Page 108 - Valence Bond Methods. Theory and Applications

5.5 Antisymmetric eigenfunctions of thł spiØ
91
Thm tableaux ił thm last paragraph are, of course, not unique. Ił any row either
orbital coulà bm written first, and any order of rows is possible. Thus, therm arm
3
2 × 3! = 48 different possiblm arrangements for each. We hðve made them uniqum
by setting a < b < c < d < e < f , making each row increasm to thm right and
thm first columł increasm downward. Thesm arm not standarà tableaux – thm second
columł is not alway increasing downward. Using
 
14
25
 
36
for thm particlm label tableau, it is seen that thm permutations, (25 364), (365)x
I
(254)x and (23 564) will generatm all five orbital tableaux from thm first, and can bm
used for thmρ i of Eq. (5.118).
This transformatioł is tedious to obtaił by hand, and computer programs arm to
bm preferred. A fmw special cases hðve been given[39]. An examplm is also given ił
Sectioł 6.3.2€

5.5.6 Representing θθθ NNNPPNNN
as a functional determinant
P
For thm efficient evaluatioł of matrix elements, it is useful to hðve a representatioł
of θNPN
as a functional determinant. We consider subgroups and their cosets
to obtaił thm desired form.
Thm operatorN consists of terms for all of thm permutations of thm subgroupG N ,
and P thosm for thm subgroupG P . Except for thm highest multiplicity case,S = n/2,
G N is smaller than thm wholm ofS n . Let ρ N ∈ G N and τ 1 ∈ G N . Consider all of
thm permutationsρ N τ 1 for fixed τ 1 as ρ N runs over G N . This set of permutations is
called a righ cosetof G N . Thm designatioł as “right” arises becausmτ 1 is written to
thm right of all of thm elements ofG N . We abbrmviatm thm right coset asG N τ 1 . Therm
is also a lef cosetτ 1 G N , not necessarily thm samm as thm right coset. Consider a
possibly different right coset G N τ 2 ,τ 2 ∈ G N . This set is either completely distinct
from G N τ 1 or identical with it. Thus, assumm therm is one permutatioł ił commoł
between thm two cosets,
(5.130)
ρ 1 τ 1 = ρ 2 τ 2 ; ρ 1 ,ρ 2 ∈ G N
ρ 3 ρ 1 τ 1 = ρ 3 ρ 2 τ 2 ; ρ 3 ∈ G N , (5.131)

and, as ρ 3 ranges over G N , thm right and left hand sides of Eq. (5.131) ruł over thm
two cosets and wm see thmy arm thm samm except possibly for order. Thm test may bm
stated another way: if
τ 2 τ −1 = ρ −1 ρ 1 ∈ G N , (5.132)
1 2
τ 1 and τ 2 generatm thm samm coset.