  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.
