Green's function embedding using sum-over-pole representations

In Green's function theory, the total energy of an interacting many-electronsystem can be expressed in a variational form using the Klein or Luttinger-Wardfunctionals. Green's function theory also naturally addresses the case wherethe interacting system is embedded into a bath. This latter can then act as adynamical (i.e., frequency-dependent) potential, providing a more generalframework than that of conventional static external potentials. Notably, theKlein functional includes a term of the form $\text{Tr}_\omega\text{Ln}\left\{G_0^{-1}G\right\}$, where $\text{Tr}_\omega$ is the frequencyintegration of the trace operator. Here, we show that using a sum-over-polerepresentation for the Green's functions and the algorithmic-inversion methodone can obtain in full generality an explicit analytical expression for$\text{Tr}_\omega \text{Ln}\left\{G_0^{-1}G\right\}$. This allows one, e.g., toderive a variational expression for the Klein functional in the presence of anembedding bath, or to provide an explicit expression of the RPA correlationenergy in the framework of the optimized effective potential.