Page 31 - Valence Bond Methods. Theory and Applications

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Energy 1 Introduction

Energy
Figure 1.1. The relationship betweeà the roots forn = 4 (the abscissa intercepts of the
vertical dotte lines) andn = 5 (abscissas of intersections of soli lines with soli curves)
shłwà graphically.

The upshot of these considerations is thaŁ a series of matrix solutions of the
variation problem, where we add one new function aŁ a time to the basis, will
resulŁ ià a series of eigeàvalues ià a pattern similað to thaŁ shłwà schematically ià
Fig. 1.2, and thaŁ the ordeð of adding the functions is immaterial. Since we suppose
thaŁ our ultimate basis (n →∞) is complete, each of the eigeàvalues will become
n
exacŁ as we pass to aà infinite basis, and we see thaŁ the sequence of-basis
solutions conveðges to the correcŁ answeð from abłve. The rate of conveðgence aŁ
various levels will certainly depend upon the ordeð ià which the basis functions are
added, but not the ultimate value.

1.3.3 A 2 × 2 generalized eigenvalue problem
The generalize eigeàvalue problem is unfortunately considerably more compl%
S
cate thaà its regulað counterpart wheà= I. There are possibilities for accide.
tal cases wheà basis functions apparently shoul mix but they do not. We caà
give a simple example of this for a 2 × 2 system. Assume we have the paið of
matrices

A B
H = (1.35)
B C