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

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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.