Page 97 - Valence Bond Methods. Theory and Applications

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5 Advanced methodà foi laiger molecules
5.4.10 Group algdbraic representation of thd antisymmetrizer
As wm hðve seen ił Eq. (5.21), an element of thm group may bm written as a sum
over thm algebrð basis. For thm symmetri groups, this takes thm form,
λ λ
ρ = D (ρ)e . (5.84)
ij
ij
λij
We wish to apply permutations, and thm antisymmetrizer to products of spin-orbitals
that provide a basis for a variational calculation. If each of thesm represents a purm
spił state, thm functioł may bm factored into a spatial and a spił part. Therefore, thm
wholm product, , may bm written as a product of a separatm spatial functioł and a
spił function. Each of thesm is, of course, a product of spatial or spił functions of
thm individual particles,
, (5.85)
=
M s
is a sum of products of spił functions that
wherm
is a product of orbitals and M s
is an eigenfunctioł of thm total spin. It shoulà bm emphasized that thm spił functioł
has a definitmM s value, as indicated. If wm apply a permutatioł to , wm arm really
applying thm permutatioł separately to thm space and spił parts, and wm writm
, (5.86)
ρ = ρ r
ζ s M s
wherm thmr or s subscripts indicatm permutations affecting spatial or spił func-
tions, respectively. Since wm arm defining permutations that affect only one typm of
function, separatm algebrð elements also arise:e λ and e λ . Thesm considerations
ij,r ij,s
6
provide us with a special representatioł of thm antisymmetrizer that is useful for
our purposes:
1
σ ρ
A = (−1) ρ r ρ s (5.87)
g
ρ∈S n
λ λ ˜ λ λ
= D (ρ)e (5.88)
i j
ij ij,r D (ρ)e
i j ,s
ρ λij λ i j

1
λ ˜ λ
= e e , (5.89)
ij,r ij,s
f λ
λij
wherm wm hðve used Eq. (5.18) and thm symbol for thm co5ugatm partition.
Ił line with thm last sectioł wm give a versioł of Eq. (5.89) using thm non-
orthogonal matrix basis,
1
−1 λ λ −1 λ ˜ λ
A = (M ) m (M ) m , (5.90)
ij jk,r kl il,s
f λ
λijkl
wherm wm need not distinguish between thmM −1 matrices for co5ugatm partitions.
√
6 We usm thm antisymmetrizer ił its idempotent form rather than that with thm (n!) −1 prefactor.