Approximate energy functionals for one-body reduced density matrix functional theory from many-body perturbation theory

We develop a systematic approach to construct energy functionals of the one-particle reduced density matrix (1RDM) for equilibrium systems at finite temperature. The starting point of our formulation is the grand potential $\Omega [\mathbf{G}]$ regarded as variational functional of the Green's function $G$ based on diagrammatic many-body perturbation theory and for which we consider either the Klein or Luttinger-Ward form. By restricting the input Green's function to be one-to-one related to a set on one-particle reduced density matrices (1RDM) this functional becomes a functional of the 1RDM. To establish the one-to-one mapping we use that, at any finite temperature and for a given 1RDM $\mathbf{\gamma}$ in a finite basis, there exists a non-interacting system with a spatially non-local potential $v[\mathbf{\gamma}]$ which reproduces the given 1RDM. The corresponding set of non-interacting Green's functions defines the variational domain of the functional $\Omega$. In the zero temperature limit we obtain an energy functional $E[\mathbf{\gamma}]$ which by minimisation yields an approximate ground state 1RDM and energy. As an application of the formalism we use the Klein and Luttinger-Ward functionals in the GW-approximation compute the binding curve of a model hydrogen molecule using an extended Hubbard Hamiltonian. We compare further to the case in which we evaluate the functionals on a Hartree-Fock and a Kohn-Sham Green's function. We find that the Luttinger-Ward version of the functionals performs the best and is able to reproduce energies close to the GW energy which corresponds to the stationary point.