Fast construction of the Kohn--Sham response function for molecules

TL;DR

This paper presents a fast, efficient method for constructing the Kohn--Sham response function in molecules using a localized orbital product basis, enabling accurate molecular spectra calculations with reduced computational cost.

Contribution

The authors develop a localized basis for orbital products that exponentially reduces residual error, enabling efficient computation of the Kohn--Sham response function with linear scaling in system size.

Findings

The method computes the response function in N^2 N_ω operations, significantly faster than traditional methods.

The constructed response function yields molecular spectra in good agreement with previous calculations.

The approach is promising for GW self-energy calculations and excitonic analysis in molecules.

Abstract

The use of the LCAO (Linear Combination of Atomic Orbitals) method for excited states involves products of orbitals that are known to be linearly dependent. We identify a basis in the space of orbital products that is local for orbitals of finite support and with a residual error that vanishes exponentially with its dimension. As an application of our previously reported technique we compute the Kohn--Sham density response function $\chi_{0}$ for a molecule consisting of $N$ atoms in $N^{2}N_{\omega}$ operations, with $N_{\omega}$ the number of frequency points. We test our construction of $\chi_{0}$ by computing molecular spectra directly from the equations of Petersilka--Gossmann--Gross in $N^{2}N_{\omega}$ operations rather than from Casida's equations which takes $N^{3}$ operations. We consider the good agreement with previously calculated molecular spectra as a validation of our construction of $\chi_{0}$. Ongoing work indicates that our method is well suited for the computation of the GW self-energy $\Sigma=\mathrm{i}GW$ and we expect it to be useful in the analysis of exitonic effects in molecules.