Second-order optimization for GHF

We can formulate the GHF optimization as an application of orbital optimization, but using the specific form of the HF density matrices. We don’t have to explicitly consider all variations of all orbital rotation generators, however, since some orbital rotation generators are redundant in GHF theory.

Using the specific and simple form of the HF density matrices, the HF Fockian matrix can be calculated as: \[\begin{align} \require{physics} \mathscr{F}_{PQ}(\hat{\mathcal{H}}) \rightarrow \mathscr{F}_{IP}(\hat{\mathcal{H}}) &= h_{PI} + \sum_J^N (g_{PIJJ} - g_{PJJI}) = h_{PI} + \sum_{J} \tilde{g}_{PIJJ} \\ &= {}^{i} F_{PI} \thinspace , \end{align}\] where the first index \(I\) must be occupied to yield a non-zero result and where \({}^{i} F_{PI}\) is the inactive (G)HF Fock matrix and \(\tilde{g}_{PQRS}\) are antisymmetrized two-electron integrals.

The \(\mathscr{Z}_{PQRS}\)-tensor must sum only over occupied indices, and the first index must be occupied: \[\begin{equation} \mathscr{Z}_{PQRS}( \hat{g} ) \rightarrow \mathscr{Z}_{IPQR}( \hat{g} ) = \sum_{JK}^N ( g_{PJRK} d_{IJQK} - g_{PQJK} d_{IRJK} - g_{PJKQ} d_{IJKR} ) \thinspace . \end{equation}\] In the calculation of the orbital Hessian, we will encounter the following types of expressions, which are already worked out for convenience. The first type is when the third index is occupied: \[\begin{equation} \mathscr{Z}_{IAJB}( \hat{g} ) = \tilde{g}_{AIBJ} \thinspace , \end{equation}\] and the second type is when the fourth index is occupied: \[\begin{equation} \mathscr{Z}_{IABJ}( \hat{g} ) = -\delta_{IJ} \sum_K \tilde{g}_{ABKK} - \tilde{g}_{AIJB} \thinspace . \end{equation}\]

Through conjugate pairs

Using the conjugate pairs approach to orbital optimization, we use the orbital rotation generators as the wave function model parameters and variationally optimize them, i.e. require the orbital gradient to vanish. The only non-redundant parameters are those for occupied-virtual rotations, i.e. \(\{ \kappa_{AI} \}\).

The elements of the orbital gradient simplify to \[\begin{align} \eval{ \displaystyle \pdv{ E(\boldsymbol{\kappa}) }{ {\kappa}_{AI} } }_{ \boldsymbol{\kappa}_0 } &= \ev*{ \comm*{ \hat{E}_{AI} }{ \hat{\mathcal{H}} } }{ \text{core} } \\ &= - \mathscr{F}_{IA}^*( \hat{\mathcal{H}} ) \thinspace . \end{align}\]

The objects \(\vb{A}\) and \(\vb{B}\) that constitute the orbital Hessian are also particularly simple in HF theory. We find: \[\begin{align} A_{AIBJ} &= - \ev*{ \comm*{ \hat{E}_{IA} }{ \comm*{ \hat{E}_{BJ} }{ \hat{\mathcal{H}} } } }{ \text{core} } \\ &= \delta_{IJ} {}^i F_{AB} - \delta_{AB} {}^i F_{IJ}^* + \tilde{g}_{AIJB} \thinspace , \end{align}\] with \({}^i F_{PQ}\) the inactive Fock matrix, and: \[\begin{align} B_{AIBJ} &= \frac{1}{2} (1 + P_{AI,BJ}) \ev*{ \comm*{ \hat{E}_{IA} }{ \comm*{ \hat{E}_{JB} }{ \hat{\mathcal{H}} } } }{ \text{core} } \\ &= \tilde{g}_{IAJB} \thinspace . \end{align}\]

Through real parameters

If we explicitly express the orbital rotations generators \(\kappa_{AI}\) in two sets of real parameters \(\{ {}^{R} \kappa_{AI} \}\) and \(\{ {}^{I} \kappa_{AI} \}\), the elements of the GHF orbital gradient are: \[\begin{equation} \eval{ \displaystyle \pdv{ E(\boldsymbol{\kappa}) }{ {}^{\text{R}}{\kappa}_{AI} } }_{ \boldsymbol{\kappa}_0 } = - \qty[ {}^{i} F_{AI} + {}^{i} F_{IA} ] \end{equation}\] and \[\begin{equation} \eval{ \displaystyle \pdv{ E(\boldsymbol{\kappa}) }{ {}^{\text{I}}{\kappa}_{AI} } }_{ \boldsymbol{\kappa}_0 } = i \qty[ {}^{i} F_{AI} - {}^{i} F_{IA} ] \thinspace . \end{equation}\] Requiring the orbital gradient to vanish is therefore equivalent to requiring the inactive Fock matrix to be diagonal, i.e. \[\begin{equation} F_{AI} = 0 \thinspace . \end{equation}\]

For the elements of the orbital Hessian, we have: \[\begin{equation} \eval{ \pdv{ E(\boldsymbol{\kappa}) }{ {}^{\text{R}}{\kappa_{AI}} }{ {}^{\text{R}}{\kappa_{BJ}} } }_{ \boldsymbol{\kappa}_0 } = \delta_{IJ} ( {}^{i} F_{AB} + {}^{i} F_{BA} ) - \delta_{AB} ( {}^{i} F_{IJ} + {}^{i} F_{JI} ) + (1 + P_{AI}) (1 + P_{BJ}) \tilde{g}_{AIBJ} \thinspace , \end{equation}\]

\[\begin{equation} \eval{ \pdv{ E(\boldsymbol{\kappa}) }{ {}^{\text{R}}{\kappa_{AI}} }{ {}^{\text{I}}{\kappa_{BJ}} } }_{ \boldsymbol{\kappa}_0 } = - i (1 - P_{IJ,AB}) \tilde{g}_{AIBJ} \thinspace , \end{equation}\] and: \[\begin{equation} \eval{ \pdv{ E(\boldsymbol{\kappa}) }{ {}^{\text{I}}{\kappa_{AI}} }{ {}^{\text{I}}{\kappa_{BJ}} } }_{ \boldsymbol{\kappa}_0 } = \delta_{IJ} ( {}^{i} F_{AB} + {}^{i} F_{BA} ) - \delta_{AB} ( {}^{i} F_{IJ} + {}^{i} F_{JI} ) - (1 - P_{AI}) (1 - P_{BJ}) \tilde{g}_{AIBJ} \thinspace . \end{equation}\]