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Linear-scaling quantum Monte Carlo calculations

Andrew Williamson, Randolph Q. Hood, Jeffrey C. Grossman

Published Jun 18, 2007• 0 citations
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Abstract

Abstract This study demonstrates linear scaling with system size in fixednode diffusion QMC calculations with pseudopotentials for carbon fullerenes and for hydrogenated silicon clusters with nearly 1000 valence electrons. The range covered is C20 to C18o and SiH4 to Si211H140, The calculations were carried out with standard pseudopotentials (leaving four electrons per C or Si atom) and Slater Jastrow trial functions with orbitals from LDA density functional calculations. The key elements leading to linear scaling were constructions of sparse determinants and simplification of orbital functions. The LDA orbitals were obtained with a plane-wave basis set which was transformed to yield Wannier functions with maximum localization. These functions were then fitted to cubic spline expressions with a limited number of grid points and cut-off distances matching those of the plane-wave orbitals. The computational effort for evaluating trial functions of this type was clearly seen to behave in a linear fashion. Exploration of the effects of variation of cut-off radius revealed that energies, sensitive only to node structures, increased rapidly for cut-offs less than 7 bohr but were unaffected for cutoffs greater than 7 bohr. The accuracies of the calculations were checked in other ways. These included comparison calculations for SiH3 and Si5H12 using Gaussian, original plane-wave, and Wannier function basis sets which gave close agreement for energies.

DOI
10.1093/oso/9780195310108.003.00134

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A method is presented for using truncated, maximally localized Wannier functions to introduce sparsity into the Slater determinant part of the trial wave function in quantum Monte Carlo calculations. When combined with an efficient numerical evaluation of these localized orbitals, the dominant cost in the calculation, namely, the evaluation of the Slater determinant, scales linearly with system size. This technique is applied to accurate total energy calculation of hydrogenated silicon clusters and carbon fullerenes containing 20-1000 valence electrons.
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Quantum Monte Carlo (QMC) techniques are used to calculate the one-body density matrix and excitation energies for the valence electrons of bulk silicon. The one-body density matrix and energies are obtained from a Slater-Jastrow wave function with a determinant of local-density approximation (LDA) orbitals. The QMC density matrix evaluated in a basis of LDA orbitals is strongly diagonally dominant. The natural orbitals obtained by diagonalizing the QMC density matrix resemble the LDA orbitals very closely. Replacing the determinant of LDA orbitals in the wave function by a determinant of natural orbitals makes no significant difference to the quality of the wave function's nodal surface, leaving the diffusion Monte Carlo energy unchanged. The extended Koopmans's theorem for correlated wave functions is used to calculate excitation energies for silicon, which are in reasonable agreement with the available experimental data. A diagonal approximation to the theorem, evaluated in the basis of LDA orbitals, works quite well for both the quasihole and quasielectron states. We have found that this approximation has an advantageous scaling with system size, allowing more efficient studies of larger systems.
Abstract Results of fixed-node diffusion QMC calculations for 17 small molecules made up of atoms H, C, N, 0, and F were compared in this paper to experimental values for atomization energies, bond lengths, and bond angles, as well as corresponding values from coupled cluster calculations. The QMC calculations were of the Ornstein-Uhlenbeck type,a with trial functions for importance sampling and node locations based on linear combinations of determinants with molecular orbitals composed of floating spherical Gaussian orbitals. Initial geometries were taken from density functional calculations and optimized to find the minima given by the QMC calculations. Zero-point energies were calculated using the energy gradients given by the same calculations. The calculated atomization energies for the 14 species for which experimental data were available were in excellent agreement with the experimental values and with values given by CCSD(T) with a cc-pVQZ basis set. The mean absolute deviation from experimental atomization energies was 0.16 kcal/mol for the QMC and 0.21 kcal/mol for the CCSD(T) calculations. Geometric parameters were in agreement with each other and with experimental values within their uncertainties. Overall, the results provide a solid confirmation of the accuracy of fixed-node diffusion QMC calculations.
For atomic and molecular quantum Monte Carlo calculations, most of the computational effort is spent in the evaluation of the local energy. We describe a scheme for reducing the computational cost of the evaluation of the Slater determinants and correlation function for the correlated molecular orbital (CMO) ansatz. A sparse representation of the Slater determinants makes possible efficient evaluation of molecular orbitals. A modification to the scaled distance function facilitates a linear scaling implementation of the Schmidt- Moskowitz-Boys-Handy (SMBH) correlation function that preserves the efficient matrix multiplication structure of the SMBH function. For the evaluation of the local energy, these two methods lead to asymptotic linear scaling with respect to the molecule size.