Nonlinear approximations for electronic structure calculations

We present a new method for electronic structure calculations based on novel algorithms for nonlinear approximations. We maintain a functional form for the spatial orbitals as a linear combination of products of decaying exponentials and spherical harmonics centred at the nuclear cusps. Although such representations bare some resemblance to the classical Slater-type orbitals, the complex-valued exponents in the representations are dynamically optimized via recently developed algorithms, yielding highly accurate solutions with guaranteed error bounds. These new algorithms make dynamic optimization an effective way to combine the efficiency of Slater-type orbitals with the adaptivity of modern multi-resolution methods. We develop numerical calculus suitable for electronic structure calculations. For any spatial orbital in this functional form, we represent its product with the Coulomb potential, its convolution with the Poisson kernel, etc., in the same functional form with optimized parameters. Algorithms for this purpose scale linearly in the number of nuclei. We compute electronic structure by casting the relevant equations in an integral form and solving for the spatial orbitals via iteration. As an example, for several diatomic molecules we solve the Hartree-Fock equations with speeds competitive to those of multi-resolution methods and achieve high accuracy using a small number of parameters.