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Propagators for the time-dependent Kohn-Sham equations

Authors: Alberto Castro and M.A.L. Marques

Ref.: in Time-dependent density functional theory, ed. by M.A.L. Marques, C. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E.K.U. Gross, Lecture Notes in Physics, Vol. 706, Springer, Berlin, 197-210 (2006)

Abstract: The main practical result of the Runge-Gross theorem are the time-dependent Kohn-Sham (TDKS) equations: a set of coupled one-particle Schroedinger-like equations.

During the last years, most applications of TDDFT were performed within linear response theory, where the response properties of the system are usually obtained in frequency domain. One may, however, work directly in the time-domain, propagating Eqs.~(1). This has the advantage of allowing the inclusion of intense external perturbations, beyond the linear response regime. Of course, this "real-time" formulation of TDDFT requires the use of an algorithm to propagate Schroedinger-like equations.

Not surprisingly, the study of efficacious algorithms for this purpose has a long history, and multiple answers. We are concerned with a very general problem, yet we must beware of general purpose solutions: one expects that the efficiency depends strongly on the characteristics of the time-independent part of the Hamiltonian, on the time-dependent perturbation, and also on the initial state. From all possible approaches, we focus in this chapter on the ones most relevant to the propagation of the TDKS equations.

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