Page 29 - Valence Bond Methods. Theory and Applications

12
(n+1)
(n+1)
are modifie to
and S
and H
(n)

h
1
1 n+1
(n+1)
H
0


···
¯
¯
2
2 n+1


V H (n+1) V = H ¯ (n+1) = 1 Introduction h 0 (n) ··· H ¯ ¯ (n+1)  (1.28)
†
.
.
.
.


 . . . . . . . . 
 
¯
¯
H (n+1) H (n+1) ··· H n+1
n+11 n+12 n+1 n+1
and
 (n+1) 
1 0 ··· S ¯
1 n+1
 0 1 S ¯ (n+1) 
¯
¯

† (n+1)
V S V = S ¯ (n+1) =  ··· 2 n+1  . (1.29)

. . .
. . . .
 . 
 . . . . 
S ¯ (n+1) S ¯ (n+1) ··· 1
n+11 n+12
Thus Eq. (1.26) becomes
(n) (n+1) (n+1) (n+1) ¯
¯
h 1 − W 0 ··· H 1 n+1 − W S (n+1)
1 n+1
(n) (n+1)
(n+1) ¯ (n+1) ¯ (n+1)
0 h − W ··· H − W S
2 2 n+1 2 n+1

0 = . . . . .
. . . . .
. . .
¯ (n+1) − W (n+1) ¯ (n+1) H ¯ (n+1) − W (n+1) ¯ (n+1) ··· H n+1 − W (n+1)

S
S
H
n+11 n+11 n+12 n+12 n+1 n+1
(1.30)
We modify the determinant ià Eq. (1.30) by using columà operations. Multiply the
th
i columà by
¯
S
H (n+1) − W (n+1) ¯ (n+1)
iŁ+1 iŁ+1
(n) (n+1)
h − W
i
th
th
n
and subtracŁ iŁ from the (+ 1) column. This is seeà to cancel thei row element
ià the lasŁ column. Performing this action for each of the firsŁ columns, the
n
determinant is converte to lłweð triangulað form, and its value is jusŁ the producŁ
of the diagonal elements,
0 = D (n+1) W (n+1)
(n)
n
(n+1)
= h − W
i
i=1
(n+1) (n+1) 2

n ¯
S
H − W (n+1) ¯
(n)
¯
× H − W (n+1) − iŁ+1 iŁ+1 . (1.31)
n+1 n+1 (n) (n+1)
i=1 h i − W
Examination shłws thaŁ D (n+1) (W (n+1) ) is a polynomial iàW (n+1) of degree n + 1,
as iŁ shoul be.