The correlation energy as an explicit functional of the one-particle density matrix from a determinantal reference state

Using an approach based on many body perturbation theory, the correlation energy $\cEco$ is expressed as an explicit functional of $\rho_1$, $v$, and $v_s$, where $\rho_1$ is the one-particle density matrix from the noninteracting, or reference, determinantal-state; $v$ is the external potential from the interacting, or target, state; $v_s$ is the (kernel of the) external potential from the noninteracting determinantal-state. In other words we have $\cEco[\rho_1,v,v_s]$. Anther possibility is the following explicit functional: $\cEco[\rho_1,v_{\text{co}},v_s]$, where $v_{\text{co}}$ is the (kernel of the) correlation potential from the noninteracting Hamiltonian. The proposed method can, in principle, be used to compute $\cEco$ in a very accurate and efficient manner, since, like the Kohn--Sham approach, there are no virtual orbitals to consider. However, in contrast to the Kohn--Sham approach, $\cEco$ is a known, explicit functional that can be approximated in a systematic manner. For simplicity, we only consider noninteracting closed-shell states and target states that are nondegenerate, singlet ground-states; so, in that case, $\rho_1$ denotes the spin-less one-particle density matrix from the determinantal reference state.