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

Parker, William David

Abstract Details

2010, Doctor of Philosophy, Ohio State University, Physics.

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.

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.

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.

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 10-2 hartree-1 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.

John W. Wilkins (Advisor)
Nandini Trivedi (Committee Member)
Jay A. Gupta (Committee Member)
Michael G. Poirier (Committee Member)
Dhabaleswar Panda (Committee Member)
96 p.

Recommended Citations

Citations

  • Parker, W. D. (2010). Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1284495775

    APA Style (7th edition)

  • Parker, William. Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations. 2010. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1284495775.

    MLA Style (8th edition)

  • Parker, William. "Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations." Doctoral dissertation, Ohio State University, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=osu1284495775

    Chicago Manual of Style (17th edition)