Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations

Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations

Bibliographic Details
Main Author: Parker, William David
Language:English
Published: The Ohio State University / OhioLINK 2010
Subjects:
Materials Science
Physics
quantum Monte Carlo
polynomial approximation
approximation error
Si interstitial defects
Online Access:http://rave.ohiolink.edu/etdc/view?acc_num=osu1284495775
id ndltd-OhioLink-oai-etd.ohiolink.edu-osu1284495775
record_format oai_dc
collection NDLTD
language English
sources NDLTD
topic Materials Science
Physics
quantum Monte Carlo
polynomial approximation
approximation error
Si interstitial defects
spellingShingle Materials Science
Physics
quantum Monte Carlo
polynomial approximation
approximation error
Si interstitial defects
Parker, William David
Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations
author Parker, William David
author_facet Parker, William David
author_sort Parker, William David
title Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations
title_short Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations
title_full Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations
title_fullStr Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations
title_full_unstemmed Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations
title_sort speeding up and quantifying approximation error in continuum quantum monte carlo solid-state calculations
publisher The Ohio State University / OhioLINK
publishDate 2010
url http://rave.ohiolink.edu/etdc/view?acc_num=osu1284495775
work_keys_str_mv AT parkerwilliamdavid speedingupandquantifyingapproximationerrorincontinuumquantummontecarlosolidstatecalculations
_version_ 1719429374032216064
spelling ndltd-OhioLink-oai-etd.ohiolink.edu-osu12844957752021-08-03T06:00:46Z Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations Parker, William David Materials Science Physics quantum Monte Carlo polynomial approximation approximation error Si interstitial defects <p>Quantum theory has successfully explained the mechanics of much of the microscopic world. However, Schrödinger's equations are difficult to solve for many-particle systems. Mean-field theories such as Hartree-Fock and density functional theory account for much of the total energy of electronic systems but fail on the crucial correlation energy that predicts solid cohesion and material properties.</p><p>Monte Carlo methods solve differential and integral equations with error independent of the number of dimensions in the problem. Variational Monte Carlo (VMC) applies the variational principle to optimize the wave function used in the Monte Carlo integration of Schrödinger's time-independent equation. Diffusion Monte Carlo (DMC) represents the wave function by electron configurations diffusing stochastically in imaginary time to the ground state.</p><p>Approximations in VMC and DMC make the problem tractable but introduce error in parameter-controlled and uncontrolled ways. The many-electron wave function consists of single-particle orbitals. The orbitals are combined in a functional form to account for electron exchange and correlation. Plane waves are a convenient basis for the orbitals. However, plane-wave orbitals grow in evaluation cost with basis-set completeness and system size. To speed up the calculation, polynomials approximate the plane-wave sum. Four polynomial methods tested are: Lagrange interpolation, pp-spline interpolation, B-spline interpolation and B-spline approximation. The polynomials all increase speed by an order of the number of particles. B-spline approximation most consistently maintains accuracy in the seven systems tested. However, polynomials increase the memory needed by a factor of two to eight. B-spline approximation with a separate approximation for the Laplacian of the orbitals increases the memory by a factor of four over plane waves.</p><p>Polynomial-based orbitals enable larger calculations and careful examination of error introduced by approximations in VMC and DMC. In silicon bulk and interstitial defects, tens of variational parameters in the wave function converge the VMC energy. A basis set cutoff ≅1000 eV converges the VMC energy to within 10 meV. Controlling the population of electron configurations representing the DMC wave function does not bias the energy above 24 configurations. An imaginary time step for the configurations of 10<sup>-2</sup> hartree<sup>-1</sup> introduces no error above the 10 meV level. Finite-size correction methods on the 16-atom cell size with difference up to 2 eV error and 1 eV discrepancy between 16- and 64-atom cells indicate finite-size error is still significant. Pseudopotentials constructed with and without scalar relativistic correction agree in DMC energy differences at the 100 meV level, and mean-field calculations with and without pseudopotentials suggest a correction of 50-100 meV. Using the VMC wave function to evaluate the nonlocal portion of the pseudopotential introduces an error on the 1 meV level. DMC energies using orbitals produced with varying mean-field approximations produce a 1 eV range in the defect formation energies while applying a backflow transformation to the electron coordinates reduces Monte Carlo fluctuations. The backflow-transformed average also permits an extrapolation to zero fluctuation. The extrapolated value estimates the formation energy unbiased by the starting wave function to be 4.5-5 eV.</p> 2010-11-01 English text The Ohio State University / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=osu1284495775 http://rave.ohiolink.edu/etdc/view?acc_num=osu1284495775 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.

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