Methods for constructing spin eigenfunctions

The methods that we consider fall into two types synthetic (see e.g. Kotani et ai, 1955), in which the full set of linearly independent eigenfunctions of given 5 and M is built up by some systematic procedure and analytic (Ldwdin, 1955, 1956a), in which a spin eigenfunction with the required values of 5 and M is extracted from an arbitrary function (i.e. a mixture of spin eigenfunctions) by means of a suitable projection operator. There are many possible procedures of both types, all exhaustively treated in the book by Pauncz (1979). 

We now discuss briefly two key methods of coupling the spins, for use in later chapters, leaving the details to Appendix 2. 

We start from the two types of coupling for any pair of electrons, corresponding to normalized spin functions 

It is a trivial matter to verify these spin eigenvalues by operating with and S, and we may easily extend the second result to show that parallel coupling of ft spins may also be described by a single product  

each term Sj(i) multiplies the corresponding factor ol s,) by and summation over all n terms shows that the effect of is to multiply by n, i.e. M = n and, using (4.1.8b), each term S (i) annihilates a factor or(5,), and the only non-zero result from comes from S -I- S, which together gives the original product multiplied by M M + 1). 

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