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arXiv:physics/0409087 · Physical Review Letters

Accurate, Efficient, and Simple Forces Computed with Quantum Monte Carlo Methods

Simone Chiesa, David M. Ceperley, Shiwei Zhang

Published Jan 24, 2005• 58 citations
arXivHTMLPDF

Abstract

Computation of ionic forces using quantum Monte Carlo methods has long been a challenge. We introduce a simple procedure, based on known properties of physical electronic densities, to make the variance of the Hellmann-Feynman estimator finite. We obtain very accurate geometries for the molecules H(2), LiH, CH(4), NH(3), H(2)O, and HF, with a Slater-Jastrow trial wave function. Harmonic frequencies for diatomics are also in good agreement with experiment. An antithetical sampling method is also discussed for additional reduction of the variance.

DOI
10.1103/physrevlett.94.036404

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Computation of ionic forces using quantum Monte Carlo methods has long been a challenge. We introduce a simple procedure, based on known properties of physical electronic densities, to make the variance of the Hellmann-Feynman estimator finite. We obtain very accurate geometries for the molecules H(2), LiH, CH(4), NH(3), H(2)O, and HF, with a Slater-Jastrow trial wave function. Harmonic frequencies for diatomics are also in good agreement with experiment. An antithetical sampling method is also discussed for additional reduction of the variance.
In order to overcome the difficulty of optimizing molecular geometry using quantum Monte Carlo methods, we introduce various approximations to the exact force expectation value. We follow Pulay’s suggestion [Mol. Phys. 17, 153 (1969)] to correct the Hellmann–Feynman estimator by introducing the contributions due to the changes in the wave function with respect to the nuclear positions. When used in conjunction with energy-optimized explicitly correlated trial wave functions for H2 and LiH, these approximations appear to yield accurate forces using both the variational and diffusion Monte Carlo methods. Also, the accuracy of the second-order estimate of the Hellmann–Feynman force estimator was investigated employing our energy-optimized trial wave functions, and an erratic behavior was uncovered for some of the studied bond lengths. The additional computational cost required to compute the corrections to the Hellmann–Feynman estimator was found to be only a small fraction of the cost for a simple mean energy calculation. The same approach could be exploited also in computing the derivative of other energy-dependent quantum-mechanical observables.
This work investigates the accuracy of real-space quantum Monte Carlo (QMC) methods for calculating molecular geometries. We present the equilibrium bond lengths of a test set of 30 diatomic molecules calculated using variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC) methods. The effect of different trial wavefunctions is investigated using single determinants constructed from Hartree-Fock (HF) and Density Functional Theory (DFT) orbitals with LDA, PBE, and B3LYP functionals, as well as small multi-configurational self-consistent field (MCSCF) multi-determinant expansions. When compared to experimental geometries, all DMC methods exhibit smaller mean-absolute deviations (MADs) than those given by HF, DFT, and MCSCF. The most accurate MAD of 3 ± 2 × 10(-3) Å is achieved using DMC with a small multi-determinant expansion. However, the more computationally efficient multi-determinant VMC method has a similar MAD of only 4.0 ± 0.9 × 10(-3) Å, suggesting that QMC forces calculated from the relatively simple VMC algorithm may often be sufficient for accurate molecular geometries.
Quantum Monte Carlo methods are accurate and promising many body techniques for electronic structure calculations which, in the last years, are encountering a growing interest thanks to their favorable scaling with the system size and their efficient parallelization, particularly suited for the modern high performance computing facilities. The ansatz of the wave function and its variational flexibility are crucial points for both the accurate description of molecular properties and the capabilities of the method to tackle large systems. In this paper, we extensively analyze, using different variational ansatzes, several properties of the water molecule, namely, the total energy, the dipole and quadrupole momenta, the ionization and atomization energies, the equilibrium configuration, and the harmonic and fundamental frequencies of vibration. The investigation mainly focuses on variational Monte Carlo calculations, although several lattice regularized diffusion Monte Carlo calculations are also reported. Through a systematic study, we provide a useful guide to the choice of the wave function, the pseudopotential, and the basis set for QMC calculations. We also introduce a new method for the computation of forces with finite variance on open systems and a new strategy for the definition of the atomic orbitals involved in the Jastrow-Antisymmetrised Geminal power wave function, in order to drastically reduce the number of variational parameters. This scheme significantly improves the efficiency of QMC energy minimization in case of large basis sets.
First-order derivatives of energies with respect to atomic coordinates are widely computed and used in quantum chemistry simulations. The rapidly emerging technology of quantum computing offers a new paradigm for solving relevant quantum chemistry equations. In this work, we have achieved analytical calculations of atomic forces based on the Hellmann-Feynman theorem within the framework of the variational quantum eigensolver. The accuracy of the approach is demonstrated by calculating the atomic forces of H2, LiH, H2O, and NH3 molecules, which are in excellent agreement with values obtained from full configuration interaction calculations. In particular, for systems with degenerate molecular orbitals, the analytical approach has a significant accuracy advantage over finite-difference-based methods and will not involve additional computational effort on a quantum computer. The calculated forces are further used to optimize the geometries of NH3 and CH4 molecules and to perform ab initio molecular dynamics simulations for the umbrella inversion of NH3, demonstrating the feasibility of the approach in practical quantum chemistry simulations.
quantum Monte Carlo (QMC) is a state-of-the-art numerical approach for evaluating accurate expectation values of many-body wave functions. However, one of the major drawbacks that still hinders widespread QMC applications is the lack of an affordable scheme to compute unbiased atomic forces. In this study, we propose an efficient method to obtain unbiased atomic forces and pressures in the variational Monte Carlo (VMC) framework with the Jastrow-correlated Slater determinant ansatz or the Jastrow antisymmetrized geminal power ansatz, exploiting the gauge-invariant and locality properties of their geminal representation. We demonstrate the effectiveness of our method for H 2 and Cl 2 molecules and for the cubic boron nitride crystal. Our framework has a better algorithmic scaling with the system size than the traditional finite-difference method and, in practical applications, is as efficient as single-point VMC calculations. Thus, it paves the way to study dynamical properties of materials, such as phonons, and is beneficial for pursuing more reliable machine-learning interatomic potentials based on unbiased VMC forces. Published by the American Physical Society 2024