Propagators for the time-dependent Kohn-Sham equations

Journal of Chemical Physics 121, 3425 - 3433 (2004)

Propagators for the time-dependent Kohn-Sham equations

A. Castro, M.A.L. Marques, A. Rubio

In this paper we address the problem of the numerical integration of the time-dependent Schrödinger equation i [partial-derivative]tphi= H-hat phi. In particular, we are concerned with the important case where H-hat is the self-consistent Kohn–Sham Hamiltonian that stems from time-dependent functional theory. As the Kohn–Sham potential depends parametrically on the time-dependent density, H-hat is in general time dependent, even in the absence of an external time-dependent field. The present analysis also holds for the description of the excited state dynamics of a many-electron system under the influence of arbitrary external time-dependent electromagnetic fields. Our discussion is separated in two parts: (i) First, we look at several algorithms to approximate exp(Â), where  is a time-independent operator [e.g.,  = –iDeltatH-hat(tau) for some given time tau]. In particular, polynomial expansions, projection in Krylov subspaces, and split-operator methods are investigated. (ii) We then discuss different approximations for the time-evolution operator, such as the midpoint and implicit rules, and Magnus expansions. Split-operator techniques can also be modified to approximate the full time-dependent propagator. As the Hamiltonian is time dependent, problem (ii) is not equivalent to (i). All these techniques have been implemented and tested in our computer code OCTOPUS, but can be of general use in other frameworks and implementations.

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