Methods for calculating forces within quantum Monte Carlo simulations

Response properties in phaseless auxiliary field quantum Monte Carlo

We present a method for calculating first-order response properties in phaseless auxiliary field quantum Monte Carlo by applying automatic differentiation (AD). Biases and statistical efficiency of the resulting estimators are discussed. Our approach demonstrates that AD enables the calculation of reduced density matrices with the same computational cost scaling per sample as energy calculations, accompanied by a cost prefactor of less than four in our numerical calculations. We investigate the role of self-consistency and trial orbital choice in property calculations. We find that orbitals obtained using density functional theory perform well for the dipole moments of selected molecules compared to those optimized self-consistently.

Towards structural optimization of gold nanoclusters with quantum Monte Carlo

We study the prospects of using quantum Monte Carlo techniques (QMC) to optimize the electronic wavefunctions and atomic geometries of gold compounds. Complex gold nanoclusters are widely studied for diverse biochemical applications, but the dynamic correlation and relativistic effects in gold set the bar high for reliable, predictive simulation methods. Here we study selected ground state properties of few-atom gold clusters by using density functional theory (DFT) and various implementations of the variational Monte Carlo (VMC) and diffusion Monte Carlo. We show that the QMC methods mitigate the exchange-correlation (XC) approximation made in the DFT approach: the average QMC results are more accurate and significantly more consistent than corresponding DFT results based on different XC functionals. Furthermore, we use demonstrate structural optimization of selected thiolated gold clusters with between 1 and 3 gold atoms using VMC forces. The optimization workflow is demonstrably consistent, robust, and its computational cost scales with nb, where b < 3 and n is the system size. We discuss the implications of these results while laying out steps for further developments.

Machine Learning Diffusion Monte Carlo Forces

Diffusion Monte Carlo (DMC) is one of the most accurate techniques available for calculating the electronic properties of molecules and materials, yet it often remains a challenge to economically compute forces using this technique. As a result, ab initio molecular dynamics simulations and geometry optimizations that employ Diffusion Monte Carlo forces are often out of reach. One potential approach for accelerating the computation of "DMC forces" is to machine learn these forces from DMC energy calculations. In this work, we employ Behler-Parrinello Neural Networks to learn DMC forces from DMC energy calculations for geometry optimization and molecular dynamics simulations of small molecules. We illustrate the unique challenges that stem from learning forces without explicit force data and from noisy energy data by making rigorous comparisons of potential energy surface, dynamics, and optimization predictions among ab initio density functional theory (DFT) simulations and machine-learning models trained on DFT energies with forces, DFT energies without forces, and DMC energies without forces. We show for three small molecules─C₂, H₂O, and CH₃Cl─that machine-learned DMC dynamics can reproduce average bond lengths and angles within a few percent of known experimental results at one hundredth of the typical cost. Our work describes a much-needed means of performing dynamics simulations on high-accuracy, DMC PESs and for generating DMC-quality molecular geometries given current algorithmic constraints.

General Analytical Nuclear Forces and Molecular Potential Energy Surface from Full Configuration Interaction Quantum Monte Carlo

The full configuration interaction quantum Monte Carlo (FCIQMC) is a state-of-the-art stochastic electronic structure method, providing a methodology to compute FCI-level state energies of molecular systems within a quantum chemical basis. However, especially to probe dynamics at the FCIQMC level, it is necessary to devise more efficient schemes to produce nuclear forces and potential energy surfaces (PES) from FCIQMC. In this work, we derive the general formula for nuclear forces from FCIQMC, and clarify different contributions of the total force. This method to obtain FCIQMC forces eliminates previous restrictions and can be used with frozen core approximation and free selection of orbitals, making it promising for more efficient nuclear forces calculations. After some numerical checks of this procedure on the binding curve of N₂ molecule, we use the FCIQMC energy and force to obtain the full-dimensional ground state PES of the water molecule via Gaussian processes regression. The new water FCIQMC PES can be used as the basis for H₂O ground state nuclear dynamics, structure optimization, and rotation-vibrational spectrum calculation.

Energy Derivatives in Real-Space Diffusion Monte Carlo

We present unbiased, finite-variance estimators of energy derivatives for real-space diffusion Monte Carlo calculations within the fixed-node approximation. The derivative d_λE is fully consistent with the dependence E(λ) of the energy computed with the same time step. We address the issue of the divergent variance of derivatives related to variations of the nodes of the wave function both by using a regularization for wave function parameter gradients recently proposed in variational Monte Carlo and by introducing a regularization based on a coordinate transformation. The essence of the divergent variance problem is distilled into a particle-in-a-box toy model, where we demonstrate the algorithm.

Toward quantum Monte Carlo forces on heavier ions: Scaling properties

Quantum Monte Carlo (QMC) forces have been studied extensively in recent decades because of their importance with spectroscopic observables and geometry optimization. Here, we benchmark the accuracy and computational cost of QMC forces. The zero-variance zero-bias (ZVZB) force estimator is used in standard variational and diffusion Monte Carlo simulations with mean-field based trial wavefunctions and atomic pseudopotentials. Statistical force uncertainties are obtained with a recently developed regression technique for heavy tailed QMC data [P. Lopez Rios and G. J. Conduit, Phys. Rev. E 99, 063312 (2019)]. By considering selected atoms and dimers with elements ranging from H to Zn (1 ≤ Z_eff ≤ 20), we assess the accuracy and the computational cost of ZVZB forces as the effective pseudopotential valence charge, Z_eff, increases. We find that the costs of QMC energies and forces approximately follow simple power laws in Z_eff. The force uncertainty grows more rapidly, leading to a best case cost scaling relationship of approximately Z_eff ^6.5(3) for diffusion Monte Carlo. We find that the accessible system size at fixed computational cost scales as Z_eff ^-2, insensitive to model assumptions or the use of the "space warp" variance-reduction technique. Our results predict the practical cost of obtaining forces for a range of materials, such as transition metal oxides where QMC forces have yet to be applied, and underscore the importance of further developing force variance-reduction techniques, particularly for atoms with high Z_eff.

Tail-regression estimator for heavy-tailed distributions of known tail indices and its application to continuum quantum Monte Carlo data

Standard statistical analysis is unable to provide reliable confidence intervals on expectation values of probability distributions that do not satisfy the conditions of the central limit theorem. We present a regression-based estimator of an arbitrary moment of a probability distribution with power-law heavy tails that exploits knowledge of the exponents of its asymptotic decay to bypass this issue entirely. Our method is applied to synthetic data and to energy and atomic force data from variational and diffusion quantum Monte Carlo calculations, whose distributions have known asymptotic forms [J. R. Trail, Phys. Rev. E 77, 016703 (2008)PLEEE81539-375510.1103/PhysRevE.77.016703; A. Badinski et al., J. Phys.: Condens. Matter 22, 074202 (2010)JCOMEL0953-898410.1088/0953-8984/22/7/074202]. We obtain convergent, accurate confidence intervals on the variance of the local energy of an electron gas and on the Hellmann-Feynman force on an atom in the all-electron carbon dimer. In each of these cases the uncertainty on our estimator is 45% and 60 times smaller, respectively, than the nominal (ill-defined) standard error.

Chaotic versus nonchaotic stochastic dynamics in Monte Carlo simulations: a route for accurate energy differences in N-body systems

We present a method to efficiently evaluate small energy differences of two close N-body systems by employing stochastic processes having a stability versus chaos property. By using the same random noise, energy differences are computed from close trajectories without reweighting procedures. The approach is presented for quantum systems but can be applied to classical N-body systems as well. It is exemplified with diffusion Monte Carlo simulations for long chains of hydrogen atoms and molecules for which it is shown that the long-standing problem of computing energy derivatives is solved.