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 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: \[\begin{align} \vb{f} &= \vb{C}^\dagger \begin{pmatrix} \vb{f}^{\alpha \alpha} & \vb{f}^{\alpha \beta} \\ \vb{f}^{\beta \alpha} & \vb{f}^{\beta \beta} \\ \end{pmatrix} \vb{C} \\ &= \begin{pmatrix} \vb{C}^\dagger & \vb{0} \\ \vb{0} & \vb{C} \end{pmatrix} \begin{pmatrix} \vb{f}^{\alpha \alpha} & \vb{f}^{\alpha \beta} \\ \vb{f}^{\beta \alpha} & \vb{f}^{\beta \beta} \\ \end{pmatrix} \begin{pmatrix} \vb{C} & \vb{0} \\ \vb{0} & \vb{C} \end{pmatrix} \\ &= \begin{pmatrix} \vb{C}^\dagger \vb{f}^{\alpha \alpha} \vb{C} & \vb{C}^\dagger \vb{f}^{\alpha \beta} \vb{C} \\ \vb{C}^\dagger \vb{f}^{\beta \alpha} \vb{C} & \vb{C}^\dagger \vb{f}^{\beta \beta} \vb{C} \\ \end{pmatrix} \thinspace , \end{align}\] 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 a spin-separated spinor basis. 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 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 \[\begin{equation} \hat{e}_{pqrs} = % \sum_{\sigma \tau} % \hat{a}^\dagger_{p \sigma} % \hat{a}^\dagger_{r \tau} % \hat{a}_{s \tau} % \hat{a}_{q \sigma} % \thinspace , \end{equation}\] so that the two-electron part of the Hamiltonian can simply be written as \[\begin{equation} \hat{g} = \frac{1}{2} \sum_{pqrs}^K \hat{e}_{pqrs} % \thinspace . \end{equation}\] Again, but now for spatial orbitals, we can introduce a chemist’s notation and a physicist’s notation: \[\begin{equation} g_{pqrs} = (pq|rs) = \braket{pr}{qs} % \thinspace , \end{equation}\] which are symmetric in a switching of the pair indices: \[\begin{equation} g_{rspq} = g_{pqrs} \end{equation}\] and whose complex conjugates are related in the following way: \[\begin{equation} \label{eq:g_pqrs_complex_conjugate} g_{qpsr} = g_{pqrs}^* % \thinspace . \end{equation}\]
The molecular Hamiltonian, in abscence of any magnetic field, thus takes the following form: \[\begin{equation} \hat{\mathcal{H}} % = \sum_{pq}^K h_{pq} \hat{E}_{pq} % + \frac{1}{2} \sum_{pqrs}^K g_{pqrs} \hat{e}_{pqrs} % \label{eq:Hamiltonian_spin_separated} \thinspace , \end{equation}\] which is the form that is used in Helgaker’s book (T. Helgaker, Jørgensen, and Olsen 2000). If we then also introduce the effective one-electron integrals \(k_{pq}\) as \[\begin{equation} \label{eq:effective_one_electron_integrals} k_{pq} = h_{pq} - \frac{1}{2} \sum_{r}^K g_{prrq} % \thinspace , \end{equation}\] we can alternatively write the Hamiltonian as \[\begin{equation} \hat{\mathcal{H}} = % \sum_{pq}^K k_{pq} \hat{E}_{pq} % + \frac{1}{2} \sum_{pqrs}^K g_{pqrs} \hat{E}_{pq} \hat{E}_{rs} % \thinspace . \end{equation}\]