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Journal of Physics A Mathematical and Theoretical

Explicit Magnus expansions for solving the time-dependent Schrödinger equation

Ochbadrakh Chuluunbaatar, Vladimir Leonardovich Derbov, Artbazar Galtbayar, Alexander A. Gusev, M. S. Kaschiev, Sergue Il'ich Vinitsky, Tugal Zhanlav

Published Jun 26, 2008• 12 citations
Publisher

Abstract

The symmetric implicit operator-difference multi-layer schemes for solving the time-dependent Schr?dinger equation based on decomposition of the evolution operator via the explicit Magnus expansion up to the sixth order of accuracy with respect to the time step are presented. Reduced schemes for solving the set of coupled time-dependent Schr?dinger equations with respect to the hyper-radial variable are devised by using the Kantorovich expansion of the wave packet over a set of appropriate parametric basis angular functions. Further discretization of the resulting problem with symmetric operators is implemented by means of the finite-element method. The convergence and efficiency of the numerical schemes are demonstrated in benchmark calculations of the exactly solvable models of a one-dimensional time-dependent oscillator, a two-dimensional oscillator in time-dependent electric field by using the conventual angular basis, and the inexactly solvable model of a three-dimensional kicked hydrogen atom in a magnetic field by using a parametric basis of the angular oblate spheroidal functions developed in our previous paper (Chuluunbaatar O et al 2007 J. Phys. A: Math. Theor. 40 11485?524).

DOI
10.1088/1751-8113/41/29/295203

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The symmetric implicit operator-difference multi-layer schemes for solving the time-dependent Schr?dinger equation based on decomposition of the evolution operator via the explicit Magnus expansion up to the sixth order of accuracy with respect to the time step are presented. Reduced schemes for solving the set of coupled time-dependent Schr?dinger equations with respect to the hyper-radial variable are devised by using the Kantorovich expansion of the wave packet over a set of appropriate parametric basis angular functions. Further discretization of the resulting problem with symmetric operators is implemented by means of the finite-element method. The convergence and efficiency of the numerical schemes are demonstrated in benchmark calculations of the exactly solvable models of a one-dimensional time-dependent oscillator, a two-dimensional oscillator in time-dependent electric field by using the conventual angular basis, and the inexactly solvable model of a three-dimensional kicked hydrogen atom in a magnetic field by using a parametric basis of the angular oblate spheroidal functions developed in our previous paper (Chuluunbaatar O et al 2007 J. Phys. A: Math. Theor. 40 11485?524).
Operator-difference multilayer schemes for solving the time-dependent Schrödinger equation up to sixth order of accuracy in the time step are presented. Reduced schemes for solving a set of coupled time-dependent Schrödinger equations with respect to the hyper-radial variable are devised using expansion of a wave packet over the set of appropriate basis angular functions. Further discretization of the resulting problem is realized by means of the finite-element method. The convergence of the expansion with respect to the number of basis functions and the efficiency of the numerical schemes are demonstrated in the exactly solvable model of an electric-field-driven two-dimensional oscillator (or a charged particle in a constant uniform magnetic field), in which we explicitly observed an effect of the periodical focusing and defocusing of the probability density flux.
The operator-difference multi-layer (ODML) schemes for solving the time-dependent Schrödinger equation (TDSE) till six order accuracy by a time step are presented. The reduced schemes for solving a set of the coupled TDSE’s are devised by using a set of appropriate basis angular functions and a finite element method with respect to a hyper-radial variable. Convergence by a number of the basis functions and efficiency of the numerical schemes are demonstrated in the case of an exactly solvable model of the two-dimensional oscillator in time-depended electric fields.
A symbolic algorithm to generate the multilayer operator-difference schemes for solving the evolution problem of the time-dependent Schrödinger equation is elaborated. An additional gauge transformation of operator-difference schemes to make good use of the finite-element discretization is applied. The efficiency of the generated numerical schemes until the sixth order with respect to the time step and until the seventh order with respect to the spatial step is demonstrated by calculations of some finite-dimensional quantum systems in external fields.
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 geometrical structure of the equations. 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 free Magnus expansions. These have been developed, however, for linear systems, yet our problem is nonlinear: formally, the full quantum-classical system can be rewritten as a nonlinear Schrödinger equation, i.e., one in which the Hamiltonian depends on the system itself. The Magnus expansion algorithms for linear systems require the application of the Hamiltonian at intermediate points in a given propagating interval. For nonlinear systems, this poses a problem as this Hamiltonian is unknown due to its dependence on the state. We approximate it by employing a higher order extrapolation using previous steps as input. The resulting technique can then be regarded as a multistep technique or, alternatively, as a predictor corrector formula.
The use of a time-dependent basis set for solving the time-dependent Schrödinger equation is proposed. The basis set is a generalization of the harmonic oscillator functions. The proposed method is similar to the approaches formulated by Lee and Heller and by Coalson and Karplus. However, in the present method the time dependence of the basis functions is determined differently. Numerical studies on the quartic oscillator and on the Morse potential are carried out to investigate the performance of the various methods.