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The Journal of Chemical Physics

A diffusion quantum Monte Carlo study of geometries and harmonic frequencies of molecules

Shih‐I Lu

Published Dec 22, 2003• 10 citations
Publisher

Abstract

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.

DOI
10.1063/1.1630022

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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.
This article accesses the performance of the Ornstein-Uhlenbeck diffusion quantum Monte Carlo with regard to the calculation of molecular geometries and harmonic frequencies of H2, LiH, HF, Li2, LiF, CO, N2, and F2 molecules. A comparison of the results for the eight first-row diatomic molecules from experiments, CCSD(T)/6-311G(3df,3pd) and CCSD(T)/cc-pV5Z levels of theory as well as our work is given. The results presented show that quantum Monte Carlo is becoming powerful tools for ab initio electronic structure calculations.
Abstract Results of fixed-node diffusion QMC calculations for 17 small molecules made up of atoms H, C, N, 0, and F were compared in this paper to experimental values for atomization energies, bond lengths, and bond angles, as well as corresponding values from coupled cluster calculations. The QMC calculations were of the Ornstein-Uhlenbeck type,a with trial functions for importance sampling and node locations based on linear combinations of determinants with molecular orbitals composed of floating spherical Gaussian orbitals. Initial geometries were taken from density functional calculations and optimized to find the minima given by the QMC calculations. Zero-point energies were calculated using the energy gradients given by the same calculations. The calculated atomization energies for the 14 species for which experimental data were available were in excellent agreement with the experimental values and with values given by CCSD(T) with a cc-pVQZ basis set. The mean absolute deviation from experimental atomization energies was 0.16 kcal/mol for the QMC and 0.21 kcal/mol for the CCSD(T) calculations. Geometric parameters were in agreement with each other and with experimental values within their uncertainties. Overall, the results provide a solid confirmation of the accuracy of fixed-node diffusion QMC calculations.
We report a new method for fixed-node diffusion quantum Monte Carlo calculation in order to improve the time step-size convergence and the fluctuation problems associated with nodes of wave functions. It adopts floating spherical Gaussian orbitals instead of Slater-type orbitals to guide random walkers (psips). They establish harmonic velocity fields under which psips undergo random walks which are simulated exactly according to the Ornstein–Uhlenbeck process. The nodes of this guiding function are used to guide psips in a novel way; as psips approach to a node, they are swept away therefrom. Explicit enforcement of a sink boundary condition is avoided to some extent, and the behavior of the random walks near nodes is efficiently simulated. The method is illustrated by applying to the ground-state energies of H2, LiH, and HF. The results are excellent compared to previous calculations.
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.
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.
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.