Reviews in Computational Chemistry

144 Spin–Orbit Coupling in Molecules

that is, the components of ^ ðkÞ are transformed among themselves upon a

T

coordinate rotation. The coefficients DðqqkÞ0 ðRÞ are elements of a 2k þ 1 dimensional irreducible matrix representation DðkÞ of the full rotation group in three dimensions, Oþð3Þ or SOð3Þ. Equivalently, it can be stated that the 2k þ 1

components ^ ðkÞ with q ranging from k to k form a basis for the irreduci-

Tq

ble representation DðkÞ of Oþð3Þ. Actually, the coefficients DðqqkÞ0 ðRÞ are the same as those obtained from a rotation operation transferring the spherical

harmonic Ykqðy; fÞ (see Eq. [25]) to Ykq0 ðy; fÞ.				^ ð0Þ
The simplest example is a scalar operator T0					with the transformation
property
^	^ ð0Þ ^	1	¼	^ ð0Þ	½148&
UðRÞT0 UðRÞ				T0

This means that scalar operators are invariant with respect to rotations in coordinate or spin space. An example for a scalar operator is the Hamiltonian, i.e., the operator of the energy.

A first-rank tensor operator ^ ð1Þ is also called a vector operator. It has

T

three components,	^ ð1Þ	and	^ ð1Þ	. Operators of this type are the angular
	T0		T 1

momentum operators, for instance. Relations between spherical and Cartesian components of first-rank tensor operators are given in Eqs. [36] and [37].

Operating with the components of an arbitrary vector operator ^ ð1Þ on

P

an eigenfunction	j	uMP	i	of the corresponding operators	^ ð1Þ	Þ	2	and	^ ð1Þ	yields
		P			ðP				P0

the following results:

^ ð1Þ

P0

^ ð1Þ

P 1

juPMP i ¼ hMPjuPMP i					i
j	uPMP	i ¼	q& j	uPMP 1
			h 21 ½PðP þ 1Þ ðMP 1ÞMP

½149&

½150&

^ ð1Þ					^	ð1Þ	increases MP by one
P	0	conserves the projection MP of P on the z axis, P				þ1
			^ ð1Þ	decreases MP
unit, and			P 1		by one unit. The prefactors in Eq. [150] differ

slightly from those obtained in Eq. [74] by operating with step operators on juMJ i (see the earlier section on step/shift/ladder and tensor operators).

A general Cartesian second-rank tensor operator is represented by a 3 3 matrix.

		T^ xxð2Þ T^ xyð2Þ		T^ xzð2Þ	1
T^ ð2Þ x; y; z		0 T^ yxð2Þ	T^ yyð2Þ	T^ yzð2Þ		151
		@			A
ð	Þ ¼	B T^ ð2Þ	T^ ð2Þ	T^ ð2Þ	C	½ &
		B zx	zy	zz	C

If the Cartesian representation corresponds to an irreducible second-rank

nine entries are not independent: The four conditions				^ ð2Þ	^ ð2Þ
tensor, its 2	Þ			Tij	¼ Tji
^ ð		¼ 0 with i; j 2 fx; y; zg	reduce the number of	independent
and Pi Tii

Symmetry 145

components to five. Spherical and Cartesian components of a second-rank irreducible tensor are related by

^ ð2Þ

^

2

^ 2

^ 2

½152&

T 2

¼ ðTxxð Þ Tyyð Þ

2iTxyð ÞÞ=2

^ ð2Þ

¼

^ ð2Þ

i ^ ð2Þ

½

153

&

T 1

Txz

Tyz

^ ð2Þ

^ 2

^ 2

^ 2

p

6

½154&

T0

¼ 2Tzzð Þ Txxð Þ Tyyð Þ

=

Symmetric, but reducible, second-rank tensor operators

( ^ ð2Þ

^ ð2Þ and

P

^ ð2Þ

Tij

¼ Tji

i Tii

¼6 0) with six independent components are widespread: the polariz-

ability and the hyperfine coupling tensors are well-known representatives.

Compound Tensor Operators

, each component of

^ ðkÞ

is to be multi-

In a tensor product

^

ðkÞ

^ð jÞ

g

fP

Q

P

2

j

1 -dimen-

plied with each component of

^ð jÞ. The resulting 2 k

þ

1

Þ

ð

þ

Q

ð

Þ

sional tensor is in general not irreducible. Reduction yields irreducible tensor operators with ranks ranging from k þ j, jk þ j 1j; ; jk jj.

As an example, consider the product of two arbitrary first-rank tensor

operators ^ ð1Þ and ^ð1Þ. It is nine-dimensional and can be reduced to a sum

P Q

of compound irreducible tensor operators of ranks 2, 1, and 0, respectively. Operators of this type play a role in spin–spin coupling Hamiltonians. In terms

of spherical and Cartesian components of ^ and ^, the resulting irreducible

P Q tensors are given in Tables 8 and 9, respectively.70

We are mainly interested in compound tensor operators of rank zero (i.e., scalar operators such as the Hamiltonian). To form a scalar from two ten-

sor operators ^ ðkÞ and ^ð jÞ, their ranks k and j have to be equal. Further, the

P Q

þq component of ^ ðkÞ has to be combined with the q component of ^ðkÞ and

P Q

Table 8 Irreducible Spherical Compound Tensor Operators Resulting from a Product

of Two First-Rank Tensor Operators ^ ð1Þ and ^ð1Þ

P Q

Compound Tensor

m

Spherical Component

^ ð1Þ

^ð1Þ

ð2Þ

þ

2

^ ð1Þ ^ð1Þ

fP

Q

gm

Pþ1Qþ1

þ P0

Q0

þ

þ1Q 1Þ

ð

^

1Qþ1

þ1

ðPþ1Q0

þ P0

Qþ1

Þ=

2

ð1Þ

^ð1Þ

p

1

P^

ð1ÞQ^

ð1Þ

P^

ð1ÞQ^

ð1Þ

=p2

0

^

ð1Þ

^

ð1Þ

2

^ ð1Þ ^ð1Þ

^

=

6

P

P

2

P^ ð1ÞQ^ð1Þ

þ

Þ

ð

1

0

0

1

^ ð1Þ

^ð1Þ

ð1Þ

1

1

1

P 1Qþ1

Þ=

ðPþ1Q 1

fP

Q

gm

þ

ðPþ1Q0

P0

Qþ1

Þ

ðP0

Q 1

P 1Q0

Þ=

P^ ð1Þ

Q^ð1Þ

mð Þ

0

^

ð1Þ

^

ð1Þ

^

ð1Þ

^

ð1Þ

p2

0

P^

ð ÞQ^

ð Þ

P^

ð ÞQ^

ð Þ

P^ ð ÞQ^ð Þ =p3

f

g

1

ð

^

ð1Þ

^

ð1Þ

^

ð1Þ ð1Þ

p2

1Þ

1

þ1

0

0

þ

þ1

0

1

1

1

1

1

1

146 Spin–Orbit Coupling in Molecules

Table 9 Irreducible Cartesian Compound Tensor Operators Resulting from a Product

of Two First-Rank Tensor Operators ^ ð1Þ and ^ð1Þ

P Q

Compound Tensor

m

Cartesian Component

^ ð1Þ

^

ð1Þ ð2Þ

þ

2

^ ð1Þ

^ð1Þ

^ ð1Þ ^ð1Þ

þ

i ^ ð1Þ

^ð1Þ

þ

i ^ ð1Þ ^ð1Þ

=2

fP

Q

gm

ðPx

Qx

Py

Qy

Px

Qy

Py

Qx

Þ

þ

1

^ ð1Þ ^ð1Þ

^

ð1Þ ^

ð1Þ

þ

i ^ ð1Þ ^ð1Þ

i ^

ð1Þ ^

ð1Þ

=2

1

ðPx

Qz

þ Pz

Qx

Py Qz

Þ

þ Pz Qy

Þ

ðP^ xð1zÞQ^zð1zÞ

P^ zð1ÞQ^xð1Þ

iP^ yð1ÞQ^zð1Þ

iP^ zð1ÞQ^yð1Þ

=2

0

^ 1

^ 1

^

1

^

1

^ 1

^ 1

p

2Pð

ÞQð Þ

Pxð

ÞQxð Þ

Pyð ÞQyð Þ = 6

Þ

ð

þ

2

^ ð1Þ

^ð1Þ

^ ð1Þ ^ð1Þ

i ^ ð1Þ

^ð1Þ

i ^ ð1Þ ^ð1Þ

=2

^ ð1Þ

ð1Þ ð1Þ

ðPx

Qx

Py

Qy

Px

Qy

Py

Qx

Þ

^

þ

1

^ ð1Þ ^ð1Þ

^

ð1Þ ^

ð1Þ

þ

i ^ ð1Þ ^ð1Þ

i ^

ð1Þ ^

ð1Þ

=2

fP

Q

gm

1

ðPx

Qz

Pz

Qx

Þ

Py Qz

Pz Qy

Þ

ðP^ zð1ÞQ^xð1Þ

P^ xð1ÞQ^zð1Þ

iP^ zð1ÞQ^yð1Þ

iP^ yð1ÞQ^zð1Þ

=2

0

^ 1

^ 1

Þ

^ 1

^ 1

p

i Pxð

ÞQyð

Pyð ÞQxð Þ = 2

þ

Þ

fPð

Þ

Qð

Þgm

ð

þ Pzð

ÞQzð

ðPxð ÞQxð Þ

þ Pyð ÞQyð Þ

ÞÞ=

^ 1

^

1

ð0Þ

0

^

1 ^

1

^

1 ^

1

^ 1 ^ 1

p3

vice versa. Two notations are found in the literature for these tensor products differing only in normalization constants and phase factors. The properly normalized product of two tensor operators forming a scalar operator reads71

^ k	^ k	ð0Þ		X	k	q ^ k ^ k
fPð	Þ Qð	Þg0	¼ ð2k þ 1Þ	q¼ k	ð 1Þ	Pð qÞ Qqð Þ	½155&

Instead, we utilize the simplified expression that differs from Eq. [155] by a
factor of p2k þ 1:
	^ ðkÞ	^ðkÞ			þk	1		k q ^ ðkÞ ^ðkÞ		156
				Þ ¼	ð		Þ		½		&
	ðP	Q						P q Qq
					q¼ 1
					X
Tensor Properties of Magnetic Interaction Terms
The operators	^	and		^	are compound tensor operators of rank
	HSO		HSS

zero (scalars) composed of vector (first-rank tensor) operators and matrix (sec- ond-rank tensor) operators. We will make use of this tensorial structure when it comes to selection rules for the magnetic interaction Hamiltonians and

symmetry	relations between their	matrix elements. Similar considerations
apply to	the molecular rotation	^	and hyperfine splitting interaction
		Hrot

Hamiltonians ^ hfs.

H

Spin–Orbit Coupling For the derivation of selection rules, it is sufficient to employ a simplified Hamiltonian. To this end, we rewrite each term in the microscopic spin–orbit Hamiltonians in form of a scalar product between an

~

appropriately chosen spatial angular momentum ^ and a spin angular

L

~

momentum S^

^			~	~
		ASO r	^	^	½157&
HSO	¼		L	S
		ð Þ

Symmetry 147

The operator [157] is a phenomenological spin–orbit operator. In addition to being useful for symmetry considerations, Eq. [157] can be utilized for setting up a connection between theoretically and experimentally determined finestructure splittings via the so-called spin–orbit parameter ASO (see the later section on first-order spin–orbit splitting). In terms of its tensor components, the phenomenological spin–orbit Hamiltonian reads

^	~	~
	^	^						½158&
HSO ¼ ASOðrÞL		S
	^	^	^	^	^	^	Þ	½159&
	¼ ASOðrÞðL0S0		Lþ1S 1		L 1Sþ1
	^	^	^	^	^ ^			½160&
	¼ ASOðrÞðLxSx		þ LySy þ LzSzÞ

The phenomenological spin–orbit Hamiltonian ought not to be used for computing spin–orbit matrix elements, though. An example for a failure of such a procedure will be discussed in detail in the later subsection on a word of caution.

Spin–Spin Coupling Although we focus on spin–orbit coupling (SOC), we need to consider the tensorial structure of electronic spin–spin coupling: Second-order SOC mimics perfectly first-order spin–spin coupling and vice versa, so that they cannot be told apart (see the later section on second-order

spin–orbit splitting).

The dipolar spin–spin coupling operators are scalar operators of the form

^ ð1Þ

^

ð2Þ

^ð1Þ. The tensorial structure of ^

becomes apparent if we write

P

T

Q

HSS

the Breit–Pauli spin–spin coupling operator as

2

h

2

X

8p

~~

~ ~

~

ð2Þ~

H^ SSBP

e

d ~^rij

~~^si^sj

^si^sj

^sið3^rij

^rijÞ

^sjÞ

161

¼

2me2c2

(

3

þ ^rij3

^rij5

)

½

&

i¼6 j

ð

Þ

e2h2

X

8p

2

~

~~

~

ð

Þ~

¼ 2me2c2

i¼6 j

d ^rij

^si^sj

^siD

^sj

½

162

&

3

ð

Þ

þ

ij

)

The first term is a tensor of rank zero involving only spin variables. It does not contribute to the multiplet splitting of an electronic state but yields only a (small) overall shift of the energy and is, henceforth, neglected. The operator

^ ð2Þ

Dij is a traceless (irreducible) second-rank tensor operator, the form of which in Cartesian components is

D^ ð2Þ

1

0

^rij2 3x^ij2

3x^ij^yij

3x^ij^zij

1

3x^ij^yij

^r2

3^y2

3^yij^zij

163

ij

¼

^r5

B

ij

C

½ &

@

A

ij B

3x^ij^zij

3^yij^zij

^rij2 3^zij2 C

148 Spin–Orbit Coupling in Molecules

For the definition of a phenomenological electronic spin–spin operator, one makes use of this tensorial structure

^	^1 ^ ð2Þ ^1 ð0Þ	½	164	&
HSS ¼ nSð ÞDSS Sð Þo

An overall coupling to a scalar poses again certain conditions on the combina-

tion of the tensor components. All products ^ ð1Þ ^ ð2Þ ^ð1Þ, which fulfill the con-

Pp Tt Qq

dition that p þ t þ q ¼ 0, may contribute to the compound scalar operator. Actual prefactors of the terms depend on the order in which the three tensor operators are coupled. In general there is no unique way to combine three angular momenta j1, j2, and j3 to a total angular momentum J. The 6j symbols allow a conversion between two possible coupling schemes involving direct products, for example, between ffj1 j2g j3g ¼ J and fj1 fj2 j3gg ¼ J.72–74 In the electronic spin–spin Hamiltonian, the two spin operators are coupled first to form a second-rank tensor operator, which is then combined

^ ð2Þ
with Dij	to form a scalar, that is,
	~	~	ð2Þ	^ ð2Þ	g	ð0Þ	½165&
	ff^si ^sjg			Dij

Matrix Elements

The Wigner–Eckart Theorem

The main reason for working with irreducible tensor operators stems from an important theorem, known as the Wigner–Eckart Theorem (WET)75,76 for matrix elements of tensor operators:

ha0j0m0jT^ qðkÞjajmi ¼ ð 1Þj 0 m0	jm0	0 q	m	ha0j0kT^ ðkÞkaji	½166&
		k	j	| {z }:
		:
| {z }
geometrical part				physical part
3j	symbol		reduced matrix element

Here, j and j0 represent angular momentum quantum numbers of the initial

and final wave functions related to the tensor ^ ðkÞ, and m and m0 denote

T

the corresponding magnetic quantum numbers. Note that a and a0 do not mean spin states in this context but stand for all other quantum numbers.

From the WET, Eq. [166], it is obvious that the reduced matrix element (RME) depends on the specific wave functions and the operator, whereas it is independent of magnetic quantum numbers m. The 3j symbol depends only on rotational symmetry properties. It is related to the corresponding vector

Symmetry

149

addition coefficient or Clebsch–Gordan (CG) coefficient by77

j0

k

j

1Þj 0 k m

j0km0q

j0kjm

167

m0

q

m

¼

ð 2j þ 1

h

i

½

&

j

p

The general selection rules emerging from the properties of the 3j symbols are

m ¼ m0 þ q 4ðjkj0Þ

½168&

where the triangle condition 4ðjkj0Þ is fulfilled if j þ j0 k jj j0j. Specifically, for tensor operators of rank 0, 1, or 2, one obtains:

^ ð0Þ

j

0

jajmi ¼ 0

unless

( m¼

0

½169&

ha0j0m0jT0

¼

<

j

0;

1

a0j0m0

T^ qð1Þ

ajm

0

unless

m¼

0; 1

170

h

j

i ¼

8

½

&

j

>

¼

:

>

j jþ j00;

11;

2

a0j0m0

T^ qð2Þ

ajm

0

unless

<

m¼

0; 1;

2

171

h

j

j

i ¼

>

¼

½

&

:

j þ j0

2

>

The condition j þ j0 1 for a matrix element of a first rank tensor operator implies, e.g., that there is no first-order SOC of singlet wave functions. Two doublet spin wave functions may interact via SOC, but the selection rule

j þ j0

2 for T^ qð2Þ (Eq. [171]) tells us that electronic spin–spin interaction

does not contribute to their fine-structure splitting in first order.

Besides providing us with selection rules, the WET can be employed to

considerably

reduce the

computational

effort: if a

single

matrix

element

ha

0j0m0

j

T^ ðkÞ

ja

jm

is known, all possible

ð

2j0

þ

1

Þð

2k

þ

1

Þð

2j

þ

1

Þ

matrix ele-

q

i k

ments ha0j0m000jT^ qð

Þjajm00i can be calculated with the aid of 3j symbols:

0j

0m000

^ ðkÞ jm00

1 m000 m0

m0 000

q0

m00

0j0m0

^ ðkÞ

jm

172

j

k

j

ha

jTq0

ja

i¼ð

Þ

ha

jTq

ja

i ½

&

j

k

j

m0

0

q

m

Equation [172] or related expressions (Table 10) are applied extensively when evaluating the spin part of spin–orbit matrix elements, for configuration interaction (CI) wave functions. The latter are usually provided for a single MS component only.

150 Spin–Orbit Coupling in Molecules

Table 10 Spin-Coupling Coefficients

S

MS0

MS

ð 1ÞS0 MS0 MS00

q0

MS

S00

q

S

S

1

S

S

1

S

1

S0 ¼ S > 0

M

M

M S 1

1=2

S

1

S0

S

1

M 1

M

ððS M þ 1ÞðS MÞ=2Þ

¼

þ

M

M

ðð

S

M

1 S

þ

M

1 =2

1=2

þ Þð

þ Þ Þ

ðð2S þ 1ÞðS þ 1Þ=2Þ 1=2

M 1

M

ððS M þ 1ÞðS M þ 2Þ=4Þ1=2

ðð2S þ 1ÞðS þ 1Þ=2Þ 1=2

Numerical values for the 3j symbols can be looked up in tables.72,77

Alternatively, they can be computed using analytic formulas revised by Roothaan.78,79 Here, only some of their symmetry properties shall be men-

tioned.

j1	j2	j3		j3	j1	j2
even permutations: m1	m2	m3		¼ m3	m1	m2					½173&
				¼ ð 1Þj1þj2þj3		j2		j1	j3
odd permutations:						m2		m1	m3		½174&
				m	m	m
interchange of rows:				¼ j11	j22	j33					½175&
				¼ ð 1Þj1þj2þj3		j1			j2	j3
m changed to m:						m1		m2		m3
											½176&

For the special case of a first-rank tensor operator and m0 ¼ j0 and m ¼ j, analytical formulas for the symmetry related factors in Eq. [172] have been worked out by McWeeny70 and by Cooper and Musher.80 Note, however, that the formulas in both publications contain typos concerning a sign or a square root.

The WET can in principle be applied to both the spatial and spin parts of a spin–orbit coupling matrix element. In molecular applications, however, the question arises how to use the WET, since L is not a good quantum number.c There is a way out: We can work with Cartesian spatial functions and spherical spin functions and apply relations [36] and [37] for transforming back and forth!

cIn quantum theory and spectroscopy, a ‘‘good’’ quantum number is one that is independent of the theoretical model.

Symmetry 151

Tips and Tricks

So far we know the selection rules for spin–orbit coupling. Further, given a reduced matrix element (RME), we are able to calculate the matrix elements (MEs) of all multiplet components by means of the WET. What remains to be done is thus to compute RMEs. Technical procedures how this can be achieved for CI wave functions are presented in the later section on Computational Aspects. Regarding symmetry, often a complication arises in this step: CI wave functions are usually determined only for a single spin component, mostly MS ¼ S. The MS quantum numbers determine the component of the spin tensor operator for which the spin matrix element hS0jS^qjSi does not vanish. The component q does not always match, however, the selection rules dictated by the spatial part of the ME.

Given a molecule that possesses C2v symmetry, let us try to figure out

3	^	3	B1i from wave functions with MS ¼ 1. The cou-
how to calculate h	A2jHSOj

pling of an A2 and a B1 state requires a spatial angular momentum operator of

						~
						^
B2 symmetry. From Table 11, we read that this is just the x component of L. A
	3			^	^ 3	1i yields zero because the
direct computation of h		A2; MS ¼ 1jLxSxj B1; MS ¼
integral of	^		M	1 functions is zero. Instead, one calculates
3	Sx with two		3 S ¼		^
	^	^				stands symbolically for
ME ¼ h A2; MS ¼ 1jLxS0j B1; MS ¼ 1i, where Lx

the spatial part of the microscopic spin–orbit Hamiltonian with x symmetry

^

correspondingly for the zero-component of the spin tensor. This is

and S0

the only nonzero matrix element for the given wave functions.

The RME (only spin) is then given by

3

^

^

3

RME

¼

h

A2; MS ¼ 1jLxS0j

B1; MS ¼ 1i

½

177

&

1

1

0

1

1

1

Table 11 Irreps of Singlet (S) and Triplet (T) Spin

~

Functions, the Angular Momentum Operators ð ^ and

L

~

S^), an Irreducible Second-Rank Tensor Operator ^ ,

D

and the Position Operators ^ ^ ^ in C Symmetry

X; Y; Z 2v

Function/Operator						Irreducible Representation
SingletMS¼0ðS0Þ						A1
TripletMS¼0ðT0Þ						A2
TripletMS¼1ðT 1Þ						fB1; B2g
^			^			B2
Lx;			Sx; Triplet Tx
^			^			B1
Ly;			Sy; Triplet Ty
^			^		Triplet Tz	A2
Lz;			Sz;
^		2		2	2	A1
D
^ 2z			x		y	A1
D	2	y		2
^ x						A2
Dxy
^						B2
Dxz
^						B1
Dyz
^						B1
X
^						B2
Y
^						A1
Z

152 Spin–Orbit Coupling in Molecules

with
	1	1		1
	11 0	1		¼ p6

Since the spatial wave functions are coupled by ^ , a spin operator of x sym-

Lx

metry has to be used also. The operator S^x contributes to both S^ 1 and S^þ1.

Thus, nonzero matrix elements are expected for hMS ¼ 1jS^xjMS ¼ 0i, hMS ¼ 0jS^xjMS ¼ 1i, hMS ¼ 0jS^xjMS ¼ 1i, and hMS ¼ 1jS^xjMS ¼ 0i. The 3j symbols for the coupling of these spin functions by S^ 1 and S^þ1 are easily

evaluated as

					1	1		1	¼		1		1		1
					1 1 0						0 1 1
				¼	1	1		1	¼		1		1			1		1	½178&
					0		1	1			1			1		0
																		¼ p6
Applying			relations [37]			and [172], the							following					interaction	matrix for
3	^ ^ 3
h	A2jLxSxj B1i is obtained:
				MS ¼			1						0					1
			S ¼	0		þRME/p2				þ			0					þRME/p2
		M												p
				1			0					RME/				2		0
			1				0			þRME/p2								0

Note, that it is not possible to compute the nonvanishing MEs of a triplet– triplet coupling by using MS ¼ 0 wave functions, because the spin part of

; M

0

~

~

0

; M

0

is zero and cannot serve for determining the

h

S ¼

j

L^

^

S ¼

i

Sj

RME.

A Word of Caution

Phenomenological operators should be utilized with great care. Do not oversimplify!

As an example, let us look at the spin–orbit coupling between a 3 and a 1 þ state of a linear molecule. This example is of some importance for the understanding of spin-forbidden transitions in the O2 molecule which exhibits a 3 g ground state and a low-lying excited 1 þg state. The coupling is symmetry allowed, that is, the matrix element h3 0 jHSOzj1 þ0 i is different from zero.

~ ~

To see this, we may employ the phenomenological operator A ^ S^

SOL

(Eq. [157]) in the symmetry analysis (the spin–orbit parameter ASO was introduced in the earlier section on tensor properties of magnetic interaction terms):

Symmetry 153

1. The ^ transforms like in the symmetry group of a linear molecule and

Lz ðgÞ

thus allows an interaction between and þ states.

2.The MS ¼ 0 component of a triplet has symmetry, S^z transforms likeðgÞ, and singlets are totally symmetric, resulting again in a symmetryallowed interaction.

However, if h3 0 jHSOzj1 þ0 i is evaluated using the phenomenological opera-

tor ([Eq. 157]) explicitly, we obtain A h j ^ j þihT jS^ jS i ¼ 0:

SO Lz 0 z 0

1.Lz operating on a state gives 0.

2.The same holds true for the action of S0 on a state with MS ¼ 0.

The resolution of this discrepancy is closely related to another question:

~	~
^	^

How is an operator such as S, when combined with L, capable of coupling

electronic states of different multiplicities while, according to Eqs. [149] and

[150],	~
	S^as a first rank tensor operator is only able to change the MS quantum
number of a state, but not its S value.
To understand these seemingly opposite facts, we have to leave the glo-
~	~
^	^
bal L S expression and rather write the spin–orbit Hamiltonian as a sum of
one-particle operators
	~	~	~	~	~	~
	^	^	^		^		½179&
	L S ¼ ‘ð1Þ^sð1Þ þ ‘ð2Þ^sð2Þ

As we shall see, it is the MS ¼ 0 level of the triplet state that couples to the singlet state in this case, and the coupling is brought about by the z component of the spin–orbit coupling operator. The wave functions of the 3 g and 1 þg states of O2 can be written as

h1 þj ¼				1	h½pxð1Þpxð2Þ þ pyð1Þpyð2ÞÞðað1Þbð2Þ bð1Það2Þ&j																½180&
				2
	3			1		1	2		1		2		1		2		1		2			181
j			i ¼	2 j½pxð				Þ pyð		Þpxð		ÞÞðað		Þbð		Þ þ bð		Það		Þ&i	½		&
							Þpyð

or in a short-hand notation

1	þ		1						182
h		j ¼						½		&
			2 hpxpx pxpx þ pypy pypyj
			1	/
j3 i ¼			2	/pxpy þ pxpy pypx pypxi				½183&

where unbarred and barred entities denote a and b electrons, respectively, and the particle label has been suppressed. The (one-electron part of the) spin– orbit matrix element h3 0 jHSOzj1 þ0 i is given by

1

4 hpxpx pxpx þ pypy pypyj½‘0ð1Þs0ð1Þ þ ‘0ð2Þs0ð2Þ&

jpxpy þ pxpy pypx pypxi				½184&