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Journal of Computational Chemistry

Extrapolation of the time‐step bias in diffusion quantum Monte Carlo by a differential sampling technique

David R. Garmer

Published Mar 1, 1989• 4 citations
Publisher

Abstract

Abstract A new method of eliminating the finite‐time‐step error inherent in diffusion quantum Monte Carlo is presented, utilizing an improved version of the existing differential techniques. An implementation is described and results of several small but representative calculations are discussed. The pertinent computation requirements on these systems were reduced by up to a factor of five by the new algorithm. It is speculated that this method may be easily applied to other quantum Monte Carlo and discretized path integral Monte Carlo techniques having related finite step‐size errors with a possibility of obtaining similar good results.

DOI
10.1002/jcc.540100205

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Abstract We compare the accuracy and precision of the wave function correction method to that of traditional diffusion quantum Monte Carlo (traditional DQMC) simulations. Direct calculation of corrections to trial wave functions reduces time step error for Be and Ne electronic ground state energies when compared to traditional DQMC results. The fixed‐node method is utilized for these fermionic systems, so error due to incorrect node locations is not addressed. This correction DQMC method achieves more accurate energies at larger time steps, promising decreased computational cost to reach a given accuracy. For an optimized Be trial wave function, the statistical uncertainty per unit of computational effort is also reduced relative to traditional DQMC. Correction DQMC utilizes walker feed integrals and growth factors, as reported previously. We illustrate the method in a simple harmonic oscillator example, and we report high accuracy energies for Be and Ne at several time steps.
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Quantum Monte Carlo (QMC) methods are powerful, stochastic techniques for computing the properties of interacting electrons and nuclei with an accuracy comparable to the standard post-Hartree--Fock methods of quantum chemistry. Whilst the favourable scaling of QMC methods enables a quantum, many-body treatment of much larger systems, the lack of accurate and efficient total energy derivatives, required to compute atomic forces, has hindered their widespread adoption. The work contained within this thesis provides an efficient procedure for calculating exact derivatives of QMC results. This procedure uses the programming technique of algorithmic differentiation (AD), which allows access to the derivatives of a computed function by applying chain rule differentiation to the underlying source code. However, this thesis shows that a straightforward differentiation of a stochastic function fails to capture the important contribution to the derivative from probabilistic decisions. A general approach for calculating the derivatives of a stochastic function is presented, where a similar adaptation of AD applied to the diffusion Monte Carlo (DMC) algorithm yields exact DMC atomic forces. The approach is validated by performing the largest ever DMC force calculations, which demonstrate the feasibility of treating systems containing thousands of electrons. The efficiency of AD also enables molecular dynamics simulations driven entirely by DMC, adding new functionality to the QMC toolkit. Another focus of this thesis is using the phenomenon of stochastic coherence to correlate DMC simulations, allowing finite difference derivatives to be obtained with a small error. Whilst this method is far easier to implement than AD, preliminary results show an instability when treating larger systems. A different approach is obtained from extrapolating this method to a finite difference step size of zero, producing algebraic expressions for a direct differentiation of the DMC algorithm.
A Makri-Miller approximation to the exact propagator and the improved split-operator propagator proposed by Drozdov are implemented within the diffusion Monte Carlo method for the simulation of boson systems, and confronted with the Trotter formula and with the importance sampling technique. As a preliminary approach, we compute analytically the time step bias of the mean energy for the different propagators in the simple case of the harmonic oscillator. These results indicate the improved split-operator propagator as the most accurate. Simulations on one- and three-dimensional model systems confirm the analytical results showing that this propagator is very efficient in reducing the time step bias, therefore improving the efficiency of the algorithm.