Wolf Summation Research Articles

Improved cutoff functions for short-range potentials and the Wolf summation

ABSTRACT A class of radial, polynomial cutoff functions for short-ranged pair potentials or related expressions is proposed. Their derivatives up to order n and n + 1 vanish at the outer cutoff and an inner radius , respectively. Moreover, and . It is shown that the used order n can qualitatively affect results: stress and bulk moduli of ideal crystals are unavoidably discontinuous with density for n = 0 and n = 1, respectively. Systematic errors on energies and computing times decrease by 20–50% for Lennard-Jones with n = 2 or n = 3 compared to standard cutting procedures. Another cutoff function turns out beneficial to compute Coulomb interactions using the Wolf summation, which is shown to not properly converge when local charge neutrality is obeyed only in a stochastic sense. However, for all investigated homogeneous systems with thermal noise (ionic crystals and liquids), the modified Wolf summation, despite being infinitely differentiable at , converges similarly quickly as the original summation. Finally, it is discussed how to reduce the computational burden of numerically exact Monte Carlo simulations using the Wolf summation even when it does not properly converge.

Assessment of the Wolf method using the Stillinger-Lovett sum rules: From strong electrolytes to weakly charged colloidal dispersions.

The Ewald method has been the cornerstone in molecular simulations for modeling electrostatic interactions of charge-stabilized many-body systems. In the late 1990s, Wolf and collaborators developed an alternative route to describe the long-range nature of electrostatic interactions; from a computational perspective, this method provides a more efficient and straightforward way to implement long-range electrostatic interactions than the Ewald method. Despite these advantages, the validity of the Wolf potential to account for the electrostatic contribution in charged fluids remains controversial. To alleviate this situation, in this contribution, we implement the Wolf summation method to both electrolyte solutions and charged colloids with moderate size and charge asymmetries in order to assess the accuracy and validity of the method. To this end, we verify that the proper selection of parameters within the Wolf method leads to results that are in good agreement with those obtained through the standard Ewald method and the theory of integral equations of simple liquids within the so-called hypernetted chain approximation. Furthermore, we show that the results obtained with the original Wolf method do satisfy the moment conditions described by the Stillinger-Lovett sum rules, which are directly related to the local electroneutrality condition and the electrostatic screening in the Debye-Hückel regime. Hence, the fact that the solution provided by the Wolf method satisfies the first and second moments of Stillinger-Lovett proves, for the first time, the reliability of the method to correctly incorporate the electrostatic contribution in charge-stabilized fluids. This makes the Wolf method a powerful alternative compared to more demanding computational approaches.

Cutoff-Based Modeling of Coulomb Interactions for Atomistic-to-Continuum Multiscale Methods

Atomic interactions in a large class of functional materials, essentially—all dielectrics, polarizable solids and ionic solids, involve long-range Coulomb interactions. Yet atomistic-to-continuum multiscale methods, such as the quasicontinuum method, are currently applicable only in cases where the atomic interactions are short-ranged. This restriction on the nature of atomic-level interactions for multiscale methods excludes numerous materials that are central to various scientific and industrial applications. A number of studies have pointed out unphysical artifacts in molecular simulations upon using the direct cutoff-based truncated sum to evaluate Coulomb interactions in ionic solids. However, recently it is understood that the artifacts of the direct cutoff-based truncated sum can be significantly minimized if a suitable correction term is added. In this work we examine whether or not the Wolf summation method, one of the prominent cutoff-based methods, is suitable to carry out the accumulation of the Coulomb interactions within the context of atomistic-to-continuum multiscale methods. In this regard, we choose the quasicontinuum method from the existing collection of multiscale methods for demonstration. It is found that the Wolf summation method can be applied if the atomistic system has charge ordering; otherwise, the error in the accumulation of the Coulomb interactions could be large.

A Modified Shifted Force Approach to the Wolf Summation.

The Wolf method for calculation of electrostatic interactions in molecular simulations is known to describe the energy well, whereas the forces have discontinuities. For a more reliable description of the forces this method can be extended with a shifted force approach. This leads to a good description of the forces and precise molecular dynamics simulation, but the description of the energy becomes poorer. In this study we propose a modification of a shifted force extension to describe the energy as well as the forces in better agreement to reference data as determined from the Ewald summation. We show that vapor-liquid phase equilibria (VLE) calculated with Monte Carlo simulations in the grand canonical ensemble and dynamic properties calculated with molecular dynamics simulations can be calculated reliably using this modification to describe the electrostatic interactions.

Modification of the Wolf Method and Evaluation for Molecular Simulation of Vapor–Liquid Equilibria

A modification of the Wolf method [ Wolf et al. J. Chem. Phys. 1999 , 110 , 8254 - 8282 ], a spherically truncated pairwise summation to evaluate electrostatic interactions efficiently, is proposed. This method achieves better results for the energy, and the approach is used for determining phase equilibria in the grand canonical ensemble. To assess optimal parameters for the Wolf summation we propose a simple iterative approach using the Ewald summation as a reference. We show that phase equilibrium properties for pure components as well as mixtures can be calculated accurately using the Wolf summation. Our study considers molecular fluids with partial charges but without net charge.

Treating electrostatics with Wolf summation in combined quantum mechanical and molecular mechanical simulations.

The Wolf summation approach [D. Wolf et al., J. Chem. Phys. 110, 8254 (1999)], in the damped shifted force (DSF) formalism [C. J. Fennell and J. D. Gezelter, J. Chem. Phys. 124, 234104 (2006)], is extended for treating electrostatics in combined quantum mechanical and molecular mechanical (QM/MM) molecular dynamics simulations. In this development, we split the QM/MM electrostatic potential energy function into the conventional Coulomb r(-1) term and a term that contains the DSF contribution. The former is handled by the standard machinery of cutoff-based QM/MM simulations whereas the latter is incorporated into the QM/MM interaction Hamiltonian as a Fock matrix correction. We tested the resulting QM/MM-DSF method for two solution-phase reactions, i.e., the association of ammonium and chloride ions and a symmetric SN2 reaction in which a methyl group is exchanged between two chloride ions. The performance of the QM/MM-DSF method was assessed by comparing the potential of mean force (PMF) profiles with those from the QM/MM-Ewald and QM/MM-isotropic periodic sum (IPS) methods, both of which include long-range electrostatics explicitly. For ion association, the QM/MM-DSF method successfully eliminates the artificial free energy drift observed in the QM/MM-Cutoff simulations, in a remarkable agreement with the two long-range-containing methods. For the SN2 reaction, the free energy of activation obtained by the QM/MM-DSF method agrees well with both the QM/MM-Ewald and QM/MM-IPS results. The latter, however, requires a greater cutoff distance than QM/MM-DSF for a proper convergence of the PMF. Avoiding time-consuming lattice summation, the QM/MM-DSF method yields a 55% reduction in computational cost compared with the QM/MM-Ewald method. These results suggest that, in addition to QM/MM-IPS, the QM/MM-DSF method may serve as another efficient and accurate alternative to QM/MM-Ewald for treating electrostatics in condensed-phase simulations of chemical reactions.

Accelerating electrostatic pair methods on graphical processing units to study molecules in supercritical carbon dioxide.

Traditionally, electrostatic interactions are modelled using Ewald techniques, which provide a good approximation, but are poorly suited to GPU architectures. We use the GPU versions of the LAMMPS MD package to implement and assess the Wolf summation method. We compute transport and structural properties of pure carbon dioxide and mixtures of carbon dioxide with either methane or difluoromethane. The diffusion of pure carbon dioxide is indistinguishable when using the Wolf summation method instead of PPPM on GPUs. The optimum value of the potential damping parameter, α, is 0.075. We observe a decrease in accuracy when the system polarity increases, yet the method is robust for mildly polar systems. We anticipate the method can be used for a number of techniques, and applied to a variety of systems. Substitution of PPPM can yield a two-fold decrease in the wall-clock time.

On Molecular Statics Simulation of Ferroelectric Barium Titanate

AbstractThe present contribution deals with the atomistic modelling of ferroelectric barium titanate. In this context a core‐shell model is implemented in a Molecular Static algorithm. Furthermore, Coulomb forces are simulated by the Wolf summation method in order to allow for a small cut off radius. We discuss the core‐shell model, molecular statics as a finite elements approach and present some numerical results. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Ab initio based polarizable force field generation and application to liquid silica and magnesia

We extend the program potfit, which generates effective atomic interaction potentials from ab initio data, to electrostatic interactions and induced dipoles. The potential parametrization algorithm uses the Wolf direct, pairwise summation method with spherical truncation. The polarizability of oxygen atoms is modeled with the Tangney-Scandolo interatomic force field approach. Due to the Wolf summation, the computational effort in simulation scales linearly in the number of particles, despite the presence of electrostatic interactions. Thus, this model allows to perform large-scale molecular dynamics simulations of metal oxides with realistic potentials. Details of the implementation are given, and the generation of potentials for SiO(2) and MgO is demonstrated. The approach is validated by simulations of microstructural, thermodynamic, and vibrational properties of liquid silica and magnesia.

Comparison of the Ewald and Wolf methods for modeling electrostatic interactions in nanowires

AbstractIonic compounds pose extra challenges with the appropriate modeling of long‐range coulombic interactions. Here, we study the mechanical properties of zinc oxide (ZnO) nanowires using molecular dynamic simulations with Buckingham potential and determine the suitability of the Ewald (Ann. Phys. 1921; 19) and Wolf (J. Chem. Phys. 1999; 110(17):8254–8282) summation methods to account for the long‐range Coulombic forces. A comparative study shows that both the summation methods are suitable for modeling bulk structures with periodic boundary conditions imposed on all sides; however, significant differences are observed when nanowires with free surfaces are modeled. As opposed to Wolf's prediction of a linear stress–strain response in the elastic regime, Ewald's method predicts an erroneous behavior. This is attributed to the Ewald method's inability to account for surface effects properly. Additionally, Wolf's method offers highly improved computational performance as the model size is increased. This gain in computational time allows for modeling realistic nanowires, which can be directly compared with the existing experimental results. We conclude that the Wolf summation is a superior technique when modeling non‐periodic structures in terms of both accuracy of the results and computational performance. Copyright © 2010 John Wiley & Sons, Ltd.

Atomic Charges Derived from Electrostatic Potentials for Molecular and Periodic Systems

We present a method for fitting atomic charges to the electrostatic potential (ESP) of periodic and nonperiodic systems. This method is similar to the method of Campa et al. [ J. Chem. Theory Comput. 2009, 5, 2866]. We compare the Wolf and Ewald long-range electrostatic summation methods in calculating the ESP for periodic systems. We find that the Wolf summation is computationally more efficient than the Ewald summation by about a factor of 5 with comparable accuracy. Our analysis shows that the choice of grid mesh size influences the fitted atomic charges, especially for systems with buried (highly coordinated) atoms. We find that a maximum grid spacing of 0.2−0.3 A is required to obtain reliable atomic charges. The effect of the exclusion radius for point selection is assessed; we find that the common choice of using the van der Waals (vdW) radius as the exclusion radius for each atom may result in large deviations between the ESP generated from the ab initio calculations and that computed from the fitted charges, especially for points closest to the exclusion radii. We find that a larger value of exclusion radius than commonly used, 1.3 times the vdW radius, provides more reliable results. We find that a penalty function approach for fitting charges for buried atoms, with the target charge taken from Bader charge analysis, gives physically reasonable results.

Direct Wolf summation of a polarizable force field for silica

We extend the Wolf direct, pairwise r(-1) summation method with spherical truncation to dipolar interactions in silica. The Tangney-Scandolo interatomic force field for silica takes regard of polarizable oxygen atoms whose dipole moments are determined by iteration to a self-consistent solution. With Wolf summation, the computational effort scales linearly in the system size and can easily be distributed among many processors, thus making large-scale simulations of dipoles possible. The details of the implementation are explained. The approach is validated by estimations of the error term and simulations of microstructural and thermodynamic properties of silica.

Dissociative Water Potential for Molecular Dynamics Simulations

A new interatomic potential for dissociative water was developed for use in molecular dynamics simulations. The simulations use a multibody potential, with both pair and three-body terms, and the Wolf summation method for the long-range Coulomb interactions. A major feature in the potential is the change in the short-range O-H repulsive interaction as a function of temperature and/or pressure in order to reproduce the density-temperature curve between 273 K and 373 at 1 atm, as well as high-pressure data at various temperatures. Using only the change in this one parameter, the simulations also reproduce room-temperature properties of water, such as the structure, cohesive energy, diffusion constant, and vibrational spectrum, as well as the liquid-vapor coexistence curve. Although the water molecules could dissociate, no dissociation is observed at room temperature. However, behavior of the hydronium ion was studied by introduction of an extra H+ into a cluster of water molecules. Both Eigen and Zundel configurations, as well as more complex configurations, are observed in the migration of the hydronium.