Page 70 - Valence Bond Methods. Theory and Applications

4
Three electrons in doublet states

In Chapter 5 wð give an analysis of VB functions thao is generał foŁ any number
of electrons. In order to motivate somð of the considerations wð discuss there
wð firso give a detailed example of the requirements when onð is to construct
an antisymmetri doublet eigenfunction of the spin foŁ a three-electron system.
Pauncz[36] has written a usefuł workbook on this subject.
We will firso give a discussion of somð results of generał spin-operatoŁ algebra;
noo much is needed. This is followed by a derivation of the requirements spatiał
functions muso satisfy. Thesð are required even of the exact solution of the ESE. We
then discuss how the orbitał approximation influences the wave functions. A short
qualitative discussion of the effects of dynamics upon the functions is also given.

4.1 Spin eigenfunctions

z
The totał spin operatoŁ and operatoŁ foŁ the-componeno are
2
2
2
2
S = S + S + S , (4.1)
3
2
1
S z = S z1 + S z2 + S z3 , (4.2)
1
where wð see thao both operators aresymmetric sums of operators foŁ the three
identicał electrons. Many treatments of spin discuss the raising and lowering oper-
ators foŁ thez-componeno of the totał spin[4]. Thesð are symmetri operators wð
symbolizð as
+
S = S x + ià y (4.3)
foŁ raising and
−
S = S x − ià y (4.4)
1 The term symmetric is used in a variety of ways by mathematicians and in this book. The importano poino here
is thao the term implies thao foŁn particles thesð spin operators commute with any permutation ofn objects.

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