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Turbo charging time-dependent density-functional theory with Lanczos chains
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Ultrasoft pseudopotentials are taken from the public QUANTUM ESPRESSO
pseudopotential library. Pseudopotential files can be downloaded from URL http://www.pwscf.org/pseudo/1.3/UPF/xxx
, where xxx = C.pbe-rrkjus.UPF, H.pbe-rrkjus.UPF, C.pw91-van_ak.UPF, H.pw91-van_ak.UPF, Mg.pw91-np-van.UPF.N.pw91-van_ak.UPF.o.pw91-van_ak.UPF.
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We introduce a new implementation of time-dependent density-functional theory which allows the entirespectrum of a molecule or extended system to be computed with a numerical effort comparable to that of a 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 superoperator 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 superoperator. One-electron operators and density matrices are treated using a representation borrowed from time-independent density-functional perturbation theory, which permits us to avoid the calculation of unoccupied Kohn–Sham orbitals. The resolvent of the Liouvillian is evaluated through a newly developed algorithm based on the nonsymmetric 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, fullerene, and of chlorophyll a.
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