Approximating a Wavefunction as an Unconstrained Sum of Slater Determinants

The wavefunction for the multiparticle Schr\"odinger equation is a function of many variables and satisfies an antisymmetry condition, so it is natural to approximate it as a sum of Slater determinants. Many current methods do so, but they impose additional structural constraints on the determinants, such as orthogonality between orbitals or an excitation pattern. We present a method without any such constraints, by which we hope to obtain much more efficient expansions, and insight into the inherent structure of the wavefunction. We use an integral formulation of the problem, a Green's function iteration, and a fitting procedure based on the computational paradigm of separated representations. The core procedure is the construction and solution of a matrix-integral system derived from antisymmetric inner products involving the potential operators. We show how to construct and solve this system with computational complexity competitive with current methods.Comment: 30 page

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A fast transform for spherical harmonics

Acknowledgements and Notes. I would like to thank my thesis advisor, R.R. Coifman, for his help and guidance. Spherical Harmonics arise on the sphere S 2 in the same way that the (Fourier) exponential functions {e ikθ}k∈Z arise on the circle. Spherical Harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform. Without a fast transform, evaluating (or expanding in) Spherical Harmonic series on the computer is slow—for large computations prohibitively slow. This paper provides a fast transform. For a grid of O(N 2) points on the sphere, a direct calculation has computational complexity O(N 4), but a simple separation of variables and Fast Fourier Transform reduce it to O(N 3) time. Here we present algorithms with times O(N 5/2 log N) and O(N 2 (log N) 2). The problem quickly reduces to the fast application of matrices of Associated Legendre Functions of certain orders. The essential insight is that although these matrices are dense and oscillatory, locally they can be represented efficiently in trigonometric series.