Fockian operators
In Helgaker (T. Helgaker, Jørgensen, and Olsen 2000), we find a very thorough chapter on Hartree-Fock theory, where Fock operators and super-Fock operators are discussed in chapter 10.8, but in a formalism that only allows for spatially restricted spin-separated spinors. I feel that the discussion presented there could be elaborated further, which is the goal of this section.
In a general spinor basis
The one-electron excitation operators \(\hat{E}_{PQ}\) play a very fundamental role in the algebraic theory of electronic structure. Not only do they form an operator basis in which every one-electron operator can be expressed, they form a unitary algebra: \[ \require{physics} \begin{equation} \comm{ % \hat{E}_{PQ} % }{ % \hat{E}_{RS}% } = \delta_{RQ} \hat{E}_{PS} % - \delta_{PS} \hat{E}_{RQ} % \thinspace , \end{equation} \] which means that they generate the unitary group. Since orbital rotations are characterized by unitary transformations, we can already appreciate that these one-electron excitation operators will play a very fundamental role when describing orbital rotations. If we are not working in the full Fock space, it makes sense to look for the `optimal’ set of spinors that describe the system given a limited Fock subspace. Using the unconstrained parametrization (i.e. with the exponential of the orbital rotation generator), the energy as a function of the anti-Hermitian orbital rotation parameters \(\kappa_{PQ}\) is given by: \[\begin{equation} E(\boldsymbol{\kappa}) = % \ev{ % \exp(\hat{\kappa}) % \hat{\mathcal{H}} % \exp(-\hat{\kappa}) % }{\Psi} % \thinspace . \end{equation}\] Applying the BCH expansion yields (up to second order): \[\begin{equation} E(\boldsymbol{\kappa}) % \approx \ev{ % \hat{\mathcal{H}}% }{\Psi} % + \ev*{ % \comm*{ \hat{\kappa} }{ \hat{\mathcal{H}} } % }{\Psi} % + \frac{1}{2} \ev*{ % \comm*{ \hat{\kappa} }{ % \comm*{ \hat{\kappa} }{ \hat{\mathcal{H}}} % } % }{\Psi} \thinspace , \end{equation}\] such that we can see that the commutator \(\comm{ \hat{\kappa} }{ \hat{\mathcal{H}} }\) will play a very substantial role in the corresponding derivations. We can then see that the main ingredient to calculate this commutator is found in the more elementary commutator \[\begin{equation} \comm{ % \hat{E}_{PQ} % }{ % \hat{\mathcal{H}} % } % \thinspace . \end{equation}\]
Filling in the definition of \(\hat{E}_{PQ}\), we find: \[\begin{equation} \comm{ % \hat{E}_{PQ} }{ % \hat{O} % } = % \hat{a}^\dagger_P % \comm{ % \hat{a}_Q % }{ % \hat{O} % } + \comm{ % \hat{a}^\dagger_P % }{ % \hat{O} % } \hat{a}_Q % \thinspace . \end{equation}\] In fact, these expressions will turn out to be so important that we will define the Fockian (operator) of a Hermitian operator \(\hat{O}\) as: \[\begin{equation} \hat{\mathscr{F}}_{PQ}( \hat{O} ) % = \hat{a}^\dagger_P % \comm{ % \hat{a}_Q % }{ % \hat{O} % } % \thinspace , \end{equation}\] with the Hermitian adjoint: \[\begin{equation} \hat{\mathscr{F}}^\dagger_{PQ}( \hat{O} ) % = - \comm{ % \hat{a}^\dagger_Q % }{ % \hat{O} % } % \hat{a}_P % \thinspace , \end{equation}\] such that we can write the initial commutator as: \[\begin{equation} \comm{ % \hat{E}_{PQ} }{ % \hat{O} % } = % \hat{\mathscr{F}}_{PQ}( \hat{O} ) % - \hat{\mathscr{F}}^\dagger_{QP}( \hat{O} ) % \thinspace . \end{equation}\] We then have the following elementary Fockians: \[\begin{align} & \hat{\mathscr{F}}_{PQ}( \hat{E}_{RS} ) % = \delta_{QR} \hat{E}_{PS} \\ % & \hat{\mathscr{F}}_{PQ}( \hat{e}_{RSTU} ) % % = \delta_{QR} \hat{e}_{PSTU} % + \delta_{QT} \hat{e}_{PURS} \\ & \phantom{\hat{\mathscr{F}}_{PQ}( \hat{e}_{RSTU} )} % = (1 + P_{RS,TU}) \delta_{QR} \hat{e}_{PSTU} % \thinspace , \end{align}\] such that we can calculate the Fockian of a one-electron operator \(\hat{f}\) and a two-electron operator \(\hat{g}\) (including the prefactor \(1/2\)) as: \[\begin{align} & \hat{\mathscr{F}}_{PQ}( \hat{f} ) = % \sum_R^M % f_{QR} % \hat{E}_{PR} \\ % & \hat{\mathscr{F}}_{PQ}( \hat{g} ) = % \sum_{RST}^M % g_{QRST} % \hat{e}_{PRST} % \thinspace . \end{align}\]
Concerning second-order commutators, we are now interested in the following commutator: \[\begin{equation*} \comm{ % \hat{E}_{PQ} % }{ % \comm{ % \hat{E}_{RS} % }{ % \hat{O} % } % } \thinspace , \end{equation*}\] which can be written as \[\begin{equation} \comm{ % \hat{E}_{PQ} % }{ % \comm{ % \hat{E}_{RS} % }{ % \hat{O} % } % } % = \hat{\mathscr{G}}_{PQRS}( \hat{O} ) % + \hat{\mathscr{G}}^\dagger_{QPSR}( \hat{O} ) \end{equation}\] if we introduce the super-Fockian operator as: \[\begin{align} \hat{\mathscr{G}}_{PQRS}( \hat{O} ) % &= \hat{a}^\dagger_P % \comm{ % \hat{a}_Q % }{ % \comm{ % \hat{E}_{RS} % }{ % \hat{O} % } % } \\ &= \hat{\mathscr{F}}_{PQ} % \qty( % \comm{ % \hat{E}_{RS} % }{ % \hat{O} % } % ) % \thinspace , \end{align}\] which has the Hermitian adjoint: \[\begin{align} \hat{\mathscr{G}}_{PQRS}^\dagger( \hat{O} ) % &= \comm{ % \hat{a}^\dagger_Q % }{ % \comm{ % \hat{E}_{SR} % }{ % \hat{O} % } % } \hat{a}_P \\ &= - \hat{\mathscr{F}}^\dagger_{PQ} % \qty( % \comm{ % \hat{E}_{SR} % }{ % \hat{O} % } % ) % \thinspace . \end{align}\] The super-Fockian of a one-electron operator \(\hat{f}\) and a two-electron operator \(\hat{g}\) (again including the prefactor \(1/2\)) then become: \[\begin{align} & \hat{\mathscr{G}}_{PQRS}( \hat{f} ) % = \delta_{QR} \hat{\mathscr{F}}_{PS}( \hat{f} ) % - f_{QR} \hat{E}_{PS} \\ % & \hat{\mathscr{G}}_{PQRS}( \hat{g} ) % = \delta_{QR} \hat{\mathscr{F}}_{PS}( \hat{g} ) % + \hat{\mathcal{Z}}_{PQRS}( \hat{g} ) \thinspace , \end{align}\] where we have introduced the operator \(\hat{\mathcal{Z}}_{PQRS}( \hat{g} )\) as: \[\begin{equation} \hat{\mathcal{Z}}_{PQRS}( \hat{g} ) = % \sum_{TU}^M (% g_{QTSU} \hat{e}_{PTRU} % - g_{QRTU} \hat{e}_{PSTU} % - g_{QTUR} \hat{e}_{PTUS} % ) \thinspace . \end{equation}\] The super-Fockian of the electronic Hamiltonian, which is the combination of a one-electron and a two-electron operator, then becomes \[\begin{equation} \hat{\mathscr{G}}_{PQRS}( \hat{\mathcal{H}} ) % = \delta_{QR} \hat{\mathscr{F}}_{PS}( \hat{\mathcal{H}} ) % - h_{QR} \hat{E}_{PS} % + \hat{\mathcal{Z}}_{PQRS}( \hat{g} ) % \thinspace . \end{equation}\]
Note that the previous formulas hold in the most general, complex case, with the only restriction on the operators \(\hat{O}, \hat{f}\) and \(\hat{g}\) is that they must be Hermitian.
Since we have here presented an overview of the Fockian (and super-Fockian) operators, we may use them in expectation values.
In a ‘restricted’ spin-orbital basis
In a restricted spin-orbital basis, we introduce the Fockian of an anti-Hermitian operator \(\hat{O}\) as: \[\begin{equation} \hat{\mathscr{F}}_{pq}(\hat{O}) = \sum_{\sigma} \hat{a}^\dagger_{p \sigma} \comm{ \hat{a}_{q \sigma} }{ \hat{O} } \thinspace , \end{equation}\] with the Hermitian adjoint: \[\begin{equation} \hat{\mathscr{F}}^\dagger_{pq}(\hat{O}) = - \sum_{\sigma} \comm{ \hat{a}^\dagger_{q \sigma} }{ \hat{O}^\dagger } \hat{a}_{p \sigma} \thinspace . \end{equation}\] We can then rewrite the following commutator: \[\begin{equation} \comm{ \hat{E}_{pq} }{ \hat{O} } = \hat{\mathscr{F}}_{pq}(\hat{O}) - \hat{\mathscr{F}}^\dagger_{qp}(\hat{O}^\dagger) \thinspace . \end{equation}\]
From these definitions, the Fockian \(\hat{\mathscr{F}}_{pq}\) is a linear operator function, while its Hermitian adjoint \(\hat{\mathscr{F}}^\dagger_{pq}\) is an antilinear operator function. The most elementary Fockians are those of the singlet one- and two-electron operators: \[\begin{align} & \hat{\mathscr{F}}_{pq}(\hat{E}_{rs}) = \delta_{qr} \hat{E}_{ps} \\ % & \hat{\mathscr{F}}_{pq}(\hat{e}_{rstu}) = \delta_{qr} \hat{e}_{pstu} + \delta_{qt} \hat{e}_{purs} \\ & \phantom{\hat{\mathscr{F}}_{pq}(\hat{e}_{rstu})} = (1 + P_{rs,tu}) \delta_{qr} \hat{e}_{pstu} \thinspace , \end{align}\] which can be used to find the Fockians of one-electron operators \(\hat{f}\) and two-electron operators \(\hat{g}\) (including the prefactor \(1/2\)): \[\begin{align} & \hat{\mathscr{F}}_{pq}(\hat{f}) = \sum_{r}^K f_{qr} \hat{E}_{pr} \\ % & \hat{\mathscr{F}}_{pq}(\hat{g}) = \sum_{rst}^K g_{qrst} \hat{e}_{prst} \thinspace . \end{align}\]
If we then introduce the super-Fockian operator \[\begin{align} \hat{\mathscr{G}}_{pqrs}(\hat{O}) &= \sum_{\sigma} \hat{a}^\dagger_{p \sigma} \comm{\hat{a}_{q \sigma}}{ \comm{\hat{E}_{rs}}{\hat{O}} } \\ &= \hat{\mathscr{F}}_{pq} \qty(\comm{\hat{E}_{rs}}{\hat{O}}) \end{align}\] and its Hermitian adjoint: \[\begin{align} \hat{\mathscr{G}}^\dagger_{pqrs}(\hat{O}) &= \sum_{\sigma} \comm{\hat{a}^\dagger_{q \sigma}}{ \comm{\hat{E}_{sr}}{\hat{O}^\dagger} } \hat{a}_{p \sigma} \\ &= \hat{\mathscr{F}}^\dagger_{pq} \qty( \comm{\hat{E}_{rs}}{\hat{O}}^\dagger ) \thinspace , \end{align}\] we can rewrite the following commutator: \[\begin{equation} \comm{\hat{E}_{pq}}{ \comm{\hat{E}_{rs}}{\hat{O}} } = \hat{\mathscr{G}}_{pqrs}(\hat{O}) + \hat{\mathscr{G}}^\dagger_{qpsr}(\hat{O}^\dagger) \thinspace . \end{equation}\]
The super-Fockian of the one-electron operators \(\hat{f}\) and two-electron operators \(\hat{g}\) then become: \[\begin{align} & \hat{\mathscr{G}}_{pqrs}(\hat{f}) = \delta_{qr} \hat{\mathscr{F}}_{ps}(\hat{f}) - f_{qr} \hat{E}_{ps} \\ % & \hat{\mathscr{G}}_{pqrs}(\hat{g}) = \delta_{qr} \hat{\mathscr{F}}_{ps}(\hat{g}) + \hat{\mathcal{Z}}_{pqrs}(\hat{g}) \thinspace , \end{align}\] where we have introduced the operator \(\hat{\mathcal{Z}}(\hat{g})\) as: \[\begin{equation} \hat{\mathcal{Z}}_{pqrs}(\hat{g}) = \sum_{tu}^K ( g_{qtsu} \hat{e}_{ptru} - g_{qrtu} \hat{e}_{pstu} - g_{qtur} \hat{e}_{ptus} ) \thinspace . \end{equation}\] Note that the previous formulas hold in the most general, complex case.
Given a normalized reference state, we can use the Fockian and super-Fockian in expectation values.