Quantum Dynamics with Trajectories

德克萨斯大学奥斯汀分校的罗伯特·怀亚特由理论和计算化学计划支持，开发分析和计算方法和算法，以生成可用于解决量子动力学中的多维问题的量子轨迹。随着弹道组合的发展，将采取从头开始的方法来开发所需的所有东西。重点将是使用轨迹作为基本的计算工具来执行与时间相关的薛定谔方程的数值精确积分。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，将发展以下方法并将其纳入多维问题的新代码中：(1)直积和非直积(球对称)分布近似泛函(DAFS)；(2)计算运动方程中出现的导数的新的变分移动最小二乘方法；(3)在开始计算导数之前过滤输入函数的新的局域规范变换方法；以及(4)几种类型的网格自适应，旨在绕过与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)在拉格朗日图像中对对流-平流-扩散(CAD)方程进行“动态”积分，以及(2)可能使用复值轨迹来避免节点问题。电子的非绝热跃迁将被应用于有或没有曲线交叉的碰撞中，以及分子内重排，如亚偏二烯异构化反应。本研究中所寻求的含时薛定谔方程的解是一个非常普遍的问题，在化学、物理和生物学中都有重要的分支。这项研究最终可能导致计算机代码的开发，使人们能够对复杂分子系统的重要性质有新的理解。

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.