Spin-Paired Spin Eigenfunctions

Abstract

In the branching-diagram scheme the last function can be characterized by the following branching-diagram symbol:

$$ {B_{f}} = (1212 \cdots 1211 \cdots 1) $$

The meaning of this spin-coupling scheme is that we couple the spins of the pairs of electrons (1, 2), (3, 4), ..., (2g — 1,2g) to singlets and the remaining (N — 2g) electrons are coupled to the maximum spin multiplicity: S = ½(N — 2g). This type of spin coupling has special importance in the valence bond (VB) method. We can build up a system of spin eigenfunctions in which we use the same type of spin-coupling scheme but the pairs to be coupled to singlets are chosen in different ways. Such a function can be written in the form

$$ {V_{k}} = {2^{{ - 1/2}}}[\alpha (i)\beta (j) - \alpha (j)\beta (i)] \cdots {2^{{1/2}}}[\alpha (m)\beta (n) - \beta (m)\alpha (n)]\alpha (p)\alpha (q) \cdots \alpha (u) $$

((5.1))

It is easy to verify that the function is a spin eigenfunction and it belongs to the spin quantum number S = ½(N — 2g). The proof is based on the repeated application of Eq. (3.30). We shall call these functions spin-paired spin eigenfunctions or valence bond spin functions.