Dissertation/Thesis Abstract

A Study of the Errors of the Fixed-Node Approximation in Diffusion Monte Carlo
by Rasch, Kevin M., Ph.D., North Carolina State University, 2012, 172; 3538537
Abstract (Summary)

Quantum Monte Carlo techniques stochastically evaluate integrals to solve the many-body Schrödinger equation. QMC algorithms scale favorably in the number of particles simulated and enjoy applicability to a wide range of quantum systems. Advances in the core algorithms of the method and their implementations paired with the steady development of computational assets have carried the applicability of QMC beyond analytically treatable systems, such as the Homogeneous Electron Gas, and have extended QMC’s domain to treat atoms, molecules, and solids containing as many as several hundred electrons.

FN-DMC projects out the ground state of a wave function subject to constraints imposed by our ansatz to the problem. The constraints imposed by the fixed-node Approximation are poorly understood. One key step in developing any scientific theory or method is to qualify where the theory is inaccurate and to quantify how erroneous it is under these circumstances.

I investigate the fixed-node errors as they evolve over changing charge density, system size, and effective core potentials. I begin by studying a simple system for which the nodes of the trial wave function can be solved almost exactly. By comparing two trial wave functions, a single determinant wave function flawed in a known way and a nearly exact wave function, I show that the fixed-node error increases when the charge density is increased. Next, I investigate a sequence of Lithium systems increasing in size from a single atom, to small molecules, up to the bulk metal form. Over these systems, FN-DMC calculations consistently recover 95% or more of the correlation energy of the system. Given this accuracy, I make a prediction for the binding energy of Li4 molecule. Last, I turn to analyzing the fixed-node error in first and second row atoms and their molecules. With the appropriate pseudo-potentials, these systems are iso-electronic, show similar geometries and states. One would expect with identical number of particles involved in the calculation, errors in the respective total energies of the two iso-electronic species would be quite similar. I observe, instead, that the first row atoms and their molecules have errors larger by twice or more in size. I identify a cause for this difference in iso-electronic species. The fixed-node errors in all of these cases are calculated by careful comparison to experimental results, showing that FN-DMC to be a robust tool for understanding quantum systems and also a method for new investigations into the nature of many-body effects.

Indexing (document details)
Advisor: Mitas, Lubos
Commitee: Brown, David, Buongiorno-Nardelli, Marco, Whitten, Jerry L.
School: North Carolina State University
Department: Physics
School Location: United States -- North Carolina
Source: DAI-B 74/07(E), Dissertation Abstracts International
Source Type: DISSERTATION
Subjects: Physical chemistry, Quantum physics, Physics
Keywords: Ab initio electronic structures, Diffusion Monte Carlo, Fixed-node approximation, Many-body quantum mechanics
Publication Number: 3538537
ISBN: 978-1-303-01562-5
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