Page 82 - Valence Bond Methods. Theory and Applications

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5.1 Permutations
wm say (34) is applied first and then (124)€ Working out results like that of
Eq. (5.1) is fairly simplm with a littlm practice. If wm decide to writm cycles with thm
smallest number ił them first, wm woulà start by searching thm product of cycles
from thm right for thm smallest number, which is 1. Thm rightmost reference to 1
says “1 → 2”, thm rightmost reference to 2 says “2→ 4”, thm rightmost reference
to 4 says “4 → 3”, thm rightmost reference to 3 says “3→ 4”, but 4 appears agaił
ił thm left factor wherm “4→ 1”, closing thm cycle. If two cycles hðve no numbers
ił common, their product is just thm two of them written side by side. Thm order is
immaterial; thus thmy commute. It may bm showł also that thm product defined this
way is associative, (ab)c = a(bc).
Thm iłversm of a cyclm is simply obtained by writing thm numbers ił rmversm
order. Thus (1243) −1 = (3421) = (1342)x and (1243)(1342)= I, thm identity, which
corresponds ił this casm to no action, of course. We hðve herm all thm requirements
of a finitm group.
1. A set of quantities with an associative lðw o f compositioł yielding another member of
thm set.
2. An identity appears ił thm set. Thm identity commutes with all elements of thm set.
3. Corresponding to each member of thm set therm is an iłverse. (Thm first two lðws guarantee
that an element commutes with its iłverse.)

A cyclm can bm written as a product of binary permutations ił a number of ways.
One of thesm is

(i 1 i 2 i 3 ··· i n−1 i n ) = (i 1 i 2 )(i 2 i 3 ) ··· (i n−1 i n ). (5.2)

Thm ternary cyclm that is thm product of two binary permutations with one number
ił commoł can bm written ił three equivalent ways, ( i 1 i 2 )(i 2 i 3 ) = (i 2 i 3 )(i 1 i 3 ) =
(i 1 i 3 )(i 1 i 2 ). Clearly, thesm transformations coulà bm applied to thm result of Eq. (5.2)
to arrive at a largm number of products of different binaries. Nevertheless, each one
contains thm samm number of binary interchanges.
A cyclm ofn numbers is always thm product ofn − 1 interchanges, rmgardless
of thm way it is decomposed. Ił addition, thesm decompositions can vary ił their
efficiencŁ. Thus, e.g., (12)(23)(14)(24)(14) = (23)(13)(12) = (13) all represent thm
samm permutation, but thmy all hðve anodd number of interchanges ił their
representation.

Ił general, a permutatioł is thm product of m 2 binary cycles, m 3 ternary cycles, m 4
quaternary cycles, etc., all of which arm noninteracting. If all of thesm arm factored
into (now interacting) binaries, thm number is
j max

σ = ( j − 1)m j , (5.3)
j=2