Turbo charging time-dependent density-functional theory with Lanczos chains

TL;DR

This paper presents a novel, efficient implementation of time-dependent density-functional theory (TDDFT) that computes entire molecular spectra with computational effort comparable to a ground-state calculation, suitable for large systems.

Contribution

The authors introduce a Lanczos-based algorithm for TDDFT that significantly reduces computational cost and avoids unoccupied orbital calculations, enabling large-scale spectral computations.

Findings

Efficiently computes spectra of large molecules like C60 and chlorophyll.

Reduces computational effort to a few times that of ground-state calculations.

Demonstrates applicability to complex molecular systems.

Abstract

We introduce a new implementation of time-dependent density-functional theory which allows the \emph{entire} spectrum of a molecule or extended system to be computed with a numerical effort comparable to that of a \emph{single} standard ground-state calculation. This method is particularly well suited for large systems and/or large basis sets, such as plane waves or real-space grids. By using a super-operator formulation of linearized time-dependent density-functional theory, we first represent the dynamical polarizability of an interacting-electron system as an off-diagonal matrix element of the resolvent of the Liouvillian super-operator. One-electron operators and density matrices are treated using a representation borrowed from time-independent density-functional perturbation theory, which permits to avoid the calculation of unoccupied Kohn-Sham orbitals. The resolvent of the Liouvillian is evaluated through a newly developed algorithm based on the non-symmetric Lanczos method. Each step of the Lanczos recursion essentially requires twice as many operations as a single step of the iterative diagonalization of the unperturbed Kohn-Sham Hamiltonian. Suitable extrapolation of the Lanczos coefficients allows for a dramatic reduction of the number of Lanczos steps necessary to obtain well converged spectra, bringing such number down to hundreds (or a few thousands, at worst) in typical plane-wave pseudopotential applications. The resulting numerical workload is only a few times larger than that needed by a ground-state Kohn-Sham calculation for a same system. Our method is demonstrated with the calculation of the spectra of benzene, C$_{60}$ fullerene, and of chlorofyll a.