Gallup G.A. - Valence Bond Methods, Theory and applications (CUP, 2002)

74		5 Advanced methods for larger molecules
We stated above that there is an inequivalent irreducible				representation of			S n
associated with each		partition of	n , and	we	use the symbol	fλ to	represent the
number of standard tableaux corresponding to the partition,						λ. Using induction on
n , Young proved the theorem
			fλ2 = n !,				(5.38)
			λ
which should be compared with Eq. (5.15).
Young also derived a formula for			fλ, but, as will be seen, we need only a small
number	of partitions	for our work with fermions	like electrons. These are either			n − 2k ≥ 0. In fact, the
{n − k,k} or		{2k,1 n −2k} for all	k = 0, 1,		. . . , such that
shapes of the tableaux corresponding to these two partitions are closely related,							{3,2} may
being	transposes	of one another. Letting		n	= 5 and k = 2, the shape of
be symbolized with dots as
			•	•	•
			•	•

If we interchange rows and columns in this shape, we obtain

• •

• •

•

which is seen to be the shape of the partition

{22,1 }. Partition shapes and tableaux

related this way are said to be

conjugates

, and we use the symbol

˜

λ to represent the

partition conjugate to

λ.

It should be reasonably self-evident

that

the

conjugate of

a standard tableau

is a standard tableau of the conjugate shape. Therefore,

fλ = fλ˜ ,

and irreducible

representations corresponding to conjugate partitions are the same size. In fact, the

irreducible representations are closely related. If

D λ(ρ) is

one of the

irreducible

representation matrices for partition

λ, one has

˜

λ(ρ),

(5.39)

D λ(ρ) = (−1) σρ D

where

σρ is the signature of

ρ.

As we noted above, Young derived a general expression for

fλ for any shape.

For the partitions we need there is, however, some simplification of the general

expression, and we have for either

{n − k,k} or

{2k,1 n −2k}

fλ

=

n

− 2k + 1

n + 1

,

(5.40)

n

+ 1

k

p

p !

q

=

.

(5.41)

q!( p

− q)!

5.4

Algebras of symmetric groups

75

5.4.5 The linear independence of

Ni Pi and

Pi Ni

The relationships expressed in Eq. (5.31) can be used to prove the very important

result that the set of algebra

elements,

Ni Pi ,

is linearly independent. First,

from

Eq. (5.30) we have seen that they are not zero, so we suppose there is a relation

i

a i Ni Pi

= 0.

(5.42)

We multiply Eq. (5.42) on the right, starting with the final one

Nf , and, because of

Eq. (5.31), we obtain

a f Nf Pf Nf

= 0.

(5.43)

Therefore, either

a f

or Nf Pf Nf , or both must be 0. We observe, however, that

[[i

i ]]

(5.45)

[N[i Pi Ni ]]= Ni

2Pi

(5.44)

=

g

N

N P

= g N ,

(5.46)

where

g N is the

order

of the subgroup of

N, and this is true

for any

i . Thus,

Nf

Pf Nf is not zero and

a

f in Eq. (5.43) must be. Now that we know

a f −1

a f is zero,

we may multiply Eq. (5.42) on the right by

N1

Nf −1

, and see that

must also be

zero. Proceeding this way until we reach

, we see that all of the

a i

are zero, and

the result is proved.

Permutations are unitary operators as seen in Eq. (5.27). This tells us how to take the Hermitian conjugate of an element of the group algebra,

†

† =

(5.47)

x

π

x π π

,

π −1 ,

(5.48)

=

x

π

π

π −

(5.49)

=

x

1 π.

π

In passing we note that

N and P are Hermitian, since the coefficients are real and

equal for inverse permutations.

N P

=

In general

P N

but is its Hermitian conjugate, since (ρπ

)†

i i is not equal to

i

i

π †ρ†. Therefore, it should be reasonably obvious that the

P N

operators are also

i i

linearly independent. We note that

an

alternative,

but

very

similar,

proof that all

P j ; j =

a i = 0 in Eq.

(5.42) could

be

constructed

by multiplying

on

the left

by

1, 2, . . . , f sequentially.

It is now fairly easy to see that we could form a new set of linearly independent

quantities

x i Ni Pi ;

i = 1, 2, . . . , f,

(5.50)

Ni Pi x i ). As is probably

x i Ni Pi

76	5 Advanced methods for larger molecules

where	x i is any set	of elements of the algebra that do not result in
Corresponding results are true for right multiplication (i.e.,
not surprising there are parallel results for right or left multiplication on
important application of this result (for left multiplication) is an algebra element like
x i = ρPi , where		ρ is any operation of the group, with corresponding expressions
for the other cases.		Pi and	Pj differ only in being based upon a different arrangement
The operators
of the numbers in the standard tableau they are associated with. Therefore, there					Pj with the relation
exists a permutation,		πi j that will interconvert		Pi and

πi j Pj = Pi πi j ,

= 0.

Pi Ni . An

(5.51)

with a similar	expression	for	Ni and		Nj . The theorems of this section		can	thus
be stated in a different way. For example, we see that the quantities,								P1 N1 π1 j =
π1 j Pj Nj ,	satisfy the	definition of Eq. (5.50), and are thus linearly independent.
Three similar results pertain for the other three possible combinations of the ordering
of the products	of	P and	N on either	side of		the equation. Explicitly, for	one	of
these cases, we may write that the relation
			i	P1	N1	π1 i a i = 0		(5.52)
implies that all	a i	= 0, with similar implications for the other cases.

5.4.6 Von Neumann’s theorem

Von Neumann proved a very useful theorem for our work (quoted by Rutherford[7]).

Using our notation it can be written

		Px N = [P[x N]PN]	,	(5.53)
where	x is any element of the algebra and	N and P are based upon the same tableau.
A similar expression holds for		N x P.
	5.4.7 Two Hermitian idempotents of the group algebra
We will choose arbitrarily to work with the first of the standard tableaux				5 of a given
partition. With this we can form the two		Hermitian	algebra elements
		u = θ PNP		(5.54)

5 Any tableau would do, but we only need one. This choice serves.

5.4 Algebras of symmetric groups

77

and

u = θ NPN,

(5.55)

where

θ and

θ

are real. We must work out what to set these values to so that

u

2 = u

and

u

2 = u .

f /g )NP or ( f /g )PN, g

= n

We stated at the end of Section 5.4.3 that (

!, will

serve as an idempotent element of the algebra associated with the partition upon

which they are based, although these are, of course, not Hermitian. This means that

NPNP =

g

NP.

(5.56)

f

Thus, observing that

P2 = g PP, we have

(PNP )2 = PNP2NP,

(5.57)

= g PPNPNP,

(5.58)

=

gg P

PNP,

(5.59)

f

where

g P is the order of the subgroup of the

f

P operator. Thus, we obtain

u

=

PNP

(5.60)

gg P

as an idempotent of the algebra that is Hermitian. A very similar analysis gives

u =

f

NPN,

(5.61)

gg

N

where

g N is the order of the subgroup of the

N operator. Although portions of the

following analysis could be done with the original non-Hermitian Young idempo-

tents, the operators of Eqs. (5.60) and (5.61) are required near the end of the theory

and, indeed, simplify many of the intervening steps.

5.4.8 A matrix basis for group algebras of symmetric groups

In

the

present

section

we will give a construction

of the matrix basis only for

the u

= θ PNP operator. The treatment for the other Hermitian operator above is

identical and may be supplied by the reader.

Consider now the quantities,

m i j = πi 1 uπ 1

j ,

(5.62)

=

(m

j i )†,

(5.63)

−1 uπ

;

(5.64)

=

π

j

π

j =

π

1

j

,

i

78

5

Advanced methods for larger molecules

none of which is zero. Since u is based upon the first standard tableau, from now

on we suppress the “1” subscript in these equations. This requires us, however, to

use the inverse symbol, as seen. Now familiar methods are

easily

used

to show

that m

i j

=0 for all

i and

j . In fact, the above results show that the

m i j constitute

f 2 linearly independent elements of the group algebra that, because of Young’s

results, completely span the space associated with the irreducible representation

labeled with the partition. Thus, because of Eq. (5.38) we have found a complete

set of linearly independent elements of the whole group algebra.

We now determine the multiplication rule for

m

i j and

m

kl ,

m

i j m

kl

= πi−1 u π j πk−1

u πl .

(5.65)

Examining the inner factors of this product, we see that

u π

j

π

−1

u

=

θ

2

PNP

π

j

π −1

PNP

,

(5.66)

k

k

θ =

f

.

(5.67)

gg

P

We now apply Eq. (5.53) to some inner factors and obtain

P

π

π −1

PN

π π −1

,

(5.68)

j

k

= π j

πk−1

PNP PN,

(5.69)

=

P

j

k

PN PN

u π j

π −

1

u

π j

−

1

(5.70)

k

=

θ 2

−

π

−

,

θ

π j

k

PNP PNPNP

= M

2 g

1

π

1

PNP PNPPNP

,

(5.71)

k jPu ,

k

(5.72)

=

−P

1 πk−1

PNPπ j .

(5.73)

M

k j

= g

Putting these transformations together,

m

i j m

kl

= M

k j m

il .

(5.74)

All of the coefficients in

PNP are

real and

the

matrix

M

is thus real

symmetric

(and Hermitian). Since

the

m

i j

are

linearly independent,

M

must be nonsingular.

In addition,

P

PNP

]] is equal to 1, so the diagonal elements of

M

are all 1.

M

g −1

[[

is essentially an overlap matrix due to the non-orthogonality of the

m

i j .

We note that if the matrix

M

were the identity, the

m i j

would satisfy Eq. (5.20). An

orthogonalization transformation of

M

may easily

be effected

by the

nonsingular

matrix

N

N † M N

=

I ,

(5.75)

	5.4	Algebras of symmetric groups				79
as we saw in Section 1.4.2. It should be recalled that					N is not unique; added condi-
tions are required to make it so. We wish				m 11	to be unchanged by the transformation,
and an upper triangular	N will accomplish both of these goals. If we require all of
the diagonal elements of	N	also to	be positive, it becomes uniquely			determined.
Making the transformations we have
					†)i k N l j m kl ,
		e i j	=	( N		(5.76)
			kl
		e i j e kl	= δ j k e il ,			(5.77)
		e 11	= m 11	,		(5.78)
as desired. These	e i j s constitute a real		matrix	basis	for the symmetric	group and,

clearly, generate a real unitary representation through the use of Eq. (5.23).

5.4.9 Sandwich representations

The reader might ask: “Is there a parallel to Eq. (5.23) for the nonorthogonal matrix basis we have just described?” We answer this in the affirmative and show the results.

Clearly, we can define matrices

		T(ρ )i j		= [ρ[m i j ],]						(5.79)
and it is seen that a normal unitary representation may be obtained from
	D (ρ )i j =			( N †)i k T(ρ )kl N l j ,						(5.80)
			kl
where we have used Eq. (5.76). The upshot of these considerations is that the										T(ρ )
matrices satisfy
		T(ρ )M	−1 T(π )		=	T(ρπ ),				(5.81)
and these have been called		sandwich	representations, because				of a fairly	obvious
analogy. In arriving at Eq. (5.81) we have used
			N N † = M		−1 ,					(5.82)
which is a consequence of Eq. (5.75).
We may also derive a result analogous to Eq. (5.21),
			−1 )λ T(ρ )λ ( M				−1 )λ m
ρ	=	( M						λ	,	(5.83)
				ki	kl		l j	i j

λi j kl

where we have added a partition label to each of our matrices and summed over it.

80	5 Advanced methods for larger molecules

5.4.10 Group algebraic representation of the antisymmetrizer

As we have seen in Eq. (5.21), an element of the group may be written as a sum over the algebra basis. For the symmetric groups, this takes the form,

ρ =
	D iλj (ρ)e iλj .	(5.84)

λi j

We wish to apply permutations, and the antisymmetrizer to products of spin-orbitals that provide a basis for a variational calculation. If each of these represents a pure spin state, the function may be factored into a spatial and a spin part. Therefore, the

whole product, , may be written as a product of a separate spatial function and a spin function. Each of these is, of course, a product of spatial or spin functions of

the individual particles,

where

is a product of orbitals and

= M s ,

(5.85)

M s is a sum of products of spin functions that

is an eigenfunction of the total spin. It should be emphasized that the spin function

has a definite

M

s

value, as indicated. If we apply a permutation to

, we are really

applying the permutation separately to the space and spin parts, and we write

ρ = ρr ρs M s ,

(5.86)

where

the

r or

s

subscripts indicate

permutations affecting spatial

or spin

func-

tions, respectively. Since we are defining permutations that affect only one type of

function, separate algebra elements also arise:

e λ

and

e λ

,s

. These considerations

i j ,r

i j

6 that is useful

for

provide us with a special representation of the antisymmetrizer

our purposes:

1

(−1) σρ ρr ρs

(5.87)

A =

g

ρ S n

˜

= D iλj (ρ)e iλj ,r D

(5.88)

iλ j (ρ)e iλ j ,s

ρλi j λ i j

1

˜

(5.89)

=

e iλj ,r

e iλj ,s ,

λi j

f

λ

where we have used Eq. (5.18) and the symbol for the conjugate partition.

In line with the last section we give a version of Eq. (5.89) using the non-

orthogonal matrix basis,

1

( M

−1 )λ m

( M

−1 )λ m

˜

(5.90)

A =

λ

λ

,

il ,s

λi j kl fλ

i j

j k,r

kl

where we need not distinguish between the

M

−1

matrices for conjugate partitions.

6 We use the antisymmetrizer in its idempotent form rather than that with the (

√

)−1

prefactor.

n !

5.5

Antisymmetric eigenfunctions of the

spin

81

5.5 Antisymmetric eigenfunctions of the spin

In this section

we

investigate

the connections

between

the symmetric

groups

and spin eigenfunctions. We have briefly outlined properties of spin operators in

Section 4.1. The reader may wish to review the material there.

One of the important properties of all of the spin operators is that they are

symmetric.

The

total

vector spin

operator is

a sum

of

the

vector

operators

for

individual electrons

n

S i ,

S =

(5.91)

i =1

indicating that the electrons are being treated

equivalently

in

these expressions.

This means

that every

π S n

must commute with

the

total vector

spin

operator.

Since all of the other operators,

S 2, raising, and lowering operators,

are algebraic

functions of the components of

S , they also commute with every permutation. We

use this result heavily below.

5.5.1

Two simple eigenfunctions of the spin

Consider

an

n

electron system in a pure

spin state

S

. The

associated

partition

is

{n /2 + S , n

/2 − S }, and the first standard tableau is

1

· · ·

n /2

−

S

· · ·

n /2

+

S

,

n /2 + S + 1

· · ·

n

where we have written the partition in terms of the

S quantum

number

we have

targeted. We consider also an array of individual spin functions with the same shape

and all

η1/2

in the first row and

η−1/2 in the second

α

· · ·

α

· · ·

α

,

β

· · ·

β

where we have used the common abbreviations

α = η1/2 and

β = η−1/2 . Associat-

ing symbols in corresponding positions of these two graphical shapes generates a

product of

αs and

βs with specific particle labels,

= α(1) · · · α(n /2 + S )β(n

/2 + S

+ 1) · · · β(n

),

(5.92)

S z = M

S ,

(5.93)

M

S = S .

(5.94)

If now we operate upon

with

N (corresponding to

{n

/2 + S , n /2 − S }) we obtain

a function with

n

/2 − S antisymmetric products of the [αβ

− βα] sort,

N = [α(1)β (n /2 + S + 1) − α(n

/2 + S

+ 1)β (1)] · · · α(n

/2 + S

).

(5.95)

82

5

Advanced methods for larger molecules

The spin raising operator (see Eq. (4.3)) may now be applied to this result, and we

obtain

S +N = N S + ,

= 0,

(5.96)

where we have used the commutation of

S + with all permutations. The following

argument indicates why the zero results. The terms of

S + give zero with each

α

encountered

but turn

each

β encountered

into

an

α. Thus

S + is n /2 − S

terms

of products, each one of which has no more than

n

/2 − S

− 1

β functions

in

it.

Considering how these would fit into the tableau shape, we see that there would

have to be, for each term, one column in the tableau that has an

α in

both

rows.

This column, with its corresponding factor from

N, would thus appear as

[I − (i j )]α(i )α( j ),

which is clearly zero. Eq. (5.96) is the consequence.

S 2 because of Eq. (4.5),

Thus,

N is an eigenfunction of

S 2N = S

( S + 1) N ,

(5.97)

and has total spin quantum number

S

(also the

M

S

value for this function). Other

values of

M

S

are available with

S − should they be needed.

We now investigate the behavior of

when we apply our two simple Hermitian

idempotents discussed earlier,

P N P

= θ PNP ,

(5.98)

= g P θ PN ,

(5.99)

N P N

= θ NPN .

(5.100)

Since

S

+ and

S z

both commute with

N

and

P

, both

P N P

and

N P N

are eigen-

functions of the

S 2 operator with total spin

S

and

M

S = S .

λ = {n /2 + S ,

Heretofore in this section we have been working with the partition

n /2 − S

}, but references to it in the equations have been suppressed. We now write

λ

and

λ

. Applying the antisymmetrizer to the function of both space and

P N P

N P N

,

spin that contains

λ

P N P

A PλN P =		1	e iλ˜j ,r e iλj ,s λP N P .
		fλ
	λ i j

If the antisymmetrizer has been conditioned (see Eqs. (5.75)–(5.78)) so that is θ PNPλ , we obtain

e iλj ,s λP N P	= δ1 j δλλ e iλ1 ,s λP N P ,
because of the orthogonality of the	e λ	for different	λs.
	i j

(5.101)

e λ

11 ,s

(5.102)

5.5 Antisymmetric eigenfunctions of the spin 83

We make a small digression and note that the

spin-degeneracy problem

we have

alluded to before is evident in Eq. (5.102). It will be observed that

i

= 1, . . . , fλ

in the

index of

e λ

λ

, and

these

functions

are

linearly

independent

since

the

i 1,s

P N P

e iλj ,s

are. There are, thus,

fλ linearly independent spin eigenfunctions of eigenvalue

S ( S

+ 1). Each of these has

a full

complement

of

M

S

values,

of

course. In

view

of Eq. (5.40) the number of spin functions increases rapidly with the number of

electrons. Ultimately, however, the dynamics of a system governs if many or few

of these are important.

Returning to our antisymmetrized function, we see it is now

1

i

˜

e iλ1,s λP N P

(5.103)

A P N P

=

e

iλ1, r

,

fλ

and we are in a position to examine its properties with regard to the Rayleigh

quotient.

Considering first the denominator, we have

A P N P

|A P N P

= fλ−2

i j

e

˜

˜

iλ1, r

e λj 1, r

e

λ

λ

e

λ

λ

,

(5.104)

× 1

i 1,s

λ˜P N P

j 1,s

P N P

λ

λ

,

(5.105)

=

f

λ−

e

11,

r

P N P

e

11,s

P N P

since

e

˜

˜

=

e

˜

e

˜

(5.106)

iλ1, r e λj 1, r

1λi,r

λj 1, r ,

δ

˜

,

(5.107)

e λ

i j

with a

very similar

expression

for the

spin

=

11, r

the

integral. Since

the

Hamiltonian

of

ESE commutes with all permutations and symmetric group algebra elements, the same reductions apply to the numerator, and we obtain

A P N P	|H |A P N P	= fλ−1 H	e	˜	λP N P
				11λ, r

This result should be carefully compared to that of Eq. (4.37), where there were two functions that have the same integral. Here we have

Our final expression for the Rayleigh quotient is

e	11λ,s λP N P	.	(5.108)

fλ of them.	7

E

=

A P N P

|H

|A P N P

,

(5.109)

A P N P

|A P N P

=

e

11λ

˜

|H

|e

11λ

.

(5.110)

˜

|

7 We may note in passing that the partition for three electrons in a doublet state is

{2,1} and

fλ for this is 2. That

is why we found two functions in our work in Chapter 4.