Spin-orbital bases

Spin-orbital bases arise naturally for simplified Pauli Hamiltonians.

Expanding the \(\alpha\)-components of the spinors in a space of \(K_\alpha\) scalar functions (and similarly for \(K_\beta\)), the total number of spin-separated spinors \(M\) we can construct is still \(M = K_\alpha + K_\beta\), but the total \((M \times M)\)-coefficient matrix can now be written as \[\begin{equation} \label{eq:coefficient_matrix_spin_separated} \require{physics} \vb{C} = \begin{pmatrix} \vb{C}^\alpha & \vb{0} \\ \vb{0} & \vb{C}^\beta \end{pmatrix} \thinspace , \end{equation}\] in which \(\vb{C}^\alpha\) is now a \((K_\alpha \times K_\alpha)\)-block that describes the expansion of the spin-separated \(\alpha\)-spinors in terms of its underlying scalar orbitals (basis functions). Similarly, we have that \(\vb{C}^\beta\) collects the expansion coefficients of the spin-separated \(\beta\)-spinors.

We should emphasize that the total dimension of the coefficient matrix \(\vb{C}\) remains \((M \times M)\), but that the effective sub-blocks \(\vb{C}^\alpha\) and \(\vb{C}^\beta\) are square, of dimension \((K_\alpha \times K_\alpha)\) and \((K_\beta \times K_\beta)\), respectively. The total number of complex expansion coefficients for spin-separated spinors has reduced from \((K_\alpha + K_\beta)^2\) to \(K_\alpha^2 + K_\beta^2\), since \(2 K_\alpha K_\beta\) parameters (the off-diagonal blocks) have been set to zero.

Since the coefficient matrix must always obey \[\begin{equation} \vb{C}^\dagger \vb{S} \vb{C} = \vb{I} \thinspace , \end{equation}\] which becomes in the spin-separated case: \[\begin{equation} \begin{pmatrix} \vb{C}^{\alpha, \dagger} & \vb{0} \\ \vb{0} & \vb{C}^{\beta, \dagger} \end{pmatrix} \begin{pmatrix} \vb{S}^{\alpha \alpha} & \vb{0} \\ \vb{0} & \vb{S}^{\beta \beta} \\ \end{pmatrix} \begin{pmatrix} \vb{C}^\alpha & \vb{0} \\ \vb{0} & \vb{C}^\beta \end{pmatrix} = \begin{pmatrix} \vb{I} & \vb{0} \\ \vb{0} & \vb{I} \\ \end{pmatrix} \thinspace , \end{equation}\] the \(\alpha\)- and \(\beta\)-spinors must be orthonormal within each spin set: \[\begin{equation} \vb{C}^{\sigma, \dagger} \vb{S}^{\sigma \sigma} \vb{C}^{\sigma} = \vb{I} \thinspace . \end{equation}\] Equivalently, we can write: \[\begin{equation} \int \dd{\vb{r}} \phi^*_{p \sigma}(\vb{r}) \phi_{q \tau}(\vb{r}) = \delta_{\sigma \tau} \delta_{pq} \thinspace , \end{equation}\] where we have made the convention to use smaller case indices when working in a spin-orbital basis. This orthonormality of the spin-orbitals has a large consequence on the fermionic field operator.