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

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.