Multiscale Computational Methods for Semiclassical Schroedinger Equations with Non-Adiabatic Effects

具有非绝热效应的半经典薛定谔方程的多尺度计算方法

• 批准号: 1522184
• 负责人: Qin Li
• 金额: $ 27万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-15 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522184&HistoricalAwards=false
• 关键词: Multiscale; Computational; Methods; Semiclassical; Schroedinger; Multiscale; Computational; Methods; Semiclassical; Schroedinger

The project addresses some fundamental issues in scientific computation in the modern age: multiscale modeling and simulation, which play essential roles in nanotechnology, communications, material sciences, and other areas of science and technology. The research will develop state-of-the-art computational methods for systems involving multiscale quantum-classical coupling. The methods under development have a wide range of applications to problems arising in solid-state physics, semiconductor device modeling, quantum chemistry, and materials science. Several graduate students will be trained through these research activities. Some of the research results also will be incorporated in the graduate curriculum to better train the next generation of researchers in modern applied mathematics.The investigator will develop efficient semiclassical and multiscale computational methods for some problems in quantum dynamics with non-adiabatic effects. The non-adiabatic effects are important since they correspond to quantum transitions between different potential energy surfaces that are important to describe quantum dynamic behavior in chemical reactions and semiconductors. Specifically, the project will investigate surface hopping and quantum dynamics in periodic lattices. In the adiabatic cases only the diagonal entries of the Wigner matrix need to be considered in the semiclassical limit, which often results in classical Liouville equations. To account for the non-adiabatic effects, one needs to also follow the dynamics of the off-diagonal entries of the Wigner matrix in order to adequately describe the quantum transition between different energy surfaces. These yield coupled, inhomogeneous, oscillatory partial differential equations that will be numerically solved by Gaussian beam and multiscale methods.

该项目解决了现代科学计算中的一些基本问题：多尺度建模和仿真，在纳米技术，通信，材料科学以及其他科学技术领域中起着重要作用。 该研究将开发涉及多尺度量子古典耦合的系统的最新计算方法。 所开发的方法对在固态物理，半导体设备建模，量子化学和材料科学中产生的问题有广泛的应用。 将通过这些研究活动对几名研究生进行培训。 一些研究结果还将在研究生课程中纳入，以更好地培训现代应用数学中的下一代研究人员。研究人员将开发有效的半经典和多尺度计算方法，以解决具有非绝热效应的量子动力学问题。非绝热效应很重要，因为它们对应于不同势能表面之间的量子转变，这对于描述化学反应和半导体中的量子动态行为很重要。具体而言，该项目将研究周期性晶格中的表面跳跃和量子动力学。在绝热的情况下，仅需要在半经典限制中考虑Wigner矩阵的对角线条目，这通常会导致经典的Liouville方程。为了说明非绝热效应，还需要遵循Wigner矩阵的非对角线条目的动力学，以充分描述不同能量表面之间的量子过渡。这些产量与高斯梁和多尺度方法在数值上求解，耦合，不均匀的振荡偏微分方程。