Spin Properties of Hartree-Fock Wavefunctions

The spin operators $\hat{S}_{z}$ and $\hat{S}^{2}$ both commute with the non-relativistic Hamiltonian, and therefore eigenfunctions of the Hamiltonian can be found which are also eigenfunctions of these spin operators. The permutation operator (equation (1.37)) commutes with $\hat{S}_{z}$ so single determinants are eigenfunctions of $\hat{S}_{z}$. Unfortunately this is not the case for the $\hat{S}^{2}$ operator. It can be shown [17] that
$$\begin{align}\langle \hat{S}^{2} \rangle = \left(\frac{N^{\alpha} - N^{\beta}}{2... ...1\right) + N^{\beta} - \sum_{ij}\vert S^{\alpha\beta}_{ij}\vert^{2}, \end{align}$$
where $N^{\alpha}$ and $N^{\beta}$ are the number of $\alpha$ and $\beta$ electrons ( $N^{\alpha} \ge N^{\beta})$, and
$$\begin{align}S^{\alpha\beta}_{ij} = \int \psi_{i}^{\alpha} \psi_{j}^{\beta} \, d{\bf r}. \end{align}$$
For RHF (open and closed shell) the occupied $\beta$ orbitals lie within the $\alpha$ orbital's space; therefore,
$$\begin{align}N^{\beta} = \sum_{ij}\vert S^{\alpha\beta}_{ij}\vert^{2}. \end{align}$$
Thus the determinants are eigenfunctions of $\hat{S}^{2}$. However, for unrestricted determinants, the $\beta$ orbitals are not constrained to lie within the $\alpha$ space; therefore,
$$\begin{align}N^{\beta} \ge \sum_{ij}\vert S^{\alpha\beta}_{ij}\vert^{2}. \end{align}$$
These determinants will not be eigenfunctions of $\hat{S}^{2}$ and are termed spin-contaminated -- they contain higher spin multiplicity components. This spin-contamination can allow the UHF function to give the correct dissociation behaviour, as the $\alpha$ and $\beta$ electrons are no longer forced to occupy the same orbital. However, for methods which build on Hartree-Fock, spin-contamination can have a disastrous effect [25,26,27,28].