR Hamiltonian. Available at: http://www.pascal-
CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 243
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
man.com/tensor-quadrupole-interaction/mathematica/
tenseurNMRhamiltonian.nb. Accessed on February
2014.
143. Grant DM, Halling MD. 2009. Metric spaces in
NMR crystallography. Concepts Magn Reson Part A
34A:217–237.
BIOGRAPHIES
Pascal P. Man b 1952. Ph.D., 1982, Mate-
rials Science, Sc.D., 1986, Physics, Uni-
versit�e Pierre et Marie Curie, Paris, France.
Introduced to NMR by H. Zanni. Postdoc-
toral work at University of Cambridge
under the direction of J. Klinowski. Uni-
versit�e Pierre et Marie Curie, 1988 -pres-
ent. Approx. 90 publications. Current
research speciality: solid state NMR on
quadrupolar nuclei and rotation.
244 MAN
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
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