A significant portion of the energy functional of the one-particle reduced density matrix (1-RDM) is elucidated through geminal functional theory (GFT) [D.A. Mazziotti, J. Chem. Phys. 112 (2000) 10125]. We optimize the functional through an iterative solution of the 1,3-contracted Schrodinger equation (1,3-CSE). The method yields energies which are (i) above the true energy, (ii) more accurate than Hartree-Fock, and (iii) exact for a general family of correlated Hamiltonians. By applying a learning algorithm to these results, we determine correlation energies within 10% for atoms and molecules.