Interatomic Coulomb interaction

Two tilings liave developed since that calculation was made. First, the overlap interaction was evaluated and appears much too small to account for the angle (Cliapter 7), and second, the covalent and polar energies have been related to pseudopotentials (Chapter 18), suggesting that IT, is much less sensitive to distortions than one would guess from interatomic Coulomb interactions. Wc shall thercrorc not repeat the full analysis made by Pantelides and Harrison (1976) but extract only the parts that now seem most relevant. 

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 . 

Interatomic Coulombic decay (ICD) is an electronic decay process that is particularly important for those inner-shell or inner-subshell vacancies that are not energetic enough to give rise to Auger decay. Typical examples include inner-valence-ionized states of rare gas atoms. In isolated systems, such vacancy states are bound to decay radiatively on the nanosecond timescale. A rather different scenario is realized whenever such a low-energy inner-shell-ionized species is let to interact with an environment, for example, in a cluster. In such a case, the existence of the doubly ionized states with positive charges residing on two different cluster units leads to an interatomic (or intermolecular) decay process in which the recombination part of the two-electron transition takes part on one unit, whereas the ionization occurs on another one. ICD [73-75] is mediated by electronic correlation between two atoms (or molecules). In clusters of various sizes and compositions, ICD occurs on the timescale from hundreds of femtoseconds [18] down to several femtoseconds [76-79]. 

Another reactive force field that is dependent on bond-order was developed by van Duin, Dasgupta, Loran, and Goddard [183] for hydrocarbons. The configurational energy is described as the sum of energy contributions from internal modes as well as non-bonding van der Waals and Coulombic interactions, but the parameters of the functions that describe each contribution is dependent upon the bond order of atoms involved in each description. It is assumed that the bond order between an atom pair is dependent on the interatomic separation. While this model has been used to predict bond dissociation energies, heats of formation and structures of simple hydrocarbons, it was not applied to predict condensed phase properties. However, the form of the potential should allow for condensed phase studies. 

Their weak interatomic interaction is responsible for the condensation of the normal inert gases into solids. Atoms of normal inert gas are brought together until the repulsive terms in the overlap interaction prevent further contraction. The attraction favors a close-packed structure, and all of the normal inert gases form face-centered cubic lattices. These two contributions to the total interaction will remain almost the same in the ionic crystals, but with added Coulomb interactions, so it is desirable to understand all of these contributions with some care. 

The first term is the difference in the sum of the one-electron energies deduced from band structure calculations for the B32 and B2 structures, respectively. The second term is a coulombic interaction of the charge densities inside the muffin-tin spheres, AE . is an exchange-correlation correction, and AEpu, is an interatomic energy difference which... 

If one succeeds in transforming not only the occupied but also the virtual MOs to a set of well-localized MOs, such that one can associate q (occupied and virtual) MOs with each atom, then one can argue that for the description of intraatomic correlation only excitations in this g-dimensional (and hence n-independent) subspace need to be considered. For interatomic correlations between a pair of neighboring atoms excitations with the 2q dimensional space of the MOs of the two atoms are necessary, and so forth. Correlations beyond next-nearest neighbors may be regarded as unimportant. The number of pairs of atoms to be considered scales with n, so the overall computational demands should scale with n as well, provided that also takes advantage of fast multipole expansion [190, 191] for the Coulomb interaction. 

The basic assumptions underlying the use of most atom-atom potential calculations are that only central forces operate between pairs of atoms and that the total interaction energy is the sum of the interactions between all pairs of atoms—the additivity assumption. The individual atom-atom interaction energies include a repulsive term with a steep rise in the energy at small interatomic distances, an attractive term designed to allow for London-type dispersion attractions and, sometimes, an additional coulombic interaction as well. With an exponential function as the repulsive term, the interaction energy between a pair of atoms can be written as... 

Fig. I. Plot of the givimd-slate energy (harlree) for the dissociation of NaCl in the gas-phase (A) and in aqueous solution (B) against the interatomic distance (A). For the molecule in solution the function is a free energy. In Figure lb the plot for the curve B is shown on an enhanced scale for a better visualization. The dashed curve C refers to the Coulombic interaction between Na and Cl. ...

As soon as covalency comes into play, small interatomic distances with small coordination numbers are preferred. This is because covalent interactions do not fall off as but much faster. For example, the overlap integral Sij between two atomic orbitals goes to zero quite rapidly because the atomic valence functions themselves scale as as has been indicated in Section 2.2. Thus, if the cooalent contribution to the cohesive energy increases, smaller coordination numbers (4) are favored over larger coordination numbers (6) although both would be equally suitable for a purely Coulomb interaction. A more detailed discussion has been given by Pearson [201]. 

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. 

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