Spin expectation values for GHF

Given the GHF wave function model, we will discuss the different expectation values of the spin operators. Since the GHF density matrices are particularly simple, the expectation values can be straightforwardly calculated as: \[\begin{align} S_x &= \ev{\hat{S}_x}{\text{core}} \\ &= \frac{1}{2} \sum_{I}^N \qty( \braket{\phi_{I \alpha}}{\phi_{I \beta}} + \braket{\phi_{I \beta}}{\phi_{I \alpha}} ) \\ &= \sum_I^M \mathfrak{R} \qty( \braket{\phi_{I \alpha}}{\phi_{I \beta}} ) \thinspace , \end{align}\]

\[\begin{align} S_y &= \ev{\hat{S}_y}{\text{core}} \\ &= - \frac{i}{2} \sum_{I}^N \qty( \braket{\phi_{I \beta}}{\phi_{I \alpha}} - \braket{\phi_{I \alpha}}{\phi_{I \beta}} ) \\ &= \sum_I^M \mathfrak{I} \qty( \braket{\phi_{I \beta}}{\phi_{I \alpha}} ) \thinspace , \end{align}\] and \[\begin{align} S_z &= \ev{\hat{S}_z}{\text{core}} \\ &= \frac{1}{2} \sum_{I}^N \qty( \braket{\phi_{I \alpha}}{\phi_{I \alpha}} - \braket{\phi_{I \beta}}{\phi_{I \beta}} ) \thinspace . \end{align}\]

The spin-operator \(\hat{S}^2 = \hat{S}^2_x + \hat{S}^2_y + \hat{S}^2_z\) can be rewritten using the ladder operators \(\hat{S}_+ = \hat{S}_x + i\hat{S}_y\) and \(\hat{S}_- = \hat{S}_x - i\hat{S}_y\) and their commutator relations as \[\begin{equation} \hat{S}^2 = \hat{S}^2_z + \frac{1}{2}\left(\hat{S}_+\hat{S}_- + \hat{S}_-\hat{S}_+\right) = \hat{S}^2_z + \hat{S}_z + \hat{S}_-\hat{S}_+. \end{equation}\] Two terms in this equation are formally a product of operators. Hence, for \(\hat{S}_z^2\) we find \[\begin{align} \hat{S}_z^2 = \sum_{PQ} [S_z^2]_{PQ} ~\hat{a}^{\dagger}_{P} \hat{a}_{Q} + \sum_{PQRS} S_{z,PQ} S_{z,RS} ~\hat{a}^{\dagger}_{P} \hat{a}^{\dagger}_{Q} \hat{a}_{R} \hat{a}_{S}. \end{align}\] This may be rewritten as \[\begin{multline} \hat{S}_z^2 = \frac{1}{4}\Biggl( \sum_{PQ} \left(\braket{\phi_{P,\alpha}}{\phi_{Q,\beta}} + \braket{\phi_{P,\beta}}{\phi_{Q,\alpha}}\hat{a}^{\dagger}_{P} \hat{a}_{Q}\right)\\ + \sum_{PQRS} \biggl(\left( \braket{\phi_{P,\alpha}}{\phi_{Q,\alpha}} - \braket{\phi_{P,\beta}}{\phi_{Q,\beta}} \right) \left( \braket{\phi_{R,\alpha}}{\phi_{S,\alpha}} - \braket{\phi_{R,\beta}}{\phi_{S,\beta}}\right)\hat{a}^{\dagger}_{P} \hat{a}^{\dagger}_{Q} \hat{a}_{R} \hat{a}_{S} \biggr). \Biggr) \end{multline}\] Here \([S_z^2]_{PQ}\) was calculated similarly to eq. \(\ref{eqn:sigmaxPQinspinor}\), and using \[\begin{equation} \hat{S}_z^2 = \hat{S}_z \cdot \hat{S}_z = \left[\frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \right] \left[\frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \right] = \frac{1}{4}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. \end{equation}\]

For \(\hat{S}_-\hat{S}_+\), we similarly find \[\begin{equation} \hat{S}_-\hat{S}_+ = \sum_{PQ} \braket{\phi_{P,\beta}}{\phi_{Q,\beta}} \hat{a}^{\dagger}_{P} \hat{a}_{Q} + \sum_{PQRS} \braket{\phi_{P,\beta}}{\phi_{Q,\alpha}}\braket{\phi_{R,\alpha}}{\phi_{S,\beta}} \hat{a}^{\dagger}_{P} \hat{a}^{\dagger}_{Q} \hat{a}_{R} \hat{a}_{S}. \end{equation}\]

With these expressions and Wick’s theorem, the expectation value of \(\hat{S}^2\) can be calculated. For the constituent parts, we find: \[\begin{equation} \ev{\hat{S}_z^2}{\text{core}} = S_z^2 + \frac{1}{4} [ \sum_I^N \qty( \braket{\phi_{I \alpha}} + \braket{\phi_{I \beta}} ) - \sum_{IJ}^N | \braket{\phi_{I \alpha}}{\phi_{J \alpha}} - \braket{\phi_{I \beta}}{\phi_{J \beta}} |^2 ] \end{equation}\] and \[\begin{equation} \ev{\hat{S}_- \hat{S}_+}{\text{core}} = \sum_I^N \braket{\phi_{I \beta}} + \sum_{IJ}^N \qty( \braket{\phi_{I \beta}}{\phi_{I \alpha}} \braket{\phi_{J \alpha}}{\phi_{J \beta}} - \braket{\phi_{I \beta}}{\phi_{J \alpha}} \braket{\phi_{J \alpha}}{\phi_{I \beta}} ) \thinspace . \end{equation}\]

Summing up all contributions, we find an expression for the expectation value of \(\hat{S}^2\): \[\begin{align} \ev{\hat{S}^2}{\text{core}} = &S_z (S_z + 1) + \frac{1}{4} \qty[ \sum_I^N \qty( \braket{\phi_{I \alpha}} + \braket{\phi_{I \beta}} ) - \sum_{IJ}^N | \braket{\phi_{I \alpha}}{\phi_{J \alpha}} - \braket{\phi_{I \beta}}{\phi_{J \beta}} |^2 ] \\ &+ \sum_I^N \braket{\phi_{I \beta}} + \sum_{IJ}^N \qty( \braket{\phi_{I \beta}}{\phi_{I \alpha}} \braket{\phi_{J \alpha}}{\phi_{J \beta}} - \braket{\phi_{I \beta}}{\phi_{J \alpha}} \braket{\phi_{J \alpha}}{\phi_{I \beta}} ) \thinspace . \end{align}\]