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

Myrta Grüning, Andrea Marini, Xavier Gonze

TL;DR

This work addresses the computational challenge of solving large non-Hermitian RPA eigenproblems by leveraging the pseudo-Hermiticity of the RPA Hamiltonian. By redefining the inner product with the appropriate metric, the authors cast the problem into a Hermitian-like Lanczos framework that preserves stability and reduces computational cost, and they tailor it to calculate the macroscopic dielectric function within TD-DFT and BS formalisms. They derive the pseudo-Hermitian Lanczos algorithm, analyze its accuracy and efficiency on BS-based optical spectra of molecular systems, and show substantial speedups over exact diagonalization while maintaining accuracy. The approach enables robust, scalable treatment of excitations in complex systems such as molecules and nanostructures, where full RPA solutions are essential for reliable optical properties.

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 [Grüning 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.

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

TL;DR

This work addresses the computational challenge of solving large non-Hermitian RPA eigenproblems by leveraging the pseudo-Hermiticity of the RPA Hamiltonian. By redefining the inner product with the appropriate metric, the authors cast the problem into a Hermitian-like Lanczos framework that preserves stability and reduces computational cost, and they tailor it to calculate the macroscopic dielectric function within TD-DFT and BS formalisms. They derive the pseudo-Hermitian Lanczos algorithm, analyze its accuracy and efficiency on BS-based optical spectra of molecular systems, and show substantial speedups over exact diagonalization while maintaining accuracy. The approach enables robust, scalable treatment of excitations in complex systems such as molecules and nanostructures, where full RPA solutions are essential for reliable optical properties.

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 [Grüning 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.

Paper Structure

This paper contains 15 sections, 29 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Mathematical background
  3. Lanczos method for Hermitian eigenproblems
  4. Lanczos method for non-Hermitian eigenproblems
  5. Pseudo-Hermicity
  6. Pseudo-Hermicity of the RPA Hamiltonian
  7. Redefinition of the inner product for RPA Hamiltonian
  8. Lanczos recursive method for RPA Hamiltonians
  9. Special case of calculation of the macroscopic dielectric function
  10. Results
  11. Effect of the particle-hole hole-particle coupling
  12. Accuracy of the pseudo-Hermitian Lanczos algorithm
  13. Efficiency of the pseudo-Hermitian Lanczos algorithm
  14. Summary
  15. Off-diagonal resolvent matrix elements in the Lanczos basis

Figures (9)

  • Figure 1: Hermitian Lanczos algorithm
  • Figure 2: Non-Hermitian Lanczos algorithm [Eqs. (\ref{['eq:lnczsn']}-\ref{['eq:abcxprE']})]. The steps that are added with respect to the Hermitian case in Fig. (\ref{['fg:lncznH']}), are tagged with a primed letter. See text for further explanations.
  • Figure 3: Pseudo-Hermitian Lanczos algorithm
  • Figure 4: Pseudo-Hermitian Lanczos algorithm modified to account for a starting vector $|P\rangle$ with the anti-Hermitian property given by Eq. (\ref{['eq:Pkprop']})
  • Figure 5: [Color online] Optical spectra of TCB isomers: 1,3,5-TCB (top panel), 1,2,3-TCB (middle panel), 1,2,4-TCB (bottom panel). The imaginary part of the macroscopic dielectric function (in arbitrary unit) calculated within BSE using both full (black solid line) and TDA (red dashed line) are compared with the gas-phase experimental absorption cross section spectra (blue dotted line). The energy and relative strength of the excitations for full BSE (black squares) and TDA-BSE (red circles) obtained by exact diagonalization are reported in linear scale (the largest intensity is normalized to 1).
  • ...and 4 more figures