atomic tests show that the approximation is good numerically. These approximate forms then lead to a ladder of approximations where the COHSEX is the simplest one possible. Third, we show, analytically and with numerical examples for atoms, that if one restricts the Green’s function to be of non-interacting form generated by Hermitian non-local potentials, the GW -RPA correlation energy has no lower bound over this set of Green’s functions; nor does it have an extremum. Going forward, the exact rewriting and its good convergence properties — coupled with algorithmic development — should pave the wave for wider application of the GW -RPA correlation functional to materials systems. The multitude of approximate forms we present here will hopefully broaden and improve the types of approximate self-energies used in self-consistent band structure methods that go beyond the usual LDA or GGA treatments. Simultaneously, having explicit energy functionals on hand means one can construct the self- energies in a variational manner: the solution of the self-consistent equation for a particular self-energy will optimize the total energy functional from which it was derived. However, the result on the unboundedness of the GW -RPA correlation energy over the space of all non-interacting Green’s functions means that applying these Green’s function approaches is not yet a rote exercise in applying standard unconstrained optimization algorithms. Rather, the choice of non-local potential that generates the non-interacting Green’s function must be constrained in some manner so as to avoid the pathological negative infinite correlation energy and to produce extrema. The theoretical question is then to understand which constraints succeed and also to devise metrics to compare their quality and accuracy. We wish to acknowledge helpful discussions or comments from Kris Delaney, Kieron Burke, Douglas A. Stone, John Tully and Mark van Schilfgaarde. This work has been supported primarily by the National Science Foundation under Grant No. MRSEC DMR 0520495. Acknowledgements [1] P. Hohenberg and W. Kohn, Phys. Rev. 136 , B864 (1964). [2] W. Kohn and L. Sham, Phys. Rev. 140 , A1133 (1965). [3] J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). [4] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, Phys. Rev. B 46, 6671 (1992). [5] Theory of the Inhomogeneous Electron Gas , edited by S. Lundqvist and N. H. March (Plenum, New York 1983) and references therein. [6] M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 (1986). [7] V. I. Anisimov, F. Aryasetiawan and A. I. Lichtenstein, J. Phys. Cond. Matt. 9 , 767 (1997). [8] A. Georges, G. Kotliar, W. Krauth and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). 36