Page 38 - Valence Bond Methods. Theory and Applications

21
1.4 Weights of nonorthgonal functions
elements equal to 1. We write iŁ ià partitione form as

s
1
(1.52)
,
S n−k =
S
†
s

where [1 s] is the firsŁ row of the matrix LeŁ n−k be aà uppeð triangulað matrix
M
partitione similarly,

1 q
M n−k = , (1.53)
0 B
and we determine q and B sł thaŁ

1 q + sB
†
(M n−k ) S n−k M n−k = , (1.54)
†
(q + sB) † B (S − s s)B

†

1 0
= , (1.55)
0 S n−k−1
where these equations may be satisfie withB the diagonal matrix
2 −1/2 2 −1/2

B = diag 1 − s 1 − s ··· (1.56)
1 2
and
q =−sB. (1.57)
The iàverse ofM n−k is easily determined:

(M n−k ) −1 = 1 s −1 , (1.58)
0 B
and, thus, N k+1 = N k Q k , where

0
I k
Q k = . (1.59)
0 M n−k
The unreduce portion of the problem is now transforme as follłws:
−1
−1
†
†
†
(C n−k ) S n−k C n−k = [(M n−k ) C n−k ] (M n−k ) S n−k M n−k [(M n−k ) C n−k ].
(1.60)
Writing

C 1
C n−k = , (1.61)
C
we have

C 1 + sC
−1
[(M n−k ) C n−k ] = −1 , (1.62)
B C

C 1 + sC
= . (1.63)
C n−k−1
Putting these togetheð, we arrive aŁ the totalN as Q 1 Q 2 Q 3 ··· Q n−1 .