Quantum Dynamics with Trajectories

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Robert Wyatt of the University of Texas at Austin is supported by the Theoretical and Computational Chemistry Program to develop analytical and computational methods and algorithms for generating quantum trajectories that can be used for solving multidimensional problems in quantum dynamics. An ab initio approach will be taken to develop all that is needed "on the fly" as trajectory ensembles evolve. The focus will be on performing numerically accurate integration of the time-dependent Schrodinger equation using trajectories as the underlying computational tool. The techniques that will be explored are the quantum trajectory method (QTM) and a second trajectory method based on an implementation of stochastic mechanics. For the QTM, the following methods will be developed and incorporated into a new code for multidimensional problems: (1) both direct product and non-direct product (spherical symmetry) distributed approximating functionals (DAFs), (2) a new variational moving least squares method to evaluate derivatives that appear in the equations of motion, (3) a new local gauge transformation method to filter input functions before derivative evaluation is initiated, and (4) several types of grid adaptation intended to bypass problems associated with regions near nodes in the wavefunction. In stochastic mechanics, the following topics will be investigated: (1) "on the fly" integration of the convection-advection-diffusion (CAD) equations in the Lagrangian picture, and (2) the possible use of complex-valued trajectories to avoid the node problem. Applications will be made to electronic nonadiabatic transitions in collisions with and without curve crossing, and intramolecular rearrangements such as the vinylidene isomerization reaction. The solution of the time-dependent Schrodinger equation sought in this research is a very general problem which has important ramifications in chemistry, physics, and biology. This research could ultimately lead to the development of computer codes that will permit new understanding of important properties of complex molecular systems.