Ewald summation versus direct summation of shifted-force potentials for the calculation of electrostatic interactions in solids: A quantitative study

The calculated Madelung energies and Madelung forces of the electrostatic interaction for nine crystal structures are reported. The method of direct summation with two different shifted-force potentials is compared to the Ewald summation. There is a considerable difference in the convergence of the energy and the force for the two shifted-force potentials regarding the cutoff radius. The convergence depends not only on the potential itself, but also on the crystal structure. One of the shifted-force potentials used is implemented in the CHARMM force field. The energy calculated with this potential shows a good convergence for small cutoff radii. With the other shifted-force potential, the force shows a better convergence for small cutoff radii. The number of pair interactions for obtaining the Madelung limit using the Ewald summation and the direct summation of a shifted-force potential is also reported. For complex structures like zeolites, the number of relevant pair interactions is smaller using the direct summation of a shifted-force potential. For simple structures such as cesium chloride, the number of significant pair interactions is smaller using the Ewald summation. © 1997 by John Wiley & Sons, Inc.     

Fast and spectrally accurate Ewald summation for 2-periodic electrostatic systems

A new method for Ewald summation in planar/slablike geometry, i.e., systems where periodicity applies in two dimensions and the last dimension is "free" (2P), is presented. We employ a spectral representation in terms of both Fourier series and integrals. This allows us to concisely derive both the 2P Ewald sum and a fast particle mesh Ewald (PME)-type method suitable for large-scale computations. The primary results are: (i) close and illuminating connections between the 2P problem and the standard Ewald sum and associated fast methods for full periodicity; (ii) a fast, O(N log N), and spectrally accurate PME-type method for the 2P k-space Ewald sum that uses vastly less memory than traditional PME methods; (iii) errors that decouple, such that parameter selection is simplified. We give analytical and numerical results to support this.     

Fast calculation of the electrostatic potential in ionic crystals by direct summation method

An efficient real space method is derived for the evaluation of the Madelung's potential of ionic crystals. The proposed method is an extension of Evjen's method. It takes advantage of a general analysis of the potential convergence in real space. Indeed, we show that the series convergence is exponential as a function of the number of canceled multipolar moments in the unit cell. The method proposed in this work reaches such an exponential convergence rate. Its efficiency is comparable to Ewald's method. However, unlike the latter, it uses only simple algebraic functions.     

Errors in the Ewald summation method with electrostatic layer correction for Coulomb interactions in slab geometry

New theoretical expressions for cut-off errors in 2D reciprocal-space summation of the electrostatic layer correction (ELC) term in energy and forces are derived, and a procedure to determine optimal parameters of the method is proposed. The procedure is tested in numerical calculations for charges distributed uniformly in a cubic box and charges located in two layers near the box basis. The summation errors for conventional Ewald method can be used to find out optimal values of the convergence parameter, and real- and reciprocal-space cut-off radii, whereas the ELC errors give possibility to choose an optimal value of an empty space gap in the simulation box.     

Fragment-based quantum mechanical methods for periodic systems with Ewald summation and mean image charge convention for long-range electrostatic interactions

We describe an Ewald-summation method to incorporate long-range electrostatic interactions into fragment-based electronic structure methods for periodic systems. The present method is an extension of the particle-mesh Ewald technique for combined quantum mechanical and molecular mechanical (QM/MM) calculations, and it has been implemented into the explicit polarization (X-Pol) potential to illustrate the computational details. As in the QM/MM-Ewald method, the X-Pol-Ewald approach is a linear-scaling electrostatic method, in which the short-range electrostatic interactions are determined explicitly in real space and the long-range Ewald pair potential is incorporated into the Fock matrix as a correction. To avoid the time-consuming Fock matrix update during the self-consistent field procedure, a mean image charge (MIC) approximation is introduced, in which the running average with a user-chosen correlation time is used to represent the long-range electrostatic correction as an average effect. Test simulations on liquid water show that the present X-Pol-Ewald method takes about 25% more CPU time than the usual X-Pol method using spherical cutoff, whereas the use of the MIC approximation reduces the extra costs for long-range electrostatic interactions by 15%. The present X-Pol-Ewald method provides a general procedure for incorporating long-range electrostatic effects into fragment-based electronic structure methods for treating biomolecular and condensed-phase systems under periodic boundary conditions.     

Ab initioLCAO-MO-SCF calculation of the electrostatic molecular potential of chlorpromazine and promazine

The electrostatic potential V( r ) arising from the ab initio LCAO-MO-SCF wave functions of chlorpromazine (CPZ ) and promazine (PZ ) has been calculated and discussed. In this approximation, the most probable sites of attack and reaction paths of electrophilic reagents are pointed out and compared. The analysis of V( r ) shows that the phenothiazine group has strong nucleophilic properties which are influenced by the phenothiazine substituent and that the electrostatic reactivity of CPZ and PZ is decidedly different near the phenothiazine substituent and similar near the side chain N atom. The dependence of V( r ) on the accuracy of the wave function has also been discussed by comparing some ab initio results on pyrrole, pyrazole, and imidazole obtained with a large basis set with an ab initio minimum basis set and with CNDO calculations.     

芬太尼类化合物的分子静电势研究

本文应用INDO波函数计算了三个芬太尼类化合物的分子静电势。酰胺氧原子周围均存在一个势阱很深的宽广的负电势区域, 是最重要的负电中心。 哌啶环4-位引入甲氧甲基后,增加了新的负电势区域。哌啶环1-位芳环周围具有宽广的正电势区域。 哌啶氮原子和酰胺氮原子附近存在较小的负电势区域。 基于计算出的静电势推测了三个化合物的镇痛作用的可能机理及药物结构与毒性的关系。     

Ewald summation and multiple time step methods for molecular dynamics simulation of biological molecules

Methods by which to determine conditions for a molecular dynamics (MD) simulation of biological molecules were investigated. Derivation of the optimal parameters of the Ewald summation was described so as to give same precision to the real space, the reciprocal space summations and the van der Waals interaction. Later, the procedure by which to determine the condition of the multiple time step method by RESPA (REference System Propagator Algorithm; Tuckerman et al., 1992, J. Chem. Phys., 97, 1990) was described as exemplified by MD simulations of a solvated β-sheet peptide. The conservation of the total energy in a microcanonical ensemble was measured to investigate the stability of the simulation conditions. The most feasible respective combinations of the time steps were: 0.25 fs for bond, angle and torsion interactions; 2 fs for van der Waals interaction and Ewald real-space summation; and 4 fs for Ewald reciprocal-space summation. Though it retained an acceptable accuracy, this condition accelerated the simulation ten-fold compared to that in which a simple velocity-Verlet method with a time step of 0.25 fs was used. The update of the correction term due to excluded neighbors was then investigated. Better results were obtained when the correction was updated with the real-space than when it was updated with the reciprocal-space summation. Finally, an MD simulation as long as 50 ps performed under the optimal Ewald and RESPA parameters was thus determined. The trajectory showed a good stability, indicating the feasibility of the parameters.     

Long-range electrostatic interactions in hybrid quantum and molecular mechanical dynamics using a lattice summation approach

A method based on a lattice summation technique for treating long-range electrostatic interactions in hybrid quantum mechanics/molecular mechanics simulations is presented in this article. The quantum subsystem is studied at the semiempirical level, whereas the solvent is described by a two-body potential of molecular mechanics. Molecular dynamics simulations of a (quantum) chloride ion in (classical) water have been performed to test this technique. It is observed that the application of the lattice summations to solvent-solvent interactions as well as on solute-solvent ones has a significant effect on solvation energy and diffusion coefficient. Moreover, two schemes for the computation of the long-range contribution to the electrostatic interaction energy are investigated. The first one replaces the exact charge distribution of the quantum solute by a Mulliken charge distribution. The long-range electrostatic interactions are then calculated for this charge distribution that interacts with the solvent molecule charges. The second one is more accurate and involves a modified Fock operator containing long-range electron-charge interactions. It is shown here that both schemes lead to similar results, the method using Mulliken charges for the evaluation of long-range interactions being, however, much more computationally efficient.     

Some problems est the calculation of ab initio potential energy curves

The non-adiabatic and adiabatic approaches to the calculation of molecular energy levels are presented. The concept of the potential energy curve, its accuracy and limitations are discussed. Simplified approaches to calculations of the intermolecular potential energy curves are also considered.     

Simple and accurate scheme to compute electrostatic interaction: Zero-dipole summation technique for molecular system and application to bulk water

The zero-dipole summation method was extended to general molecular systems, and then applied to molecular dynamics simulations of an isotropic water system. In our previous paper [I. Fukuda, Y. Yonezawa, and H. Nakamura, J. Chem. Phys. 134, 164107 (2011)], for evaluating the electrostatic energy of a classical particle system, we proposed the zero-dipole summation method, which conceptually prevents the nonzero-charge and nonzero-dipole states artificially generated by a simple cutoff truncation. Here, we consider the application of this scheme to molecular systems, as well as some fundamental aspects of general cutoff truncation protocols. Introducing an idea to harmonize the bonding interactions and the electrostatic interactions in the scheme, we develop a specific algorithm. As in the previous study, the resulting energy formula is represented by a simple pairwise function sum, enabling facile applications to high-performance computation. The accuracy of the electrostatic energies calculated by the zero-dipole summation method with the atom-based cutoff was numerically investigated, by comparison with those generated by the Ewald method. We obtained an electrostatic energy error of less than 0.01% at a cutoff length longer than 13 A for a TIP3P isotropic water system, and the errors were quite small, as compared to those obtained by conventional truncation methods. The static property and the stability in an MD simulation were also satisfactory. In addition, the dielectric constants and the distance-dependent Kirkwood factors were measured, and their coincidences with those calculated by the particle mesh Ewald method were confirmed, although such coincidences are not easily attained by truncation methods. We found that the zero damping-factor gave the best results in a practical cutoff distance region. In fact, in contrast to the zero-charge scheme, the damping effect was insensitive in the zero-charge and zero-dipole scheme, in the molecular system we treated. We discussed the origin of this difference between the two schemes and the dependence of this fact on the physical system. The use of the zero damping-factor will enhance the efficiency of practical computations, since the complementary error function is not employed. In addition, utilizing the zero damping-factor provides freedom from the parameter choice, which is not trivial in the zero-charge scheme, and eliminates the error function term, which corresponds to the time-consuming Fourier part under the periodic boundary conditions.     

Generation of OPLS-like charges from molecular electrostatic potential using restraints

A new method for generating atom-centered charges for use in condensed phase computer simulations is presented, which is based on a restrained electrostatic potential (RESP) procedure. Charges are calculated from a least-squares fit to the quantum mechanical electrostatic potential with a restraint applied to reduce their magnitude. The restraint developed here offers advantages over that used in RESP. The magnitude of the restraint is optimized to yield charges as close to the equivalent OPLS values as possible while still reproducing the molecule's electrostatic potential. A cross-validation analysis is used to show that the restraint is insensitive to the selection of OPLS molecules from which it is derived. Thus, with this method, OPLS-like charges may be produced from the electrostatic potential for atom types not in the OPLS force field. In addition, the restraint is shown to reduce the conformational dependence of the charges. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 483–498, 1999     

Simulation of liquid imidazole using a high-rank quantum topological electrostatic potential

Rigid body molecular dynamics simulations were carried out on pure liquid imidazole at four different temperatures and at 1 atm. Imidazole, which is important both in life science and materials science, is one of the simplest molecules to possess both a lone pair and a π system. These two features are known to benefit from multipolar electrostatics. Here the electrostatic interaction is governed by atomic multipole moments obtained from topologically partitioned ab initio electron densities. The non-electrostatic terms are modeled with Lennard-Jones parameters adjusted to fit the experimental liquid density. All σ values are incrementally increased by one single scaling factor. We report on how the presence of multipolar electrostatics influences the local structure, dynamics and thermodynamics of the liquid compared to electrostatics by atomic point charges. The point charge force field exaggerates the number of π-stacked dimers in the liquid, and underestimates the number of hydrogen-bonded dimers. The effect of the temperature on the local structure of liquid imidazole was analysed using radial and spatial distribution functions.     

About using the approximate fields to calculate the electrostatic potential distribution of membrane channels

By the example of gramicidine channel, a comparative analysis of different approximate representations (heavy atoms, polar protons, near-by atoms) of AMBER force field has been carried out to calculate the electrostatic potential distribution of ionic channels in biological membranes. The results obtained are compared with the potential computed in a full-atom representation. The use of approximate representations is shown to lead to estimated errors of the potential.     

Multipole expansions of the electrostatic molecular potential

Multipole expansions of the electrostatic molecular potential up to the hexadecapole terms are examined for H₂O, NH₃ and C₂H₄NH. A reasonable approximation to get unexpensive first order representations of the electrostatic potential for regions outside the van der Waals volume is found.     

胰岛素单体、双体分子静电势的计算

计算了三方二锌胰岛素晶胞不对称单位中两个胰岛素分子单体及其双体的分子静电势。结果表明:胰岛素两个单体分子的构象不同,其分子表面静电势也不相同,由于它们的构象是类似的,所以其电势分布也是接近的,单体胰岛素分子的表面静电势呈偶极型分布,双体的表面静电势基本保持单体的特征。文中还比较了两个单体分子及其集聚后的静电势变化。