Efficient dielectric matrix calculations using the Lanczos algorithm for fast many-body $G_0W_0$ implementations

TL;DR

This paper introduces a highly efficient $G_0W_0$ computational method that overcomes key bottlenecks by using the Lanczos basis, Sternheimer equations, and a model dielectric operator, achieving significant speedups and maintaining accuracy.

Contribution

The authors develop a novel $G_0W_0$ implementation that reduces computational costs by employing a Lanczos basis and other techniques without sacrificing accuracy.

Findings

Achieves a 500-fold speedup for silane molecule calculations.

Maintains accuracy comparable to traditional methods.

Demonstrates efficiency and reliability through comparison with experimental data.

Abstract

We present a $G_0W_0$ implementation that assesses the two major bottlenecks of traditional plane-waves implementations, the summations over conduction states and the inversion of the dielectric matrix, without introducing new approximations in the formalism. The first bottleneck is circumvented by converting the summations into Sternheimer equations. Then, the novel avenue of expressing the dielectric matrix in a Lanczos basis is developed, which reduces the matrix size by orders of magnitude while being computationally efficient. We also develop a model dielectric operator that allows us to further reduce the size of the dielectric matrix without accuracy loss. Furthermore, we develop a scheme that reduces the numerical cost of the contour deformation technique to the level of the lightest plasmon pole model. Finally, the use of the simplified quasi-minimal residual scheme in replacement of the conjugate gradients algorithm allows a direct evaluation of the $G_0W_0$ corrections at the desired real frequencies, without need for analytical continuation. The performance of the resulting $G_0W_0$ implementation is demonstrated by comparison with a traditional plane-waves implementation, which reveals a 500-fold speedup for the silane molecule. Finally, the accuracy of our $G_0W_0$ implementation is demonstrated by comparison with other $G_0W_0$ calculations and experimental results.