The three channels of many-body perturbation theory: $GW$, particle-particle, and electron-hole $T$-matrix self-energies

TL;DR

This paper derives explicit formulas for three key self-energies in many-body perturbation theory, enabling their computation through RPA eigenvalue problems, and compares their performance on small molecules and strongly correlated systems.

Contribution

It provides explicit expressions for the $GW$, particle-particle, and electron-hole $T$-matrix self-energies, linking them to RPA eigenvalue problems for the first time.

Findings

Computed ionization potentials for small molecules at each level of theory.

Analyzed performance of self-energies on strongly correlated systems like B2 and C2.

Demonstrated the computational approach's effectiveness for response functions.

Abstract

We derive the explicit expression of the three self-energies that one encounters in many-body perturbation theory: the well-known $GW$ self-energy, as well as the particle-particle and electron-hole $T$-matrix self-energies. Each of these can be easily computed via the eigenvalues and eigenvectors of a different random-phase approximation (RPA) linear eigenvalue problem that completely defines their corresponding response function. For illustrative and comparative purposes, we report the principal ionization potentials of a set of small molecules computed at each level of theory. The performance of these schemes on strongly correlated systems (\ce{B2} and \ce{C2}) is also discussed.