Cartesian and Spherical Tensors in NMR Hamiltonians

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

mailto:[email protected]

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.

198 MAN

Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a

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)

CARTESIAN AND SPHERICAL TENSORS IN NMR HAMILTONIANS 199


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

200 MAN


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.



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}.

202 MAN


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



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


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@


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@


3 D�� 21

D�

a1

a2

a3

0BBB@


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@


3D21D

a1

a2

a3

0BBB@


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:



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


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



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)

208 MAN


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


� �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:



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


� �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

210 MAN


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)



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


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.

212 MAN


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


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.



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).

214 MAN


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



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

216 MAN


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:



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


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)


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



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)


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


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/

220 MAN


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


� ��

5X1l

mm052l

Tlm0Dðl;AÞm0m ða;b; gÞ D


� ��:

(216)

Equation (173) allows us to simplify the above

relation:

Xl

m52l

TRlm D


� ��5X1l

m052l

Tlm0dm0m00 : (217)

Finally, we have our desired relation for covariant

spherical components:

Tlm005Xl

m52l


� ��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



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


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)


� �A105Az


� � :

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



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


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)


� �; (247)


� �; (248)

t2125e111 � e111; (249)


� �; (250)

t2051ffiffiffi6p e111 � e12112e10 � e101e121 � e111

� �;

(251)


� �; (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



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


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


(293)

A10 2f g5 V� U½ �10521ffiffiffi2p i VxUy2VyUx

� �; (294)

A161 2f g5 V� U½ �161

521


� �n o;

(295)

A20 2f g5 V� U½ �2051ffiffiffi6p 2VxUx12VzUz2VyUy

� �5


;

(296)



A261 2f g5 V� U½ �261

571


� �n o;

(297)

A262 2f g5 V� U½ �262

51


� �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


(300)

A111 1f g5 V� U½ �11151ffiffiffi2p V111U102V10U111

� �5

1


� �n o;

(301)


� �5

1ffiffiffi2p i VxUy2VyUx

� �;

(302)


� �5

1


� �n o;

(303)

A212 1f g5 V� U½ �2125V111U111

51


� �n o;

(304)


� �52

1


� �n o;

(305)

A20 1f g5 V� U½ �2051ffiffiffi6p V111U12112V10U101V121U111

� �5

1ffiffiffi6p 2VxUx12VzUz2VyUy

� �5


;

(306)


� �5

1


� �n o;

(307)

A222 1f g5 V� U½ �2225V121U121

51


� �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


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


� �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


� �n o;

(324)



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


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.



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


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)


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)


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



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


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�.



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


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



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


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.

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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




Documents

Cartesian and Spherical Tensors in NMR Hamiltonians