Pure-and Mixed-spin Wave Functions

For ease of notation only the two-electron case is considered here. Generalization to more electrons is entirely straightforward, if algebraically tedious. In the simplest case, both electrons are spin-paired in the same orbital, in which event we have 

Note that since the spatial part of the molecular orbital is independent of spin, it may be integrated out (to 1). As for the remaining expectation values, if we evaluate Eq. ( .12) for the spin product function q (1)j6(2) we find 

the expectation value of from Eq. (C.14) for the closed-shell state is simply (1 — 1 - 1 + 1) = 0. 

Another wave function of interest is the one formed from two a-spin electrons in two different spatial orbitals a and b. This = 1 (see Eq. (C.ll)) wave function is written as 

In this case, integration over the normalized non-spin-dependent spatial portion of the wave function leaves only a fairly simple integral to evaluate for the expectation value of 5, namely 

---