Reaction Field Correction Research Articles

Leveraging the sampling efficiency of RE-EDS in OpenMM using a shifted reaction-field with an atom-based cutoff.

Replica-exchange enveloping distribution sampling (RE-EDS) is a pathway-independent multistate free-energy method currently implemented in the GROMOS software package for molecular dynamics (MD) simulations. It has a high intrinsic sampling efficiency as the interactions between the unperturbed particles have to be calculated only once for multiple end-states. As a result, RE-EDS is an attractive method for the calculation of relative solvation and binding free energies. An essential requirement for reaching this high efficiency is the separability of the nonbonded interactions into solute-solute, solute-environment, and environment-environment contributions. Such a partitioning is trivial when using a Coulomb term with a reaction-field (RF) correction to model the electrostatic interactions but not when using lattice-sum schemes. To avoid cutoff artifacts, the RF correction is typically used in combination with a charge-group-based cutoff, which is not supported by most small-molecule force fields as well as other MD engines. To address this issue, we investigate the combination of RE-EDS simulations with a recently introduced RF scheme including a shifting function that enables the rigorous calculation of RF electrostatics with atom-based cutoffs. The resulting approach is validated by calculating solvation free energies with the generalized AMBER force field in water and chloroform using both the GROMOS software package and a proof-of-concept implementation in OpenMM.

Ionic mobilities of Na+ and Cl− at 25°C as a function of Ewald sum parameter: a comparative molecular dynamics simulation study

ABSTRACTWe present the result of molecular dynamics (MD) simulations to calculate the ionic mobilities μ of Na+ and Cl− ions in SPC/E water at 25°C using reaction field correction, simple truncation, and Ewald sums employing the Gear’s fifth order predictor–corrector and velocity Verlet algorithms. The static and dynamic properties of the Na+-water and Cl−-water systems are analysed and compared in terms of energetics, radial distribution functions, hydration numbers, residence times, and water diffusion to explain the behaviour of the ionic mobilities μ of Na+ and Cl− obtained from our MD simulations.

Compatibility of GROMOS-Derived Atomic Parameters for Lipopolysaccharide Membranes with the SPC/E Water Model and Alternative Long-Range Electrostatic Treatments Using Single Nonbonded Cutoff and Atom-Based Charge Schemes

Recent developments of GROMACS v.2016 ceased to support methodological approaches used in the development and validation of the GROMOS force field. We investigated the performance of a previously developed extension of the GROMOS force field for lipopolysaccharides to reproduce the structural dynamics of bacterial outer membrane (OM) using a single cutoff for nonbonded interactions and atom-based charge truncation. We further compared this setup for use with reaction field (RF) or particle mesh Ewald (PME) approximations in the presence of simple point charge (SPC) and extended simple point charge (SPC/E) water models. We find that the OM structural dynamics is well conserved in all simulated conditions, reproducing the available experimental data within the measurement uncertainty. The SPC/E model induces a small increase in OM fluidity, and when combined with the RF correction, shows a decrease in water orientation at the membrane surface. The present simulations support the compatibility of the GROMOS-derived lipopolysaccharide (LPS) parameters for use with single cutoff and atom-based charge schemes, either with RF or PME approximations, in GROMACS v.2016. Both SPC and SPC/E water models are suitable for use, but usage of SPC/E combined with the reaction field correction needs to be further investigated.

Anisotropic reaction field correction for long-range electrostatic interactions in molecular dynamics simulations.

Reaction-field (RF) methods have been extensively used in molecular dynamics simulations to efficiently compute long-range electrostatic interactions. They assume a continuous dielectric medium outside a certain cutoff, which has shown to be a reasonable approximation in many cases. However, lattice sum or fast multipole methods are nowadays often used instead, which treat long-range interactions explicitly but may introduce different artefacts. In the following work, the major issue of RFs is addressed, i.e., their inability to account for inhomogeneity even in heterogenous environments (e.g., membranes or protein binding pockets). By using a first-order Laplace series expansion of the dielectric permittivity on the cutoff sphere, local anisotropic effects can be described in a simple form. It is shown that the resulting boundary-value problem cannot be solved analytically, but instead a well-behaved approximative anisotropic reaction field (ARF) is introduced, which preserves coordinate invariance and approaches the standard RF solution for homogeneous systems. The comparison of RF to the state-of-the-art particle-particle particle-mesh (P3M) method shows a difference in the orientation of molecules close to the interface between two different dielectrics (water/chloroform). It was found that the ARF leads to a correction of the orientational distribution toward the P3M reference for planar and spherical interfaces.

Prediction of the reduction potential in transition-metal containing complexes: How expensive? For what accuracy?

Accurate computationally derived reduction potentials are important for catalyst design. In this contribution, relatively inexpensive density functional theory methods are evaluated for computing reduction potentials of a wide variety of organic, inorganic, and organometallic complexes. Astonishingly, SCRF single points on B3LYP optimized geometries with a reasonably small basis set/ECP combination works quite well--B3LYP with the BS1 [modified-LANL2DZ basis set/ECP (effective core potential) for metals, LANL2DZ(d,p) basis set/LANL2DZ ECP for heavy nonmetals (Si, P, S, Cl, and Br), and 6-31G(d') for other elements (H, C, N, O, and F)] and implicit PCM solvation models, SMD (solvation model based on density) or IEFPCM (integral equation formalism polarizable continuum model with Bondi atomic radii and α = 1.1 reaction field correction factor). The IEFPCM-Bondi-B3LYP/BS1 methodology was found to be one of the least expensive and most accurate protocols, among six different density functionals tested (BP86, PBEPBE, B3LYP, B3P86, PBE0, and M06) with thirteen different basis sets (Pople split-valence basis sets, correlation consistent basis sets, or Los Alamos National Laboratory ECP/basis sets) and four solvation models (SMD, IEFPCM, IPCM, and CPCM). The MAD (mean absolute deviation) values of SCRF-B3LYP/BS1 of 49 studied species were 0.263 V for SMD and 0.233 V for IEFPCM-Bondi; and the linear correlations had respectable R2 values (R2 = 0.94 for SMD and R2 = 0.93 for IEFPCM-Bondi). These methodologies demonstrate relatively reliable, convenient, and time-saving functional/basis set/solvation model combinations in computing the reduction potentials of transition metal complexes with moderate accuracy. © 2017 Wiley Periodicals, Inc.

Phase diagram of the ST2 model of water

We evaluate the free energy of the fluid and crystal phases for the ST2 potential with reaction field corrections for the long-range interactions. We estimate the phase coexistence boundaries in the temperature–pressure plane, as well as the gas–liquid critical point and gas–liquid coexistence conditions. Our study frames the location of the previously identified liquid–liquid critical point relative to the crystalline phase boundaries, and opens the way for exploring crystal nucleation in a model where the metastable liquid–liquid critical point is computationally accessible.

Including the Dispersion Attraction into Structure-Adapted Fast Multipole Expansions for MD Simulations.

Molecular dynamics (MD) simulations of protein–solvent systems, which are modeled by polarizable or nonpolarizable all-atom force fields and are enclosed by periodic boundaries, require accurate and efficient algorithms for the computation of the long-range interactions. A possible choice is the fast structure-adapted multipole method called SAMMp/RF (Lorenzen et al. J. Chem. Theory Comput. 2012, 8, 3628–3636). It is based on pth order Cartesian Taylor expansions of the electrostatic interactions, on an adaptive and hierarchical decomposition of a macromolecular simulation system into a quaternary tree of nested atom clusters, and on a reaction field (RF) correction originating from a distant dielectric continuum. Here, we substantially extend this method by adding qth order Taylor expansions of the dispersion attraction and by formulating an interaction acceptance criterion for cluster–cluster interactions, which is based on substance-specific accuracy estimates. As a result, we obtain with the default e...

Net charge changes in the calculation of relative ligand-binding free energies via classical atomistic molecular dynamics simulation.

The calculation of binding free energies of charged species to a target molecule is a frequently encountered problem in molecular dynamics studies of (bio-)chemical thermodynamics. Many important endogenous receptor-binding molecules, enzyme substrates, or drug molecules have a nonzero net charge. Absolute binding free energies, as well as binding free energies relative to another molecule with a different net charge will be affected by artifacts due to the used effective electrostatic interaction function and associated parameters (e.g., size of the computational box). In the present study, charging contributions to binding free energies of small oligoatomic ions to a series of model host cavities functionalized with different chemical groups are calculated with classical atomistic molecular dynamics simulation. Electrostatic interactions are treated using a lattice-summation scheme or a cutoff-truncation scheme with Barker–Watts reaction-field correction, and the simulations are conducted in boxes of different edge lengths. It is illustrated that the charging free energies of the guest molecules in water and in the host strongly depend on the applied methodology and that neglect of correction terms for the artifacts introduced by the finite size of the simulated system and the use of an effective electrostatic interaction function considerably impairs the thermodynamic interpretation of guest-host interactions. Application of correction terms for the various artifacts yields consistent results for the charging contribution to binding free energies and is thus a prerequisite for the valid interpretation or prediction of experimental data via molecular dynamics simulation. Analysis and correction of electrostatic artifacts according to the scheme proposed in the present study should therefore be considered an integral part of careful free-energy calculation studies if changes in the net charge are involved. © 2013 The Authors Journal of Computational Chemistry Published by Wiley Periodicals, Inc.

Optimizing the Accuracy and Efficiency of Fast Hierarchical Multipole Expansions for MD Simulations.

Based on p'th order Cartesian Taylor expansions of Coulomb interactions and on hierarchical decompositions of macromolecular simulation systems into hierarchies of nested, structure-adapted, and adaptively formed clusters of increasing size, fast multipole methods are constructed for rapid and accurate calculations of electrostatic interactions. These so-called SAMMp algorithms are formulated through totally symmetric and traceless tensors describing the multipole moments and the coefficients of local Taylor expansions. Simple recursions for the efficient evaluation and shifting of multipole moments are given. The required tensors are explicitly given up to order p = 4. The SAMMp algorithms are shown to guarantee the reaction principle. For systems with periodic boundaries, a reaction field (RF) correction is applied, which introduces at distances beyond the "minimum image convention" boundary a dielectric continuum surrounding each cluster at the top level of coarse graining. The correctness of the present SAMMp implementation is demonstrated by analyzing the scaling of the residuals and by checking the numerical accuracy of the reaction principle for a pair of distant molecular ions in vacuum. Molecular dynamics simulations of pure water and aqueous solutions containing artificial ions, which are enclosed by periodic boundaries, demonstrate the stability and low-noise behavior of SAMMp/RF.

An image-based reaction field method for electrostatic interactions in molecular dynamics simulations of aqueous solutions.

In this paper, a new solvation model is proposed for simulations of biomolecules in aqueous solutions that combines the strengths of explicit and implicit solvent representations. Solute molecules are placed in a spherical cavity filled with explicit water, thus providing microscopic detail where it is most needed. Solvent outside of the cavity is modeled as a dielectric continuum whose effect on the solute is treated through the reaction field corrections. With this explicit/implicit model, the electrostatic potential represents a solute molecule in an infinite bath of solvent, thus avoiding unphysical interactions between periodic images of the solute commonly used in the lattice-sum explicit solvent simulations. For improved computational efficiency, our model employs an accurate and efficient multiple-image charge method to compute reaction fields together with the fast multipole method for the direct Coulomb interactions. To minimize the surface effects, periodic boundary conditions are employed for nonelectrostatic interactions. The proposed model is applied to study liquid water. The effect of model parameters, which include the size of the cavity, the number of image charges used to compute reaction field, and the thickness of the buffer layer, is investigated in comparison with the particle-mesh Ewald simulations as a reference. An optimal set of parameters is obtained that allows for a faithful representation of many structural, dielectric, and dynamic properties of the simulated water, while maintaining manageable computational cost. With controlled and adjustable accuracy of the multiple-image charge representation of the reaction field, it is concluded that the employed model achieves convergence with only one image charge in the case of pure water. Future applications to pKa calculations, conformational sampling of solvated biomolecules and electrolyte solutions are briefly discussed.

Assessing the influence of electrostatic schemes on molecular dynamics simulations ofsecondary structure forming peptides

Electrostatic interactions play a fundamental role in determining the structure anddynamics of biomolecules in solution. However the accurate representation of electrostaticsin classical mechanics based simulation approaches such as molecular dynamics (MD) is achallenging task. Given the growing importance that MD simulation methods are taking onin the study of protein folding, protein stability and dynamics, and in structure predictionand design projects, it is important to evaluate the influence that different electrostaticschemes have on the results of MD simulations. In this paper we performed longtimescale simulations (500 ns) of two peptides, beta3 and RN24 forming differentsecondary structures, using for each peptide four different electrostatic schemes(namely PME, reaction field correction, and cut-off schemes with and withoutneutralizing counterions) for a total of eight 500 ns long MD runs. The structural andconformational features of each peptide under the different conditions were evaluated interms of the time dependence of the flexibility, secondary structure evolution,hydrogen-bonding patterns, and several other structural parameters. The degreeof sampling for each simulation as a function of the electrostatic scheme wasalso critically evaluated. Our results suggest that, while in the case of the shortpeptide RN24 the performances of the four methods are comparable, PME and RFschemes perform better in maintaining the structure close to the native one for theβ-sheet peptide beta3, in which long range contacts are mostly responsible for the definitionof the native structure.

Combining the lattice-sum and reaction-field approaches for evaluating long-range electrostatic interactions in molecular simulations

A new scheme, the lattice-sum-emulated reaction-field (LSERF) method, is presented that combines the lattice-sum (LS) and reaction-field (RF) approaches for evaluating electrostatic interactions in molecular simulations. More precisely, the LSERF scheme emulates a RF calculation (based on an atomic cutoff) via the LS machinery. This is achieved by changing the form of the electrostatic interactions in a standard LS calculation (Coulombic) to the form corresponding to RF electrostatics (Coulombic plus quadratic reaction-field correction term, truncated at the cutoff distance). It is shown (both analytically and numerically) that in the limit of infinite reciprocal-space accuracy, (i) the LSERF scheme with a finite reaction-field cutoff and a given reaction-field permittivity is identical to the RF scheme with the same parameters (and an atomic cutoff), and (ii) the LSERF scheme is identical to the LS scheme in the limit of an infinite reaction-field cutoff, irrespective of the reaction-field permittivity. This new scheme offers two key advantages: (i) from a conceptual point of view, it shows that there is a continuity between the RF and LS schemes and unifies them into a common framework; (ii) from a practical point of view, it allows us to perform RF calculations with arbitrarily large reaction-field cutoff distances for the same computational costs as a corresponding LS calculation. The optimal choice for the cutoff will be the one that achieves the best compromise between artifacts arising from the dielectric heterogeneity of the system (short cutoff) and its artificial periodicity (long cutoff). The implementation of the LSERF method is extremely easy, requiring only very limited modifications of any standard LS code. For practical applications to biomolecular systems, the use of the LSERF scheme with large reaction-field cutoff distances is expected to represent a significant improvement over the current RF simulations involving comparatively much shorter cutoffs.

From quantum chemistry and the classical theory of polar liquids to continuum approximations in molecular mechanics calculations

AbstractBiological macromolecules and other polymers belong to the class of mesoscopic systems, with characteristic length scale of the order of a nanometer. Although microscopic models would be the preferred choice in theoretical calculations, their use in computer simulations becomes prohibitive for large systems or long simulation times. On the other hand, the use of purely macroscopic models in the mesoscopic domain may introduce artifacts, with effects that are difficult to assess and that may compromise the reliability of the calculations. Here is proposed an approach with the aim of minimizing the empirical nature of continuum approximations of solvent effects within the scope of molecular mechanics (MM) approximations in mesoscopic systems. Using quantum chemical methods, the potential generated by the molecular electron density is first decomposed in a multicenter‐multipole expansion around predetermined centers. The monopole and dipole terms of the expansion at each site create electric fields that polarize the surrounding aqueous medium whose dielectric properties can be described by the classical theory of polar liquids. Debye's theory allows a derivation of the dielectric profiles created around isolated point charges and dipoles that can incorporate Onsager reaction field corrections. A superposition of screened Coulomb potentials obtained from this theory makes possible a simple derivation of a formal expression for the total electrostatic energy and the polar component of the solvation energy of the system. A discussion is presented on the physical meaning of the model parameters, their transferability, and their convergence to calculable quantities in the limit of simple systems. The performance of this continuum approximation in computer calculations of amino acids in the context of an atomistic force field is discussed. Applications of a continuum model based on screened Coulomb potentials in multinanosecond simulations of peptides and proteins are briefly reviewed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Structural manifestations of proton transfer in complexes of 2,6-dichlorophenols with pyridines.

DFT B3LYP/6-31G(d,p) calculations were performed to describe the proton transfer reaction pathway in the 2,6-dichlorophenolate of pyridine. The aim of these calculations was to establish the character of the dependence of the structure parameters on the proton transfer and comparing the results with known structures, e.g. the 2,6-dichloro-4-nitro- and pentachlorophenolates of pyridines. To make this comparison more reliable, the calculations were repeated with the use of a reaction-field correction with the Onsager radius and electric permittivity taken from the solid-state measurements. The calculations show that the second approach gives a better description of the structural modifications during the proton transfer.

A fast-Fourier transform method to solve continuum-electrostatics problems with truncated electrostatic interactions: Algorithm and application to ionic solvation and ion–ion interaction

An iterative algorithm based on fast-Fourier transforms is presented that solves the equations of continuum electrostatics for systems of heterogeneous dielectric permittivity (e.g., solute cavity in a solvent) under periodic boundary conditions. The method makes explicit use of the charge–dipole and dipole–dipole interaction tensors, and is thus applicable both to Coulombic interactions (Ewald scheme) and cutoff-based electrostatic interactions described by any polynomial function (including a Coulombic r−1 term), as commonly used in molecular dynamics simulations. The latter case includes, in particular, straight truncation of Coulombic interactions and truncation including a reaction-field correction. After testing and validation by comparison with existing methods, the algorithm is used to investigate the effect of cutoff truncation and artificial periodicity in explicit-solvent simulations of ionic solvation and ion–ion interactions. Both cutoff truncation and artificial periodicity are found to significantly affect the polarization around a spherical ion and its solvation free energy. The nature and magnitude of the two perturbations are analyzed in detail, and approximate analytical correction terms are derived to be applied to the results of explicit-solvent simulations. Cutoff truncation induces strong alterations in the potential of mean force for the interaction between two spherical ions. The present observations based on continuum electrostatics help to rationalize artifacts previously reported from explicit-solvent simulations involving cutoff truncation and, in particular, the unphysical attraction of like charges and repulsion of opposite charges, and the corresponding alterations in the relative stabilities of contact, solvent-separated, and free ion pairs.

A fast multipole method combined with a reaction field for long-range electrostatics in molecular dynamics simulations: The effects of truncation on the properties of water

We present a combination of the structure adapted multipole method with a reaction field (RF) correction for the efficient evaluation of electrostatic interactions in molecular dynamics simulations under periodic boundary conditions. The algorithm switches from an explicit electrostatics evaluation to a continuum description at the maximal distance that is consistent with the minimum image convention, and, thus, avoids the use of a periodic electrostatic potential. A physically motivated switching function enables charge clusters interacting with a given charge to smoothly move into the solvent continuum by passing through the spherical dielectric boundary surrounding this charge. This transition is complete as soon as the cluster has reached the so-called truncation radius Rc. The algorithm is used to examine the dependence of thermodynamic properties and correlation functions on Rc in the three point transferable intermolecular potential water model. Our test simulations on pure liquid water used either the RF correction or a straight cutoff and values of Rc ranging from 14 Å to 40 Å. In the RF setting, the thermodynamic properties and the correlation functions show convergence for Rc increasing towards 40 Å. In the straight cutoff case no such convergence is found. Here, in particular, the dipole–dipole correlation functions become completely artificial. The RF description of the long-range electrostatics is verified by comparison with the results of a particle-mesh Ewald simulation at identical conditions.

Application of the Linearized MD Approach for Computing Equilibrium Solvation Free Energies of Charged and Dipolar Solutes in Polar Solvents

The linearized MD technique is developed in order to treat systematically free energies of simple charged and dipolar solutes in water. The solvent electrostatic response field in the solute region is modeled by averaging instantaneous fields found in a MD computation for solvent configurations confined within a cavity that conforms to the real shape of the solute particle. At this stage, all electrostatic interactions are explicitly treated inside the cavity. The solvent in the external region (outside the cavity) is modeled in terms of a standard continuum theory. For nonspherical cavities, the present approach is more accurate than the field computation employed at the preceding MD stage, where spherically truncated Coulomb potentials are modified by the reaction field corrections. We considered two different linearization schemes based on a computation of either the average response field or of its fluctuations. Only the first algorithm proved to be successful. For a series of single-charged monatomic...

Analysis and Evaluation of Channel Models: Simulations of Alamethicin

Alamethicin is an antimicrobial peptide that forms stable channels with well-defined conductance levels. We have used extended molecular dynamics simulations of alamethicin bundles consisting of 4, 5, 6, 7, and 8 helices in a palmitoyl-oleolyl-phosphatidylcholine bilayer to evaluate and analyze channel models and to link the models to the experimentally measured conductance levels. Our results suggest that four helices do not form a stable water-filled channel and might not even form a stable intermediate. The lowest measurable conductance level is likely to correspond to the pentamer. At higher aggregation numbers the bundles become less symmetrical. Water properties inside the different-sized bundles are similar. The hexamer is the most stable model with a stability comparable with simulations based on crystal structures. The simulation was extended from 4 to 20 ns or several times the mean passage time of an ion. Essential dynamics analyses were used to test the hypothesis that correlated motions of the helical bundles account for high-frequency noise observed in open channel measurements. In a 20-ns simulation of a hexameric alamethicin bundle, the main motions are those of individual helices, not of the bundle as a whole. A detailed comparison of simulations using different methods to treat long-range electrostatic interactions (a twin range cutoff, Particle Mesh Ewald, and a twin range cutoff combined with a reaction field correction) shows that water orientation inside the alamethicin channels is sensitive to the algorithms used. In all cases, water ordering due to the protein structure is strong, although the exact profile changes somewhat. Adding an extra 4-nm layer of water only changes the water ordering slightly in the case of particle mesh Ewald, suggesting that periodicity artifacts for this system are not serious.

Calculation of the group-based pressure in molecular simulations. I. A general formulation including Ewald and particle-particle–particle-mesh electrostatics

A general formulation is given for the calculation of the isotropic or anisotropic group-based instantaneous pressure in molecular simulations under periodic boundary conditions. The equations, derived from the statistical mechanical definition of the pressure, apply to groups defined as single atoms (atomic pressure) or whole molecules (molecular pressure), but also to any other arbitrary atom grouping. Different definitions lead to different pressure fluctuations, but to the same average pressure. Two sets of equations are derived for the calculation of the group-based virial. The “traditional” set, which is the one commonly used to compute molecular pressures in simulations, has two main drawbacks: (i) it requires bookkeeping of group definitions in the inner loop of the nonbonded interaction calculation, (ii) it cannot be applied when electrostatic interactions are computed through lattice-sum methods. The “alternative” set is based on the remarkable result that any group-based virial can be computed from the atomic virial by adding a computationally inexpensive correction term to account for atom grouping. This new formalism presents the following advantages: (i) it requires no bookkeeping of group definitions in the inner loop of the nonbonded interaction calculation, (ii) the isotropic virial corresponding to each homogeneous pairwise interaction term can be computed directly from the corresponding interaction energy contribution without knowledge of the pairwise forces, (iii) application to lattice-sum electrostatics is straightforward. Traditional and alternative virial expressions are derived for all terms typical of interaction functions used in molecular simulations, namely covalent, Lennard-Jones (and long-range correction), truncated electrostatic (and reaction-field correction), and lattice-sum electrostatic (Ewald and particle-particle–particle-mesh including self-energy) terms.

Modeling ion–ion interaction in proteins: A molecular dynamics free energy calculation of the guanidinium-acetate association

The twin nitrogen–twin oxygen, C2v association of guanidinium and acetate in water is examined by means of molecular dynamics free energy calculations, using three approaches for handling Coulomb electrostatic interactions: (i) an Ewald lattice summation, (ii) a generalized reaction field correction, and, (iii) a smoothed spherical truncation. The potential of mean force obtained from the Ewald simulation exhibits the expected minima characteristic of a contact and a solvent-separated ion pair. In contrast, the repulsive and, therefore, physically unrealistic profile resulting from a spherical truncation of the electrostatic interactions at 12 Å calls into question the validity of such a protocol for simulating charged proteins. The introduction of a generalized reaction field improves the description of the solution significantly, in spite of an artifactual behavior towards the edge of the cutoff sphere. Whereas this approach, compared to a conventional spherical truncation, implies virtually no additional computational effort, it rapidly becomes costly for large ion–water and water–water cutoff radii. At 12 Å, the computational investment is comparable to that of an Ewald simulation with an appropriately chosen number of k-vectors and separation parameter, making the latter more cost-effective.