Energy as a functional of the density matrix

A variational principle is presented for the ground state energy in terms of the one-particle density matrix. The functional expression of the energy in terms of the density matrix is obtained as the limit of zero temperature of the free energy. This is achieved by adding a nonlocal external potential to the N-body Hamiltonian and obtaining a renormalized equation for the density matrix instead of the Green's function. By eliminating the external source it was possible to express the energy as a functional of the density matrix.