Showing papers by "Gregory Beylkin published in 2007"

Approximating a Wavefunction as an Unconstrained Sum of Slater Determinants

Abstract: 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.

Grids and transforms for band-limited functions in a disk

Abstract: We develop fast discrete Fourier transforms (and their adjoints) from a square in space to a disk in the Fourier domain. Since our new transforms are not unitary, we develop a fast inversion algorithm and derive corresponding estimates that allow us to avoid iterative methods typically used for inversion. We consider the eigenfunctions of the corresponding band-limiting and space-limiting operator to describe spaces on which these new transforms can be inverted and made useful. In the process, we construct polar grids which provide quadratures and interpolation with controlled accuracy for functions band-limited within a disk. For rapid computation of the involved trigonometric sums we use the unequally spaced fast Fourier transform, thus yielding fast algorithms for all new transforms. We also introduce polar grids motivated by linearized scattering problems which are obtained by discretizing a family of circles. These circles are generated by using a single circle passing through the origin and rotating this circle with the origin as a pivot. For such grids, we provide a fast algorithm for interpolation to a near optimal grid in the disk, yielding an accurate adjoint transform and inversion algorithm.

MADNESS applied to density functional theory in chemistry and nuclear physics

Abstract: We describe some recent mathematical results in constructing computational methods that lead to the development of fast and accurate multiresolution numerical methods for solving quantum chemistry and nuclear physics problems based on Density Functional Theory (DFT). Using low separation rank representations of functions and operators in conjunction with representations in multiwavelet bases, we developed a multiscale solution method for integral and differential equations and integral transforms. The Poisson equation, the Schrodinger equation, and the projector on the divergence free functions provide important examples with a wide range of applications in computational chemistry, nuclear physics, computational electromagnetic and fluid dynamics. We have implemented this approach along with adaptive representations of operators and functions in the multiwavelet basis and low separation rank (LSR) approximation of operators and functions. These methods have been realized and implemented in a software package called Multiresolution Adaptive Numerical Evaluation for Scientific Simulation (MADNESS).

Preliminary results on approximating a wavefunction as an unconstrained sum of Slater determinants

Abstract: A multiparticle wavefunction, which is a solution of the multiparticle Schrodinger equation, satisfies the antisymmetry con- dition, thus making it natural to approximate it as a sum of Slater determinants. Many current methods do so but, in addition, they impose structural constraints on the Slater determinants, such as orthogonality between orbitals or a particular excitation pattern. By removing these constraints, we hope to obtain much more efficient expansions. We use an integral formulation of the problem, a Green's function iteration, and a fitting procedure based on the computa- tional paradigm of separated representations. For constructing and solving a matrix-integral system of equations derived from antisymmetric inner products, we develop new algorithms with computational complexity competitive with current methods. We describe preliminary numerical results and make some observations. Given the difficulties of solving the multiparticle Schrodinger equation, current numerical methods in quantum chem- istry/physics are remarkably successful. Part of their success comes from efficiencies gained by imposing structural con- straints on the wavefunction to match physical intuition. However, such methods scale poorly to high accuracy, and are biased to only reveal structures that were part of their own construction. In (3) we develop a method that allows better scaling to high accuracy and an unbiased exploration of the structure of the wavefunction by approximating it as an unconstrained sum of Slater determinants. Motivated by the physical intuition that electrons may be excited into higher energy states, the Configuration Interaction (CI) family of methods choose a set of determinants with predetermined orbitals, and then optimize the coefficients used to combine them. When it is found insufficient, methods to optimize the orbitals, work with multiple reference states, etc., are introduced. A common feature of all these methods is that they impose some structural constraints on the Slater determinants, such as orthogonality of orbitals or an excitation pattern. As the requested accuracy increases, these structural constraints trigger an explosion in the number of determinants used, making the computation intractable for high accuracy. The a priori structural constraints present in CI-like methods also force the wavefunction to comply with such structure, whether or not it really is the case. For example, if you use a method that approximates the wavefunction as a linear combination of a reference state and excited states, you could not learn that the wavefunction is better approximated as a linear combination of several non-orthogonal, near-reference states. Thus, the choice of numerical method is not just a computational issue; it can help or hinder our understanding of the wavefunction. Our goal is to construct an adaptive numerical method without imposing a priori structural constraints besides that of antisymmetry. In (3) we derive and present an algorithm for approximating a wavefunction with an unconstrained sum of Slater determinants, with fully-adaptive single-electron functions. In particular, we discard the notions of reference state and excitation of orbitals. The functions comprising the Slater determinants need not come from a particular basis set, be orthogonal, or follow some excitation pattern. They are computed so as to optimize the overall representation. In this respect we follow the philosophy of separated representations (1, 2), which allow surprisingly accurate expansions with remarkably few terms. Our construction generates a solution using an iterative procedure based on nonlinear approximations via separated repre- sentations. We derive a system of integral equations that describe the fully-correlated many-particle problem. The computa- tional core of the method is the repeated construction and solution of a matrix-integral system of equations. Specifically, the following are distinctive features of our approach. We use: