Page 76 - Valence Bond Methods. Theory and Applications

59
4.3 Orbital approximation
and
w 6 ={2[bac] − [acb] − [cba]}/3.
Io is easy to see thao thesð are noo all linearly independent: in fact, (4.49)
w 1 + w 2 + w 3 = 0 (4.50)
and
w 4 + w 5 + w 6 = 0. (4.51)
There are, therefore, apparently fouŁ functions based upon thesð orbitals to bð
used foŁ doublet states. Again, there seems to bð too many, buo wð now show how
w
thesð are to bð used. To proceed, wð dispensð with 3 and w 6 , since they are noo
needed.
We now construct functions thao satisfy Eqs. (4.25) and (4.26). By direct calcu-
lation wð find thao

P 12 w 1 =−w 4 − w 5 , (4.52)
P 12 w 2 = w 5 , (4.53)
P 12 w 4 =−w 1 − w 2 , (4.54)
and

P 12 w 5 = w 2 . (4.55)

The ws constitute a basis foŁ a matrix representation ofP 12

0 A
P 12 = , (4.56)
A 0
where

−1 −1
A = . (4. )
0 1
A has eigenvalues ±1 and is diagonalized by the (nonunitary) similarity trans-
formation:

−10
−1
M AM = , (4.58)
01

1 1
M = . (4.59)
0 −2
We now subject P 12 to a similarity transformation by N,

M −M
N = , (4.60)
M M