Nonlinear Approximations for Electronic Structure Calculations

In this talk I will present a new method for electronic structure calculations. This approach maintains a functional form for the spatial orbitals that involves (linear combinations of) products of exponentials and spherical harmonics centered at the nuclear cusp locations. Although such representations bear some resemblance to classical Slater-type orbitals, the (complex-valued) exponents in the representations are dynamically optimized throughout the course of the computation using recently developed algorithms. These algorithms make such dynamic optimization a practical --- and, in fact, efficient --- way to combine the compactness of Slater-type orbitals with the ability of modern multi-resolution methods to compute highly accurate solutions with guaranteed error bounds. 

I will also discuss a numerical calculus based on such representations that is suitable for solving electronic structure calculations. This calculus is used for electronic structure calculations by casting the (e.g. Hartree-Fock or Kohn-Sham) equations in integral form via the Lippmann-Schwinger reformulation, and solving for the orbitals through iteration; after operations such as multiplying spatial orbitals or convolution with the Poisson kernel, the algorithms developed here are used to maintain the basic functional form using a small number of parameters. As an example of this approach, I will present numerical experiments on the Hartree-Fock equations for several diatomic molecules. These numerical experiments demonstrate that high accuracy can be achieved using a small number of parameters and with speeds that are competitive with multiresolution methods.