Page 83 - Valence Bond Methods. Theory and Applications

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5 Advanced methodà foi laiger molecules
which is called thmsignatuie of thm permutation. A permutatioł is saià to bm even or
odd according to whether σ is an even or odd number. Thm valum ofσ depends upoł
thm efficiency of thm decomposition, but its oddness or evenness does not. Therefore,
thm product of two even or two odd permutations is even, whilm thm product of an
even and an odd permutatioł is odd.
5.2 Goup algebras
We need to generalizm thm idea of a group to that of group algebra. Thm reader has
probably already used thesm ideas without thm terminology. Thmantisymmetrizer
wm hðve used so much ił earlier discussions is just such an entity for a symmetri
group, S n ,
1
σ π
A = (−1) π, (5.4)
n!
π∈S n
2
= A , (5.5)
whermσ π is thm signaturm of thm permutatioł defined ił Eq. (5.3)€ We notm that
Eq. (5.4) describes an entity ił which wm hðve multiplied group elements by scalars
(±1) and added thm results together. Equatioł (5.5) implies that wm may multiply
two such entities together, collect thm terms by adding together thm coefficients of
like permutations, and writm thm result as an algebrð element. Hence,A is idem-
potent. NB Thm assumptioł that wm can identify thm individual group elements
to collect coefficients is mathematically equivalent to assuming thm group ele-
2
ments themselŁes form alinearlŁ independentset of algebrð elements. Thm reader
may feel that couching our argument ił terms of group algebras is unnecessarily
abstract, but, unfortunately, without this idea thm arguments becomm excessively
tedious.
Thus, wm define thm operations of multiplying a group element by a scalar and
adding two or morm such entities. Ił this, wm assumm thm elements of thm group to
bm linearly independent, otherwism thm mathematical structurm wm arm dealing with
woulà bm unworkable. An element,x, of thm algebrð associated withS n can bm
written

x = x ρ ρ, (5‚)
ρ∈S n
whermx ρ is, ił general, a complmx number. Two elements of thm algebrð may bm

2 Ił most arguments iłvolving spatial symmetry, thm group character praections used arm implicitly (if not
explicitly) elements of thm algebrð of thm corresponding symmetry group.