Propagators for Quantum-Classical Models: Commutator-Free Magnus Methods.

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.

Exponential propagators for the Schrödinger equation with a time-dependent potential.

We consider the numerical integration of the Schrödinger equation with a time-dependent Hamiltonian given as the sum of the kinetic energy and a time-dependent potential. Commutator-free (CF) propagators are exponential propagators that have shown to be highly efficient for general time-dependent Hamiltonians. We propose new CF propagators that are tailored for Hamiltonians of the said structure, showing a considerably improved performance.

Symplectic time-average propagators for the Schrödinger equation with a time-dependent Hamiltonian.

Several symplectic splitting methods of orders four and six are presented for the step-by-step time numerical integration of the Schrödinger equation when the Hamiltonian is a general explicitly time-dependent real operator. They involve linear combinations of the Hamiltonian evaluated at some intermediate points. We provide the algorithm and the coefficients of the methods, as well as some numerical examples showing their superior performance with respect to other available schemes.

Solving the Schrödinger eigenvalue problem by the imaginary time propagation technique using splitting methods with complex coefficients.

The Schrödinger eigenvalue problem is solved with the imaginary time propagation technique. The separability of the Hamiltonian makes the problem suitable for the application of splitting methods. High order fractional time steps of order greater than two necessarily have negative steps and cannot be used for this class of diffusive problems.

Fourier methods for the perturbed harmonic oscillator in linear and nonlinear Schrödinger equations.

We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since the system can be split into the kinetic and remaining part, and each part can be solved efficiently using fast Fourier transforms. Splitting the system into the quantum harmonic-oscillator problem and the remaining part allows us to get higher accuracies in many cases, but it requires us to change between Hermite basis functions and the coordinate space, and this is not efficient for time-dependent frequencies or strong nonlinearities.

Symplectic splitting operator methods for the time-dependent Schrodinger equation.

We present a family of symplectic splitting methods especially tailored to solve numerically the time-dependent Schrodinger equation. When discretized in time, this equation can be recast in the form of a classical Hamiltonian system with a Hamiltonian function corresponding to a generalized high-dimensional separable harmonic oscillator. The structure of the system allows us to build highly efficient symplectic integrators at any order.

Comment on "Structure of positive decompositions of exponential operators".

An elementary proof is shown on the necessary existence of negative coefficients in splitting methods of order p > or = 3.

Symplectic maps for approximating polynomial Hamiltonian systems.

We study how to approximate polynomial Hamiltonian systems by composition of symplectic maps. Recently, a number of methods preserving the symplectic character have appeared. However, they are not completely satisfactory because, in general, they are computationally expensive, very difficult to obtain or their accuracy is relatively low.