dorsal/arxiv
View SchemaZero-Variance Zero-Bias Principle for Observables in quantum Monte Carlo: Application to Forces
| Authors | Roland Assaraf, Michel Caffarel |
|---|---|
| Categories | |
| ArXiv ID | physics/0310035 |
| URL | https://arxiv.org/abs/physics/0310035 |
| DOI | 10.1063/1.1621615 |
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 observable by an improved or ``renormalized'' estimator. Using this estimator more accurate averages are obtained: Not only the statistical fluctuations are reduced but also the 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 distance) and $\omega_e$ (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%.
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"abstract": "A simple and stable method for computing accurate expectation values of\nobservable with Variational Monte Carlo (VMC) or Diffusion Monte Carlo (DMC)\nalgorithms is presented. The basic idea consists in replacing the usual\n``bare\u0027\u0027 estimator associated with the observable by an improved or\n``renormalized\u0027\u0027 estimator. Using this estimator more accurate averages are\nobtained: Not only the statistical fluctuations are reduced but also the\nsystematic error (bias) associated with the approximate VMC or (fixed-node) DMC\nprobability densities. It is shown that improved estimators obey a\nZero-Variance Zero-Bias (ZVZB) property similar to the usual Zero-Variance\nZero-Bias property of the energy with the local energy as improved estimator.\nUsing this property improved estimators can be optimized and the resulting\naccuracy on expectation values may reach the remarkable accuracy obtained for\ntotal energies. As an important example, we present the application of our\nformalism to the computation of forces in molecular systems. Calculations of\nthe entire force curve of the H$_2$,LiH, and Li$_2$ molecules are presented.\nSpectroscopic constants $R_e$ (equilibrium distance) and $\\omega_e$ (harmonic\nfrequency) are also computed. The equilibrium distances are obtained with a\nrelative error smaller than 1%, while the harmonic frequencies are computed\nwith an error of about 10%.",
"arxiv_id": "physics/0310035",
"authors": [
"Roland Assaraf",
"Michel Caffarel"
],
"categories": [
"physics.chem-ph",
"cond-mat"
],
"doi": "10.1063/1.1621615",
"title": "Zero-Variance Zero-Bias Principle for Observables in quantum Monte Carlo: Application to Forces",
"url": "https://arxiv.org/abs/physics/0310035"
},
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