Spin contamination in UHF
Comparing the expectation values of \(\hat{S}^2\) for RHF: \[\begin{align} \require{physics} \ev{\hat{S}^2}{\text{RHF}} &= 0 \\ &= S_z(S_z + 1) \end{align}\] and UHF: \[\begin{equation} \ev{\hat{S}^2}{\text{UHF}} = S_z(S_z + 1) + N_\beta - \sum_{i}^{N_\alpha} \sum_{j}^{N_\beta} | \braket{\phi_{i \alpha}}{\phi_{j \beta}} |^2 \thinspace , \end{equation}\] suggests the definition of the UHF spin contamination as: (Tsuchimochi and Scuseria 2010) \[\begin{equation} \delta_{\hat{S}^2} = N_\beta - \sum_{i}^{N_\alpha} \sum_{j}^{N_\beta} | \braket{\phi_{i \alpha}}{\phi_{j \beta}} |^2 \thinspace , \end{equation}\] whose origin lies solely in the expectation value for \(\ev{\hat{S}_- \hat{S}+}{\text{UHF}}\): \[\begin{equation} \delta_{\hat{S}^2} = \ev{\hat{S}_- \hat{S}+}{\text{UHF}} \thinspace . \end{equation}\]
The term can also be calculated using the AO density matrices \(\vb{P}^\alpha\) and \(\vb{P}^\beta\), and the AO overlap matrix \(\vb{S}^{\text{AO}}\): \[\begin{equation} \delta_{\hat{S}^2} = N_\beta - \tr \qty[ (\vb{P}^{\alpha})^\text{T} \vb{S}^{\text{AO}} (\vb{P}^{\beta})^\text{T} \vb{S}^{\text{AO}} ] \thinspace . \end{equation}\]
Alternatively, we may transform the AO density matrices to the basis of the \(\alpha\) spin-orbitals, leading to: \[\begin{align} & \vb{D}^\alpha = \vb{C}^{\alpha, -1} (\vb{P}^{\alpha})^\text{T} (\vb{C}^{\alpha, -1})^\dagger \\ % & \vb{D}^{' \beta} = \vb{C}^{\alpha, -1} (\vb{P}^{\beta})^\text{T} (\vb{C}^{\alpha, -1})^\dagger \thinspace , \end{align}\] in which we have made clear with the prime in \(\vb{D}^{' \beta}\) that it is the \(\beta\)-density matrix in the basis of the \(\alpha\) spin-orbitals. The UHF spin contamination may then be expressed as \[\begin{equation} \delta_{\hat{S}^2} = N_\beta - \tr\qty[ \vb{D}^\alpha \vb{D}^{' \beta} ] \thinspace . \end{equation}\]