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##### Propagators for the time-dependent Kohn-Sham equations

**Authors**: A. Castro, M.A.L. Marques, and A. Rubio

**Ref.**: J. Chem. Phys **121**, 3425-3433 (2004)

**Abstract**: In this article we address the problem of the numerical integration of the time-dependent Schroedinger equation i d_t phi = H phi. In particular, we are concerned with the important case where H is the selfconsistent Kohn-Sham Hamiltonian that stems from time-dependent functional theory. As the Kohn-Sham potential depends parametrically on the time-dependent density, H 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 any-electron system under the influence of arbitraty external time-dependent electromagnetic fields. Our discussion is separated in two parts: i) First, we look at several algorithms to approximate exp(A), where A is a time-independent operator [e.g. A =-i Delta t H(tau) for some given time tau. In particular, polynomial expansions, projection in Krylov subspaces, and split-operator methods are nvestigated. ii)~We then discuss different approximations for the time-evolution operator, like the mid-point 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.

**Citations**: 399 (Google scholar)

**DOI**: 10.1063/1.1774980

**URL**: arxiv.org

**Bibtex**:

@article{Castro_2004, doi = {10.1063/1.1774980}, url = {https://doi.org/10.1063%2F1.1774980}, year = 2004, month = {aug}, publisher = {{AIP} Publishing}, volume = {121}, number = {8}, pages = {3425--3433}, author = {Alberto Castro and Miguel A. L. Marques and Angel Rubio}, title = {Propagators for the time-dependent Kohn{\textendash}Sham equations}, journal = {The Journal of Chemical Physics} }