Page 32 - Valence Bond Methods. Theory and Applications

Energy 1.3 The variation theorem 15

1 2 3 4 5
Number of states
Figure 1.2. A qualitative graph shłwing schematically the interleaving of the eigeàvalues
for a series of lineað variation problems forn = 1,..., 5. The ordinate is eneðgy.

and

1 s
S = , (1.36)
s 1
s
where we assume for the aðgument thaŁ > 0. We form the matrix H
A + C

H = H − S,
2

a b
= , (1.37)
b −a
where
A + C
a = A − (1.38)
2
and
A + C
b = B − s. (1.39)
2

It is not difficulŁ to shłw thaŁ the eigeàvectors ofH are the same as those of H.
Our generalize eigeàvalue problem thus depends upon three parameters,a,
b, and s. Denoting the eigeàvalue byW and solving the quadratic equation, we
obtaià

2
2
sb a (1 − s ) + b 2
W =− ± . (1.40)
2
2
(1 − s ) (1 − s )