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ACS symposium series

Electronic Quantum Monte Carlo Calculations of Energies and Atomic Forces for Diatomic and Polyatomic Molecules

Myung Won Lee, Massimo Mella, Andrew M. Rappe

Published Dec 31, 2006• 0 citations
Publisher

Abstract

We calculated the energies and atomic forces of first-row monohydrides, carbon monoxide, and small organic polyatomic molecules using quantum Monte Carlo (QMC) method. Accurate forces were obtained with the method of Casalegno, Mella, and Rappe, combining the Hellmann- Feynman theorem forces calculated by the Assaraf-Caffarel method with a many-body Pulay correction. Improved algorithms for the minimization of the variational integral were useful in the force calculations.

DOI
10.1021/bk-2007-0953.ch006

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We calculated the energies and atomic forces of first-row monohydrides, carbon monoxide, and small organic polyatomic molecules using quantum Monte Carlo (QMC) method. Accurate forces were obtained with the method of Casalegno, Mella, and Rappe, combining the Hellmann- Feynman theorem forces calculated by the Assaraf-Caffarel method with a many-body Pulay correction. Improved algorithms for the minimization of the variational integral were useful in the force calculations.
An improved method of moving the parameters in variational quantum Monte Carlo (VMC) calculations is described, and then applied to the calculation of atomic forces for first-row hydrides. We propose use of Newton's method with singular-value decomposition (SVD) to update the variational parameters, combined with steepest descent (SD) updates along directions rejected by the SVD. Using this approach to minimize the energy expectation value, we compute atomic forces [J. Chem. Phys. 118, 7193 (2003)] for first-row monohydrides in VMC and fixed-node diffusion Monte Carlo (DMC). The forces agree quantitatively with potential-energy surfaces, demonstrating the accuracy of this force procedure. Properties obtained from these calculations, such as harmonic vibrational frequencies and anharmonicity constants, agree well with experimental values.
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.
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.
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.
Quantum Monte Carlo (QMC) is one of the most promising methods for solving quantum many-body problems. QMC combines known analytical properties of wave functions, results from other methods, and stochastic techniques, into a powerful tool for investigation of the electronic structure of real materials. Furthermore, the method has a favorable scaling ( O[N3] in the number of electrons), perfect scalability on parallel architectures, and can be applied to a wide range of systems. Our work has focused both on expanding the capabilities of QMC for calculation of quantities beyond energies, such as interatomic forces, and on application of the method to challenging problems in nanostructure materials research. For the calculation of interatomic forces we have implemented a finite difference correlated sampling method. The correlated sampling enables us to avoid the problem of statistical noise in evaluating the energy differences and allows us to obtain forces with an average 1% error. We have analyzed the effect of the method's main approximations on the estimate of the forces, both by theoretical arguments and by performing a variety of tests on small systems. Furthermore, we have used this newly developed method for estimating equilibrium geometries in a problem where other methods have failed. QMC enabled us to treat the electron correlation effects with high accuracy and to provide accurate predictions in a number of interesting applications. In particular, for carbon rings of C4N+2 stoichiometry we have resolved a long-standing problem regarding the competition between the aromaticity and second order Jahn-Teller effects. We have found that the dominant mechanism at small and intermediate sizes, which stabilizes the bond angle and bond length alternated geometries, is the secondorder Jahn-Teller effect while the aromatic isomer is always found to be a transition state. These high accuracy calculations which have involved multi-reference wave functions, extensive variational optimizations, and the use of massively parallel platforms demonstrate clearly the unique capabilities of the QMC approach.