General Hartree-Fock theory
Generalized Hartree-Fock theory is perhaps the most fundamental approach to tackle the Hartree-Fock wave function model. The name comes from a symmetry-breaking point of view, in which is the most general form and (restricted) is the least general form. (Seeger and Pople 1977) Perhaps it is more fundamental to call GHF-theory just HF-theory with spinors. All the other HF-related theory then just deals with certain restrictions on the spinors, as discussed in section \(\ref{eq:spinor_to_spatial_orbital_rotations}\) on spinor rotations.
It is suitable as an initial guess in X2C theory, and can be used in conjunction with spin-orbit coupling approaches (Desmarais, Flament, and Erba 2019).
The HF wave function model has an exponential parametrization: \[\begin{equation} \ket{\text{HF}(\boldsymbol{\kappa})} = \exp(-\hat{\kappa}) \ket{\text{core}} \thinspace , \end{equation}\] where \(\ket{\text{core}}\) represents the Slater determinant in which the first \(N\) orbitals are considered `occupied’: \[ \require{physics} \begin{equation} \ket{\text{core}} = \qty( \prod_I^N \hat{a}^\dagger_I ) \ket{\text{vac}} \thinspace . \end{equation} \] Since the wave function model is normalized for any values of the parameters, we will use the normalized Rayleigh quotient for our energy function: \[\begin{equation} E(\boldsymbol{\kappa}) = \ev{ \exp(\hat{\kappa}) \hat{\mathcal{H}} \exp(-\hat{\kappa}) }{ \text{core} } \thinspace , \end{equation}\] which represents the energy of the rotated state in the original spinor basis as the expectation value of the molecular electronic Hamiltonian: \[\begin{equation} \hat{\mathcal{H}} = \sum_{PQ}^M h_{PQ} \hat{E}_{PQ} + \frac{1}{2} \sum_{PQRS}^M g_{PQRS} \hat{e}_{PQRS} \thinspace . \end{equation}\]
At the current orbitals, which is a situation that we represent by \(\boldsymbol{\kappa}_0 = \vb{0}\), the \(1\)- and \(2\)-DMs are given by \[\begin{equation} D_{PQ} \rightarrow D_{IJ} = \delta_{IJ} \end{equation}\] and \[\begin{equation} d_{PQRS} \rightarrow d_{IJKL} = \delta_{IJ} \delta_{KL} - \delta_{IL} \delta_{KJ} \thinspace , \end{equation}\] in which \(I\) and \(J\) are indices of spinors occupied in the Hartree-Fock core determinant \(\ket{\text{core}}\). If the indices \(PQRS\) are anything but occupied spinor indices, the HF density matrix vanishes. The HF energy can then be straightforwardly calculated (Stuber and Paldus 2003): \[\begin{equation} E = \sum_I^N h_{II} + \frac{1}{2} \sum_{IJ}^N (g_{IIJJ} - g_{IJJI}) \thinspace , \end{equation}\] which becomes \[\begin{equation} E = \frac{1}{2} \sum_{\sigma \tau} \sum_{\mu}^{K_\sigma} \sum_{\nu}^{K_\tau} P^{\sigma \tau}_{\mu \nu} \qty( h^{\sigma \tau}_{\mu \nu} + F^{\sigma \tau}_{\mu \nu} ) \thinspace , \end{equation}\] where we have used the AO density matrix \(\vb{P}\) and Fock matrix \(\vb{F}\).
The GHF wave function model can be optimized:
- through Lagrange’s method of multipliers;
- using a second-order approach using the GHF orbital gradient and Hessian.