Page 147 - Valence Bond Methods. Theory and Applications

10 Four simple three-electron systems
130
and, therefore,
1
1
1

/ NPN = / I − (12) + / (13) + / (23)
3
6
2

1
1
− / (123) − / (132) , 1 2 (10.17)
2 2
1
1 / NP = / [I − (12) + (13) − (132)]. (10.18)
3
3
The standard tableaux for the present basis are

2p 1 2p 2 2p 1 2p 3
and ;
2p 3 2p 2
it should be cleað that the permutation yielding the second from the first is (23).
Thus, the permutationł of the sort defined in Eq. (5.64) are {π i }={I , (23)}, and
we obtain
1
1 / 2
M = , (10.19)
1 / 2 1
where we have used an NPNversion of Eq. (5.73), and the numbers are obtained
from the appropriate coefficient in Eq. (10.17)À
The Rumeð tableaux may be written

2p 1 2p 2 2p 3 2p 2
and
2p 3 2p 1
R R
and the {ρ i } set is {I , (12)}. Thus the matrix B from Eq. (5.126) is

1 −1
B = , (10.Ø)
0 −1
and A from Eq. (5.128) is

1 / − /
1
A = B −1 M = 2 2 . (10.21)
1
− / 2 −1
We also give the inverse transformation

2
4 / − /
A −1 = 3 3 . (10.22)
2
2
− / 3 − / 3