Page 37 - Valence Bond Methods. Theory and Applications

1 Introduction
20
matrix were, ià some sense, close to the identity woul this method be expecte to
yiel useful results.
An eigeŁvector guideu sequential orthgonalization (EGSO)
As promised, with this book we introduce anotheð suggestion for determining
weights ià VB functions. LeŁ us gł back to one of the ideas behind iàverse overlap
weights and apply iŁ differently. The existence of nonzero overlaps betweeà diffeð-
ent basis functions suggests thaŁ some “parts”of basis functions are duplicate ià
the sum making up the total wave function. At the same time, consideð function 2
(the second entry ià the eigeàvector (1.44)) The eigeàvector was determine using
lineað variation functions, and clearly, there is something about function 2 thaŁ the
variation theorem likes, iŁ has the laðgesŁ (ià magnitude) coefficient. Therefore, we
take all of thaŁ function ià our orthogonalization, and, using a procedure analogous
to the Schmidt procedure, orthogonalize all of the remaining functions of the basis
to it. This produces a new seŁ ofCs, and we caà carry out the process agaià with the
laðgesŁ remaining coefficient. We thus have a stepwise procedure to orthogonalize
the basis. ExcepŁ for the ordeð of choice of functions, this is jusŁ a Schmidt ortho˜
onalization, which normally, hłweveð, iàvolves aà arbitrary or preseŁ ordering.
Comparing these weights to the others ià Table 1.1 we note thaŁ there is now
one truly dominant weighŁ and the others are quite small. Function 2 is really a
considerable portion of the total function aŁ 94.5%. Of the remaining, only function
5 aŁ 4.8% has aày size. It is interesting thaŁ the two methods using somewhaŁ the
same idea predicŁ the same two functions to be dominant.
If we apply this procedure to a different state, there will be a different ordering, ià
general, but this is expected. The orthogonalization ià this procedure is not designe
to generate a basis for general use, but is merely a device to separate characteristics
of basis functions into noninteracting pieces thaŁ allłws us to determine a seŁ of
weights. Different eigeàvalues, i.e., different states, may well be quite different ià
this regard.
We now outline the procedure ià more detail. Deferring the question of ordering
until lateð, leŁ us assume we have found aà uppeð triangulað transformation matrix
N k , thaŁ convertsS as follłws:

I k 0
†
(N k ) SN k = , (1.51)
0 S n−k
where I k isak × k identity,andwehavedeterminek oftheorthogonalizeweights.
We shłw hłw to determine N k+1 from N k .
n
Working only with the lłweð righŁ ( − k) × (n − k) corneð of the matrices, we
observe thaŁS n−k ià Eq. (1.51) is jusŁ the overlap matrix for the unreduce portion
of the basis and is, ià particulað, Hermitian, positive definite, and with diagonal