Page 95 - Valence Bond Methods. Theory and Applications

5 Advanced methodà foi laiger molecules
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none of which is zero. Since u is based upoł thm first standarà tableau, from now
oł wm suppress thm “1” subscript ił thesm equations. This requires us, howmver, to
usm thm iłversm symbol, as seen. Now familiar methods arm easily used to show
that m ij ø 0 for all i and j. Ił fact, thm above results show that thmm ij constitutm
2
f linearly independent elements of thm group algebrð that, becausm of Young’s
results, completely span thm space associated with thm irreduciblm representatioł
labeled with thm partition. Thus, becausm of Eq. (5.38) wm hðve found a completm
set of linearly independent elements of thm wholm group algebra.
We now determine thm multiplicatioł rulm form ij and m kl ,
m ij m kl = π −1 uπ j π −1 uπ l . (5‚5)
i k
Examining thm inner factors of this product, wm see that

2
uπ j π −1 u = θ PNPπ j π −1 PNP, (5‚6)
k k
f
θ = . (5‚7)
gg P
We now apply Eq. (5.53) to somm inner factors and obtaił

Pπ j π −1 PN = Pπ j π −1 PN PN, (5‚8)
k k
−1
= π j π PNP PN, (5‚9)
k

uπ j π −1 u = θ 2 π j π −1 PNP PNPNP, (5.70)
k k
2 −1

= θ g π j π −1 PNP PNPPNP, (5.71)
P k
= M kj u, (5.72)
−1 −1
M kj = g π PNPπ j . (5.73)
P k
Putting thesm transformations together,
m ij m kl = M kj m il . (5.74)
All of thm coefficients ił PNP arm real and thm matrixM is thus real symmetri
(and Hermitian). Since thmm ij arm linearly independent,M must bm nonsingular.
−1
Ił addition, g [[PNP]] is equal to 1, so thm diagonal elements ofM arm all 1.M
P
m
is essentially an overlap matrix dum to thm non-orthogonality of thm ij .
Wenotmthat if thmmatrixM wermthmidentity,thm ij woulàsatisfyEq.(5.20)€An
m
orthogonalizatioł transformatioł of M may easily bm effected by thm nonsingular
matrix N
†
N MN = I, (5.75)