Computational methods for materials science, high frequency wave propagation, and quantum mechanics

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This project involves modeling, simulation, and efficient computation of technologically important and intellectually interesting problems. In the area of material science the research investigates two new off-lattice methods for the computational modeling of the evolution of crystalline materials in which the lattice structure may change. This is needed when defect formation is important, for example. The project will also address the efficient computation of high frequency wave propagation in nonhomogeneous materials. This classic problem in computational science and applied mathematics has attracted much attention because it is useful for problems in geophysics and optics and it is a mathematically rich problem requiring insight into the behavior of wave equation. The PI also plans to develop numerical methods for studying quantum dynamics of molecules. The importance of quantum mechanics in understanding chemical reactions cannot be overstated, and this research offers the potential of providing detailed insight and predictive power for a variety of photo-induced chemical reactions. The project also involves training of graduate students.The first project involves developement of off-lattice computation models for the evolution of crystalline materials. In one case, the PI plans to formulate an off-lattice kinetic Monte Carlo method for the simulation of epitaxial growth and in other an alternative to the phase field crystal method is proposed. In both cases the underlying energy of the system is a suitable intermolecular potential. In addition, in both cases one will be computing on time scales orders of magnitude larger than molecular dynamics. Another project involves the computation of high frequency solutions of the wave equation. The goal is to exploit the asymptotic structure of high frequency solutions to develop efficient numerical methods that do not have the errors that result from asymptotic approximations that occur when one uses methods that rely on geometrical optics or Gaussian beams. The PI also plans to develop numerical methods for the studying quantum dynamics of molecules when conical intersections play an important role. A full quantum mechanical treatment would mean solving the Schroedinger equation in 20 to 30 dimensions even for small molecules. Our plan is to identify the important modes by using surface hopping methods as a probe of the energy landscape. A full quantum treatment can then be accurately obtained just using this smaller set of modes.