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Molecular Simulation

A Comparison of Different Numerical Propagation Schemes for Solving the Time-Dependent Schrödinger Equation in the Position Representation in One Dimension for Mixed Quantum-and Molecular Dynamics Simulations

Salomon R. Billeter, Wilfred F. Van Gunsteren A

Published Nov 1, 1995• 17 citations
Publisher

Abstract

Various numerical integration schemes to calculate the propagation of a state following the time-dependent Schrodinger equation in the one dimensional position representation are presented and compared to each other. Three potentials have been used: a harmonic, a double-well and a zero potential. Eigenstates and a coherent state have been chosen as initial states. Special attention has been given to the long-time stability of the algorithms. These are: kinetic referenced split operator (KRSO), kinetic referenced Cayley (KRC), distributed approximating functions (DAF), Chebysheff expansion (CH), residuum minimization (RES), second-order differencing (SOD), an eigenstate expansion (EE) and a corrected kinetic referenced split operator (CKRSO). In addition, a speedup of the KRC and KRSO methods is presented which is specially suited when very few grid points are used. Numerical results are compared to analytically calculated values. Mixed classical/quantum mechanical simulations require a representation of the quantum state on a limited number of grid points, classical integration time steps of about one femtosecond and compatibility with methods to solve the time-ordering problem. For the considered potentials which differ quite essentially from the potentials used for scattering problems in particle physics, the EE method has been found to be faster, more accurate and more stable than the other methods if only a few grid points are required. Otherwise, good results have been obtained with KRC, KRSO, CH, DAF and RES. SOD has been found to be too slow, and CKRSO is not stable enough for long simulation times.

DOI
10.1080/08927029508022343

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Various numerical integration schemes to calculate the propagation of a state following the time-dependent Schrodinger equation in the one dimensional position representation are presented and compared to each other. Three potentials have been used: a harmonic, a double-well and a zero potential. Eigenstates and a coherent state have been chosen as initial states. Special attention has been given to the long-time stability of the algorithms. These are: kinetic referenced split operator (KRSO), kinetic referenced Cayley (KRC), distributed approximating functions (DAF), Chebysheff expansion (CH), residuum minimization (RES), second-order differencing (SOD), an eigenstate expansion (EE) and a corrected kinetic referenced split operator (CKRSO). In addition, a speedup of the KRC and KRSO methods is presented which is specially suited when very few grid points are used. Numerical results are compared to analytically calculated values. Mixed classical/quantum mechanical simulations require a representation of the quantum state on a limited number of grid points, classical integration time steps of about one femtosecond and compatibility with methods to solve the time-ordering problem. For the considered potentials which differ quite essentially from the potentials used for scattering problems in particle physics, the EE method has been found to be faster, more accurate and more stable than the other methods if only a few grid points are required. Otherwise, good results have been obtained with KRC, KRSO, CH, DAF and RES. SOD has been found to be too slow, and CKRSO is not stable enough for long simulation times.
The numerical evaluation of coherent-state path integrals for quantum dynamical problems is discussed for one-dimensional examples. To propagate an initial state, we use the normal and antinormal ordered coherent-state path integrals combined with a split-operator technique dividing the Hamiltonian into harmonic and anharmonic parts. For numerical purposes integrations must be approximated by quadrature formulae. This leads to a matrix multiplication scheme which is systematically tested for the double-well and Morse potentials. The method is accurate for propagation times much longer than the natural time scale of the system, and it allows for short as well as long time steps without loss of stability.
This chapter focuses on widely used approximate method: the numerical integration of the Schrodinger equation, restricting ourselves to one dimension for simplicity. There is a plethora of approaches that yield highly accurate numerical eigenvalues and eigenfunctions. In quantum mechanics one sometimes seeks after bound stationary states that satisfy given physical boundary conditions. Boundary conditions at infinity play a relevant role in quantum mechanics, in which case one looks for bound states that vanish asymptotically at the end points of coordinate interval. Solving the propagation matrix equation exactly for every coordinate pair in the physical interval is as difficult as solving the original differential equation. However, sometimes the former is preferable to develop approximate numerical algorithms for the accurate calculation of eigenvalues and eigenfunctions. The perturbation method based on the Taylor expansion only requires derivatives of the potential energy function, which one obtains easily in most cases and appears to be preferable for the systematic improvement of the approximate propagation matrix.
We compare efficiency of two methods for numerical solution of the time‐dependent Schrödinger equation, namely the Chebyshev method and the recently introduced generalized Crank‐Nicholson method. As a testing system the free propagation of a particle in one dimension is used. The space discretization is based on the high‐order finite diferences to approximate accurately the kinetic energy operator in the Hamiltonian. We show that the choice of the more effective method depends on how many wave functions must be calculated during the given time interval to obtain relevant and reasonably accurate information about the system, i.e. on the choice of the time step.
The time-dependent Schrödinger Equation (TDSE) is a parabolic partial differential equation (PDE) comparable to a diffusion equation but with imaginary time. Due to its first order time derivative, exponential integrators or propagators are natural methods to describe evolution in time of the TDSE, both for time-independent and time-dependent potentials. Two splitting methods based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracies in the time step △t. The unitary factorization of exponentials to high order accuracy depends on commutators of kinetic energy operators with potentials. Fourth-order accuracy propagators can involve negative or complex time steps, or real time steps only but with gradients of potentials, i.e. forces. Extending the propagators of TDSE's to imaginary time allows to also apply these methods to classical many-body dynamics, and quantum statistical mechanics of molecular systems.
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 detailed calculation of the dynamics of a chemical system is usually not considered due to the size and cost of the computation. It is thus useful to examine various approximation methods. Such methods first need to be tried out on simple systems, like one-dimensional motion. Here two approaches to approximating the solutions of the Schrödinger and von Neumann equations by single time-dependent Gaussians are explored and contrasted, explicitly for one-dimensional barrier penetration. The first approach, in which no tunneling occurs, is local in nature and characterized by an expansion of the equations of motion to second order about the average position of the Gaussian wave packet or about the average position and momentum of the Gaussian Wigner function. This approach was first introduced by Heller [E. J. Heller, J. Chem. Phys. 62, 1544 (1975)]. Here both Heller’s approach and a more general truncation method are considered. Indeed tunneling can be incorporated if second-order terms in the quantal von Neumann equation are included. However, the resulting dynamics is unstable for kinetic energies where the exact wavepacket (and Wigner function) splits into nonnegligible parts that represent reflection and transmission. In contrast, the second approach is a global method which is obtained by applying appropriate closure approximations to the equations of motion for the first- and second-order position and momentum expectation values. This method allows tunneling and is stable at all kinetic energies. It is also possible to approximate the global equations in such a way that the local equations are obtained.