There are various ways that functionals can be employed to generate auxiliary systems and useful approximations by limiting the range of the functions input to the functional. Each of the functionals of the density, Green's function, or self-energy are defined over a specified domain D, e.g., ФG] and F[∑] are defined for all functions that have the required analytic properties for a Green's function or self-energy (see Sec. H.1). Here we consider limiting the domain to a subset of D denoted by D′. The first examples below derive expressions that provide insights concerning the link of non-locality and frequency dependence, and that lead to useful approximations in the spirit of optimized effective potentials. This is followed by an approach to find an interacting auxiliary system where the full many-body problem can be solved with no approximations, thus providing a self-energy that is exact within the restricted domain of the auxiliary system.

Auxiliary system to reproduce selected quantities

Let us first consider an auxiliary system that reproduces exactly some quantity of interest. This is similar to the Kohn–Sham approach, but it is more general and it leads to explicit expressions for the functionals in terms of Green's functions [1163].

Suppose we are interested in a quantity P that is part of the information carried by the Green's function G, symbolically expressed as P = p{G}. An example is the density where the “part” to be taken is the diagonal of the one-particle G: p{G} = n(r) = −iG(r, r, t′−t = 0+) or −G(r, r, τ = 0−). Consider an auxiliary system that has the same bare Green's function G0 as the original one, but a different effective potential, or a self-energy ∑P that leads to an auxiliary Green's function GP.