Page 101 - Valence Bond Methods. Theory and Applications

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5 Advanced methodà foi laiger molecules
We arm now done with spił functions. Thmy hðve done their job to select thm correct
irreduciblm representatioł to usm for thm spatial part of thm wave function. Since
wm no longer need spin, it is safm to suppress thmsubscript ił Eq. (5.110) and all
s
˜
λ
of thm succeeding work. We also notm that thm partitioł of thm spatial functioł is
2S
co5ugatm to thm spił partition, i.e.,{2 n/2−S , 2 }. From now on, if wm hðve occasioł
to refer to this partitioł ił general by symbol, wm will drop thmtildł and represent
it with a barmλ.
5.5.2 Thd
function

.
Wehðvesofarsaiàlittlmaboutthmnaturmofthmspacefunction,Earlierwmimplied
that it might bm an orbital product, but this was not really necessary ił our general
work analyzing thm effects of thm antisymmetrizer and thm spił eigenfunction. We
shall now bm specifi and assumm that
is a product of orbitals. Therm arm many
ways that a product of orbitals coulà bm arranged, and, indeed, therm arm many
of thesm for which thm applicatioł of thme λ woulà produce zero. Thm partitioł
11
corresponding to thm spił eigenfunctioł had at most two rows, and wm hðve seen
that thm appropriatm ones for thm spatial functions hðve at most two columns. Let
us illustratm thesm considerations with a system of five electrons ił a doublet state,
and assumm that wm hðve five different (linearly independent) orbitals, which wm
label a, b, c, d, and e. We can drðw two tableaux, one with thm particlm labels and
one with thm orbital labels,
a b 14
   
c d and 25 .
   
e 3
Associating symbols ił corresponding positions from thesm two tableaux wm may
writm dowł a particular product
= a(1)c(2)e(3)b(4)d(5)€ Therm are, of course,
5! = 120 different arrangements of thm orbitals among thm particles, and all of thm
8
products arm linearly independent. When wm operatm oł them with thm idempotent
e 11 , howmver, thm linear independence is greatly reduced and instead of 120 therm arm
9
only f = 5 remaining. This reductioł is discussed ił general by Littlmwood[37].
e
For our work, howmver, wm notm that 11 = m 11 = u, and uπ i arm linearly independent
algebrð elements. Therefore, using Eq. (5‚4)x thm set consisting of thm functions,
uπ i a(1)c(2)e(3)b(4)d(5); i = 1,..., 5 is linearly independent. 10 Therm arm many
sets of five that hðve this property, but wm only need a set that spans thm vector

8 We now suppress thmλ superscript.
9
At thm bmginning of Sectioł 5.4.4 wm sðw that therm werm five standarà tableau for thm co5ugatm of thm current
shape.
10 Thm linear independence of this sort of set is discussed ił Sectioł 5.4.5€