Page 33 - Valence Bond Methods. Theory and Applications

16
1 Introduction
We note the possibility of aà accident thaŁ cannot happeà if= 0 and b
⊕ 0: Shoul
b =±as, one of the two values of W is eitheð±a, and one of the two diagonal
5

elements of H is unchanged. LeŁ us for definiteness assume thaŁb = as and iŁ is
a we obtain. Then, clearly the vector C 1 we obtaià is s

1
,
0
and there is no mixing betweeà the states from the application of the variation
theorem. The otheð eigeàvector is simply determine because iŁ musŁ be orthogonal
to C 1 , and we obtaià
√

−s/ 1 − s 2
C 2 = √ ,
1/ 1 − s 2
sł the otheð state is mixed. It musŁ normally be assume thaŁ this accident is
rare ià practical calculations. Solving the generalize eigeàvalue problem results
ià a nonorthogonal basis changing both directions and internal angles to become
orthogonal. Thus one basis function coul geŁ “stuck”ià the process. This shoul
be contraste with the case wheàS = I, ià which basis functions are unchange
only if the matrix was originally already diagonal with respecŁ to them.
We do not discuss it, but there is aà n × n version of this complication. If
there is no degeneracy, one of the diagonal elements of the H-matrix may be
unchange ià going to the eigeàvalues, and the eigeàvector associate with iŁ is
[0,..., 0, 1, 0,..., 0] .
†

1.4 Weights of nonorthogonal functionł
The probability interpretation of the wave function ià quantum mechanics obtaine
by forming the square of its magnitude leads naturally to a simple idea for the
weights of constituent parts of the wave function wheà iŁ is writteà as a lineað
combination of orthonormal functions. Thus, if

= ψ i C i , (1.41)
i
and ψ i |ψ j ⊕ δ ij , normalization of requires
2
|C i | = 1. (1.42)
i
If, also, each of the ψ i has a certaià physical interpretation or significance, theà
one says the wave function , or the state represente by it, consists of a fraction

5 NB We assume this not to happeà ià our discussion abłve of the conveðgence ià the lineað variation problem.