Quantizing operators in spin-orbital bases

‘Unrestricted’ spin-orbital bases

In a spin-orbital basis, the field operators take on a simplified form. Therefore, we may simplify the expressions from the quantization of one-electron operators in general spinor bases. A general one-electron operator is still a \((2 \times 2)\)-matrix operator: \[\begin{equation} \require{physics} f^c(\vb{r}) = \begin{pmatrix} f^{c, \alpha \alpha}(\vb{r}) & f^{c, \alpha \beta}(\vb{r}) \\ f^{c, \beta \alpha}(\vb{r}) & f^{c, \beta \beta}(\vb{r}) \end{pmatrix} \thinspace , \end{equation}\] but the use of the spin-orbital field operators now yields the following second-quantized operator: \[\begin{equation} \hat{f} = \sum_{\sigma \tau} \sum_{pq} f_{p\sigma, q\tau} \hat{a}^\dagger_{p \sigma} \hat{a}_{q \tau} \thinspace , \end{equation}\] where the matrix elements \(f_{p\sigma, q\tau}\) are given by: \[\begin{align} f_{p\sigma, q\tau} &= \matrixel{\phi_{p\sigma}}{f^{c, \sigma \tau}}{\phi_{q\tau}} \\ &= \int \dd{\vb{r}} \phi^*_{p \sigma}(\vb{r}) f^{c, \sigma \tau}(\vb{r}) \phi_{q \tau}(\vb{r}) \thinspace . \end{align}\] Using matrix-matrix multiplications, they can be rewritten as: \[\begin{align} \vb{f} &= \vb{C}^\dagger \begin{pmatrix} \vb{f}^{\alpha \alpha} & \vb{f}^{\alpha \beta} \\ \vb{f}^{\beta \alpha} & \vb{f}^{\beta \beta} \\ \end{pmatrix} \vb{C} \\ &= \begin{pmatrix} \vb{C}^{\alpha, \dagger} & \vb{0} \\ \vb{0} & \vb{C}^{\beta, \dagger} \end{pmatrix} \begin{pmatrix} \vb{f}^{\alpha \alpha} & \vb{f}^{\alpha \beta} \\ \vb{f}^{\beta \alpha} & \vb{f}^{\beta \beta} \\ \end{pmatrix} \begin{pmatrix} \vb{C}^\alpha & \vb{0} \\ \vb{0} & \vb{C}^\beta \end{pmatrix} \\ &= \begin{pmatrix} \vb{C}^{\alpha, \dagger} \vb{f}^{\alpha \alpha} \vb{C}^\alpha & \vb{C}^{\alpha, \dagger} \vb{f}^{\alpha \beta} \vb{C}^\beta \\ \vb{C}^{\beta, \dagger} \vb{f}^{\beta \alpha} \vb{C}^\alpha & \vb{C}^{\beta, \dagger} \vb{f}^{\beta \beta} \vb{C}^\beta \\ \end{pmatrix} \thinspace . \end{align}\]

For one-electron operators that are diagonal in their two-component structure, the integrals become \[\begin{equation} f_{p \sigma, q \tau} = \delta_{\sigma \tau} \int \dd{\vb{r}} \phi^*_{p \sigma}(\vb{r}) f^{c, \sigma \sigma}(\vb{r}) \phi_{q \sigma}(\vb{r}) \thinspace , \end{equation}\] such that the second-quantized operator takes the following simplified form: \[\begin{equation} \hat{f} = \sum_\sigma \sum_{pq} f_{p \sigma, q \sigma} \hat{a}^\dagger_{p \sigma} \hat{a}_{q \sigma} \thinspace . \end{equation}\]

Similarly to the one-electron integrals, we can simplify the second-quantized Coulomb operator to: \[\begin{equation} \hat{g} = \frac{1}{2} \sum_{\sigma \tau} \sum_{pqrs} (p \sigma q \sigma | r \tau s \tau) \hat{a}^\dagger_{p \sigma} \hat{a}^\dagger_{r \tau} \hat{a}_{s \tau} \hat{a}_{q \sigma} \thinspace . \end{equation}\]

Restricted spin-orbital bases

When we additionally impose the restriction that the same scalar orbitals should be used for both the \(\alpha\)- and \(\beta\)-components: \[\begin{equation} \require{physics} \phi_p(\vb{r}) = \phi_{p \alpha}(\vb{r}) = \phi_{p \beta}(\vb{r}) \thinspace , \end{equation}\] we are said to be working in a restricted spin-orbital formalism. Having \(K_\alpha = K_\beta = K\) basis functions to our disposal (which are obviously required to be the same for the \(\alpha\)- and \(\beta\)-components), the total coefficient matrix is still of dimension \((2K \times 2K)\): \[\begin{equation} \vb{C}_{\text{total}} = \begin{pmatrix} \vb{C} & \vb{0} \\ \vb{0} & \vb{C} \end{pmatrix} \thinspace , \end{equation}\] but the \(\alpha\) and \(\beta\) expansion coefficients are now equal: \[\begin{equation} \vb{C}^\alpha = \vb{C}^\beta = \vb{C} \end{equation}\] and collected in a \((K \times K)\)-coefficient matrix \(\vb{C}\).

This means that the matrix elements of any second-quantized one-electron operator in a spin-restricted spinor basis become: \[\begin{align} \vb{f} &= \vb{C}^\dagger \begin{pmatrix} \vb{f}^{\alpha \alpha} & \vb{f}^{\alpha \beta} \\ \vb{f}^{\beta \alpha} & \vb{f}^{\beta \beta} \\ \end{pmatrix} \vb{C} \\ &= \begin{pmatrix} \vb{C}^\dagger & \vb{0} \\ \vb{0} & \vb{C}^\dagger \end{pmatrix} \begin{pmatrix} \vb{f}^{\alpha \alpha} & \vb{f}^{\alpha \beta} \\ \vb{f}^{\beta \alpha} & \vb{f}^{\beta \beta} \\ \end{pmatrix} \begin{pmatrix} \vb{C} & \vb{0} \\ \vb{0} & \vb{C} \end{pmatrix} \\ &= \begin{pmatrix} \vb{C}^\dagger \vb{f}^{\alpha \alpha} \vb{C} & \vb{C}^\dagger \vb{f}^{\alpha \beta} \vb{C} \\ \vb{C}^\dagger \vb{f}^{\beta \alpha} \vb{C} & \vb{C}^\dagger \vb{f}^{\beta \beta} \vb{C} \\ \end{pmatrix} \thinspace , \end{align}\] such that the second-quantized representation of a one-electron operator becomes \[\begin{equation} \hat{f} = \sum_{\sigma \tau} \sum_{p}^{K_\sigma} \sum_{q}^{K_\tau} f_{p\sigma, q\tau} \hat{a}^\dagger_{p \sigma} \hat{a}_{q \tau} \thinspace , \end{equation}\] which is exactly of the same form as in the ‘unrestricted’ spin-orbital case. Only the matrix elements \(f_{p\sigma, q\tau}\) have a different form, since the \(\alpha\)-spinors are equal to the \(\beta\)-spinors: \[\begin{align} f_{p\sigma, q\tau} &= \matrixel{\phi_p}{f^{c, \sigma \tau}}{\phi_q} \\ &= \int \dd{\vb{r}} \phi^*_p(\vb{r}) f^{c, \sigma \tau}(\vb{r}) \phi_q(\vb{r}) \thinspace . \end{align}\] For one-electron operators that have a diagonal \((2 \times 2)\)-structure that is related to the identity matrix, the integrals become \[\begin{equation} f_{p \alpha, q \alpha} = f_{p \beta, q \beta} = f_{pq} \thinspace , \end{equation}\] such that the one-electron operator can be rewritten as \[\begin{equation} \hat{f} = \sum_{pq} f_{pq} \hat{E}_{pq} \end{equation}\] by introducing the one-electron singlet excitation operator \(\hat{E}_{pq}\).

For general two-electron operators (that are Coulomb-like in their tensor structure), we can write: \[\begin{equation} \hat{g} = \frac{1}{2} \sum_{pqrs}^K g_{pqrs} \sum_{\sigma \tau} \hat{a}^\dagger_{p \sigma} \hat{a}^\dagger_{r \tau} \hat{a}_{s \tau} \hat{a}_{q \sigma} \thinspace , \end{equation}\] in which \[\begin{align} g_{pqrs} &= (p\alpha q\alpha | r\alpha s \alpha) = (p\alpha q\alpha | r\beta s \beta) = (p\alpha q\alpha | r\beta s \beta) = (p\beta q\beta | r\beta s \beta) \\ % &= \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} \phi_p^*(\vb{r}_1) \phi_q(\vb{r}_1) \thinspace \frac{1}{\norm{\vb{r}_1 - \vb{r}_2}} \thinspace \phi_r^*(\vb{r}_2) \phi_{s}(\vb{r}_2) \thinspace . \end{align}\] We will finally introduce the two-electron singlet excitation operator, so that we can write: \[\begin{equation} \hat{g} = \frac{1}{2} \sum_{pqrs}^K g_{pqrs} \hat{e}_{pqrs} \thinspace . \end{equation}\]