Benchmarking the plasmon-pole and multipole approximations in the Yambo Code using the GW100 dataset

Verification and validation of electronic structure codes are essential to ensure reliable and reproducible results in computational materials science. While density functional theory has been extensively benchmarked, systematic assessments of many-body perturbation theory methods such as the GW approximation have only recently emerged, most notably through the GW100 dataset. In this work, we assess the numerical accuracy and convergence behavior of the GW implementation in the yambo code using both the Godby-Needs plasmon-pole model and the recently introduced multipole approximation. Quasiparticle energies are compared against GW100 reference data to evaluate the performance, numerical stability, and consistency of these approaches.

💡 Research Summary

This paper presents a systematic benchmark of two frequency‑dependence approximations implemented in the Yambo many‑body perturbation theory code: the Godby‑Needs plasmon‑pole model (GN‑PPA) and the recently introduced multipole approximation (MPA). Using the GW100 dataset—a curated collection of 100 closed‑shell molecules with reference ionization potentials (IPs) and electron affinities (EAs) obtained from high‑level calculations—the authors evaluate the numerical accuracy, convergence behavior, and stability of G₀W₀@PBE quasiparticle energies computed with each model.

The methodological framework follows a single‑shot G₀W₀ scheme based on Kohn‑Sham eigenvalues from a PBE density‑functional calculation. All DFT inputs are generated with Quantum ESPRESSO using plane‑wave basis sets and optimized norm‑conserving Vanderbilt pseudopotentials (ONCV) from the SG15 library. Molecules are placed in face‑centered cubic supercells (a = 13 Å) with Martyna‑Tuckerman corrections to mitigate spurious periodic image interactions, thereby reducing the vacuum volume relative to earlier GW100 studies.

In the GW step, the self‑energy Σ(ω) is split into a static exchange part Σₓ and a dynamic correlation part Σ_c(ω). The latter requires an accurate representation of the screened Coulomb interaction W_c(ω). GN‑PPA approximates W_c by a single symmetric pole, interpolated from values at ω = 0 and ω = iE_PPA (E_PPA = 30 eV). This yields a simple analytic form but may lack fidelity at higher frequencies. MPA, by contrast, expands W_c into a sum over multiple complex poles (typically around ten). The poles and residues are obtained via a non‑linear interpolation over a set of complex frequencies sampled according to a double‑parallel scheme. This approach reproduces full‑frequency (FF) behavior with far fewer sampling points, achieving FF‑level accuracy while dramatically reducing computational cost.

Convergence tests focus on two critical parameters: the number of empty states (N_b) entering the sum‑over‑states and the kinetic‑energy cutoff (G_cut) defining the size of the reciprocal‑lattice‑vector set used in the screening matrix. For the LiF molecule, the HOMO energy stabilizes within ~25 meV when N_b ≥ 7000 and G_cut ≥ 36 Ry, and similar trends are observed across the full GW100 set. The authors adopt N_b ≈ 7000 and G_cut ≈ 36 Ry as their production settings.

Benchmark results compare computed IPs and EAs against the GW100 reference values. GN‑PPA yields a mean absolute error (MAE) of 190 meV, while MPA reduces the MAE to 143 meV. These deviations are comparable to the spread observed among different full‑frequency implementations (e.g., Hybertsen‑Louie PPA, contour‑deformation, analytic continuation) and indicate that both approximations are numerically reliable. MPA consistently outperforms GN‑PPA for molecules with many high‑energy empty states (e.g., small metal clusters), where its richer pole structure captures the frequency dependence of W more faithfully.

The study also examines numerical stability. GN‑PPA can exhibit mild oscillations if the pole energy E_PPA is poorly chosen, but the authors find that the standard value (30 eV) together with sufficient empty‑state inclusion mitigates these issues. MPA’s complex‑pole framework requires careful handling of branch cuts in the complex plane; however, the automated sampling algorithm employed in Yambo robustly avoids singularities, delivering stable results across all test cases.

In conclusion, the authors demonstrate that the GN‑PPA provides a fast and reasonably accurate route for large‑scale GW screenings, whereas the MPA offers superior precision akin to full‑frequency calculations with a modest increase in computational effort. By quantifying the impact of frequency‑dependence approximations on GW quasiparticle energies, this work supplies practical guidance for selecting the appropriate method in future electronic‑structure investigations, thereby enhancing the reliability and reproducibility of many‑body perturbation theory calculations.