量子化学中非绝热问题的数学分析和计算方法

Nonadiabatic phenomenon has been a key research area in quantum chemistry and related applied mathematics, where the lack of systematic treatments to the multi-level, multiscale and high-dimensional quantum system remains the main challenges. In this project, we aim to focus on two aspects of the filed: the mixed quantum-classical dynamics, and thermal averages of multi-level quantum systems. For the dynamical part, my collaborator, Jianfeng Lu (Duke University) and I have developed a surface hopping algorithm based on frozen Gaussian approximation for semiclassical matrix Schrödinger equations, in the spirit of Tully’s fewest switches surface hopping method. The resulting algorithm can be viewed as a path integral stochastic representation of the semiclassical matrix Schrödinger equations. Although we have obtained some recognition from our quantum chemistry colleagues, there are still many remaining issues to be understood, including the possible challenges in many quantum energy levels, the implementations difficulties in some realistic scientific problems and optimizations issues due to the freedom within the algorithms, etc. For the thermal average part, Jianfeng and I have proposed a novel ring polymer representation for multi-level quantum system, and a path integral molecular dynamics with surface hopping (PIMD-SH) dynamics is also developed to sample the equilibrium distribution of ring polymer configurational space. It shares some challenges with the dynamical part, such as the difficulties in many energy level systems and realistic scientific problems, and there are some new possible research directions as well, including detailed comparison with other prevailing methods and exploring the connections between those methods.

非绝热现象一直是量子化学和相关应用数学领域的核心研究问题，但是由于多能级、多尺度以及高纬度的多方面挑战，这类问题一直没有系统严格的分析计算方法。在这个项目，我们计划主要探究这类问题的两个方面：混合量子-经典动力学问题和多能级量子系统的热力学均值问题。在热力学问题方面，我和我的合作者，杜克大学的鲁剑锋教授，已经对于准经典的多能级薛定谔方程设计出了一套基于冻结高斯逼近的能面跃迁方法。虽然我们已经得到了量子化学界的一定认可，这类方法还有很多没有被足够理解和探索的地方，包括更多数量能级系统中的数学方法上难点和实际科学问题数值实现上的挑战，还有基于算法中自由度的优化设计。在热力学均值方面，我和鲁剑锋对于多能级量子系统提出了一种ring polymer表示，对基于这种表示设计了含有能面跃迁的路径积分分子动力学方法，我们将继续研究这些方法在高位多能级中的挑战，并与其他高效算法做系统的分析比较。

非绝热现象一直是量子化学和相关应用数学领域的核心研究问题，但是由于多能级、多尺度等的多方面挑战，这类问题一直没有系统严格的分析计算方法。在热力学均值方面，我和鲁剑锋对于多能级量子系统提出了一种ring polymer的表示，并对基于这种表示设计了含有能面跃迁的路径积分分子动力学方法。本项目中我们继续开展这方面的研究。一方面，我们研究了ring polymer表示的连续性极限，并因此设计了预处理方法，避免了采样步长过小的数值约束；一方面，我们基于贝叶斯的框架，探索了路径积分分子动力学的反问题的适定性和快速求解方法；另外，我们还针对由于多能级产生的多等级采样问题，提出了多等级蒙特卡罗方法，有效降低了计算复杂度。

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