Spin expectation values for UHF
Given the UHF wave function model: \[\begin{equation} \require{physics} \ket{\text{core}} = \qty( \prod_i^{N_\alpha} \hat{a}^\dagger_{i \alpha} ) \qty( \prod_i^{N_\beta} \hat{a}^\dagger_{i \beta} ) \ket{\text{vac}} \thinspace , \end{equation}\] we will discuss its spin properties.
Owing to the particularly simple form for \(\hat{S}_z\) in a spin-separated spinor basis, we immediately have \[\begin{equation} \ev{\hat{S}_z}{\text{core}} = \frac{1}{2} (N_\alpha - N_\beta) \end{equation}\] and \[\begin{equation} \ev{\hat{S}^2_z}{\text{core}} = \frac{1}{4} (N_\alpha - N_\beta)^2 \thinspace . \end{equation}\] Furthermore, since the mixed-spin density matrices vanish for the simple UHF wave function model, we find \[\begin{align} & \ev{\hat{S}_x}{\text{core}} = 0 \\ % & \ev{\hat{S}_y}{\text{core}} = 0 \thinspace . \end{align}\]
For \(\ev{\hat{S}_- \hat{S}_+}{\text{core}}\), we find \[\begin{equation} \ev{\hat{S}_- \hat{S}_+}{\text{core}} = N_\beta - \sum_{i}^{N_\alpha} \sum_{j}^{N_\beta} | \braket{\phi_{i \alpha}}{\phi_{j \beta}} |^2 \thinspace , \end{equation}\] such that the expectation value of \(\hat{S}^2\) can be calculated as: (Szabo 1989) \[\begin{equation} \ev{\hat{S}^2}{\text{core}} = S_z(S_z + 1) + N_\beta - \sum_{i}^{N_\alpha} \sum_{j}^{N_\beta} | \braket{\phi_{i \alpha}}{\phi_{j \beta}} |^2 \thinspace . \end{equation}\]
The two terms on the right-hand side are called the UHF spin contamination. (Tsuchimochi and Scuseria 2010)