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The Journal of Chemical Physics

Linear scaling for the local energy in quantum Monte Carlo

Sebastian Manten, Arne Lüchow

Published Jul 3, 2003• 29 citations
Publisher

Abstract

The scaling of the diffusion quantum Monte Carlo method can be greatly improved when localized orbitals and short-range correlation functions are employed as recently suggested by the authors. The local diffusion quantum Monte Carlo method is described in detail with a careful analysis of errors. The new method achieves near linear scaling in the calculation of the local energy. Results demonstrating the improved performance are presented.

DOI
10.1063/1.1580474

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The scaling of the diffusion quantum Monte Carlo method can be greatly improved when localized orbitals and short-range correlation functions are employed as recently suggested by the authors. The local diffusion quantum Monte Carlo method is described in detail with a careful analysis of errors. The new method achieves near linear scaling in the calculation of the local energy. Results demonstrating the improved performance are presented.
Abstract This paper describes the use of localized orbitals and shortrange correlation functions to achieve near linear scaling in fixednode diffusion QMC calculations. The success of the method is demonstrated with a series of all-electron calculations for linear hydrocarbons in the range of C2H6 to C2sHs2 with 18 to 202 electrons. The procedure used was not very different from that of earlier all-electron calculations for systems of these sizes. Some changes were needed for evaluation of the trial wave functions and their derivatives. The molecular orbitals for the determinantal wavefunction were composed of contracted Gaussian functions decaying rapidly with distance but limited by a cut-off radius. The resulting determinants were sparse and could be evaluated by a standard matrix LU decomposition package. The standard Schmidt Moskowitz correlation function was modified with use of a short-range scaled distance to allow short cut-off radii.
We have reformulated the quantum Monte Carlo (QMC) technique so that a large part of the calculation scales linearly with the number of atoms. The reformulation is related to a recent alternative proposal for achieving linear-scaling QMC, based on maximally localized Wannier orbitals (MLWO), but has the advantage of greater simplicity. The technique we propose draws on methods recently developed for linear-scaling density functional theory. We report tests of the new technique on the insulator MgO, and show that its linear-scaling performance is somewhat better than that achieved by the MLWO approach. Implications for the application of QMC to large complex systems are pointed out.
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.
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.
We derive an automatic procedure for generating a set of highly localized, non-orthogonal orbitals for linear scaling quantum Monte Carlo calculations. We demonstrate the advantage of these orbitals in calculations of the total energy of both semiconducting and metallic systems by studying bulk silicon and the homogeneous electron gas. For silicon, the improved localization of these orbitals reduces the computational time by a factor five and the memory by a factor of six compared to localized, orthogonal orbitals. For jellium, we demonstrate that the total energy is converged for orbitals truncated within spheres with radii 7-8 rsr_srs​, opening the possibility of linear scaling QMC calculations for realistic metallic systems.
A reformulation of the fixed-node diffusion quantum Monte Carlo method (FN-DQMC) in terms of the N-particle density matrix is presented, which allows us to reduce the computational effort to linear for the evaluation of the local energy. The reformulation is based on our recently introduced density matrix-based approach for a linear-scaling variational QMC method [J. Kussmann et al., Phys. Rev. B. 75, 165107 (2007)]. However, within the latter approach of using the positive semi-definite N-particle trial density (rhoN T(R)=mid R:Psi(T)(R)mid R:(2)), the nodal information of the trial function is lost. Therefore, a straightforward application to the FN-DQMC method is not possible, in which the sign of the trial function is usually traced in order to confine the random walkers to their nodal pockets. As a solution, we reformulate the FN-DQMC approach in terms of off-diagonal elements of the N-particle density matrix rhoN T(R;R'), so that the nodal information of the trial density matrix is obtained. Besides all-electron moves, a scheme to perform single-electron moves within N-PDM QMC is described in detail. The efficiency of our method is illustrated for exemplary calculations.