Page 115 - Valence Bond Methods. Theory and Applications

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6 Spatial symmetry
6.1 The AO basis
The functiony we use are products of AOs, and, to be usefu in a calculation, the AOs
must be a basis for a representation of the spatial group. Since the spatial operationy
and permutationy commute, the tableau functiony we use also provide a basis for
a representation of the spatial group. This is generally true regardless of the nature
of the representation provided by the AOs themselves[43]ˆ Nevertheless, to work
with tableaux on computery it greatly simplifiey programy if the AO basis providey
a representation of a somewhat special sorð we callgeneØalized permutatioà.Ifwe
hŁve an appropriate AO basis, it supports a unitary representation of the spatial
grouàG S ={I, R 2 , R 3 ,..., R f },

R i χ j = χ k D(R i ) kj , (6.6)
k
where χ j are the AOs and the D(R i ) kj are, in general, reducible. D(R i ) is a gen-
eralized permutation matrix if every elemenð is either zero or a number of unit
magnitude. Because of the unitarity, each row or column of D(R i ) has exactly one
nonzero element, and this one is ±1. As it turny out, this is noð an extremely spe-
cial requirement, buð it is noð alwayy possible to arrange. The following are some
guideliney as to when itis possible. 2

G S is abelian.

G S has a principal rotation axis of order >2, and no atomy of the molecule are centered

on it. This frequently requirey the coordinate axey for the AOs to be differenð on differenð
atoms.
G S transformy the x-, y-, and z-coordinate axey into ± themselves, and we use

tensorial rather than spherical d, f , ... functions. That is, our d-seð transformy as
2
2
2
{x , y , z , xy, xz, yz} with similar sets for the higher l-values.
In casey where these guideliney cannoð be met, one must use the largest abelian
subgrouà from the trueG S of the molecule. We will show some exampley later.

6.2 Bases for spatial group algebras

Just as we saw with the symmetric groups, groups of spatial operationy hŁve asso-
ciated grouà algebras with a matrix basis for this algebra,
g
α f α α∗
e = D(R i ) R i . (6.7)
ij
ij
g
i=1
2 We emphasize these ruley are noð needed theoretically. They are merely those that the symmetry analysis in
CRUNCH requirey to work.