Mathematics Colloquia and Seminars

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Description

The Kohn-Sham density functional theory (KSDFT) is the most widely used theory for studying electronic properties of molecules and solids. It allows us to reduce the need to solve a many-body Schrodinger's equation to the task of solving a system of single electron equations coupled by the electron density. These equations can be viewed as a nonlinear eigenvalue problem. Although they contain far fewer degrees of freedom, they are more difficult to solve due to their nonlinear properties. In this talk, I will give an overview on efficient algorithms for solving this type of problem. I will describe discretization techniques that can potentially reduce the number of degrees of freedom required to represent the Kohn-Sham Hamiltonian. I will also present recently developed algebraic techniques for reducing the computational complexity of the Kohn-Sham DFT calculation. A key concept that is important for understanding these algorithms is a nonlinear map known as the Kohn Sham map. The ground state electron density is a fixed point of this map. I will examine properties of this map and its Jacobian. These properties can be used to develop effective strategies for accelerating Broyden's method for finding the optimal solution. They can also be used to reduce the computational complexity associated with the evaluation of the Kohn Sham map, which is often the most expensive step in a Broyden iteration.