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International Journal of Quantum Chemistry

Two aspects of quantum monte carlo: Determination of accurate wavefunctions and determination of potential energy surfaces of molecules

Cyrus J. Umrigar

Published Jun 19, 2009• 32 citations
Publisher

Abstract

Two aspects of quantum Monte Carlo are discussed. First, we review a procedure for obtaining trial wavefunctions for use in quantum Monte Carlo simulations that have both smaller statistical errors and improved expectation values than commonly used functions. Second, we present a correlated sampling approach for calculating energy differences in variational Monte Carlo much more accurately than the values of the energies. This method is used to calculate the potential energy surfaces of H2 and BH. In the case of BH, at bond lengths 10% different from the reference geometry, the reduced statistical error in the relative energy results in a factor of 3000 saving in computer time.

DOI
10.1002/qua.560360826

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Two aspects of quantum Monte Carlo are discussed. First, we review a procedure for obtaining trial wavefunctions for use in quantum Monte Carlo simulations that have both smaller statistical errors and improved expectation values than commonly used functions. Second, we present a correlated sampling approach for calculating energy differences in variational Monte Carlo much more accurately than the values of the energies. This method is used to calculate the potential energy surfaces of H2 and BH. In the case of BH, at bond lengths 10% different from the reference geometry, the reduced statistical error in the relative energy results in a factor of 3000 saving in computer time.
Most methods of quantum chemistry calculate total energies rather than directly the energy differences that are of interest to chemists. In the case of statistical methods like quantum Monte Carlo the statistical errors in the absolute values need to be considerably smaller than their difference. The calculation of small energy differences is therefore particularly time consuming. Correlated sampling techniques provide the possibility to compute directly energy differences by simulating the underlying systems with the same stochastic process. The smaller the energy difference the smaller its statistical error. Correlated sampling is well established in variational quantum Monte Carlo, but it is much more difficult to implement in diffusion quantum Monte Carlo because of the fixed node approximation. A correlated sampling formalism and a corresponding algorithm based on a transformed Schrödinger equation having the form of a Kolmogorov’s backward equation is derived. The numerical verification of the presented algorithm is given for the harmonic oscillator. The extension of the algorithm to electron structure calculations is discussed.
Abstract The paper describes several extensions of fixed-node diffusion in the areas of obtaining analytic derivatives of the energies of molecules, calculating expectation values for quantities other than the energy, and treating excited states. The calculation of energy derivatives is illustrated with a calculation of the first derivative of the energy of the ground state of H2 with respect to the internuclear distance, without using finite differences. With importance sampling the energy is obtained from a weighted average of local energies expressed in analytic form. These may be differentiated to obtain their derivatives with respect to any of several variables, including the internuclear distance, and their weighted averages may be used to determine the desired derivatives. The results for the H2 energy derivatives were in agreement with independent determinations. For calculation of expectation values of the energy, sampling of the (true wavefunction)(trial wavefunction) ‘1/J’l/Jr distribution is sufficient, but other quantities usually require sampling the ‘ljJ2 distribution, which is not so readily available in QMC calculations. Extrapolations of ‘!/Jf and ‘ljJ’l/Jr results, from variational and diffusion calculations, to ‘!jJ2 results are illustrated for various expectation values for the case of ground-state H2.
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.
An alternative approach to conventional quantum chemistry techniques for molecular studies is described.This approach uses the quantum Monte Carlo method, which was developed and used primarily in nuclear and condensed-matter physics.In this approach the many-body Schrodinger equation is re-interpreted as a diffusion equation.Simulation of the appropriate random walk process allows one to calculate expectation values of molecular properties.In principle these expectation values can be calculated exactly, subject only to statistical errors (which may be made arbitrarily small).In preliminary work on small molecules (H 2 , LiH, Li 2 , and H 2 0), we use a simple but accurate approximation to ease the treatment of Fermi statistics.In that approximation, the calculated total energy remains an upper bound to the true energy.How good the bound is, as well as the magnitude of the statistical error, depends on the quality of an "importance function" which guides the diffusion into more probable regions of phase space.With relatively simple importance functions, we obtain from 75-100% of the correlation energy of the above-mentioned molecules.
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 mechanics for many-body systems may be reduced to the evaluation of integrals in 3N dimensions using Monte Carlo, providing the Quantum Monte Carlo ab initio methods. Here we limit ourselves to expectation values for trial wave functions, that is to variational quantum Monte Carlo. Almost all previous implementations employ samples distributed as the physical probability density of the trial wave function, and assume the central limit theorem to be valid. In this paper we provide an analysis of random error in estimation and optimization that leads naturally to new sampling strategies with improved computational and statistical properties. A rigorous lower limit to the random error is derived, and an efficient sampling strategy presented that significantly increases computational efficiency. In addition the infinite variance heavy tailed random errors of optimum parameters in conventional methods are replaced with a Normal random error, strengthening the theoretical basis of optimization. The method is applied to a number of first row systems and compared with previously published results.