Interconnection between functional derivative and effective operator approaches to ab initio density functional theory

In the recent implementation of the second-order Optimized Effective Potential method ( R. J. Bartlett, I. Grabowski, S. Hirata, S. Ivanov, J. chem. Phys. ( submitted).), the effective potential is determined by requiring that the perturbative contribution to the reference density vanishes through second order. We show that in general, such a density condition for any. finite order leads to an OEP equation equivalent to that obtained via the functional derivative of the corresponding order energy. We develop diagrammatic tools for functional differentiation to facilitate the transition between many-body diagrammatic expressions and those of density functional theory.