Page 35 - Valence Bond Methods. Theory and Applications

1 Introduction
18
Table 1.1. Weights for nonorthgonal basis functions
Chiðgwin– by various methods.
Symmetric
Invers€
Coulson overlap orthogon. EGSO a
0.266 999 0.106 151 0.501 707 0.004 998
0.691 753 0.670 769 0.508 663 0.944 675
–0.000 607 0.000 741 0.002 520 0.000 007
0.016 022 0.008 327 0.042 909 0.002 316
0.019 525 0.212 190 0.051 580 0.047 994
0.006 307 0.001 822 0.000 065 0.000 010
a EGSO = eigeàvector guide sequential orthogonalization.

1.4.1 Weights without orthogonalization

The method of ChiłgwiŁ and Coulson
These workers[23] suggesŁ thaŁ one use

w i = C ∗ S ij C j , (1.45)
i
j
although, admittedly, they propose iŁ only ià cases where the quantities were real.
As written, this w i is not guarantee eveà to be real, and wheà theC i and S ij are real,
iŁ is not guarantee to be positive. Nevertheless, ià simple cases iŁ caà give some
idea for weights. We shłw the results of applying this method to the eigeàvector
and overlap matrix ià Table 1.1 abłve. We see thaŁ the relative weights of basis
functions 2 and 1 are fairly laðge and the others are quite small.

Invełse overlapweights
Norbeck and the author[24] suggeste thaŁ ià cases where there is overlap, the
basis functions each caà be considere to have a unique portion. The “length”of
this may be shłwà to be equal to the reciprocal of the diagonal of the S −1 matrix
corresponding to the basis function ià question. Thus, if a basis function has a
unique portion of very short length, a laðge coefficient for iŁ means little. This
suggests thaŁ a seŁ ofelative weights coul be obtaine from
r
2
−1
w i ∝àC i | /(S ) ii , (1.46)
where these w i do not generally sum to 1. As implemented, these weights are
renormalize słthaŁthey do sumto1 toprovideconvenient fractionsorpercentages.
This is aà awkward feature of this method and makes iŁ behave nonlinearly ià some
contexts. Although these firsŁ two methods agree as to the mosŁ important basis
function they transpose the next two ià importance.