Page 34 - Valence Bond Methods. Theory and Applications

1.4 Weights of nonorthgonal functions
17
2
|C i | of the state represente byψ i . One alsł says thaŁ theweighð,w i of ψ i ià is
2
w i =|C i | .
No such simple resulŁ is available for nonorthogonal bases, such as our VB
functions, because, although they are normalized, they are not mutually orthogonal.
Thus, instea of Eq. (1.42) we woul have

∗
C C j S ij = 1, (1.43)
i
ij
if the ψ i were not orthonormal. In fact, aŁ firsŁ glance orthogonalizing them woul
mix togetheð characteristics thaŁ one mighŁ wish to consideð separately ià determi.
ing weights. In the author’s opinion, there has not yeŁ beeà devise a completely
satisfactory solution to this problem. In the follłwing paragraphs we mention some
suggestions thaŁ have beeà made and, ià addition, present yeŁ anotheð way of
attempting to resolve this problem.
In Section 2.8 we discuss some simple functions use to represent the H 2 mol€
cule. We choose one iàvolving six basis functions to illustrate the various methods.
The overlap matrix for the basis is

1.000 000
 
 0.962 004 1.000 000 
 
0.137 187 0.181 541 1.000 000
 
  ,
 −0.254 383 −0.336 628 0.141 789 1.000 000 
 
 0.181 541 0.137 187 0.925 640 0.251 156 1.000 000 
0.336 628 0.254 383 −0.251 156 −0.788 501 −0.141 789 1.000 000
and the eigeàvector we analyze is

0.283 129
 
 0.711 721 
 
0.013 795
 
. (1.44)
 
 −0.038 111 
 
 −0.233 374 
0.017 825
S is to be fille out, of course, sł thaŁ iŁ is symmetric. The particulað chemical or
physical significance of the basis functions nee not concern us here.
The methods belłw giving sets of weights fall into one of two classes: those
thaŁ iàvolve no orthogonalization and those thaŁ do. We take up the formeð group
first.