The Pauli overlap operator

The Pauli overlap operator is represented by the \((2 \times 2)\)-identity matrix: \[\begin{equation} \require{physics} S(\vb{r}) = \begin{pmatrix} 1 % & 0 \\ 0 % & 1 \\ \end{pmatrix} \thinspace , \end{equation}\] such that the elements of the spinor overlap matrix can be calculated as \[\begin{equation} S_{PQ} = \braket{P \alpha}{Q \alpha} + \braket{P \beta}{Q \beta} \thinspace . \end{equation}\]

When expanding the spinors into their underlying scalar bases, the spinor overlap matrix can be calculated as: \[\begin{equation} S_{PQ} = \sum_{\mu \nu}^{K_\alpha} C^{\alpha *}_{\mu P} C^{\alpha}_{\nu Q} S^{\alpha \alpha}_{\mu \nu} + \sum_{\mu \nu}^{K_\beta} C^{\beta *}_{\mu P} C^{\beta}_{\nu Q} S^{\beta \beta}_{\mu \nu} \thinspace , \end{equation}\] or, equivalently: \[\begin{equation} \vb{S} = \vb{C}^\dagger \begin{pmatrix} \vb{S}^{\alpha \alpha} & \vb{0} \\ \vb{0} & \vb{S}^{\beta \beta} \\ \end{pmatrix} \vb{C} \thinspace , \end{equation}\] where \(\vb{S}^{\alpha \alpha}\) and \(\vb{S}^{\beta \beta}\) are the overlap matrices in the underlying scalar bases for the \(\alpha\)- and \(\beta\) components.