Coulomb interatomic potential

Themiodynamic stability requires a repulsive core m the interatomic potential of atoms and molecules, which is a manifestation of the Pauli exclusion principle operating at short distances. This means that the Coulomb and dipole interaction potentials between charged and uncharged real atoms or molecules must be supplemented by a hard core or other repulsive interactions. Examples are as follows. 

The covalent curve of Figure 3-5 has been drawn in the same way as for the HX curves. The ionic curve represents a Coulomb attractive potential and a repulsive potential b/R9f in which the constant b is given values which lead (with use of the Madelung constant and corresponding constant in the repulsive potential) to the correct interatomic distance in the corresponding crystal (Chap. 13). Polarization is neglected. 

Starr, T. L. and Williams, D. E. (1977 ). Coulombic nonbonded interatomic potential functions derived from crystal-lattice vibrational frequencies in hydrocarbons. Acta Crystallogr A, 33, 771-6. [153]... 

In ideal circumstances, %(r) properly moderates the Coulomb potential to describe the interaction between ions and atoms at all separation distances. For large distances, %(r) should tend to zero, while for very small distances, %(r) should tend to unity. Such features allow a single interatomic potential energy function, (2.8), to describe the entire collision process. 

T. L. Starr and D. E. Williams, Acta Crystallogr. Ser. A, A106, 771 (1977). Coulombic Non-bonded Interatomic Potential Functions Derived from Crystal-Lattice Vibrational Frequencies in Hydrocarbons. 

The above analysis can be applied to other forms of the interatomic potential. For example, in some crystals, the attractive term is of shorter range than a coulombic interaction and is often replaced by a term with an r dependence. 

An elementary approach for determining the structural energies of a solid is to eonstruct an algebraic representation of the interatomic force field. There are numerous obstacles to constructing such potentials. For example, changes in coordination, re-hybridization, charge transfer, and Jahn-Teller distortions are very difficult to incorporate in classical potentials. However, if the Coulomb forces play a dominant role in the chemical bonds present, it may be possible to obtain some useful results with interatomic potentials. This may be the case for materials subjected to high pressure situations. 

The reliability of the results of computer simulation mainly depends on the accuracy of the interatomic potential models used in the calculation[5]. The interatomic potentials most often used are generally based on the Born model of the solid, which includes a long-range Coulombic interaction, and a short-range term to model the repulsion between electron charge clouds and the van der Waals dispersive interaction[6]. 

The bond between atoms is described by the curve that traces the way the energy changes as the atomic coordinates change. The potential of the interatomic attraction is the energy as a function of the normal vibration coordinate strain, which is usually taken to be the distance between the bound atoms (Figure 2.1). Many variants exist. A Morse curve is representative for the potential between covalently bonded atoms. If the bond is between ions the interatomic potential is well described by a Lennard-Jones potential with a Coulomb term. 

MD simulations of electrolytes for lithium batteries retain the atomistic representation of the electrolyte molecules but do not treat electrons explicitly. Instead the influence of electrons on intermolecular interactions is subsumed into the description of the interatomic interactions that constitute the atomistic potential or force field. The interatomic potential used in MD simulations is made up of dispersion/ repulsion terms. Coulomb interactions described by partial atomic charges, and in some cases, dipole polarizability described by atom-based polarizabilities. The importance of explicit inclusion of polarization effects is considered below. In the most accurate force fields, interatomic potentials are informed by high-level QC calculations. Specifically, QC calculations provide molecular geometries, conformational energetic, binding energies, electrostatic potential distributions, and dipole polarizabilities that can be used to parameterize atomic force fields. 

A similar calculation can be carried out for ionic crystals (Table 8.2). In this case, the Coulomb interaction is taken into account, in addition to the van der Waals attraction and the Pauli repulsion. Although the van der Waals attraction contributes htde to the three-dimensional lattice energy, its contribution to the surface energy is significant and typically 20-40% [838]. The calculated surface energy is sensitive to the particular choice of the interatomic potential. 

Structural stability can be further examined in terms of an interatomic potential where R refers to real space, which is a rearrangement potential of the atoms at constant volume. The potential (1 ) is not a complete potential it does not contain the purely volume-dependent terms in Uq, being composed of the Coulomb repulsion Z /R and the band structure energy expressed as a spherically symmetric interatomic potential The form of... 

Since empirical force fields do not accurately estimate the true interatomic forces, it is difficult a priori to say how accurate the fast multipole approximation to the exact Coulomb potential and forces (exact in terms of the sum over partial charges) should be. Probably a good rule is to make sure that at each atom the approximate electrostatic force is within a few percent relative error of the true electrostatic force, obtained by explicitly summing over all atom pairs, i.e., IF — FJ < 0.05 F , for all atoms i, where F is the... 

It can be seen from Fig. 7 that V is a linear function of the qf This qV relation was pointed out and discussed at some length in the papers in ref. 6. It is not simple electrostatics in that it would not exist for an arbitrary set of charges on the sites, even if the potentials are calculated exactly. The charges must be the result of a self-consistent LDA calculation. The linearity of the relation and fie closeness of the points to the line is demonstrated by doing a least squares fit to the points. The sums that define the potentials V do not converge at all rapidly, as can be seen by calculating the Coulomb potential from the standard formula for one nn-shell after another. The qV relation leads to a special form for the interatomic Coulomb energy of the alloy... 

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. 

In a statistical Monte Carlo simulation the pair potentials are introduced by means of analytical functions. In the election of that analytical form for the pair potential, it must be considered that when a Monte Carlo calculation is performed, the more time consuming step is the evaluation of the energy for the different configurations. Given that this calculation must be done millions of times, the chosen analytic functions must be of enough accuracy and flexibility but also they must be fastly computed. In this way it is wise to avoid exponential terms and to minimize the number of interatomic distances to be calculated at each configuration which depends on the quantity of interaction centers chosen for each molecule. A very commonly used function consists of a sum of rn terms, r being the distance between the different interaction centers, usually, situated at the nuclei. In particular, non-bonded interactions are usually represented by an atom-atom centered monopole expression (Coulomb term) plus a Lennard-Jones 6-12 term, as indicated in equation (51). 

In conclusion, the repulsive interactions arise from both a screened coulomb repulsion between nuclei, and from the overlap of closed inner shells. The former interaction can be effectively described by a bare coulomb repulsion multiplied by a screening function. The Moliere function, Eq. (5), with an adjustable screening length provides an adequate representation for most situations. The latter interaction is well described by an exponential decay of the form of a Bom-Mayer function. Furthermore, due to the spherical nature of the closed atomic orbitals and the coulomb interaction, the repulsive forces can often be well described by pair-additive potentials. Both interactions may be combined either by using functions which reduce to each interaction in the correct limits, or by splining the two forms at an appropriate interatomic distance . 

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