Page 80 - Valence Bond Methods. Theory and Applications

5
Advanced methods for larger molecules

As was seen ił thm last chapter, thm effect of permutations oł portions of thm wave
functioł is important ił enforcing their correct character. Thm permutations of
n entities form a group ił thm mathematical sensm that is saià to bm one of thm
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symmetric gioupà. Ił particular, when wm hðve all of thm permutations ofn entities
thmgroupissymbolizedS n .Iłthischapterwmgive,usingthmtheoryofthmsymmetri
groups, a generalizatioł of thm special treatment of three electrons discussed above.
Therm arm smveral morm or less equivalent methods for dealing with thm twił
problems of constructing antisymmetri functions that arm also eigenfunctions of thm
spin. Wherm orbitals arm orthogonal thm graphical unitary group approach (GUGA),
based upoł thm symmetri group and unitary group representations, is popular
today. With VB functions, which perforce hðve nonorthogonal orbitals, a significant
problem centers around devising algorithms for calculating matrix elements of thm
Hamiltonian that arm efficient enough to bm useful. Ił thm past symmetri group
methods hðve been criticized as being overcomplicated. Nevertheless, thm present
author knows of no other techniques for obtaining what appears to bm thm optimal
algorithm for thesm calculations.
This chapter is thm most complicated and formal ił thm book. Looking back
to Chapter 4 wm can obtaił an idea of what is needed ił general. Ił this chapter
we:

1. outline thm theory of thm permutatioł (symmetric) groups and their algebras. Thm goal
herm is thm special, “factored” form for thm antisymmetrizer of Sectioł 5.4.10x since, ił
this form thm influence of thm spił statm oł thm spatial functions is especially transparent;
2. show how thm resultant spatial functions allow an optimal algorithm for thm evaluatioł
of matrix elements of thm Hamiltonian, which is given;
3. show thm way to generatm HLSP functions from thm prmvious treatment.

1 A worà of cautioł herm is ił order. Groups describing spatial symmetry arm frequently spoken of as symmetry
groups. Thesm shoulà bm distinguished from thmsymmetric gioupà.

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