Quantizing one- and two-electron operators in general spinor bases

We can use the total field operators \(\require{physics} \hat{\psi}(\vb{r})^\dagger\) and \(\hat{\psi}(\vb{r})\) to quantize the one- and two-electron operators.

If the coordinate representation of a one-electron operator for a system of \(N\) electrons is given by \[\begin{equation} f^c(\vb{r}_1, \ldots, \vb{r}_N) = \sum_{i = 1}^N f^c(\vb{r}_i) \thinspace , \end{equation}\] then in second quantization it is represented by the operator \[\begin{equation} \hat{f} = \int \dd{\vb{r}} \hat{\psi}^\dagger(\vb{r}) \thinspace f^c(\vb{r}) \thinspace \hat{\psi}(\vb{r}) \thinspace , \end{equation}\] in which 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 in general a scalar differential operator, which means that these scalar operators can only act on scalar basis functions (i.e. spinor components) and not on spinors themselves.

Expanding the field operators, we then have \[\begin{equation} \label{eq:one-electron_operator_spin_orbitals} \hat{f} = \sum_{PQ}^M f_{PQ} \hat{E}_{PQ} \thinspace , \end{equation}\] where the integrals are given by: \[\begin{align} f_{PQ} &= \matrixel{\phi_P}{f^c}{\phi_Q} \\ &= \int \dd{\vb{r}} \phi_P^\dagger(\vb{r}) \thinspace f^c(\vb{r}) \thinspace \phi_Q(\vb{r}) \thinspace , \end{align}\] which we will now explore a little further. Every one-electron operator in Pauli theory is necessarily a \((2 \times 2)\) matrix, so we can work out its matrix elements as \[\begin{align} f_{PQ} &= \int \dd{\vb{r}} \begin{pmatrix} \phi_{P \alpha}^*(\vb{r}) & \phi_{P \beta}^*(\vb{r}) \end{pmatrix} \thinspace % \begin{pmatrix} f^{c, \alpha \alpha}(\vb{r}) & f^{c, \alpha \beta}(\vb{r}) \\ f^{c, \beta \alpha}(\vb{r}) & f^{c, \beta \beta}(\vb{r}) \end{pmatrix} \thinspace \begin{pmatrix} \phi_{Q \alpha}(\vb{r}) \\ \phi_{Q \beta}(\vb{r}) \end{pmatrix} \\ &= \sum_{\sigma \tau} f^{\sigma \tau}_{P \sigma, Q \tau} \thinspace , \end{align}\] in which we have introduced the matrix representation of the underlying scalar one-electron operators (of the one-electron operator) in the components of spinor: \[\begin{equation} f_{P \sigma, Q \tau}^{\sigma \tau} = \int \dd{\vb{r}} \phi^*_{P \sigma}(\vb{r}) f^{c, \sigma \tau}(\vb{r}) \phi_{Q \tau}(\vb{r}) \thinspace . \end{equation}\]

Using the expansion of the components of the spinor in terms of the underlying scalar bases (cfr. \(\eqref{eq:spinor_expansion_scalar_bases}\)), 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.

As an example, we note that the Pauli overlap operator is represented by the \((2 \times 2)\)-identity matrix: \[\begin{equation} S(\vb{r}) = \begin{pmatrix} 1 % & 0 \\ 0 % & 1 \\ \end{pmatrix} \thinspace , \end{equation}\] 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 are expanded into 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.

A two-electron operator in coordinate representation \[\begin{equation} g^c(\vb{r}_1, \ldots, \vb{r}_N) = \frac{1}{2} \sum_{i \neq j}^N g^c(\vb{r}_i, \vb{r}_j) \thinspace , \end{equation}\] is then represented in second quantization by \[\begin{equation} \hat{g} = \frac{1}{2} \int \int \dd{\vb{r}}_1 \dd{\vb{r}}_2 \hat{\psi}^\dagger(\vb{r}_1) \hat{\psi}^\dagger(\vb{r}_2) \thinspace g^c(\vb{r}_1, \vb{r}_2) \thinspace \hat{\psi}(\vb{r}_2) \hat{\psi}(\vb{r}_1) \thinspace , \end{equation}\] such that the second-quantized operator can be conveniently written as: \[\begin{equation} \hat{g} = \frac{1}{2} \sum_{PQRS}^M g_{PQRS} \thinspace \hat{e}_{PQRS} \end{equation}\] with the integrals \[\begin{equation} g_{PQRS} = \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} \phi_P^\dagger(\vb{r}_1) \phi_R^\dagger(\vb{r}_2) \thinspace g^c(\vb{r}_1, \vb{r}_2) \thinspace \phi_S(\vb{r}_2) \phi_Q(\vb{r}_1) \thinspace . \end{equation}\] In my view, the operations that appear in equation \(\eqref{eq:two_electron_integrals_2C}\) are such that the resulting integral can be calculated analogously to equation \(\eqref{eq:one_electron_operator_2C}\) as \[\begin{equation} g_{PQRS} = \sum_{\sigma \tau \rho \pi} g^{\sigma \tau \rho \pi}_{P \sigma Q \tau R \rho S \pi} \thinspace , \end{equation}\] which is a quadruple summation over the components of every spinor, and where \(g^{\sigma \tau \rho \pi}_{P \sigma Q \tau R \rho S \pi}\) represents the integral over one underlying scalar two-electron operator \(g^{c, \sigma \tau \rho \pi}(\vb{r}_i, \vb{r}_j)\) that appears in the tensor operator \(g^c(\vb{r}_i, \vb{r}_j)\). For non-relativistic purposes, the only two-electron operator is the electron-electron Coulomb repulsion operator, whose underlying scalar components are given by: \[\begin{equation} g^{c, \sigma \tau \rho \pi}(\vb{r}_i, \vb{r}_j) = \delta_{\sigma \tau} \delta_{\rho \pi} \thinspace \frac{1}{\norm{\vb{r}_i - \vb{r}_j}} \thinspace . \end{equation}\] The Coulomb repulsion integrals can then be calculated as \[\begin{equation} g_{PQRS} = \sum_{\sigma \tau} (P \sigma Q \sigma | R \tau S \tau) \thinspace , \end{equation}\] which, due to the specific tensor nature of the Coulomb operator, corresponds to an interaction between two Pauli distributions at \(\vb{r}_1\) and \(\vb{r}_2\): \[\begin{equation} g_{PQRS} = \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} \phi_P^\dagger(\vb{r}_1) \phi_Q(\vb{r}_1) \thinspace \frac{1}{ \norm{\vb{r}_1 - \vb{r}_2} } \thinspace \phi_R^\dagger(\vb{r}_2) \phi_S(\vb{r}_2) \thinspace . \end{equation}\] We can think of the vertical bar in the notation \(( \cdot | \cdot )\) as a way to represent the scalar operator \[\begin{equation} \frac{1}{\norm{\vb{r}_i - \vb{r}_j}} \thinspace , \end{equation}\] which means that the total symbol \(( \cdot | \cdot )\) represents a way to write down a (scalar) Coulomb interaction energy between the distributions on the left- and right-hand sides. At this point, we should also already remark that \(g_{PQRS}\) is a measure of the Coulomb interaction energy, which should not be confused with any Coulomb integrals or exchange integrals, which merely result from the single-Slater determinant picture in a mean-field theory.

Using the expansion of the components of the spinor in terms of the underlying scalar bases (cfr. \(\eqref{eq:spinor_expansion_scalar_bases}\)), the matrix elements of a two-electron operator can be calculated as: \[\begin{equation} g_{PQRS} = \sum_{\sigma \tau} \sum_{\mu \nu}^{K_\sigma} \sum_{\rho \lambda}^{K_\tau} C^{\sigma *}_{\mu P} C^{\sigma}_{\nu Q} C^{\tau *}_{\rho R} C^{\tau}_{\lambda S} (\mu \sigma \nu \sigma | \rho \tau \lambda \tau) \thinspace , \end{equation}\] in which \(\vb{C}^{\sigma}\) is the coefficient matrix for the \(\sigma\)-component and \((\mu \sigma \nu \sigma | \rho \tau \lambda \tau)\) is a Coulomb integral over scalar basis functions: \[\begin{equation} (\mu \sigma \nu \sigma | \rho \tau \lambda \tau) = \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} \chi^{\sigma *}_\mu(\vb{r}_1) \chi^{\sigma}_{\nu}(\vb{r}_1) \frac{1}{\norm{\vb{r}_1 - \vb{r}_2}} \chi^{\tau *}_{\rho}(\vb{r}_2) \chi^{\tau}_{\lambda}(\vb{r}_2) \thinspace . \end{equation}\]

The notation of the two-electron integrals is often a source of confusion, because we have Mulliken’s notation \(g_{PQRS}\), chemist’s notation \((PQ|RS)\) and physicist’s notation \(\braket{PQ}{RS}\), which are linked by \[\begin{equation} g_{PQRS} = (PQ|RS) = \braket{PR}{QS} \thinspace . \end{equation}\] Furthermore, specifically in some physics literature literature, we sometimes find another notation for the two-electron integrals. If we define an antisymmetrized integral: \[\begin{equation} g^{\mathcal{A}}_{PQRS} = g_{PQRS} - g_{PSRQ} \thinspace , \end{equation}\] the Hamiltonian can equivalently be written as \[\begin{equation} \hat{\mathcal{H}} = \sum_{PQ}^M h_{PQ} \hat{E}_{PQ} + \frac{1}{4} \sum_{PQRS}^M g^{\mathcal{A}}_{PQRS} \thinspace \hat{e}_{PQRS} + h_{\text{nuc}} \thinspace . \end{equation}\]

For complex spinors, the following symmetries hold for the one-electron integrals, due to the Hermitian nature of the Pauli Hamiltonian: \[\begin{equation} h_{QP} = h_{PQ}^* \thinspace . \end{equation}\] The two-electron Coulomb repulsion integrals obey the following symmetries: \[\begin{equation} \label{eq:two_electron_integrals_complex_conjugation} g_{QPSR} = g_{PQRS}^* \end{equation}\] and \[\begin{equation} g_{RSPQ} = g_{PQRS} \thinspace . \end{equation}\] When working with real spinors, we additionally have the following four-fold permutational symmetry: \[\begin{equation} g_{PQRS} = g_{PQSR} = g_{QPRS} = g_{QPSR} \thinspace . \end{equation}\]