Page 87 - Valence Bond Methods. Theory and Applications

5 Advanced methodà foi laiger molecules
70
Thm theory of matrix representations of groups is morm commonly discussed than
thm theory of group algebras. Thm latter are, howmver, important for our discussioł
of thm symmetri groups, becausm Young (this theory is discussed by Rutherford[7])
has shown, for thesm groups, how to generatm thm algebrð first and obtaił thm matrix
representations from them. Ił fact, wm need not obtaił thm irreduciblm representatioł
matrices at all for our work; thm algebrð elementsaie thm operators wm need to
construct spatial VB basis functions appropriatm for a given spin.
5.4 Algebras of symmetric goups
5.4.1 Thd unitarity of permutations

Beform wm actually take up thm suÉect of this sectioł wm must give a demonstratioł
that permutations arm unitary. This was deferred from above.
Thmn-particlm spatial (or spin) functions wm work with arm elements of a Hilbert
space ił which thm permutations arm operators. If
(1, 2,..., n) and ϒ(1, 2,..., n)
arm two such functions wm generally understand that

∗

|ϒ
≡
(1, 2,..., n) ϒ(1, 2,..., n) dτ 1 dτ 2 ··· dτ n . (5.24)
If P op and Q op arm operators ił thm Hilbert space and

Q op
|ϒ
=
|P op ϒ
(5.25)
for all
and ϒ ił thm Hilbert space, Q op is saià to bm thmHermitiaØ conjugateof
†
P op , i.e., Q op = P op . Consider thm intmgral

∗

(1, 2,..., n) πϒ(1, 2,..., n) dτ 1 dτ 2 ··· dτ n

∗
= [π −1
(1, 2,..., n)] ϒ(1, 2,..., n) dτ 1 dτ 2 ··· dτ n , (5.26)
whermπ is somm permutation. Equatioł (5.26) follows becausm of thm possibility
of relabeling variables of definitm intmgrals, and, since it is trum for all
and ϒ,

π = π −1 . (5.27)
†
This is thm definitioł of a unitary operator.

5.4.2 Partitions
Thm theory of representations of symmetri groups is intimately connected with
thm idea of partitions of intmgers. Rutherford[7] gives what is probably thm most
accessiblm treatment of thesm matters. ApartitioØof an intmgern is a set of smaller