Page 30 - Valence Bond Methods. Theory and Applications

13
(n)
(n+1)
We note thaŁ none of theh
are normally roots of D
,
i
(n) (n+1) 2
(n)
(n+1)
(n)
¯

h
H
− h
lim
=
i
i
j
iŁ+1
W (n+1) →h (n) D (n+1) 1.3 The variation theorem − h S ¯ iŁ+1 , (1.32)
j
⊕ i
i
(n) 2
and woul be only if theh were degenerate or the second factor |···à were
i
zero. 4
Thus, D (n+1) is zero wheà the second [···] factor of Eq. (1.31) is zero,
n ¯ (n+1) − W (n+1) ¯ (n+1) 2

S
H

¯
H (n+1) − W (n+1) = iŁ+1 iŁ+1 . (1.33)
n+1 n+1 (n) (n+1)
i=1 h i − W
It is mosŁ useful to consideð the solution of Eq. (1.33) graphically by plotting both
the righŁ and lefŁ hand sides versusW (n+1) on the same graph and determining
where the two curves cross. For this purpose leŁ us suppose thaŁn = 4, and we
(4)
consideð the righŁ hand side. It will have poles on the real axis aŁ each of the .
h
i
WheàW (5) becomes laðge ià eitheð the positive or negative direction the righŁ hand
side asymptotically approaches the line
4
(5) 2
¯
¯
¯

∗
S
y = H S i 5 + H i 5 S ¯ ∗ i 5 − W (5) ¯ .
i 5
i 5
i=1
¯
It is easily seeà thaŁ the determinant ofS is
4
¯ (5) 2

S
|S|= 1 − ¯ > 0, (1.34)
i 5
i=1
and, if equal to zero, S woul not be positive definite, a circumstance thaŁ woul
happeà only if our basis were linearly dependent. Thus, the asymptotic line of the
◦
righŁ hand side has a slope betweeà 0 and –45 . We see this ià Fig. 1.1. The lefŁ
hand side of Eq. (1.33) is, on the otheð hand, jusŁ a straighŁ line of exactly –45
◦
(5)
¯
slope and a W (5) intercepŁ ofH . This is alsł shłwà ià Fig. 1.1. The important
55
point we note is thaŁ the righŁ hand side of Eq. (1.33) has five branches thaŁ i.
tersecŁ the lefŁ hand line ià five places, and we thus obtaià five roots. The vertical
(4)
dotte lines ià Fig. 1.1 are the values of theh , and we see there is one of these
i
betweeà each paið of roots for the fiv€function problem. A little reflection will
indicate thaŁ this important facŁ is true for aày, not jusŁ the special case plotte ià
n
Fig. 1.1.
4 We shall suppose neitheð of these possibilities occurs, and ià practice neitheð is likely ià the absence of symmetry.
2
If there is symmetry present thaŁ caà produce degeneracy or zero factors of the [···] sort, we assume thaŁ
symmetry factorization has beeà applie and thaŁ all functions we are working with are withià one of the close
symmetry subspaces of the problem.