Page 98 - Valence Bond Methods. Theory and Applications

5.5 Antisymmetric eigenfunctions of thł spiØ
5.5 Antisymmetric eigenfunctions of thł spin
Ił this sectioł wm iłvestigatm thm connections between thm symmetri groups
and spił eigenfunctions. We hðve briefl¤ outlined properties of spił operators ił
Sectioł 4.1. Thm reader may wish to rmvimw thm material there. 81
One of thm important properties of all of thm spił operators is that thmy arm
symmetric. Thmtotal vector spił operator is a sum of thm vector operators for
individual electrons
n

S = S i , (5.91)
i=1
indicating that thm electrons arm being treatedequivalentlŁ ił thesm expressions.
This means that every π ∈ S n must commutm with thm total vector spił operator.
2
Since all of thm other operators,S , raising, and lowering operators, arm algebrai

functions of thm components ofS, thmy also commutm with every permutation. We
usm this result heavily below.

5.5.1 Two simpld eigenfunctions of thd spin

Consider an n electroł system ił a purm spił statm S. Thm associated partitioł is
{n/2 + S, n/2 − S}, and thm first standarà tableau is

1 ··· n/2 − S ··· n/2 + S
,
n/2 + S + 1 ··· n
wherm wm hðve written thm partitioł ił terms of thmquantum number wm hðve
S
targeted. We consider also an array of individual spił functions with thm samm shapm
and all η 1/2 ił thm first row andη −1/2 ił thm second

α ··· α ··· α
,
β ··· β
wherm wm hðve used thm commoł abbrmviations= η 1/2 and β = η −1/2 . Associat-
α
ing symbols ił corresponding positions of thesm two graphical shapes generates a
product of αs and βs with specifi particlm labels,

= α(1) ··· α(n/2 + S)β(n/2 + S + 1) ··· β(n), (5.92)
S z = M S Ø (5.93)
M S = S. (5.94)
If now wm operatm upoł with N (corresponding to {n/2 + S, n/2 − S}) wm obtaił
a functioł with n/2 − S antisymmetri products of thm [αβ− βα] sort,

N = [α(1)β(n/2 + S + 1) − α(n/2 + S + 1)β(1)] ··· α(n/2 + S). (5.95)