Redundant orbital rotation generators in GHF
(T. Helgaker, Jørgensen, and Olsen 2000). For (G)HF theory, we can find the redundant orbital rotation generators by inspecting which terms in \[\begin{equation*} - \hat{\kappa} \ket{\Psi} \end{equation*}\] are automatically zero due to the effect of the elementary operators: if an operator in \(\hat{\kappa}\) itself annihilates the state \(\ket{\Psi}\), the parameter that is associated to it will never have any effect. For (G)HF theory, we find:
\[ \require{physics} \begin{align} \hat{\kappa} \ket{\text{core}} &= \sum_{PQ} \kappa_{PQ} \hat{E}_{PQ} \ket{\text{core}} \\ &= \qty( \sum_I^N \kappa_{II} + \sum_I^N \sum_{A=N+1}^M \kappa_{AI} \hat{E}_{AI} ) \ket{\text{core}} \thinspace , \end{align} \] where we see that only the occupied-virtual rotations (parametrized by \(\kappa_{AI}\)) are non-redudant, together with the in-place rotations. We should note that, since the orbital rotation generators \(\boldsymbol{\kappa}\) form an anti-Hermitian matrix, its diagonal values \(\kappa_{II}\) (related to these in-place rotations) must be purely imaginary.
Furthermore, we can show that these parameters are actually redundant, considering the effect on the energy expression due to these diagonal parameters. Their energy contribution is calculated through: \[\begin{align} E_{\text{diagonal}}( \boldsymbol{\kappa}_{\geq}, \boldsymbol{\kappa}^*_{\geq} ) = &\sum_I^N \kappa_{II} \ev*{ \comm*{ \hat{N}_I }{ \hat{\mathcal{H}} } }{ \text{core} } + \frac{1}{2} \sum_{IJ}^N \kappa_{II} \kappa_{JJ} \ev*{ \comm*{ \hat{N}_I }{ \comm*{ \hat{N}_J }{ \hat{\mathcal{H}} } } }{ \text{core} } \\ &+ \frac{1}{2} \sum_{AI} \sum_J^N \kappa_{AI} \kappa_{JJ} \qty[ \ev*{ \comm*{ \hat{E}_{AI} }{ \comm*{ \hat{N}_J }{ \hat{\mathcal{H}} } } }{ \text{core} } + \ev*{ \comm*{ \hat{N}_J }{ \comm*{ \hat{E}_{AI} }{ \hat{\mathcal{H}} } } }{ \text{core} } ] \\ &- \frac{1}{2} \sum_I^N \sum_{BJ} \kappa_{II} \kappa^*_{BJ} \qty[ \ev*{ \comm*{ \hat{N}_I }{ \comm*{ \hat{E}_{JB} }{ \hat{\mathcal{H}} } } }{ \text{core} } + \ev*{ \comm*{ \hat{E}_{JB} }{ \comm*{ \hat{N}_I }{ \hat{\mathcal{H}} } } }{ \text{core} } ] \thinspace . \end{align}\]
We can work out the above commutators whose first argument is \(\hat{N}_I\) by first writing explicitly the commutator and then realizing that it vanishes because the mentioned number operator is an identity operation on the HF core state: \[\begin{equation} \hat{N}_I \ket{\text{core}} = \ket{\text{core}} \thinspace . \end{equation}\]
The other terms also vanish, but for a more complicated reason. Let us investigate, for example, the term \[\begin{equation} \ev*{ \comm*{ \hat{E}_{AI} }{ \comm*{ \hat{N}_J }{ \hat{\mathcal{H}} } } }{ \text{core} } \thinspace . \end{equation}\] By invoking the Jacobi identity for a doubly-nested commutator, we find: \[\begin{equation} \ev*{ \comm*{ \hat{E}_{AI} }{ \comm*{ \hat{N}_J }{ \hat{\mathcal{H}} } } }{ \text{core} } = - \ev*{ \comm*{ \hat{\mathcal{H}} }{ \comm*{ \hat{E}_{AI} }{ \hat{N}_J } } }{ \text{core} } - \ev*{ \comm*{ \hat{N}_J }{ \comm*{ \hat{\mathcal{H}} }{ \hat{E}_{AI} } } }{ \text{core} } \thinspace , \end{equation}\] whose last term vanishes by a similar reasoning as above. Using a commutator property for the excitation operators and the number operator, we find: \[\begin{align} \ev*{ \comm*{ \hat{\mathcal{H}} }{ \comm*{ \hat{E}_{AI} }{ \hat{N}_J } } }{ \text{core} } &= \delta_{IJ} \ev*{ \comm*{ \hat{\mathcal{H}} }{ \hat{E}_{AI} } }{ \text{core} } \\ &= - \delta_{IJ} \eval{ \pdv{ E( \boldsymbol{\kappa}_{\geq}, \boldsymbol{\kappa}^*_{\geq} ) }{\kappa_{AI}} }_{\boldsymbol{\kappa}_0} \thinspace , \end{align}\] which vanishes when the current orbitals \(\boldsymbol{\kappa}_0\) is considered to be an optimal point.
As a conclusion, we have shown that the diagonal contribution to the orbital rotation energy vanishes identically at the optimal point \(\boldsymbol{\kappa}_0\), so we may treat the in-place rotations for HF theory as redundant.