Spin in general spinor bases
Using a general spinor basis, the individual components of the spin operator \(\require{physics} \hat{\vb{S}}\) are given by \[\begin{equation} \hat{S}_i = \frac{1}{2} \sum_{PQ}^M \sigma_{i, PQ} \hat{E}_{PQ} \thinspace , \end{equation}\] in which we have introduced the second-quantized matrix elements of the Pauli matrices: \[\begin{equation} \sigma_{i, PQ} = \int \dd{\vb{r}} \phi_P(\vb{r})^\dagger \thinspace \sigma_i \thinspace \phi_Q(\vb{r}) \thinspace . \end{equation}\] These integrals can be calculated by expanding \(\phi_P^\dagger\), \(\phi_Q\), and \(\sigma_i\) in spinor basis. For example, for \(\sigma_{x,PQ}\) may be calculated as \[\begin{align} \sigma_{x,PQ} &= \int d\textbf{r}~ \begin{pmatrix} \phi_{P\alpha}^{\dagger} & \phi_{P\beta}^{\dagger} \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \phi_{Q\alpha} \\ \phi_{Q\beta} \end{pmatrix} \\ &= \int d\textbf{r}~ \begin{pmatrix} \phi_{P\alpha}^{\dagger} & \phi_{P\beta}^{\dagger} \end{pmatrix} \begin{pmatrix} \phi_{Q\beta} \\ \phi_{Q\alpha} \end{pmatrix} \\ &= \int d\textbf{r}~ \left[ \left(\phi_{P\alpha}^{\dagger} \phi_{Q\beta} \right) + \left(\phi_{P\beta}^{\dagger} \phi_{Q\alpha} \right) \right] = \braket{\phi_{P\alpha}}{\phi_{Q\beta}} + \braket{\phi_{P\beta}}{\phi_{Q\alpha}}. \end{align}\]
Using this framework, the spin operators \(\hat{S}_i, i \in [x,y,z]\) may be written as (Cassam-Chenaï 2015) \[\begin{align} & \sigma_{x, PQ} = \braket{\phi_{P \alpha}}{\phi_{Q \beta}} + \braket{\phi_{P \beta}}{\phi_{Q \alpha}} \\ % & \sigma_{y, PQ} = i \braket{\phi_{P \beta}}{\phi_{Q \alpha}} - i \braket{\phi_{P \alpha}}{\phi_{Q \beta}} \\ % & \sigma_{z, PQ} = \braket{\phi_{P \alpha}}{\phi_{Q \alpha}} - \braket{\phi_{P \beta}}{\phi_{Q \beta}} \thinspace , \end{align}\] we can calculate the second-quantized representations of the components of the spin operator as \[\begin{align} & \hat{S}_x = \frac{1}{2} \sum_{PQ}^M \qty( \braket{\phi_{P \alpha}}{\phi_{Q \beta}} + \braket{\phi_{P \beta}}{\phi_{Q \alpha}} ) \hat{E}_{PQ} \\ % & \hat{S}_y = \frac{i}{2} \sum_{PQ}^M \qty( \braket{\phi_{P \beta}}{\phi_{Q \alpha}} - \braket{\phi_{P \alpha}}{\phi_{Q \beta}} ) \hat{E}_{PQ} \\ % & \hat{S}_z = \frac{1}{2} \sum_{PQ}^M \qty( \braket{\phi_{P \alpha}}{\phi_{Q \alpha}} - \braket{\phi_{P \beta}}{\phi_{Q \beta}} ) \hat{E}_{PQ} \\ % \end{align}\]
Equivalently, the matrix representations of the components of the spin operator are given by: \[\begin{align} & \vb{S}_x = \frac{1}{2} \vb{C}^\dagger \begin{pmatrix} \vb{0} & \vb{S}^{\alpha \beta} \\ \vb{S}^{\beta \alpha} & \vb{0} \end{pmatrix} \vb{C} \\ % & \vb{S}_y = \frac{i}{2} \vb{C}^\dagger \begin{pmatrix} \vb{0} & -\vb{S}^{\alpha \beta} \\ \vb{S}^{\beta \alpha} & \vb{0} \end{pmatrix} \vb{C} \\ % & \vb{S}_z = \frac{1}{2} \vb{C}^\dagger \begin{pmatrix} \vb{S}^{\alpha \alpha} & \vb{0} \\ \vb{0} & -\vb{S}^{\beta \beta} \end{pmatrix} \vb{C} \thinspace , \end{align}\] with \(\vb{C}\) the coefficient matrix.
Concerning the spin raising and spin lowering operators, we find: \[\begin{equation} \hat{S}_+ = \sum_{PQ}^M \braket{\phi_{P \alpha}}{\phi_{Q \beta}} \hat{E}_{PQ} \end{equation}\] and \[\begin{equation} \hat{S}_- = \sum_{PQ}^M \braket{\phi_{P \beta}}{\phi_{Q \alpha}} \hat{E}_{PQ} \thinspace . \end{equation}\]
The matrix representations are given by: \[\begin{align} & \vb{S}_+ = \vb{C}^\dagger \begin{pmatrix} \vb{0} & \vb{S}^{\alpha \beta} \\ \vb{0} & \vb{0} \end{pmatrix} \vb{C} \\ % & \vb{S}_- = \vb{C}^\dagger \begin{pmatrix} \vb{0} & \vb{0} \\ \vb{S}^{\beta \alpha} & \vb{0} \end{pmatrix} \vb{C} \end{align}\]
Do the usual commutation relations for the spin angular momentum also hold in the (projected) second-quantized formalism? For this, we should verify if indeed the usual commutator also holds for the matrix representations. We’ll take a look at one example and try to verify if indeed: \[\begin{equation} \comm{\vb{S}_x}{\vb{S}_y} \overset{?}{=} i \vb{S}_z \thinspace . \end{equation}\]
Combining the expressions for \(\vb{S}_x\) and \(\vb{S}_y\) and realizing that \(\vb{C} \vb{C}^\dagger = \vb{S}^{-1}\), we find: \[\begin{equation} \comm{\vb{S}_x}{\vb{S}_y} = \frac{i}{2} \vb{C}^\dagger \begin{pmatrix} \vb{S}^{\alpha \beta} \qty(\vb{S}^{\beta \beta})^{-1} \vb{S}^{\beta \alpha} & 0 \\ 0 & -\vb{S}^{\beta \alpha} \qty(\vb{S}^{\alpha \alpha})^{-1} \vb{S}^{\alpha \beta} \end{pmatrix} \vb{C} \thinspace . \end{equation}\] Unfortunately, we cannot a priory assume that the upper-left corner yields the required result: \[\begin{equation} \vb{S}^{\alpha \beta} \qty(\vb{S}^{\beta \beta})^{-1} \vb{S}^{\beta \alpha} \not\equiv \vb{S}^{\alpha \alpha} \thinspace , \end{equation}\] and similarly for the lower-right corner. In fact, it is possible that this relation is only satisfied for two different underlying scalar bases if they are both complete. We should therefore conclude that, using different underlying scalar bases for the \(\alpha\)- and \(\beta\)-component yields quantized spin operators that do not anymore reflect an angular momentum and as such this should be avoided.
Fortunately, we can escape this pecularity by using the same scalar functions for the expansion of the \(\alpha\)- and \(\beta\)-components of general spinor bases. We then immediately find: \[\begin{align} \comm{\vb{S}_x}{\vb{S}_y} &= \frac{i}{2} \vb{C}^\dagger \begin{pmatrix} \vb{S}^{\text{AO}} \qty(\vb{S}^{\text{AO}})^{-1} \vb{S}^{\text{AO}} & 0 \\ 0 & -\vb{S}^{\text{AO}} \qty(\vb{S}^{\text{AO}})^{-1} \vb{S}^{\text{AO}} \end{pmatrix} \vb{C} \\ &= \frac{i}{2} \vb{C}^\dagger \begin{pmatrix} \vb{S}^{\text{AO}} & 0 \\ 0 & -\vb{S}^{\text{AO}} \end{pmatrix} \vb{C} \\ &= i \vb{S}_z \thinspace , \end{align}\] where \(\vb{S}^{\text{AO}}\) is the overlap matrix of the underlying scalar orbitals.