Zero-Variance Zero-Bias Principle for Observables in quantum Monte Carlo: Application to Forces

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%.

References in corpus (7)

• The role of electronic correlation in the Si(100) reconstruction: a quantum Monte Carlo study
• Unbiased estimators in Quantum Monte Carlo methods: application to liquid ${}^4$He
• Zero-variance principle for Monte Carlo algorithms
• Electron correlation in C_(4N+2) carbon rings: aromatic vs. dimerized structures
• Correlated sampling in quantum Monte Carlo: a route to forces
• Carbon clusters near the crossover to fullerene stability
• Quantum Monte Carlo calculations of H${}_2$ dissociation on Si(001)

Cited by in corpus (52)

• Continuum variational and diffusion quantum Monte Carlo calculations
• Recurrent Neural Network Wave Functions
• Correlated geminal wave function for molecules: an efficient resonating valence bond approach
• Energy and variance optimization of many body wave functions
• Variational and Diffusion Quantum Monte Carlo Calculations with the CASINO Code
• Ab-initio quantum chemistry with neural-network wavefunctions
• Algorithmic differentiation and the calculation of forces by quantum Monte Carlo
• Variational Neural Annealing
• Zero Variance Markov Chain Monte Carlo for Bayesian Estimators
• Accurate, efficient and simple forces with Quantum Monte Carlo methods
• Simple formalism for efficient derivatives and multi-determinant expansions in quantum Monte Carlo
• Heavy-tailed random error in quantum Monte Carlo
• Molecular properties by Quantum Monte Carlo: an investigation on the role of the wave function ansatz and the basis set in the water molecule
• Methods for calculating forces within quantum Monte Carlo simulations
• Transition metal oxides using quantum Monte Carlo
• Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density
• Electronic Quantum Monte Carlo Calculations of Atomic Forces, Vibrations, and Anharmonicities
• Calculation of interatomic forces and optimization of molecular geometry with auxiliary-field quantum Monte Carlo
• Efficient Monte Carlo Calculations of the One-Body Density
• Analytic nuclear forces and molecular properties from full configuration interaction quantum Monte Carlo
• Total forces in the diffusion Monte Carlo method with nonlocal pseudopotentials
• Optimized structure and vibrational properties by error affected potential energy surfaces
• Quantum Monte Carlo for minimum energy structures
• Towards Quantum Monte Carlo Forces on Heavier Ions: Scaling Properties
• Post-Processing of MCMC
• Space-warp coordinate transformation for efficient ionic force calculations in quantum Monte Carlo
• Interatomic force from neural network based variational quantum Monte Carlo
• Accurate quantum Monte Carlo forces for machine-learned force fields: Ethanol as a benchmark
• The Fermion Monte Carlo revisited
• A tail-regression estimator for heavy-tailed distributions of known tail indices and its application to continuum quantum Monte Carlo data
• Alternative sampling for variational quantum Monte Carlo
• van der Waals forces stabilize low-energy polymorphism in B2O3: Implications for the crystallization anomaly
• Computation of forces and stresses in solids: Towards accurate structural optimization with auxiliary-field quantum Monte Carlo
• TurboGenius: Python suite for high-throughput calculations of ab initio quantum Monte Carlo methods
• Direct evaluation of the force constant matrix in quantum Monte Carlo
• Theoretical S1 <- S0 absorption energies of the anionic forms of oxyluciferin by Variational Monte Carlo and Many Body Green's Function Theory
• Benchmark Quantum Monte Carlo calculations of the ground-state kinetic, interaction, and total energy of the three-dimensional electron gas
• Radial distribution function in a diffusion Monte Carlo simulation of a Fermion fluid between the ideal gas and the Jellium model
• Highly Accurate Real-space Electron Densities with Neural Networks
• A structural optimization algorithm with stochastic forces and stresses
• Deep learning lattice gauge theories
• Control functionals for Monte Carlo integration
• Efficient calculation of unbiased atomic forces in ab initio Variational Monte Carlo
• Deep learning quantum Monte Carlo for solids
• Introduction to the variational and diffusion Monte Carlo methods
• Hellmann and Feynman theorem versus diffusion Monte Carlo experiment
• Diffusion Monte Carlo evaluation of disiloxane linearization barrier
• Encoding strongly-correlated many-boson wavefunctions on a photonic quantum computer: application to the attractive Bose-Hubbard model
• Self-consistency error correction for accurate machine learning potentials from variational Monte Carlo
• Zero Variance and Hamiltonian Monte Carlo Methods in GARCH Models
• On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes
• Fast Evaluation of Unbiased Atomic Forces in ab initio Variational Monte Carlo via the Lagrangian Technique