The $GW$ Approximation: A Quantum Chemistry Perspective

Abstract

We provide an in-depth examination of the $GW$ approximation of Green's function many-body perturbation theory by detailing both its theoretical and practical aspects in the realm of quantum chemistry. First, the quasiparticle context is introduced before delving into the derivation of Hedin's equations. From these, we explain how to derive the well-known $GW$ approximation of the self-energy. In a second time, we meticulously explain each step involved in a $GW$ calculation and what type of physical quantities can be computed. To illustrate its versatility, we consider two contrasting systems: the water molecule, a weakly correlated system, and the carbon dimer, a strongly correlated system. Each stage of the process is thoroughly detailed and explained alongside numerical results and illustrative plots. We hope that the contribution will facilitate the dissemination and democratization of Green's function-based formalisms within the computational and theoretical quantum chemistry community.

Figures (7)

• Figure 1: Schematic representation of an electron removal (left) and electron addition (right) process within the $GW$ approximation: the bare hole (white) and the bare electron (blue) are "dressed" by the RPA neutral excitations and then become a quasihole (green) and a quasielectron (red).
• Figure 2: Low-frequency part of the absorption spectrum of H2O (orange) and C2 (purple) computed at the RPA level with the aug-cc-pVTZ basis. See main text for more details.
• Figure 3: Pseudo-algorithm for each $GW$ scheme: $G_0W_0$@HF, lin$G_0W_0$@HF, ev$GW$@HF, and qs$GW$.
• Figure 4: Fundamental gap $E_\text{g}^\text{fun} = I^N - A^N$, where $I^N$ and $A^N$ are the principal ionization potential and the principal electron affinity of the $N$-electron system.
• Figure 5: Self-energy (black curves) associated with the HOMO (top) and LUMO (bottom) orbitals of H2O (left) and C2 (right) computed at the $G_0W_0$ level of theory with the aug-cc-pVTZ basis and $\eta = d-3\hartree$. The solutions of the quasiparticle equation are given by the intersection of the black and colored curves. In the central panels, the spectral function (in ^-1) associated with the HOMO and LUMO orbitals are represented.