The inverse Hartree–Fock (iHF) method calculates the Hartree–Fock Hamiltonian of the cluster by measuring its one-particle reduced density matrix (1RDM). This approach is referred to as the inverse Hartree–Fock method in order to avoid confusion with self-consistent Hartree-Fock.

The Hartree–Fock equations for a term which is quartic in fermionic creation \(\hat{c}^\dagger\) and annihilation \(\hat{c}\) operators is obtained by the mean-field decoupling:

\[\begin{split} \hat{c}^\dagger_i \hat{c}^\dagger_j \hat{c}^{}_k \hat{c}^{}_\ell \rightarrow & \; \hphantom{+} \braket{\hat{c}^\dagger_i \hat{c}^{}_\ell} \,\hat{c}^\dagger_j \hat{c}^{}_k \,+\, \braket{\hat{c}^\dagger_j \hat{c}^{}_k } \, \hat{c}^\dagger_i \hat{c}^{}_\ell \,-\, \braket{ \hat{c}^\dagger_i \hat{c}^{}_\ell} \braket{ \hat{c}^\dagger_j \hat{c}^{}_k} \\ & - \braket{\hat{c}^\dagger_i \hat{c}^{}_k} \,\hat{c}^\dagger_j \hat{c}^{}_\ell \,-\, \braket{\hat{c}^\dagger_j \hat{c}^{}_\ell } \, \hat{c}^\dagger_i \hat{c}^{}_k \,+\, \braket{\hat{c}^\dagger_i \hat{c}^{}_k} \braket{ \hat{c}^\dagger_j \hat{c}^{}_\ell} \\ & +\underbrace{\braket{\hat{c}^\dagger_i \hat{c}^\dagger_j}}_{={\braket{\hat{c}^{}_j \hat{c}^{}_i}}^\dagger} \,\hat{c}^{}_k \hat{c}^{}_\ell \,+\, \braket{\hat{c}^{}_k \hat{c}^{}_\ell } \, \hat{c}^\dagger_i \hat{c}^\dagger_j \,-\, \braket{\hat{c}^\dagger_i \hat{c}^\dagger_j } \braket{ \hat{c}^{}_k \hat{c}^{}_\ell }\end{split}\]

where the third term of each r.h.s expression accounts for the double counting of the interaction. Note that we have included the spin degree of freedom in the site index. Due to the fermionic algebra we have \(i\ne j\) and \(k \ne \ell\). However, we may still have \(i = \ell\) and/or \(j = k\)

Restricting ourselves to a density-density interaction, \(\hat{n}_\ell = \hat{c}^\dagger_\ell \hat{c}^{}_\ell\), or setting \(i = \ell\) and \(j = k\) in the HF equations, we obtain

\[\begin{split}\hat{n}_i \hat{n}_j \rightarrow& \;\hphantom{+} \braket{\hat{n}_j}\, \hat{n}_i \,+\, \braket{\hat{n}_i}\, \hat{n}_j \,-\, \braket{\hat{n}_i} \braket{\hat{n}_j} \\& - \braket{ \hat{c}^\dagger_i \hat{c}^{}_j }\, \hat{c}^\dagger_j \hat{c}^{}_i \,-\, \braket{ \hat{c}^\dagger_j \hat{c}^{}_i }\, \hat{c}^\dagger_i \hat{c}^{}_j \,+\, \braket{ \hat{c}^\dagger_i \hat{c}^{}_j } \braket{ \hat{c}^\dagger_j \hat{c}^{}_i } \\&+ \braket{ \hat{c}^\dagger_i \hat{c}^\dagger_j }\, \hat{c}^{}_j \hat{c}^{}_i \,+\, \braket{ \hat{c}^{}_j \hat{c}^{}_i }\, \hat{c}^\dagger_i \hat{c}^\dagger_j \,-\, \braket{ \hat{c}^\dagger_i \hat{c}^\dagger_j } \braket{ \hat{c}^{}_j \hat{c}^{}_i }\end{split}\]

Note that we have extended the standard HF description with a BCS decoupling, so that we are actually performing a HF/BCS decoupling. Here, the first line of the r.h.s denotes the Hartree contribution, the second line the Fock contribution, and the third line the BCS contribution. The BCS contribution is currently not enabled in the SCCE software, however its implementation is planned for the future.

Within the iHF approach we replace the expectation values in the above expression with the measured 1RDM \(\rho\),

\[\rho_{p,q} = \braket{ \hat{c}^\dagger_p \hat{c}^{}_q}\]