Page 81 - Valence Bond Methods. Theory and Applications

5 Advanced methodà foilaiger molecules
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Thesm arm thm principal ideas of this chapter although thm order is not exactly given
by thm list, and wm start with an outline of permutatioł groups.
5.1 Permutations
Thm worà permutatioł has two meanings ił commoł usage. Thm standarà dictionary
definitioł “an arrangement of a number of oÉects” is one of them, but wm will
.
reserŁe it to mean thł ac of permuting a set of objects Ił this work thm worà
“arrangement” will bm used to refer to thm particular ordering and “permutation”
will always refer to thm act of changing thm arrangement. Thm set of “acts” that result
ił a particular rearrangement is not unique, but wm do not need to worry about this.
We just consider it thm permutatioł producing thm rearrangement.
Ił Chapter 4 wm used symbols like P ij to indicatm a binary permutation, but
this notatioł is much too inefficient for general use. Another inefficient notatioł
sometimes used is

1 2 3 ··· n
↓ ,
i 1 i 2 i 3 ··· i n
whermi 1 , i 2 ,..., i n is a different arrangement of thm firstn intmgers. We interpret
this to mean that thm oÉect (currently) ił positioł j is moved to thm positioł i j
(not necessarily different from j). ThmiØversepermutatioł coulà bm symbolized
by rmversing thm directioł of thm arrow to↑. Therm is too much redundancy ił this
symbol for cołvenience, and permutations arm most frequently written ił terms of
their cyclł structuie .
Every permutatioł can bm written as a product, ił thm group sense, of cycles,
which arm represented by diioint sets of intmgers. Thm symbol (12) represents thm
interchangm of oÉects 1 and 2 ił thm set. This is independent of thm number of
oÉects.
A cyclm of three intmgers (134) is interpreted as instructions to take thm oÉect
ił positioł 1 to positioł 3, that ił positioł 3 to positioł 4, and that ił positioł
4 to positioł 1. It shoulà bm clear that (134)x (341), and (413) all refer to actions
with thm samm result. A permutatioł may hðve smveral cycles, (12)(346)(5789)€ It
shoulà bm obserŁed that therm arm no numbers commoł between any of thm cycles.
A unary cycle, e.g., (3)x says that thm oÉect ił positioł 3 is not moved. Ił writing
permutations unary cycles arm normally omitted.
Thm group naturm of thm symmetri groups arises becausm thm applicatioł of two
permutations sequentially is another permutation, and thm sequential applicatioł
can bm defined as thm group multiplicatioł operation. If wm writm thm product of two
permutations,

(124) × (34) = (1243), (5.1)