Requirements of spatial functions

When dealing with spin functions it is normally convenient to arrange the bases to be orthonormal, and we obtain two functions, 

For simplicity we do not label these functions with the Ms value. Our work in VB theory and solving the ESE seldom needs any but the principal spin function with Ms = S. The S operator is always available should other Ms values be needed. 

With the spin eigenfunctions of Eqs. (4.14) and (4.15) we have an example of the spin degeneracy alluded to in Chapter 2. Unlike the single singlet function we arrived at for two electrons in Section 2.1.1 we now obtain two. Writing out the equations specifically, 

We now have a significant difiference from the case of two electrons in a singlet state, namely, we have two spin functions to combine with spatial functions for a solution to the ESE rather than only one. For a doublet three-electron system our general solution must be 

We must now investigate the effect of the binary interchange operators, P/y on the (j)i functions. We suppress the spin-label superscript for these considerations. It is straightforward to determine 

---