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arXiv:cond-mat/0404578 · Journal of Physics Condensed Matter

Linear-scaling quantum Monte Carlo technique with non-orthogonal localized orbitals

Dario Alfè, Michael J. Gillan

Published Jun 7, 2004• 38 citations
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Abstract

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.

DOI
10.1088/0953-8984/16/25/l01

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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.
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.
We derive an automatic procedure for generating a set of highly localized, nonorthogonal orbitals for linear scaling quantum Monte Carlo (QMC) calculations. We demonstrate the advantage of these orbitals for calculating 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 of 5 and the memory by a factor of 6 compared to localized, orthogonal orbitals. For jellium at typical metallic densities, we demonstrate that the total energy is converged to 3 meV per electron for orbitals truncated within spheres with radii 7rs7{r}_{s}7rs​, opening the possibility of linear scaling QMC calculations for realistic metallic systems.
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.
We present a reformulation of the variational quantum Monte Carlo (VQMC) method using the NNN-particle density matrix (NNN-PDM VQMC), which allows us to reduce the conventional cubic scaling behavior to linear for the calculation of the local energy with the number of electrons for systems with a nonvanishing highest occupied molecular orbital--lowest unoccupied molecular orbital gap. The ``perturbation'' of the electron distribution, caused by the truncation of the self-consistent-field (SCF) one-electron density matrix, is compensated by the use of the McWeeny purification transformation in the presented scheme. The performance and accuracy of the presented method are illustrated through exemplary calculations.
Quantum Monte Carlo (QMC) methods have received considerable attention over past decades due to their great promise for providing a direct solution to the many-body Schrodinger equation in electronic systems. Thanks to their low scaling with the number of particles, QMC methods present a compelling competitive alternative for the accurate study of large molecular systems and solid state calculations. In spite of such promise, the method has not permeated the quantum chemistry community broadly, mainly because of the fixed-node error, which can be large and whose control is difficult. In this Perspective, we present a systematic application of large scale multideterminant expansions in QMC and report on its impressive performance with first row dimers and the 55 molecules of the G1 test set. We demonstrate the potential of this strategy for systematically reducing the fixed-node error in the wave function and for achieving chemical accuracy in energy predictions. When compared to traditional quantum chemistry methods like MP2, CCSD(T), and various DFT approximations, the QMC results show a marked improvement over all of them. In fact, only the explicitly correlated CCSD(T) method with a large basis set produces more accurate results. Further developments in trial wave functions and algorithmic improvements appear promising for rendering QMC as the benchmark standard in large electronic systems.