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The Journal of Chemical Physics

Geometry optimization in quantum Monte Carlo with solution mapping: Application to formaldehyde

Charles A. Schuetz, Michael Frenklach, Alexander C. Kollias, William A. Lester

Published Oct 31, 2003• 7 citations
Publisher

Abstract

The solution mapping methodology is presented as a method of geometry optimization of molecules in the quantum Monte Carlo method. Applied to formaldehyde as a test system, this approach is found to yield optimized bond lengths and bond angle in the diffusion Monte Carlo method that lie within experimental error. The variational Monte Carlo optimized geometry also lies within experimental error, with the exception of the CH bond length, which is slightly underestimated. Additionally, the resulting quadratic representation of the potential energy surface in the region of the minimum is used to calculate three of the force constants and harmonic frequencies.

DOI
10.1063/1.1614212

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The Journal of Chemical Physics•Apr 11, 2003•51 citations
The solution mapping methodology is presented as a method of geometry optimization of molecules in the quantum Monte Carlo method. Applied to formaldehyde as a test system, this approach is found to yield optimized bond lengths and bond angle in the diffusion Monte Carlo method that lie within experimental error. The variational Monte Carlo optimized geometry also lies within experimental error, with the exception of the CH bond length, which is slightly underestimated. Additionally, the resulting quadratic representation of the potential energy surface in the region of the minimum is used to calculate three of the force constants and harmonic frequencies.
Abstract Since analytic gradients of the energy with respect to nuclear positions are not easily obtained in QMC calculations, determining the locations of critical points, such as energy minima and saddle points, on potential energy surfaces is not straightforward. The statistical uncertainty in QMC energies prevents the direct estimation of these derivatives, and although several techniques for overcoming the problem have been developed their applications have been limited to very small systems. This paper describes a different approach, more readily applicable, in which the number and location of points in the vicinity of a critical point are chosen in such a way as to minimize the computation effort required in predicting the location of the critical point. The method is generally known as “solution mapping” and is often referred to as “factorial design.” The procedure is illustrated with calculations to determine the minimum energy or equilibrium geometry of formaldehyde H2CO, assumed planar and symmetric about the C-0 bond, in terms of the C-0 and C-H bond lengths and the H-C-0 angle. The potential energies calculated at specified points are fit to a polynomial expression of ten terms involving the three variables as l,x1,x2,x3,XI,X ,x ,x1x2,x1x3,x2x3, and ten coeffcients. With 15 points prescribed by the factorial design the determination of the coefficients is simplified by the column orthogonality of the design matrix. Of course, statistical uncertainties in the energies lead to uncertainties in the coefficients, but the method minimizes their effects.
This article describes an approach in determination of equilibrium geometries and harmonic frequencies of molecules by the Ornstein-Uhlenbeck diffusion quantum Monte Carlo method based on the floating spherical Gaussians. In conjunction with a projected and renormalized Hellmann-Feynman gradient and an electronic energy at variational Monte Carlo and diffusion quantum Monte Carlo, respectively, the quasi-Newton algorithm implemented with the Broyden-Fletcher-Goldfarb-Shanno updated Hessian was used to find the optimized molecular geometry. We applied this approach to N2 and H2O molecules. The geometry and harmonic frequencies calculated were consistent with some sophisticated ab initio calculated values within reasonable statistical uncertainty.
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.
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.
We propose an algorithm for accurate, systematic, and scalable computation of interatomic forces within the auxiliary-field quantum Monte Carlo (AFQMC) method. The algorithm relies on the Hellmann-Feynman theorem and incorporates Pulay corrections in the presence of atomic orbital basis sets. We benchmark the method for small molecules by comparing the computed forces with the derivatives of the AFQMC potential energy surface and by direct comparison with other quantum chemistry methods. We then perform geometry optimizations using the steepest descent algorithm in larger molecules. With realistic basis sets, we obtain equilibrium geometries in agreement, within statistical error bars, with experimental values. The increase in computational cost for computing forces in this approach is only a small prefactor over that of calculating the total energy. This paves the way for a general and efficient approach for geometry optimization and molecular dynamics within AFQMC.
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.