Reviews in Computational Chemistry

114 Spin–Orbit Coupling in Molecules

From Eqs. [30], we conclude that the three components of the angular momentum cannot be determined simultaneously—with the trivial exception when all three components are zero.

By contrast, the square modulus of the orbital angular momentum (see

~

Eq. [22]) commutes with all three components of ^ , that is,

L

^ ^ 2	& ¼ 0	^ ^ 2	& ¼ 0	^ ^ 2	& ¼ 0	½31&
½Lx; L		½Ly; L		½Lz; L

General Angular Momenta

As mentioned above, the definition of an angular momentum is a direct consequence of the isotropy of space, and this property leads directly to the commutation relations. Instead of relying on the special properties of the orbital angular momentum, we shall solve the eigenvalue problem of the angular momentum solely based on the commutation relations that are common to all types of angular momenta. This approach has the advantage that cases with half-integer angular momentum quantum numbers are included; these are related to spin and do not occur for pure orbital angular momenta.

Step/Shift/Ladder and Tensor Operators

For the determination of matrix elements, it is often more convenient to use linear combinations of the Cartesian components of the angular momentum operator instead of the Cartesian components themselves. In the literature, two different kinds of operators are employed. The first type is defined by

	^	^ ^	^	^	^	^ ^	½32&
	Jþ ¼ Jx þ iJy		J0	¼ Jz	J ¼ Jx iJy
^	^

Jþ and J are called step-up and step-down operators, respectively, or shift

operators for reasons to become clear soon. These operators are also denomi-

^

^

^

¼6

^

nated ladder operators. Jþ and J are not self-adjoint, that is, ðJþÞ

Jþ

^

^

and vice versa. Together

and so on; instead J

þ

is the complex conjugate of J

^

~

^

with Jz they form a linear independent set of components of J. In terms of the

^2

can be expressed as

shift operators, J

^2

^

^

^2

^

½33&

J

¼ JþJ þJz

hJz

^

^

^2

^

½34&

¼ J Jþ þJz

þ hJz

1

^

^

^ ^

^2

½35&

¼

2

ðJþJ þJ JþÞ þJz

Each of these forms will be used later.

A second set of operators, the so-called tensor operators, differ only slightly from the ladder operators. They are introduced here without further

Angular Momenta 115

explanation. Cartesian and spherical tensor components are related by

^

^

^

^

^

Jx þ iJy

^

^

^

Jx iJy

½36&

J0 ¼ Jz

Jþ1 ¼ p2

J 1 ¼ p2

or conversely

J 1

Jþ1

iðJ 1

þ Jþ1Þ

37

Jx

¼

p

Jz

¼

J0

Jy

¼

p

½ &

2

2

Phase conventions have been chosen to be consistent with those of Condon

~

and Shortley.13 In terms of tensor operators, the square modulus of ^ becomes

J

^2	^ ^	^ ^	^2	½38&
J	¼ Jþ1J 1	J 1Jþ1	þJ0

We shall come back to these operators after learning what a tensor is.

Commutation Relations

As for the orbital angular momentum, the commutation relations

~

between the Cartesian components of a general angular momentum ^ and

J

its square modulus ^2 read

J

^	^	^	^ ^	^	^ ^		^	½39&
½Jx;Jy& ¼ ihJz			½Jy;Jz& ¼ ihJx		½Jz;Jx& ¼ ihJy
and
	^ ^2	& ¼ 0	^ ^2	& ¼ 0	^ ^2	& ¼ 0		½40&
	½Jx;J		½Jy;J		½Jz;J

Using Eq. [39] and the definition of the step-up and step-down operators (Eq. [32]), one easily obtains their commutation relations

^	^	^	½41&
½Jþ;J & ¼ 2hJz
and
^	^	^	½42&
½Jz;J & ¼ hJ
From the latter, the useful relation
^	^n	^	½43&
½Jz;J & ¼ nhJ

can be derived.

116 Spin–Orbit Coupling in Molecules

Like the Cartesian components, the shift operators also commute with ^2

J

^ ^2	& ¼ 0	½44&
½J ;J

The same is true for the tensor operators.

^	^2	& ¼ 0	½45&
½J 1	;J

The commutation relations among their components differ slightly from those of the shift operators. From Eqs. [36] and [39], it follows that

^

^

^

½46&

½Jþ1;J 1& ¼ hJ0

and

^

^

^

½47&

½J0;J 1

& ¼ hJ 1

^2

^

The Eigenvalues of J

and Jz

^2

^

Because J

and Jz commute, they must have common eigenvectors that

we shall denote by jui. The eigenvectors satisfy the equations

^2

jui ¼ ah

2

jui

½48&

J

^

½49&

Jzjui ¼ Mhjui

Applying Eqs. [42] to the eigenvectors jui

^

^

^

^

^

^

½50&

JzJ jui ¼ J Jzjui hJ jui ¼ ðM 1ÞhJ jui

^

^

we see that J jui are also eigenvectors of

Jz, but with eigenvalue ðM 1Þh.

^

^2

with eigenvalue ah

2

because

Moreover, J jui is also an eigenvector of J

^2 ^

^

^2

2 ^

½51&

J J jui ¼ J J

jui ¼ ah J jui

^

^

^2

is to step up

This means that the action of J

þ

on an eigenvector of J

z

and J

^

the eigenvalue of Jz by one unit while remaining within the subset of functions

^2

^

analogously steps down the eigenvalue

belonging to the eigenvalue a of J

. J

^

of Jz by one unit.

^2

in the form Eq.

To set up a connection between a and M, we express J

[35] and note that

^

Jþ and J are Hermitian conjugates. Applying the turn-

over rule (Eq. [17]) yields

^

^

1

^

^

1

^

^

^

^

hJfjJfi ¼

2

hJ fjJ fi þ

2

hJþfjJþfi þ hJzfjJzfi

½52&

Angular Momenta 117

Each of these terms is always positive or zero—this follows from the fact that the length of a vector is always positive or zero. The fact that each of the terms in Eq. [52] is positive or zero leads us to the inequality

^2	^2	½53&
hfjJ	jfi hfjJz jfi 0

By identifying jfi with an eigenvector jui, we obtain from Eqs. [48] and [49]

	a M2 0					½54a&
or						½		&
	pa		M		pa		54b

which means that the eigenvalues M are limited from above and below; there

is a minimal value Mmin and a maximal value Mmax.

In particular, if we apply Eq. [50] to an eigenvector jumaxi belonging to

Mmax

^ ^	^	½56&
JzJþjumaxi ¼ ðMmax þ 1ÞhJþjumaxi

then this equation can only be satisfied, if

^	½57&
Jþjumaxi ¼ 0

because Mmax was assumed to be the maximal M value. Using expression [34] yields

^ ^	^2		^2	^	½58&
J Jþjumaxi ¼ ðJ			Jz	hJzÞjumaxi ¼ 0
leading to
	ah2 Mmax2	h2 Mmax h2 ¼ 0			½59&
From this, we obtain the relation between a and Mmax
	a ¼ MmaxðMmax þ 1Þ				½60&
Analogously, application of expression [33] to jumini yields
	a ¼ MminðMmin 1Þ				½61&

118 Spin–Orbit Coupling in Molecules

Finally, stepping down jumaxi repeatedly by applying Eq. [43]
^ ^n			^n	½62&
JzJ jumaxi ¼ ðMmax	nÞhJ jumaxi
we can always find a positive integer n (the largest possible) such that
Mmax n ¼ Mmin				½63&
leading to
MmaxðMmax þ 1Þ ¼ MminðMmin 1Þ ¼ ðMmax nÞðMmax n 1Þ				½64&
or
Mmax ¼		n		½65&
	2

In the following, we shall denote this maximal value Mmax by J. This maximum J can only be an integer or a half-integer

J ¼ 0;	1	; 1;	3		;	½66&
	2		2

The eigenvalue a from Eq. [48] adopts the values JðJ þ 1Þ and Mmin ¼ J. For M, we thus obtain the values

M ¼ J; J 1 J

½67&

Finally, if we denominate the corresponding eigenvectors by juMJ i, the eigenvalue equations of the angular momentum operators read

^2

M

i ¼

JðJ

þ 1Þh

2

M

½68&

J

juJ

juJ i

^

M

i ¼

M

i

½69&

JzjuJ

MhjuJ

Because ^

Hermitian and each

j

uM

belongs to a different eigenvalue, the

Jz is

M

J i

eigenvectors juJ i are orthogonal; after normalization we obtain

h

uM

uM0

i ¼

dJJ0

dMM0

½

70

J

j

J0

&

^

^

^

^2

^

The Action of J

, J

x

, and J

y

on Eigenvectors of J

and J

z

According to

^

M

^

Eqs. [50] and [51],

i is both

an eigenvector of

J juJ

^2

Jz

with eigenvalue ðM 1Þh and

an eigenvector of

J

with eigenvalue

Angular Momenta 119

JðJ þ 1Þh2. Thus ^ juMi and juM 1i are parallel, and we can determine the

J J J

proportionality constants from the normalization constraint:

^

M

^

M

M

^

^

M

½71&

hJ uJ

jJ uJ i ¼ huJ

jJ J juJ i

M

^2

^2

^

M

i

½72&

¼ huJ

jðJ

Jz

hJz

ÞjuJ

¼ JðJ þ 1Þ M2 M h2

½73&

Up to an arbitrary phase factor, this yields

^

j

uM

i ¼

p

i

h J

J

þ

1

Þ

M

M

1

uM 1

74

J

J

ð

ð

Þ j

J

½ &

Accordingly, by inverting the definition of the ladder operators in

terms of the Cartesian components, we can determine the actions of ^ and

Jx

^

Jy

^

Jx

^

Jy

	on juJMi:						p				p
		1
	juJMi ¼			hn				JðJ þ 1Þ MðM þ 1ÞjuJMþ1iþ			JðJ þ 1Þ MðM 1ÞjuJM 1io
			2
					i		hn		p		½75&
	juJMi ¼								JðJ þ 1Þ MðM þ 1ÞjuJMþ1i
					2
					p					io
									M 1		½76&
		JðJ þ 1Þ MðM 1ÞjuJ

Matrix Elements

By using the results of the last two subsections, the matrix elements of

momentum operators are easily determined. The

j

uM

i are eigen-

the angular 2

^

2

J

^

^

^

vectors of J

and Jz

. Therefore, nonvanishing integrals of J

and Jz are con-

fined to the diagonal of the matrix.

u

M

^2

u

M0

2

J

J

1

dJJ0 d

MM0

77

h

J

J

J0

i ¼

h

ð

þ

Þ

j j

½ &

and

h

u

M

^

u

M0

i ¼

hMdJJ0 d

MM0

78

J

Jz

J0

j

j

½ &

^2

^

^

have only off-diagonal

In the J

, Jz

representation, the shift operators J

matrix elements

p

uM

j

^

j

uM0

i ¼

MM0

1

h

J

J

J0

ð

þ

Þ

0ð

0

ÞdJJ0 d

½ &

120 Spin–Orbit Coupling in Molecules

^

^

The same is true for the Cartesian Jx and Jy

h

uM

j

^

j

uM0

i ¼

1

1p

MM0

1

h J

J

þ

1

Þ

M

0

ð

M

0 þ

1

ÞdJJ0 d

J

Jx

J0

2

ð

þ i

p

MM0

1

h J

ð

J

þ

1

Þ

M

0

ð

M

0

1

ÞdJJ0 d

j

2

uM

^

uM0

i ¼ i

p

MM0

1

h

h J

J

þ

1

Þ

M

0

ð

M

0

þ

1

ÞdJJ0 d

J

jJy

J0

2

ð

p

MM0

1

þ

h J

ð

J

þ

1

Þ

M

0

ð

M

0

1

ÞdJJ0 d

2

In particular, for J ¼ 21 states we obtain

M

M0

1

1

2

2

M

^2

M0

i ¼

1

3

h

2

0

huJ

jJ

juJ

2

4

21

0

43 h2

M

M0

1

1

M0

2

2

M ^

i ¼

1

1

h

0

huJ

jJzjuJ

2

2

21

0

21 h

M

M0

21

21

M

^

M0

i ¼

1

0 h

huJ

jJþjuJ

2

21

0

0

M

^

M0

i ¼

M

M0

21

21

1

0 0

huJ

jJ juJ

2

21

h

0

M

M0

1

1

M0

2

2

M

i ¼

2

0

2 h

huJ

j Jx juJ

1

1

21

21 h

0

M

M0

1

1

M0

2

2

M

^

1

i

huJ

jJyjuJ

i ¼

0

h

2

2

21

i

h

0

2

½80&

½81&

½82&

½83&

½84&

½85&

½86&

½87&

Angular Momenta 121

A Pictorial Representation
In the previous sections, we learned that the modulus of the angular
ð2J þ 1Þ-fold degenerate.				p
	~						2
momentum	^	amounts to		JðJ þ 1Þh and that the eigenvalues of		^		are
	jJj					J
			Thus, if we measure the modulus of the angular
momentum,	we	know that		the state vector is located somewhere		in		the
				^2	. If we determine also the component
2J þ 1 dimensional vector space uJ of J

~

of ^ along a direction (which we call the z axis), we will find values of Mh.

J

Unlike the components of the linear momentum, not all components of the angular momentum can be determined simultaneously. Thus, it is not possible to represent the measured values of the quantum mechanical angular momentum as an arrow. Conveniently, the angular momentum is visualized as a cone

(see Figure 8) with the axis oriented along the direction of the measured com-
is that the angular momentum vector	p
ponent (z axis), height Mh, and radius	JðJ þ 1Þ M2 h. All that we can say

lies somewhere on this cone; its x and y

components remain undetermined.

We shall frequently encounter cases with angular momentum values of J ¼ 12 and J ¼ 1. For J ¼ 12, such as for electron spin, there are only two possible orientations of the angular momentum; they are depicted in Figure 9. Commemorative of the old Bohr–Sommerfeld theory, we say that the angular momentum is oriented parallel to the z axis in the case of M ¼ þ 12 and antiparallel for M ¼ 12.

In the case of J ¼ 1, three possibilities arise (Figure 10): an upward cone for M ¼ 1, a downward cone for M ¼ 1, and a cone with height 0—thus reducing to a disk—for M ¼ 0.

Spin Angular Momentum

As mentioned earlier, we cannot make use of the correspondence principle to derive quantum mechanical spin operators, because spin has no classical analog. Instead, the spin eigenfunctions jsmsi may be identified with ju1=12=2i

z

M h

J(J+1) h

Figure 8 The cone of an angular momentum vector.

122 Spin–Orbit Coupling in Molecules

z

M = 1/2

M = −1/2

Figure 9 The two possible orientations of

J ¼ 12.

and Eqs. [82]–[87] are then employed to define a matrix representation of the spin operators.

Spinors and Spin Operators

Obviously, the spin eigenfunction jsmsi is not a function of the spatial coordinates; mathematically it is known as a spinor. Different notations are

z

M = 1

M = 0

M = −1

Figure 10 The three possible orientations of J ¼ 1.

							Angular Momenta		123
in common usage:
			8 j	21	21i	1	a
						0
			>	1	1			b
	sms		<			0			88
j		i ¼	>		2i 1				½ &
			>
			>	2
			> j
			:

The symbols a and b are the ones most familiar to chemists. For the definition of spin operators, it is convenient to utilize the representation of the spin eigenfunctions as the orthonormal basis vectors of a two-dimensional (2D) vector space. In this representation, the spin operators may be written as matrices

					^2			h2			3	0		!
							¼				4										89
												4
					S						0										½ &
												3
S^0 ¼ h		1		0	!	S^þ ¼ h					0	1			S^ ¼ h			0	0
																					½90&
		0 21									0 0							1 0
		2
^		0	1	!	^			0				i			^		1		0	!
	¼ h		2			¼ h						2				¼ h	2				½91&
Sx		1	0		Sy				i		0	!			Sz		0		1
		2						2											2

acting on the column vectors ð10Þ and ð01Þ by means of the usual matrix-vector product.

For later convenience, we also define the irreducible tensor operators

		2	!					!		0		0	!
	1	0			0	1
S^0 ¼ h	2			S^þ1 ¼ h		p			S^ 1					ð92Þ
	0	1			0	0 2				¼ h p12		0

Pauli Spin Matrices

Pauli introduced slightly different spin operators known as the spin matrices. They are defined by

	¼	1	0	¼ i 0		¼ 0	1
sx		0	1	0 i	sz	1	0
			sy

Pauli

½93&

Apart from the numerical factor 2h, they are in fact identical with the Carte-

~

^

sian spin operators in Eq. [91]. The difference between ~s and S appears to be trivial; nevertheless, it is important to recognize that the operator ~s does not qualify as an angular momentum operator because it does not fulfill the usual