Quantizing one-electron operators in general spinor bases

If the coordinate representation of a one-electron operator for a system of \(N\) electrons is given by \[\begin{equation} \require{physics} f^c(\vb{r}_1, \ldots, \vb{r}_N) = \sum_{i = 1}^N f^c(\vb{r}_i) \thinspace , \end{equation}\] then in second quantization it is represented by the operator \[\begin{equation} \hat{f} = \int \dd{\vb{r}} \hat{\phi}^\dagger(\vb{r}) \thinspace f^c(\vb{r}) \thinspace \hat{\phi}(\vb{r}) \thinspace , \end{equation}\] where \(\hat{\phi}(\vb{r})\) is a field operator and the coordinate representation of the one-electron operator \(f^c(\vb{r})\) is necessarily a \((2 \times 2)\) matrix operator: \[\begin{equation} f^c(\vb{r}) = \begin{pmatrix} f^{c, \alpha \alpha}(\vb{r}) & f^{c, \alpha \beta}(\vb{r}) \\ f^{c, \beta \alpha}(\vb{r}) & f^{c, \beta \beta}(\vb{r}) \end{pmatrix} \thinspace . \end{equation}\] Every underlying one-electron operator \(f^{c, \sigma \tau}(\vb{r})\) is a scalar operator, which means that these scalar operators can only act on scalar basis functions (i.e. spinor components) and not on two-component spinors themselves.

Expanding the field operators, we then obtain \[\begin{equation} \hat{f} = \sum_{PQ}^M f_{PQ} \hat{E}_{PQ} \thinspace , \end{equation}\] where the one-electron integrals \(f_{PQ}\) are given by: \[\begin{align} f_{PQ} &= \matrixel{\phi_P}{f^c}{\phi_Q} \\ &= \int \dd{\vb{r}} \phi_P^\dagger(\vb{r}) \thinspace f^c(\vb{r}) \thinspace \phi_Q(\vb{r}) \\ &= \sum_{\sigma \tau} f^{\sigma \tau}_{P \sigma, Q \tau} \thinspace , \end{align}\] where we have introduced the integrals of the underlying scalar one-electron operators: \[\begin{equation} f_{P \sigma, Q \tau}^{\sigma \tau} = \int \dd{\vb{r}} \phi^*_{P \sigma}(\vb{r}) f^{c, \sigma \tau}(\vb{r}) \phi_{Q \tau}(\vb{r}) \thinspace . \end{equation}\] If the one-electron operator \(f^c(\vb{r})\) is Hermitian, the one-electron integrals obey the following permutational symmetry: \[\begin{equation} f_{PQ}^* = f_{QP} \thinspace . \end{equation}\]

Using the expansion of the components of the spinor in terms of the underlying scalar bases, the matrix elements of a one-electron operator can be calculated as: \[\begin{equation} f_{PQ} = \sum_{\sigma \tau} \sum_{\mu}^{K_\sigma} \sum_{\nu}^{K_\tau} C^{\sigma *}_{\mu P} f_{\mu \nu}^{\sigma \tau} C^{\tau}_{\nu Q} \thinspace , \end{equation}\] where we have introduced the one-electron integrals in the underlying scalar basis as: \[\begin{equation} f_{\mu \nu}^{\sigma \tau} = \int \dd{\vb{r}} \chi^{\sigma *}_\mu(\vb{r}) f^{c, \sigma \tau}(\vb{r}) \chi^{\tau}_\nu(\vb{r}) \thinspace . \end{equation}\] We should note that we can rewrite equation \(\eqref{eq:one_electron_integrals_expanded}\) using matrix multiplications as \[\begin{equation} \label{eq:one_electron_integrals_expansion} \vb{f} % = \vb{C}^\dagger \begin{pmatrix} \vb{f}^{\alpha \alpha} & \vb{f}^{\alpha \beta} \\ \vb{f}^{\beta \alpha} & \vb{f}^{\beta \beta} \\ \end{pmatrix} \vb{C} \thinspace , \end{equation}\] where \(\vb{f}^{\sigma \tau}\) is the matrix representation of \(f^{c, \sigma \tau}(\vb{r})\) in terms of the underlying scalar bases for the \(\sigma\) and \(\tau\) components. We should already note that, in order to build up correct second-quantized spin operators, the \(\alpha\)- and \(\beta\) scalar bases must be equal.

There are many examples of one-electron operators:

• the Pauli overlap operator;
• the spin operators.