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The Journal of Chemical Physics

Nodal hypersurfaces and Anderson’s random-walk simulation of the Schrödinger equation

Douglas J. Klein, Herbert M. Pickett

Published Jun 1, 1976• 51 citations
Publisher
DOI
10.1063/1.432043

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Nodal hypersurfaces and Anderson’s random-walk simulation of the Schrödinger equation

Douglas J. Klein, Herbert M. Pickett

Abstract Klein and Pickett provided a discussion of the problem of determining the nodes for a fixed-node QMC calculation proposed earlier by Andersona They suggested using the nodes of a Hartree-Fock wavefunction and noted that a Jastrow wavefunctionb consisting of a Hartree-Fock function multiplied by a permutationally symmetric and positive function to correct for electron correlation could be quite accurate. Only later was it realized that the node structures given by Hartree-Fock wavefunctions were surprisingly simple even for large many-electron systems. Combinations of determinantal functions with Jastrow functions were found especially useful as trial functions for importance sampling in diffusion and Green’s function QMC calculations.

•Jun 18, 2007•1 citation

Quantum Diffusion-Limited Aggregation

David Johnson, Ordóñez, Gonzalo

Though classical random walks have been studied for many years, research concerning their quantum analogues, quantum random walks, has only come about recently. Numerous simulations of both types of walks have been run and analyzed, and are generally well-understood. Research pertaining to one of the more important properties of classical random walks, namely, their ability to build fractal structures in diffusion-limited aggregation, has been particularly noteworthy. However, only now has research begun in this area in regards to quantum random motion. The study of random walks and the structures they build has various applications in materials science. Since all processes are quantum in nature, it is important to consider the quantum variant of diffusion-limited aggregation. Recognizing that Schrödinger equation and a classical random walk are both diffusion equations, it is possible to connect and compare them. Using similar parameters for both equations, we ran various simulations aggregating particles. Our results show that particles moving according to Schrödinger equation can create fractal structures, much like the classical random walk. Furthermore, the fractal dimensions of these quantum diffusion-limited aggregates vary between 1.43 and 2, depending on the size of the initial wave packet.

arXiv:1111.0626•Nov 2, 2011•1 citation
Abstract Klein and Pickett provided a discussion of the problem of determining the nodes for a fixed-node QMC calculation proposed earlier by Andersona They suggested using the nodes of a Hartree-Fock wavefunction and noted that a Jastrow wavefunctionb consisting of a Hartree-Fock function multiplied by a permutationally symmetric and positive function to correct for electron correlation could be quite accurate. Only later was it realized that the node structures given by Hartree-Fock wavefunctions were surprisingly simple even for large many-electron systems. Combinations of determinantal functions with Jastrow functions were found especially useful as trial functions for importance sampling in diffusion and Green’s function QMC calculations.
Though classical random walks have been studied for many years, research concerning their quantum analogues, quantum random walks, has only come about recently. Numerous simulations of both types of walks have been run and analyzed, and are generally well-understood. Research pertaining to one of the more important properties of classical random walks, namely, their ability to build fractal structures in diffusion-limited aggregation, has been particularly noteworthy. However, only now has research begun in this area in regards to quantum random motion. The study of random walks and the structures they build has various applications in materials science. Since all processes are quantum in nature, it is important to consider the quantum variant of diffusion-limited aggregation. Recognizing that Schrödinger equation and a classical random walk are both diffusion equations, it is possible to connect and compare them. Using similar parameters for both equations, we ran various simulations aggregating particles. Our results show that particles moving according to Schrödinger equation can create fractal structures, much like the classical random walk. Furthermore, the fractal dimensions of these quantum diffusion-limited aggregates vary between 1.43 and 2, depending on the size of the initial wave packet.