Parquet theory for molecular systems: Formalism and static kernel parquet approximation

TL;DR

This paper introduces and implements the parquet formalism for molecular systems to go beyond the $GW$ approximation by treating all two-body correlation channels on equal footing. It develops a static-kernel parquet approximation, analyzes both frequency-space structure and spin-orbital projections, and demonstrates a self-consistent scheme that couples eh and pp channels while optionally updating one-body quantities. The results show that two-body self-consistency significantly improves principal IP predictions, bringing them close to FCI values with quasiparticle weights near unity, while one-body self-consistency has a smaller effect. Overall, parquet theory offers a balanced, symmetry-preserving framework with the potential to capture complex correlation in molecules, albeit at a higher computational cost; future work will extend beyond the static kernel and explore broader observables and scaling improvements.

Abstract

The $GW$ approximation has become a method of choice for predicting quasiparticle properties in solids and large molecular systems, owing to its favorable accuracy-cost balance. However, its accuracy is the result of a fortuitous cancellation of vertex corrections in the polarizability and self-energy. Hence, when attempting to go beyond $GW$ through inclusion of vertex corrections, the accuracy can deteriorate if this delicate balance is disrupted. In this work, we explore an alternative route that theoretically goes beyond $GW$: the parquet formalism. Unlike approaches that focus on a single correlation channel, such as the electron-hole channel in $GW$ or the particle-particle channel in $T$-matrix theory, parquet theory treats all two-body scattering channels on an equal footing. We present the formal structure of the parquet equations, which couple the one-body Green's function, the self-energy, and the two-body vertex. We discuss the approximations necessary to solve this set of equations, the advantages and limitations of this approach, outline its implementation for molecular systems, and assess its accuracy for principal ionization potentials of small molecular systems.

Figures (11)

• Figure 1: Diagrammatic representation of the full one-body vertex, as defined in Eq. \ref{['eq:tilde_Sig']}.
• Figure 2: Perturbation expansion of the full one-body vertex $\tilde{\Sigma}$.
• Figure 3: Diagrammatic representation of the Dyson equation, as defined in Eq. \ref{['eq:dyson_eq']}
• Figure 4: Diagrammatic representation of the full two-body vertex, as defined in Eq. \ref{['eq:G2_F']}.
• Figure 5: The three different types of two-particle reducibility lead to three topologically distinct ways of partitioning a diagram into two parts by cutting two propagator lines: eh (top left), $\teh$ (right), and pp (bottom left).