Born-Mayer interactions

It is possible to infer these structures, that of lowest energy and others of higher energy as well, from simple models for some kinds of clusters. This is particularly so for atomic clusters whose binding forces are well represented by pairwise interactions such as the Lennard-Jones potentials that approximate van der Waals interactions, or the Born-Mayer interactions that describe the forces between the ions in alkali halide clusters. Many molecular clusters also behave much like their simple models, models... 

Equation 5 is often used to decribe the interaction between the incoming ion and the target atoms. The interaction between two target atoms generally occurs at low energy where the Thomas-Fermi potential overestimates the interaction. Under this situation a Born-Mayer potential is more appropriate , i.e. ... 

For species with strong polar bonds (e.g., metal oxides or halides) the interaction between ions can be described as the sum of Coulomb and short-range pair interactions. The latter is usually written in the Born-Mayer form... 

The interaction between dissolved ions and the dipole molecules of water in aqueous solutions is called - hydration. (See also - Born equation, -> Born-Haber cycle, -> Born-Mayer equation, -> hydrated ion, -> hydration number.)... 

As mentioned earlier, the shell model is closely related to those based on polarizable point dipoles in the limit of vanishingly small shell displacements, they are electrostatically equivalent. Important differences appear, however, when these electrostatic models are coupled to the nonelectrostatic components of a potential function. In particular, these interactions are the nonelectrostatic repulsion and van der Waals interactions—short-range interactions that are modeled collectively with a variety of functional forms. Point dipole-and EE-based models of molecular systems often use the Lennard-Jones potential. On the other hand, shell-based models frequently use the Buckingham or Born-Mayer potentials, especially when ionic systems are being modeled. 

The first term is the classical interaction between point charges and the second term is a repulsive Born-Mayer potential. 

Here qi is the effective charge of an atom is a dispersion interaction constant and Ay and bij are parameters of the Born - Mayer atom-atom repulsion potential. To calculate the long range Coulomb term in Eq. (1) one generally has to employ the Ewald summation technique. To obviate this inconvenience, the Coulomb term has been multiplied by the screening factor (and the dispersion term has been neglected) ... 

One of the most widely used potential function forms in the MD simulation of glass structures is a special version of the standard Born-Mayer form (Chapter 3) which has been used in many studies of ionic crystals and which attempts to relate the potential to properties of the individual interacting ions. Known as the Born-Mayer-Huggins (BMH) potential, the functional form is as follows ... 

The form that (p p, the harmonic force constant, takes depends on the nature of the interaction between atoms k and k in the crystal. Although the interactions are, in fact, quite complex, the assumption of effective two-body interactions such as a Born-Mayer potential... 

The highly ionic nature of MgO means that quite accurate empirical potentials can be constmcted. The polarizable shell model potential is the most widely used for MgO and also for a wide range of other ionic materials. It is instmctive to discuss the main elements of this potential in order to understand the nature of interactions between the ions. The dominating contribution to the interaction is electrostatic and in the simplest approximation can be represented by associating a point charge (usually the formal charge) with each ion. In addition there is a short-range repulsive term due to the overlap of electron density between the ions (Born-Mayer) and a weakly attractive... 

The Born-Mayer equation emphasizes the fact that Eq. 4.1 is designed only to match the observed phenomenon. It is not a fundamental truth like the Coulomb interaction. 

The eutectic composition of LiF-CaF2 (80.5 to 19.5 mol%) were simulated by MD. The interaction potential used in those simulations was calculated with the PIM which consists of a Born-Mayer pair potential forming together with an ionic polarisation [11]. The simulations were performed on a box containing 239 F , 39 Ca " and 161 Li+ ions, and for temperatures ranging between 770 and OtWC. 

Third-order elastic constants of LaSe at 0 K were calculated using the Born-Mayer potential model. The repulsive interaction was considered up to the second nearest neighbors. The interatomic distance To = 3.030A leads to the values in lO N/m ( lO dyn/cm ) Cin = -21.439, c°i2 = Cii6=-1.860, C123 = cJse = C144 = 0.743. The temperature dependence for is given by = + where are (in lO N-m K ) am =6.837, an2 = 3.601,... 

The intermolecular potentials, on the other hand, are commonly described by electrostatic and two-body interactions in, for example, the Born-Mayer-Huggins form ... 

In addition to these interactions the van der Waals interactions (dispersion forces) between the ions in an ionic molecule or crystal should be considered. This effect has been discussed by M. Born and J. E. Mayer, Z. Physik 75, 1 (1932), and by J. E. Mayer, J. Chem Phys. 1, 270 (1933). Multipole polarization of ions in alkali halcgtmide crystals has been discussed on the basis of a simple quantum-mechanical theory by H. L6vy, thesis, Calif. Inst. Tech., 1938. 

The theory of Born and Mayer has been extended by the work of Landshoff using the methods of quantum mechanics. Taking sodium chloride as an example, Landshoff accepts the assumption that the lattice consists of Na+ and Cl ions and calculates the ionic interaction energy on the basis of the Heitler-London theory using the known distributions of electrons in the Na+ and Cl " ions. In addition to the correction terms of Bom and Mayer, additional interactions related to the superposition of the electron clouds, the attraction between electrons and nuclei and the mutual repulsion of electrons are incorporated. The values obtained by this more exact method, however, differ from the values given in Table CXLVII by only a few kcals, the value for sodium chloride being 183 kcals. 

For the constitution of the ionic lattices also, the Van der Waals attraction has been found to be a very decisive factor. We know the forces at present much better for these ions than for the neutral molecules. Using an interaction of the form (21), Born and Mayer have calculated the lattice energy of all alkali halides for the NaCl-type and simultaneously for the CsCl-type and comparing the stability of the two types they could show quantitatively that the relatively great Van der Waals attraction between the heavy ions Cs, I , Br, Cl cf. Table II.) accounts for the fact that CsCl, CsBr, Csl, and these only, prefer a lattice structure in which the ions of the same kind have smaller distances from each other than in the NaCl-type. The contribution of the Van der Waals forces to the total lattice energy of an ionic lattice is of course a relatively small one, it varies from I per cent, to 5 per cent., but just this little amount is quite sufficient to explain the transition from the NaCl-type to the CsCl-type. 

Crystal Repulsion according to the law BJrn Repulsion according to the law exp (-ar) Van der Waal s interaction Zero-point energy Difference between E calc, from equation 13.1 Band by Born and Mayer... 

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