Zero-Variance Zero-Bias Principle for Observables in quantum Monte Carlo: Application to ForcesReport as inadecuate

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1 LCT - Laboratoire de chimie théorique

Abstract : A simple and stable method for computing accurate expectation values of observable with Variational Monte Carlo VMC or Diffusion Monte Carlo DMC algorithms is presented. The basic idea consists in replacing the usual ``bare- estimator associated with the observableby an improved or ``renormalized- estimator. Using this estimator more accurate averages are obtained:Not only the statistical fluctuations are reduced but alsothe systematic error bias associated with the approximate VMC or fixed-node DMC probability densities. It is shown that improved estimators obey a Zero-Variance Zero-Bias ZVZB property similar to the usual Zero-Variance Zero-Bias property of the energy with the local energy as improved estimator. Using this property improved estimators can be optimized and the resulting accuracy on expectation values may reach the remarkable accuracy obtained for total energies. As an important example, we present the application of our formalism to the computation of forces in molecular systems. Calculations of the entire force curve of the H$ 2$,LiH, and Li$ 2$ molecules are presented. Spectroscopic constants $R e$ equilibrium distanceand $\omega e$ harmonic frequency are also computed.The equilibrium distances are obtained with a relative error smaller than 1$\%$, while the harmonic frequencies are computed with an error of about 10$\%$.

keyword : zero-variance property improved estimators QMC atomic forces

Author: Roland Assaraf - Michel Caffarel -



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