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arXiv:0708.2896 · Journal of Mathematical Physics

Approximating a wavefunction as an unconstrained sum of Slater determinants

Gregory Beylkin, Martin J. Mohlenkamp, Fernando Pérez

Published Mar 1, 2008• 31 citations
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Abstract

The wavefunction for the multiparticle Schrödinger 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.

DOI
10.1063/1.2873123

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Approximating a Wavefunction as an Unconstrained Sum of Slater Determinants

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The wavefunction for the multiparticle Schrödinger 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.
University of Colorado APPM preprint number 554. The wavefunction for the multiparticle Schrödinger 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.
Our goal was to develop novel multiparticle computational tools for materials science based on representing operators and functions of many variables as short sums of separable functions. We have extended multiresolution separated representation of free-space Green's functions to those periodic or satisfying boundary conditions, making fast algorithms available for solving integral equations. The problem of incorporating inter-electron cusps within two-particle methods led us to the problem of approximating multivariable functions by sums of products of Gaussians. We have developed and tested a new (suboptimal) algorithm, sufficient for many practical purposes. Using this algorithm we constructed novel separated representations of non-convolutional Green's functions and spectral projectors for operators with potentials from the Rollnik class. Such potentials include all physically significant potentials considered within a finite domain. For optimal representations, we have developed an algorithm in the case of two variables; extensions to higher dimensions need to be worked on further. We started work on moving preliminary implementations of these algorithms into the Python environment where they can be used for practical computations. We have developed the algorithmic framework for a size-extensive/consistent method for the multiparticle Schrodinger equation. The central computation involved has revealed a center-of-mass principle for electrons, which gives hope for the construction of an analogue of the Fast Multipole Method for quantum mechanical systems.
Given a function w completely antisymmetric in n variables, there may exist a set of n functions of one variable such that the given function is a Slater determinant in the latter. The first problem considered is that of obtaining a criterion for this to be the case for a given function. This problem is solved by considering the function w as a mapping of the space of functions in n − 1 variables onto the space of functions of one variable. A necessary and sufficient condition for the initial function to be a Slater determinant is then shown to be that the image space be n dimensional. This criterion is converted into practical algorithms which can be employed for the determination. The application of one of these yields the theorem that an arbitrary linear combination of the n + 1 Slater determinants in n variables formed from n + 1 one-variable functions can always be written as a single Slater determinant. It is further proved that if the image space of the mapping is m(>n) dimensional, the original function can be expressed as a linear combination of m!/(m − n)!n! Slater determinants in n variables formed from m one-variable functions. Playing an important role in the analysis is the product of the mapping described above by its adjoint (the product is simply related to Dirac's density matrix for a quantum mechanical system of identical particles) as well as the eigenvectors and eigenvalues of this Hermitian positive semidefinite mapping. The latter form a basis for a systematic approximation procedure for representing a given function by a single Slater determinant or by sums of Slater determinants formed from a particular number of one-variable functions, which yields results obtained previously by Löwdin. Problems of simultaneous approximation of sets of antisymmetric functions and possible physical applications to many-fermion systems are briefly discussed.
We propose an approach to the electronic structure problem based on noninteracting electron pairs that has similar computational cost to conventional methods based on noninteracting electrons. In stark contrast to other approaches, the wave function is an antisymmetric product of nonorthogonal geminals, but the geminals are structured so the projected Schrödinger equation can be solved very efficiently. We focus on an approach where, in each geminal, only one of the orbitals in a reference Slater determinant is occupied. The resulting method gives good results for atoms and small molecules. It also performs well for a prototypical example of strongly correlated electronic systems, the hydrogen atom chain.
The center-of-mass principle is the key to the rapid computation of the interaction of a large number of classical particles. Electrons governed by the multiparticle Schrödinger equation have a much more complicated interaction mainly due to their spatial extent and the antisymmetry constraint on the total wave function of the combined electron system. We present a center-of-mass principle for quantum particles that accounts for this spatial extent, the antisymmetry constraint, and the potential operators. We use it to construct an algorithm for computing a size-consistent approximate wave function for large systems with simple geometries.