Spin in spin-orbital bases

Unrestricted spin-orbital bases

In a spin-orbital basis, the (one-electron) spin operators may be quantized as follows. For the components of \(\require{physics} \hat{\vb{S}}\), we have: \[\begin{align} & \hat{S}_x = \frac{1}{2} \sum_{p}^{K_\alpha} \sum_{q}^{K_\beta} \braket{\phi_{p \alpha}}{\phi_{q \beta}} \hat{a}^\dagger_{p \alpha} \hat{a}_{q \beta} + \frac{1}{2} \sum_{p}^{K_\beta} \sum_{q}^{K_\alpha} \braket{\phi_{p \beta}}{\phi_{q \alpha}} \hat{a}^\dagger_{p \beta} \hat{a}_{q \alpha} \\ % & \hat{S}_y = \frac{i}{2} \sum_{p}^{K_\beta} \sum_{q}^{K_\alpha} \braket{\phi_{p \beta}}{\phi_{q \alpha}} \hat{a}^\dagger_{p \beta} \hat{a}_{q \alpha} - \frac{i}{2} \sum_{p}^{K_\alpha} \sum_{q}^{K_\beta} \braket{\phi_{p \alpha}}{\phi_{q \beta}} \hat{a}^\dagger_{p \alpha} \hat{a}_{q \beta} \\ % & \hat{S}_z = \frac{1}{2} \qty( \hat{N}_\alpha - \hat{N}_\beta ) \thinspace , \end{align}\] in which something peculiar has happened in quantization of \(\hat{S}_z\). Due to its diagonal form, we now see the \(\alpha\)- and \(\beta\)-electron counting operators \(\hat{N}_\alpha\) and \(\hat{N}_\beta\) appear instead of general excitation operators \(\hat{E}^{\sigma \tau}_{pq}\). Since ONVs are eigenvectors of these number operators, this means that any ONV (for spin-separated spinors) will be an eigenvector of \(\hat{S}_z\).

The matrix elements of the spin-raising and -lowering operators have a particularly simple form: \[\begin{align} & \qty(\vb{S}_+)_{p\sigma, q\tau} = \delta_{\sigma \alpha} \delta_{\tau \beta} \braket{\phi_{p \alpha}}{\phi_{q \beta}} \\ % & \qty(\vb{S}_-)_{p\sigma, q\tau} = \delta_{\sigma \beta} \delta_{\tau \alpha} \braket{\phi_{p \beta}}{\phi_{q \alpha}} \thinspace , \end{align}\] leading to the second-quantized operators: \[\begin{align} & \hat{S}_+ = \sum_{p}^{K_\alpha} \sum_{q}^{K_\beta} \braket{\phi_{p \alpha}}{\phi_{q \beta}} \hat{a}^\dagger_{p \alpha} \hat{a}_{q \beta} \\ % & \hat{S}_- = \sum_{p}^{K_\beta} \sum_{q}^{K_\alpha} \braket{\phi_{p \beta}}{\phi_{q \alpha}} \hat{a}^\dagger_{p \beta} \hat{a}_{q \alpha} \thinspace . \end{align}\]

We should note that, similarly to the general case, in order to obtain faithful second-quantized spin operators, we must use equal underlying scalar functions for the \(\alpha\)-spinors and \(\beta\)-spinors.

Restricted spin-orbital bases

In a restricted spin-orbital bases(#restricted-spin-orbital-bases, the quantization of the components of the spin angular momentum operator turn out to be the following: \[\begin{align} \require{physics} & \hat{S}_x = \frac{1}{2} \sum_p^K \qty( \hat{a}^\dagger_{p \alpha} \hat{a}_{p \beta} + \hat{a}^\dagger_{p \beta} \hat{a}_{p \alpha} ) \\ % & \hat{S}_y = -\frac{i}{2} \sum_p^K \qty( \hat{a}^\dagger_{p \alpha} \hat{a}_{p \beta} - \hat{a}^\dagger_{p \beta} \hat{a}_{p \alpha} ) \\ % & \hat{S}_x = \frac{1}{2} \qty( \hat{N}_\alpha - \hat{N}_{\beta} ) \thinspace . \end{align}\]

The spin-raising and spin-lowering operators have the following second-quantized expressions: \[\begin{equation} \hat{S}_+ = \sum_p^K \hat{a}^\dagger_{p \alpha} \hat{a}_{p \beta} \end{equation}\] and \[\begin{equation} \hat{S}_- = \sum_p^K \hat{a}^\dagger_{p \beta} \hat{a}_{p \alpha} \thinspace . \end{equation}\]

In a spin-restricted spinor basis, the quantization of the components of the spin angular momentum operator turn out to be the following: \[\begin{align} \require{physics} & \hat{S}_x = \frac{1}{2} \sum_p^K \qty( \hat{a}^\dagger_{p \alpha} \hat{a}_{p \beta} + \hat{a}^\dagger_{p \beta} \hat{a}_{p \alpha} ) \\ % & \hat{S}_y = -\frac{i}{2} \sum_p^K \qty( \hat{a}^\dagger_{p \alpha} \hat{a}_{p \beta} - \hat{a}^\dagger_{p \beta} \hat{a}_{p \alpha} ) \\ % & \hat{S}_x = \frac{1}{2} \qty( \hat{N}_\alpha - \hat{N}_{\beta} ) \thinspace . \end{align}\]

The spin-raising and spin-lowering operators have the following second-quantized expressions: \[\begin{equation} \hat{S}_+ = \sum_p^K \hat{a}^\dagger_{p \alpha} \hat{a}_{p \beta} \end{equation}\] and \[\begin{equation} \hat{S}_- = \sum_p^K \hat{a}^\dagger_{p \beta} \hat{a}_{p \alpha} \thinspace . \end{equation}\]