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

TL;DR

This paper derives an explicit analytical expression for a key term in Green's function theory using sum-over-pole representations, enabling improved variational calculations in embedded many-electron systems.

Contribution

It introduces a general method to analytically evaluate the trace of the logarithm of Green's functions with sum-over-pole representations, enhancing variational approaches in Green's function embedding.

Findings

Derived explicit analytical expression for the trace term

Enabled variational expressions in embedded systems

Facilitated calculation of RPA correlation energy

Abstract

In Green's function theory, the total energy of an interacting many-electron system can be expressed in a variational form using the Klein or Luttinger-Ward functionals. Green's function theory also naturally addresses the case where the interacting system is embedded into a bath. This latter can then act as a dynamical (i.e., frequency-dependent) potential, providing a more general framework than that of conventional static external potentials. Notably, the Klein functional includes a term of the form $\text{Tr}_\omega \text{Ln}\left\{G_0^{-1}G\right\}$, where $\text{Tr}_\omega$ is the frequency integration of the trace operator. Here, we show that using a sum-over-pole representation for the Green's functions and the algorithmic-inversion method one 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., to derive a variational expression for the Klein functional in the presence of an embedding bath, or to provide an explicit expression of the RPA correlation energy in the framework of the optimized effective potential.