Correlation potential, nonlocal, exact

In this respect three problems seem to be worthwhile mentioning. The GGA, which has become the standard xc-functional in the nonrelativistic context by now, can neither describe negative ions nor dispersion forces and also fails to reproduce the ground state of highly correlated systems. The first aspect reflects the fact that the single-particle spectrum produced by the GGA is far from the exact KS spectrum, due to the exponential decay of the GGA potential. The spectrum, however, is not only pertinent for the existence of negative ions, but is also particularly important for the study of excitation or ionization processes. The second problem of the GGA points at its semi-local character Only a fully nonlocal functional, which can build up an attractive force even in regions where the density vanishes, is able to reproduce dispersion forces. 

We now have equations of motion for the one- and two-particle Green s functions. They depend on the Hartree-exchange-correlation self-energy. Its Hartree part is trivial, but a practical way of calculating its exchange-correlation part is needed. Hedin [36] proposed a scheme that yields to a set of coupled equations and allows in principle for the calculation of the exact self-energy. This scheme can be seen as a perturbation theory in terms of the screened interaction W instead of the bare Coulomb interaction v. We show a generalization of this derivation for the case of a nonlocal potential. 

We elaborate on recent attempts to derive the local and energy-dependent density-functional potential v from the diagrammatic structure of many-body perturbation theory for the exact exchange-correlation energy, without explicit recourse to an extremal principle. The local v can be related to the nonlocal and dynamic self-energy E obtained from perturbation theory. 

---