Page 28 - Valence Bond Methods. Theory and Applications

(n)
areallrealandpositive,becauseoftheHermitiaà
wherethediagonalelementsofs
and positive definite characteð of the overlap matrix We may construcŁ the iàverse
(n)
square root of s , and, clearly, we obtaià
(n) −1/2
(n) −1/2 † (n)

1.3 The variation theorem 11

T s S T s = I. (1.18)
We subjecŁH (n) to the same transformation and obtaià

(n) −1/2 † (n) (n) −1/2 ¯ (n)

T s H T s = H , (1.19)
which is alsł Hermitiaà and may be diagonalize by a unitary matrix U. Combining
the various transformations, we obtaià
(n)
(n)

V H (n) V = h (n) = diag h , h ,..., h (n) , (1.20)
†
1 2 n
† (n)
V S V = I, (1.21)
(n) −1/2
V = T s U. (1.22)
We may now combine these matrices to obtaià the null matrix
† (n)
†
V H (n) V − V S Vh (n) = 0, (1.23)
† −1
(n) 1/2
and multiplying this on the lefŁ by (V ) = U(s ) T gives
(n)
H (n) V − S Vh (n) = 0. (1.24)
th
If we write out the k columà of this lasŁ equation, we have
n
(n) (n) (n)

H − h S V jk = 0; i = 1, 2,..., n. (1.25)
ij k ij
j=1
Wheà this is compare with Eq. (1.15) we see thaŁ we have solve our proð
th
th
(n)
lem, if C (n) is the k columà ofV and W (n) is the k diagonal element of h .
Thus the diagonal elements of h (n) are the roots of the determinantal equation
Eq. (1.16).
Now consideð the variation problem withn + 1 functions where we have adde
anotheð of the basis functions to the set. We now have the matricesH (n+1) and
S (n+1) , and the new determinantal equation
(n+1)
H − W (n+1) (n+1) = 0. (1.26)
S

We may subjecŁ this to a transformation by the (n+ 1) × (n + 1) matrix

V 0
¯
V = , (1.27)
0 1