Set Of Point Charges Research Articles

Determination of the electric field strength of high-voltage substations

The electric field strength is one of the main factors influencing sensitive microprocessor equipment and personnel on power stations and substations, power lines. Determining its level is an important applied task for ensuring the safe operation of electrical installations. The aim is to develop calculation relationships for determining the electric field strength created by the busbar of high-voltage substations in the working areas of personnel. The solution of the problem was based on the use of the method of equivalent charges to determine the strength of the electric field created by the complex busbar of high-voltage substations. Methodology. The development was based on solving the problem of the potential of the electric field of a point charge located in a dielectric half-space for a cylindrical coordinate system. By representing the electrode in the form of a set of point charges and subsequent integration, an expression for calculating the potential is obtained, created by a busbar of arbitrary orientation of finite length in an analytical form. Using the principle of superposition of fields and the definition of the derivative, expressions were obtained for calculating the vertical component of the electric field strength at given heights. Results. Based on the obtained expressions, using Visual Basic, the simulation of the distribution of the electric field strength under a three-phase power line with a voltage of 150 kV was performed. Comparison with the known calculation results obtained on the basis of analytical expressions for infinitely long conductors showed that the obtained expressions have an error of no more than 7%. The scientific novelty lies in the fact that for the first time expressions were obtained for determining the electric field strength created by a system of electrodes of finite length, based on the analytical method for solving differential equations. Practical significance. The proposed technique is implemented as a test module of the LiGro specialized software package, which allows modeling complex busbar systems typical for power stations and substations and power lines. A test calculation was carried out for an operating substation of regional electric networks with a voltage class of 110 kV. By comparing the duration of the calculation of switchgears with a diagonal of about 500 m, it was found that the calculation time in the LiGro complex based on the analytical method is several tens of times less than the calculation based on the finite element method. In addition, a more powerful computer was used for the end element simulation.

Modeling the spectral properties of poly(x-phenylenediamine) conducting polymers using a combined TD-DFT and electrostatic embedding approach.

The absorption spectra of polymers derived from ortho, meta and para phenylenediamines (o-PDA, m-PDA and p-PDA) have been simulated combining periodic density functional theory (DFT) calculations with time-dependent DFT simulations. These latter have been carried out on finite clusters embedded in a set of point charges devised to exactly reproduce the electrostatic potential of the periodic chains. The results are compared with those obtained for solvated o-PDA, m-PDA and p-PDA oligomers of increasing sizes extracted from the periodic structures. The electronic transitions involved have been investigated by a qualitative analysis based on isodensity maps completed by a quantitative analysis based on the density-based index (DCT ). For poly-(o)- and poly-(p)- phenylenediamines the agreement with the experimental data is achieved already by modeling solvated dimers whereas the inclusion of long-range electrostatic effects is mandatory for poly-(m)-phenylenediamine highlighting the importance of an accurate treatment of the electrostatic environment when a finite cluster approach is considered.

The Equivalence of the Charge Interaction Sum and the Ionic Strength

The electrostatic interaction among a neutral and finite set of point charges is based on the sum of their pairwise charge products, zizj, yet many analyses yield terms which simply contain a sum of the squares of the separate charges, corresponding to the ionic strength, ½∑mizi2.This submission collects together a number of important instances of this result and explains their equivalence. In effect, the ionic strength-like terms provide conveniently calculated coulomb sums for systems with finite collections of charges.

The u-series: A separable decomposition for electrostatics computation with improved accuracy.

The evaluation of electrostatic energy for a set of point charges in a periodic lattice is a computationally expensive part of molecular dynamics simulations (and other applications) because of the long-range nature of the Coulomb interaction. A standard approach is to decompose the Coulomb potential into a near part, typically evaluated by direct summation up to a cutoff radius, and a far part, typically evaluated in Fourier space. In practice, all decomposition approaches involve approximations-such as cutting off the near-part direct sum-but it may be possible to find new decompositions with improved trade-offs between accuracy and performance. Here, we present the u-series, a new decomposition of the Coulomb potential that is more accurate than the standard (Ewald) decomposition for a given amount of computational effort and achieves the same accuracy as the Ewald decomposition with approximately half the computational effort. These improvements, which we demonstrate numerically using a lipid membrane system, arise because the u-series is smooth on the entire real axis and exact up to the cutoff radius. Additional performance improvements over the Ewald decomposition may be possible in certain situations because the far part of the u-series is a sum of Gaussians and can thus be evaluated using algorithms that require a separable convolution kernel; we describe one such algorithm that reduces communication latency at the expense of communication bandwidth and computation, a trade-off that may be advantageous on modern massively parallel supercomputers.

Revisiting the Potential Energy Surface of the Stacked Cytosine Dimer: FNO-CCSD(T) Interaction Energies, SAPT Decompositions, and Benchmarking

Nucleobase stacking interactions are crucial for the stability of nucleic acids. This study investigates base stacking energies of the cytosine homodimer in different configurations, including intermolecular separation plots, detailed twist dependence, and displaced structures. Highly accurate ab initio quantum chemical single point energies using an energy function based on MP2 complete basis set extrapolation ([6 → 7]ZaPa-NR) and a CCSD(T)/cc-pVTZ-F12 high-level correction are presented as new reference data, providing the most accurate stacking energies of nucleobase dimers currently available. Accurate SAPT2+(3)δMP2 energy decomposition is used to obtain detailed insights into the nature of base stacking interactions at varying vertical distances and twist values. The ab initio symmetry adapted perturbation theory (SAPT) energy decomposition suggests that the base stacking originates from an intricate interplay between dispersion attraction, short-range exchange-repulsion, and Coulomb interaction. The interpretation of the SAPT data is a complex issue as key energy terms vary substantially in the region of optimal (low energy) base stacking geometries. Thus, attempts to highlight one leading stabilizing SAPT base stacking term may be misleading and the outcome strongly depends on the used geometries within the range of geometries sampled in nucleic acids upon thermal fluctuations. Modern dispersion-corrected density functional theory (among them DSD-BLYP-D3, ωB97M-V, and ωB97M-D3BJ) is benchmarked and often reaches up to spectroscopic accuracy (below 1 kJ/mol). The classical AMBER force field is benchmarked with multiple different sets of point-charges (e.g. HF, DFT, and MP2-based) and is found to produce reasonable agreement with the benchmark data.

Investigations of Stacked DNA Base-Pair Steps: Highly Accurate Stacking Interaction Energies, Energy Decomposition, and Many-Body Stacking Effects.

The stacking energies of 10 unique B-DNA base-pair steps were calculated with highly accurate quantum chemistry and used as reference values in a thorough benchmark of (dispersion-corrected) DFT, wave function methods, tight-binding methods, and different force fields, including charge variants thereof. The reference values were computed using a focal-point energy function based on extrapolated explicitly correlated MP2-F12 and conventional CCSD(T) data at the triple-ζ level. A collection of 29 different density functionals, sometimes with multiple dispersion corrections (D3(BJ), D3M(BJ), and VV10) were evaluated, including recent functionals like B97M-V, ωB97M-V, and SCAN-D3(BJ), which perform excellently. The double-hybrid DSD-BLYP-NL was found to be the best DFT method. Common wave function methods (MP2, SCS-MP2, and MP2.5) and the SNS-MP2 protocol were tested as well, where only the latter and DLPNO-CCSD(T)/CBS were competitive with DFT. The tight-binding methods DFTB3-D3 and GFN-xTB revealed a comparatively low accuracy. The AMBER force field outperformed CHARMM and GROMOS but still showed systematic gas-phase overbinding, which could be traced back to the electrostatic term, as revealed by comparison of different sets of point charges. High-order SAPT, e.g., SAPT2 + 3δ(MP2), was not only benchmarked but also used to study the nature of the stacking interactions to high accuracy. The δ(MP2) term turned out to be crucially important to reach high accuracy. Finally, we investigated four-body stacking effects with DLPNO-CCSD(T) and DFT, which were found to be significant and strongest for the CpC base-pair step where they reached almost 30% of the total stacking energy.

Spatial averaging of electrostatic potential differences in layered nanostructures with ionic hopping conductivity

Laws of spatial averaging on nanoscale for local electrostatic potential differences in ionic conductors are of key importance for the understanding of ion-transport phenomena and effects in space-charge layers. Currently, these laws are not investigated. As a consequence, the leading software packages for nanoscience do not have any competence for modeling of space-charge formation/relaxation in nanostructures with ionic hopping conductivity. So the article addresses the theme. For a layered nanostructure {Xk}, formed by parallel crystallographic Xk planes, an average electrostatic potential difference (<Vk,k+1>) in a non-uniform field is defined in terms of normalized sums Σ of the local works Wk,k+1 for Xk→Xk+1 transitions of mobile ions in a field of a single point charge. At a weak external electrical influence on {Xk}, the <Vk,k+1> can be easily generalized for a non-uniform field created by a set of point charges. The sums ΣWk,k+1 are calculated for small average distances (~0.6 nm) between mobile ions (as in advanced superionic conductors). Computer experiments show: (i) ΣWk,k+1 sums are independent of k-indices with the accuracy ~10%, (ii) probability density functions for local potential differences Vk,k+1 are close to normal distributions and (iii) <Vk,k+1> ≈ VUk,k+1, where VUk,k+1 is a potential difference in a uniform electrostatic field. These nanoscale laws allow manipulating by an average potential difference (<Vk,k+N>) between Xk and Xk+N planes (N > 1), so the modeling of non-stationary ion-transport processes and energy flows in parallel combinations of nanostructures become possible.

Interactions between interleaving holes in a sea of unit rhombi

Consider a family of collinear, equilateral triangular holes of any even side length lying within a sea of unit rhombi. The results presented below show that as the distance between the holes grows large the interaction between them may be approximated (up to a multiplicative constant) by taking the exponential of the negative of the electrostatic energy of the system obtained by viewing the holes as a set of point charges, each with a signed magnitude given by a certain statistic. Furthermore it is shown that the interaction between a family of left pointing collinear triangular holes and a free boundary may be approximated (again up to some multiplicative constant) by taking the exponential of the negative of the electrostatic energy of the system obtained by considering the holes as a set of point charges and the boundary a straight equipotential conductor. These two differing systems of point charges can be related via the method of image charges, a well-known physical law that also surfaces in the following mathematical analysis of enumeration formulas that count tilings of certain regions of the plane by unit rhombi.

Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals

AbstractWe consider a classical system of n charged particles in an external confining potential in any dimension d ≥ 2. The particles interact via pairwise repulsive Coulomb forces and the coupling parameter is of order n−1 (mean‐field scaling). By a suitable splitting of the Hamiltonian, we extract the next‐to‐leading‐order term in the ground state energy beyond the mean‐field limit. We show that this next order term, which characterizes the fluctuations of the system, is governed by a new “renormalized energy” functional providing a way to compute the total Coulomb energy of a jellium (i.e., an infinite set of point charges screened by a uniform neutralizing background) in any dimension. The renormalization that cuts out the infinite part of the energy is achieved by smearing out the point charges at a small scale, as in Onsager's lemma. We obtain consequences for the statistical mechanics of the Coulomb gas: next‐to‐leading‐order asymptotic expansion of the free energy or partition function, characterizations of the Gibbs measures, estimates on the local charge fluctuations, and factorization estimates for reduced densities. This extends results of Sandier and Serfaty to dimension higher than 2 by an alternative approach. © 2016 Wiley Periodicals, Inc.

Subtleties in obtaining the electrostatic energy of continuous distributions

The mathematical steps that generalize the expression for the electrostatic energy of a set of point charges to the corresponding expression for a continuous charge distribution involve a few subtleties that are not usually discussed in standard introductory or advanced electromagnetic textbooks. In this paper, we point out such subtleties and discuss how to deal with them in cases of volume and surface charge distributions. We also show explicitly that it is not possible to define electrostatic energy for a linear charge distribution, since this energy would be divergent. Finally, we use dimensional analysis to recover our results in a simpler and more elegant form.

Effect of a chiral electrostatic cavity on product selection in a reaction with a bifurcating reaction path

The possibility of influencing the selection of product-forming branches in a reaction with a bifurcating reaction path is explored. The reaction chosen is a substituted version of the well-known ring opening of singlet cyclopropanylidene. The substituents are arranged such that the starting carbene is meso (i.e., achiral) but the products are enantiomeric allenes. The question being posed is whether optical activity can be induced in the products by choosing the reaction to occur within a chirally disposed set of point charges. Direct-dynamics trajectory calculations are used to address the problem. It is found that, on a microscopic level, the point charges have a very significant effect on the choice of bifurcation branch. However, when the individual microscopic effects are averaged, to take account of free rotation of the molecule with respect to the charge array, the net effect is small (about 3 % enantiomeric excess after 500 fs). Nevertheless, even this small effect is larger than one might have predicted on the basis of an earlier analysis of electrostatic contributions to the interactions between chiral molecules. The reason is that the present simulation involves a concentric arrangement of chiral, polar objects, whereas the earlier analysis considered only a side-by-side arrangement. Through the use of numerical simulation, it is shown that two freely rotating arrays of chiral point charges exhibit larger diastereomeric discrimination when the arrays are concentric than when they are not, even for comparable average distances between the charges.

Low Order Physical Multipoles

Point multipole expansions are widely used to gain physical insight into complex distributions of charges and to reduce the cost of computing interactions between such distributions. However, practical applications that typically retain only a few leading terms may suffer from unacceptable loss of accuracy in the near-field. We propose an alternative approach for approximating electrostatic charge distributions, Optimal Physical Multipoles (OPMs), which optimally represent the original charge distribution with a set of point charges. By construction, approximation of electrostatic potential based on OPMs retains many of the useful properties of the corresponding point multipole expansion, including the same asymptotic behavior of the approximate potential for a given multipole order. At the same time, OPMs can be significantly more accurate in the near field: up to 5 times more accurate for some of the charge distributions tested here which are relevant to biomolecular modeling. Unlike point multipoles, for point charge distributions the OPM always converges to the original point charge distribution at finite order. Furthermore, OPMs may be more computationally efficient and easier to implement into existing molecular simulations software packages than approximation schemes based on point multipoles. In addition to providing a general framework for computing OPMs to any order, closed-form expressions for the lowest order OPMs (monopole and dipole) are derived. Thus, for some practical applications Optimal Physical Multipoles may represent a preferable alternative to point multipoles.

General Purpose Electrostatic Embedding Potential

We present a method and a computer code for accurate calculation of electrostatic potential in an arbitrary crystalline lattice modeled using a finite system. The method is based on complementing a lattice unit cell with a set of point charges in order to annihilate simultaneously all components of any number of the lowest multipole moments. The positions and the values of the complementary charges are determined analytically. The electrostatic potential produced by each modified cell is short range, and the corresponding lattice series converges absolutely, which makes it convenient to use in embedded cluster calculations of solids, surfaces, and low-dimensional structures. The method is illustrated by application to the rutile TiO2 and α-quartz SiO2 lattices and to those of several complex minerals.

Effects of the Zeolite Framework on the Adsorptions and Hydrogen-Exchange Reactions of Unsaturated Aliphatic, Aromatic, and Heterocyclic Compounds in ZSM-5 Zeolite: A Combination of Perturbation Theory (MP2) and a Newly Developed Density Functional Theory (M06-2X) in ONIOM Scheme

The confinement effect on the adsorption and reaction mechanism of unsaturated aliphatic, aromatic and heterocyclic compounds on H-ZSM-5 zeolite has been investigated by the four ONIOM methods (MP2:M06-2X), (MP2:B3LYP), (MP2:HF), and (MP2:UFF). The H-ZSM-5 'nanoreactor' porous intersection, where chemical reactions take place, is represented by a quantum cluster of 34 tetrahedral units. Ethene, benzene, ethylbenzene, and pyridine are chosen to represent reactions of various adsorbates of aliphatic, aromatic and heterocyclic compounds. Among the four combined methods, (MP2:M06-2X) outperforms the others. The results confirm that the method that takes weak interactions, especially the van der Waals interaction, into account is essential for describing the confinement effect from the zeolite framework. The effects of the infinite zeolitic framework on the cluster model are also included by a set of point charges generated by the embedded ONIOM model. The energies for the adsorption of ethene, benzene, ethylbenzene, and pyridine on H-ZSM-5 from an embedded ONIOM(MP2:M06-2X) calculation are predicted to be -14.0, -19.8, -24.7, and -48.4 kcal/mol, respectively, which are very close to available experimental observations. The adsorption energy of pyridine agrees well with the experiment data of -47.6 kcal/mol. We also applied the same computational methodology on the systematic investigation of the H/H exchange reaction of benzene and ethylbenzene with the acidic H-ZSM-5 zeolite. The H/H exchange reaction was found to take place in a single concerted step. The calculated apparent activation energies for benzene and ethylbenzene are 12.6 and 4.9 kcal/mol, which can be compared to the experimental estimates of 11.0 and 6.9 kcal/mol, respectively. The confinement effect of the extended zeolite framework has been clearly demonstrated not only to stabilize the adsorption complexes but also to improve their corresponding activation energies to approach the experimental benchmark.

Ray-Tracing Simulations of Liquid-Crystal Devices

In previous work, we have introduced the Hamiltonian method which enables to calculate the ray paths of light rays in inhomogeneous anisotropic media in the geometrical-optics approach. With this method we are able to simulate the effect of optical anisotropy in three dimensions. In this manuscript, we simulate the optical properties of multiple liquid-crystal configurations. We demonstrate the resemblance between the optical properties of a gradient-index fiber and a liquid-crystal layer with a Freédericksz alignment. In addition, we investigate the optical properties of artificial liquid-crystal profiles induced by the electric field of a set of point charges. Altogether, we conclude that, in the geometrical-optics approach, we are able to assess the optical properties of any arbitrary anisotropic optical system.

Reaction Mechanisms of the Methylation of Ethene with Methanol and Dimethyl Ether over H-ZSM-5: An ONIOM Study

The mechanisms of ethene methylation with methanol and dimethyl ether have been investigated using a 128T cluster of ZSM-5 zeolite modeled by ONIOM(B3LYP/6-31G(d,p):UFF) and ONIOM(M06-2X/6-311+G(2df,2p):UFF) calculations. The effects of the infinite zeolitic framework on the model of the zeolite nanopocket, which consisted of a quantum cluster of 34 tetrahedral units and of 128 tetrahedral units modeled on the UFF level, were also included. The zeolitic Madelung potential was reproduced by a set of point charges generated by the SCREEP method. The energies for the adsorption of methanol and dimethyl ether on H-ZSM-5 from an ONIOM2(M06-2X/6-311+G(2df,2p):UFF)+SCREEP calculation are −26.3 and −29.4 kcal/mol, respectively, which are in good agreement with the experimental data. Dissociative and associative mechanisms of the ethene methylation by methanol and dimethyl ether have been considered. For the dissociative mechanism, the methylation reaction of ethene starts with the protonation of methanol or dimet...

Assessing the Role of Polarization in Docking

We describe a strategy for including ligand and protein polarization in docking that is based on the conversion of induced dipoles to induced charges. Induced charges have a distinct advantage in that they are readily implemented into a number of different computer programs, including many docking programs and hybrid QM/MM programs; induced charges are also more readily interpreted. In this study, the ligand was treated quantum mechanically to avoid parametrization issues and was polarized by the target protein, which was treated as a set of point charges. The induced dipole at a given target atom, due to polarization by the ligand and neighboring residues, was reformulated as induced charges at the given atom and its bonded neighbors, and these were allowed to repolarize the ligand in an iterative manner. The final set of polarized charges was evaluated in docking using AutoDock 4.0 on 12 protein-ligand systems against the default empirical Gasteiger charges, and against nonpolarized and partially polarized potential-derived charges. One advantage of AutoDock is that the best rmsd structure can be identified not only from the lowest energy pose but also from the largest cluster of poses. Inclusion of polarization does not always lead to the lowest energy pose having a lower rmsd, because docking is designed by necessity to be rapid rather than accurate. However, whenever an improvement in methodology, corresponding to a more thorough treatment of polarization, resulted in an increased cluster size, then there was also a corresponding decrease in the rmsd. The options for implementing polarization within a purely classical docking framework are discussed.

Electrostatic effects on cluster simulation of ionic crystals and surfaces

Local perturbations in ionic solids and surfaces are often studied with cluster models embedded in an electrostatic Madelung potential. This potential is usually represented by a finite set of point charges whose values are defined by using several different methods. Local properties seem to a large amount independent from the criteria adopted for defining the Madelung potential. In this study, the cluster electrostatic potential is defined unambiguously on the basis of a periodic calculation of the unperturbed crystal; this permits us to understand the reasons for the insensitivity of local properties to the quality of the Madelung potential

Multiconfigurational g Tensor Calculations as a Probe for the Covalency of the Copper−Ligand Bonds in Copper(II) Complexes: [CuCl4]2-, [Cu(NH3)4]2+, and Plastocyanin

Calculations of the g tensor of three copper(II) complexes [Cu(NH3)4]2+, [CuCl4]2-, and plastocyanin are presented. Two different sum-over-states-based approaches are considered, making use of the multistate CASPT2 method for excitation energies and PMCAS (perturbation modified CAS) wave functions for the computation of the angular momentum and spin-orbit coupling matrix elements. Test calculations on [Cu(NH3)4]2+ and [CuCl4]2- point to the need of including in the MS-CASPT2 treatment the specific charge-transfer state with an electron excited out of the bonding counterpart of the ground-state SOMO. The computed g shifts for these two molecules present a considerable improvement with respect to the results obtained from our previous g tensor calculations based instead on CASSCF/CASPT2. This is shown to be related to an improved description of the covalency of the Cu-L bonds. For the calculations on plastocyanin, different models are used, taken from a recent (QM/MM) DFT study by Sinnecker and Neese. The effect of the surrounding protein is taken into account by surrounding the central cluster either with a dielectric continuum (epsilon = 4) or with a set of point charges. The second approach is found to be indispensable for an accurate description of environmental effects. With this approach, the calculated g values compare to within 30 ppt with the experimental data of plastocyanin.

Core charge distribution and self assembly of columnar phases: the case of triphenylenes and azatriphenylenes

BackgroundThe relation betweeen the structure of discotic molecules and columnar properties, a crucial point for the realization of new advanced materials, is still largely unknown. A paradigmatic case is that hexa-alkyl-thio substituted triphenylenes present mesogenic behavior while the corresponding azatriphenylenes, similar in shape and chemical structure, but with a different core charge distribution, do not form any liquid crystalline mesophase. This study is aimed at investigating, with the help of computer simulations techniques, the effects on phase behaviour of changes of the charge distribution in the discotic core.ResultsWe described the shape and the pair, dispersive and electrostatic, interactions of hexa alkyl triphenylenes by uniaxial Gay-Berne discs with embedded point charges. Gay-Berne parameters were deduced by fitting the dispersive energies obtained from an atomistic molecular dynamics simulation of a small sample of hexa-octyl-thio triphenylene molecules in columnar phase, while a genetic algorithm was used to get a minimal set of point charges that properly reproduces the ab anitio electrostatic potential. We performed Monte Carlo simulations of three molecular models: the pure Gay-Berne disc, used as a reference, the Gay-Berne disc with hexa-thio triphenylene point charges, the Gay-Berne disc with hexa-thio azatriphenylene point charges. The phase diagram of the pure model evidences a rich polymorphism, with isotropic, columnar and crystalline phases at low pressure, and the appearance of nematic phase at higher pressure.ConclusionWe found that the intermolecular electrostatic potential among the cores is fundamental in sta-bilizing/destabilizing columnar phases; in particular the triphenylene charge distribution stabilizes the columnar structure, while the azatriphenylene distribution suppresses its formation in favor of the nematic phase. We believe the present model could be successfully employed as the basis for coarse-grained level simulations of a wider class of triphenylene derivatives.