Quantum Dynamics with Trajectories

德克萨斯大学奥斯汀分校的Robert Wyatt在理论和计算化学项目的支持下，开发了分析和计算方法和算法，用于生成量子轨迹，可用于解决量子动力学中的多维问题。一个从头开始的方法将被用来开发所有需要的“在飞行中”随着轨迹集成的发展。重点将是执行数值精确积分的时间依赖薛定谔方程使用轨迹作为潜在的计算工具。将探索的技术是量子轨迹方法（QTM）和基于随机力学实现的第二轨迹方法。对于QTM，将开发以下方法并将其纳入针对多维问题的新代码：(1)直接积和非直接积（球对称）分布近似函数（daf），(2)一种新的变分移动最小二乘法来评估运动方程中出现的导数，(3)一种新的局部规范变换方法在导数评估开始之前过滤输入函数，以及(4)几种类型的网格自适应旨在绕过波函数中节点附近区域相关的问题。在随机力学中，将研究以下主题：(1)拉格朗日图中对流-平流-扩散（CAD）方程的“动态”积分，以及(2)可能使用复值轨迹来避免节点问题。将应用于有或没有曲线交叉的碰撞中的电子非绝热跃迁，以及分子内重排，如偏乙烯异构化反应。本研究寻求的时变薛定谔方程的解是一个非常普遍的问题，在化学、物理和生物学中具有重要的分支。这项研究可能最终导致计算机代码的发展，这将使人们对复杂分子系统的重要特性有新的认识。

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.