Implementation and testing of Lanczos-based algorithms for Random-Phase Approximation eigenproblems

TL;DR

This paper introduces a Lanczos-based algorithm exploiting pseudo-Hermiticity to efficiently and stably solve non-Hermitian Random-Phase Approximation eigenproblems, demonstrated on optical response calculations of molecules and carbon nanotubes.

Contribution

The paper presents a novel Lanczos algorithm leveraging pseudo-Hermiticity for improved stability and efficiency in non-Hermitian RPA eigenproblems.

Findings

Algorithm achieves stability comparable to Hermitian methods.

Demonstrated efficiency in optical absorption calculations.

Applicable to Bethe-Salpeter equation frameworks.

Abstract

The treatment of the Random-Phase Approximation Hamiltonians, encountered in different frameworks, like Time-Dependent Density Functional Theory or Bethe-Salpeter equation, is complicated by their non-Hermicity. Compared to their Hermitian Hamiltonian counterparts, computational methods for the treatment of non-Hermitian Hamiltonians are often less efficient and less stable, sometimes leading to the breakdown of the method. Recently [Gr\"uning et al. Nano Lett. {\bf 8}, 2820 (2009)], we have identified that such Hamiltonians are usually pseudo-Hermitian. Exploiting this property, we have implemented an algorithm of the Lanczos type for random-Phase Approximation Hamiltonians that benefits from the same stability and computational load as its Hermitian counterpart, and applied it to the study of the optical response of carbon nanotubes. We present here the related theoretical grounds and technical details, and study the performance of the algorithm for the calculation of the optical absorption of a molecule within the Bethe-Salpeter equation framework.