Coulombic pairwise potentials

The main handicap of MD is the knowledge of the function [/( ). There are some systems where reliable approximations to the true (7( r, ) are available. This is, for example, the case of ionic oxides. (7( rJ) is in such a case made of coulombic (pairwise) interactions and short-range terms. A second example is a closed-shell molecular system. In this case the interaction potentials are separated into intraatomic and interatomic parts. A third type of physical system for which suitable approaches to [/( r, ) exist are the transition metals and their alloys. To this class of models belong the glue model and the embedded atom method. Systems where chemical bonds of molecules are broken or created are much more difficult to describe, since the only way to get a proper description of a reaction all the way between reactant and products would be to solve the quantum-mechanical problem at each step of the reaction. 

When the interaction is expressed in terms of pairwise atom-atom potential functions, all three components of the van der Waals interactions are grouped together because of their common r 6 distance dependence. A repulsive term is added, while Coulombic interactions may be accounted for separately. In the expression due to Lennard Jones, the repulsion has an r 12 dependence, to give the pairwise potential function... 

Recently, detailed molecular pictures of the interfacial structure on the time and distance scales of the ion-crossing event, as well as of ion transfer dynamics, have been provided by Benjamin s molecular dynamics computer simulations [71, 75, 128, 136]. The system studied [71, 75, 136] included 343 water molecules and 108 1,2-dichloroethane molecules, which were separately equilibrated in two liquid slabs, and then brought into contact to form a box about 4 nm long and of cross-section 2.17 nmx2.17 nm. In a previous study [128], the dynamics of ion transfer were studied in a system including 256 polar and 256 nonpolar diatomic molecules. Solvent-solvent and ion-solvent interactions were described with standard potential functions, comprising coulombic and Lennard-Jones 6-12 pairwise potentials for electrostatic and nonbonded interactions, respectively. While in the first study [128] the intramolecular bond vibration of both polar and nonpolar solvent molecules was modeled as a harmonic oscillator, the next studies [71,75,136] used a more advanced model [137] for water and a four-atom model, with a united atom for each of two... 

The electrons are free particles, which interact pairwise according to the Coulomb law. Potential energy of interaction per electron can be expressed as... 

The Fast Multipole Method (FMM) was originally introduced by Greengard and Rokhlin [127-130] for efficient simulation of Np particles interacting through a Coulomb-like potential confined in a nonperiodic cell. The FMM relies on the standard multipole expansion for the electrostatic potential (forces) by separating the pairwise interactions into two components one due to nearby particles, computed directly, and another due interaction with distant particles, approximated by their multipole expansions. 

One important linear-scaling problem concerns the summation of the Coulomb pairwise interactions in a system which is formally a O(n ) process. Of course, this is a crucial step when MM potentials are being used too, although there... 

This leads to the third virial coefficient for hard spheres. In general, the nth virial coefficient of pairwise additive potentials is related to the coefficient7) in the expansion of g(r), except for Coulombic systems for which the virial coefficients diverge and special teclmiques are necessary to resiim the series. 

An alternative to the GB, COSMO, and Poisson electrostatic calculations is to model the solution to the Poisson equation in terms of pair potentials between solute atoms this procedure is based on the physical picture that the solvent screens the intra-solute Coulombic interactions of the solute, except for the critical descreening of one part of the solute from the solvent by another part of this solute. This descreening can be modeled in an average way to a certain level of accuracy by pairwise functions of atomic positions.18, M 65 One can obtain quite accurate solvation energies in this way, and it has recently been shown that this algorithm provides a satisfactory alternative to more expensive explicit-solvent simulations even for the demanding cases of 10-base-pair duplexes of DNA and RNA in water.66... 

One of the remarkable features of Coulomb s law when applied to nuclei and electrons is its additivity. The potential energy of an assemblage of particles is just the sum of all the pairwise interaetions in the form given in Eq. (1.1). Thus, consider a system with K nuclei, a =, 2,..., K having atomic numbers Z . We also consider the molecule to have N eleetrons. If the molecule is uncharged as a whole, then = N. We will use lower ease Latin letters, /, j,k,..to label electrons and lower case Greek letters, a, f, y,..., to label nuclei. The full potential energy may then be written... 

It is evident from the above equation and Eq. (26) that only static intermolecular correlations contribute to (Usoiv and therefore to the short-time decay of C(t) Identifying solvent-pair contributions to C(f) is straightforward for pairwise-additive potentials such as the site-site Coulombic form of Eq. (7). For such potentials. 

Although the underlying approximations are too crude to obtain an accurate potential energy surface, another very important observation can be made when the London equation is compared to the energy expression for H2 the total energy is not equal to the sum of pairwise H-H interactions. Thus, E(Rab, Rac, Rbc) Z Eab + Eac + Ebg, where Eab corresponds to E+ of Eq. (3.31), and Eac and Ebc are given by similar expressions. The simple summation of pairwise H-H interactions only holds for the Coulomb integrals ... 

We consider a generic donor-acceptor complex solute at infinite dilution in a polyatomic solvent. Both the solute and solvent molecules are represented by rigid and non-polarizable ISM models. In the ISM models the potential energy of interaction between two molecules is a sum of pairwise-additive site-site terms, including Coulombic interactions between partial charges located at the molecular sites. Throughout the paper the subscript A refers to interaction sites of the solute, while the subscript aj refers to interaction site j of solvent molecule a. 

The first term (kinetic energy) is a summation over all the particles in the molecule. The second term (potential enctgy) uses Coulomb s law to calculate the interaction between every pair of particles in the molecule, where e, and cj ate the charges on particles i and j. For electrons, the charge -e, while the charge for a nucleus is Ze, where Z Is the atomic number. The summation nutation ipairwise interaction terms in the summation (e.g., eg / = e e, and should only appear in the potential energy term once). The denominator r in the. second term is the distance between particles i anil j. J. i is understood to be the electronic wave function for a many-atom system. 

We represent the cluster as composed of rigid molecules interacting through a sum of pairwise atom-atom potentials of Lennard-Jones and Coulomb types [37] ... 

The first extension of this approach to ionic energetic crystals was done by Sorescu and Thompson [104] for ADN. Using the rigid-ion approximation, the intermolecular potential used was composed by pairwise Lennard-Jones (LJ), hydrogen bonding (HB), and Coulombic (C) terms of the form... 

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