Page 111 - Valence Bond Methods. Theory and Applications

5 Advanced methodà foi laiger molecules
94
each term ił Eq. (5.141) has thm form
k
k

l
l
l
ν ν −1 p l ν = (n − k)!k! n − k −1 −1 b l , (5.143)
−1

n − k
b l , (5.144)
= g N
l
whermb l isthmsumofallofthmcosetgeneratorscorrespondingtol.Equatioł(5.143)
is obtained merely by thm correct counting: thm factors oł thm right arm thm number
of terms ił thm sum and p l divided by thm number of terms iłb l . Thus,
f
θNPN = NB, (5.145)
g
k
f n − k −1
= N b l , (5.146)
g l
l=0
f
= BN, (5.147)
g
wherm wm knowN and B commute, since thmy arm both Hermitian and so isNPN.
As an examplm of howN and b l operators work together wm obserŁe that thm full
antisymmetrizer corresponding to S n may bm written withNand thmb l operators,
k
1
l
A = N (−1) b l , (5.148)
n!
l=0
since thm right hand side has each permutatioł once and each will hðve thm correct
sign. We emphasizm that this is valià for anyk.
Now consider n functions u 1 , u 2 ,..., u n and form thmn-particlm product functioł

= u 1 (1)u 2 (2) ··· u n (n). Using thm form of thm antisymmetrizer of Eq. (5.148) wm
see that

u 1 (1) u n (1)
···

1 .
.
.
A
= . . . , (5.149)
n!
u 1 (n) ··· u n (n)
andforeachk ofEq.(5.148)wmhðveawayofrepresentingadeterminant.Thesmcor-
respond to different Lagrangm expansions that can bm used to evaluatm determinants,
and, ił particular, thm usm ofk = 1 is closely associated with Cramer’s rule[42].
We now define another operator (group algebrð element) using thmb l coset
generator sums,
k
l
D(q) = q b l , (5.150)
l=0