We examine various integration schemes for the time-dependent Kohn-Sham equations. Contrary to the time-dependent Schrödinger's equation, this set of equations is nonlinear, due to the dependence of the Hamiltonian on the electronic density. We discuss some of their exact properties, and in particular their symplectic structure. Four different families of propagators are considered, specifically the linear multistep, Runge-Kutta, exponential Runge-Kutta, and the commutator-free Magnus schemes. These have been chosen because they have been largely ignored in the past for time-dependent electronic structure calculations. The performance is analyzed in terms of cost-versus-accuracy. The clear winner, in terms of robustness, simplicity, and efficiency is a simplified version of a fourth-order commutator-free Magnus integrator. However, in some specific cases, other propagators, such as some implicit versions of the multistep methods, may be useful.
We consider the numerical propagation of models that combine both quantum and classical degrees of freedom -usually, electrons and nuclei, respectively. We focus, in our computational examples, on the case in which the quantum electrons are modeled with time-dependent density-functional theory, although the methods discussed below can be used with any other level of theory. Often, for these so-called quantum-classical molecular dynamics models, one uses some propagation technique to deal with the quantum part, and a different one for the classical equations. While the resulting procedure may in principle be consistent, it can however spoil some of the properties of the methods, such as the accuracy order with respect to the time step or the preservation of the equations geometrical structure. Few methods have been developed specifically for hybrid quantum-classical models. We propose using the same method for both the quantum and classical particles, in particular one family of techniques that proves to be very efficient for the propagation of Schrödinger-like equations: the (quasi)-commutator
We present an implementation of optimal control theory for the first-principles nonadiabatic Ehrenfest molecular dynamics model, which describes a condensed matter system by considering classical point-particle nuclei, and quantum electrons, handled in our case with time-dependent density-functional theory. The scheme is demonstrated by optimizing the Coulomb explosion of small sodium clusters: the algorithm is set to find the optimal femtosecond laser pulses that disintegrate the clusters, for a given total duration, fluence, and cutoff frequency. We describe the numerical details and difficulties of the method.
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