Born model potentials

The work of Sanders et al. (1984), Jackson and Catlow (1988) and Tomlinson et al. (1990) showed that with Born model potentials it is possible accurately to reproduce the crystal structures of silicates including zeolites. A typical example is shown in Plate I which illustrates the calculated and experimental structures of a purely siliceous (i.e. pure Si02 polymorph) zeolite, silicalite. (A closely related, isostructural material, ZSM-5, which contains a low concentration of Al is an effective isomerization and hydrocarbon synthesis catalyst.) The agreement between theory and experiment is evidently good more discussion follows in Chapter 9. 

Solids and their surfaces were modeled by the Born model potential as discussed in the following paragraph. Adsorbed molecules were modeled by the molecular mechanics approach in which energy for a covalently bonded molecule is dependant mainly on bond stretching term (E ) along other contributing terms. 

B2.2.5.8 LIST OF BORN CROSS SECTIONS FOR MODEL POTENTIALS... 

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

It should also be mentioned that a theoretical model using an empirical LEPS potential energy surface has successfully been used to reproduce the vibrational population distribution of the products of this surface reaction.40 This approach confines itself to the assumptions of the Born-Oppenheimer approximation and underscores one of the major questions remaining in this field do we just need better Born Oppenheimer potential surfaces or do we need a different theoretical approach ... 

From the point of view of associative desorption, this reaction is an early barrier reaction. That is, the transition state resembles the reactants.46 Early barrier reactions are well known to channel large amounts of the reaction exoergicity into product vibration. For example, the famous chemical-laser reaction, F + H2 — HF(u) + H, is such a reaction producing a highly inverted HF vibrational distribution.47-50 Luntz and co-workers carried out classical trajectory calculation on the Born-Oppenheimer potential energy surface of Fig. 3(c) and found indeed that the properties of this early barrier reaction do include an inverted N2 vibrational distribution that peaks near v = 6 and extends to v = 11 (see Fig. 3(a)). In marked contrast to these theoretical predictions, the experimentally observed N2 vibrational distribution shown in Fig. 3(d) is skewed towards low values of v. The authors of Ref. 44 also employed the electronic friction theory of Tully and Head-Gordon35 in an attempt to model electronically nonadiabatic influences to the reaction. The results of these calculations are shown in... 

Nevertheless, very-long-lived quasi-stationary-state solutions of Schrodinger s equation can be found for each of the chemical structures shown in (5.6a)-(5.6d). These are virtually stationary on the time scale of chemical experiments, and are therefore in better correspondence with laboratory samples than are the true stationary eigenstates of H.21 Each quasi-stationary solution corresponds (to an excellent approximation) to a distinct minimum on the Born-Oppenheimer potential-energy surface. In turn, each quasi-stationary solution can be used to construct an alternative model unperturbed Hamiltonian //(0) and perturbative interaction L("U),... 

The Born model of solvation overestimates solvation free energies but indicates the general trends correctly. Potential inversion, as observed in many other systems containing two identical oxidizable or reducible groups separated by an unsaturated bridge (Scheme 1.4), can be rationalized in the same manner. 

A common alternative is to synthesize approximate state functions by linear combination of algebraic forms that resemble hydrogenic wave functions. Another strategy is to solve one-particle problems on assuming model potentials parametrically related to molecular size. This approach, known as free-electron simulation, is widely used in solid-state and semiconductor physics. It is the quantum-mechanical extension of the classic (1900) Drude model that pictures a metal as a regular array of cations, immersed in a sea of electrons. Another way to deal with problems of chemical interaction is to describe them as quantum effects, presumably too subtle for the ininitiated to ponder. Two prime examples are, the so-called dispersion interaction that explains van der Waals attraction, and Born repulsion, assumed to occur in ionic crystals. Most chemists are in fact sufficiently intimidated by such claims to consider the problem solved, although not understood. 

The lattice energy based on the Born model of a crystal is still frequently used in simulations [14]. Applications include defect formation and migration in ionic solids [44,45],phase transitions [46,47] and, in particular, crystal structure prediction whether in a systematic way [38] or from a SA or GA approach [ 1,48]. For modelling closest-packed ionic structures with interatomic force fields, typically only the total lattice energy (per unit cell) created by the two body potential,... 

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. 

By changing the reference potential in a series of redox monitors, it is then possible to determine the dependence of the cluster potential on the nuclearity. The general trend of increasing redox potential with nuclearity is the same for all metals in solution as it is illustrated in Fig. 2 in the case of E°(AgVAg,) q. However, in gas phase, the variation of the ionization potential IV(Ag ) exhibits the opposite trend versus the nuclearity n. Indeed, since the Fermi potential of the normal hydrogen electrode (NHE) in water is 4.5 eV, and since the solvation free energy of Ag decreases with size as deduced from the Born model, one can explain the two opposite variations with size of F°(Ag /AgJ q and IP (AgJ as illustrated in Fig. 2. 

Figure 14. Model ground-state Born-Oppenheimer potential energy surface. 

There have been previous model studies of these systems [61]. These studies, while including the effects of environment, did not address the question of the effect of a promoting vibration. These reactions are inherently electronically nonadia-batic, while the formulation we have thus far presented included evolution only on a single Born-Oppenheimer potential energy surface. We have developed a model system to allow the extension of the Quantum Kramers methodology to such systems, and we now describe that model. 

Dolan et al. [ 176] and Rinzler et al. [ 142] presented a semi-empirical approach based on a single configuration coordinate model (see Sect. 3.1) for separating the effects of local compressibihty and crystal field strength on pressure-induced changes in the T2 energy of Cr + in several fluoride elpasohte systems. Their approach is based on an empirical form, motivated by the Born-Mayer potential, for the local compressibility k(P) ... 

The shell model has its origin in the Born theory of lattice dynamics, used in studies of the phonon dispersion curves in crystals.70,71 Although the Born theory includes the effects of polarization at each lattice site, it does not account for the short-range interactions between sites and, most importantly, neglects the effects of this interaction potential on the polarization behavior. The shell model, however, incorporates these short-range interactions.72,73 The earliest applications of the shell model, as with the Born model, were to analytical studies of phonon dispersion relations in solids.74 These early applications have been well reviewed elsewhere.71,75-77 In general, lattice dynamics applications of the shell model do not attempt to account for the dynamics of the nuclei and typically use analytical techniques to describe the statistical mechanics of the shells. Although the shell model continues to be used in this fashion,78 lattice dynamics applications are beyond the scope of this chapter. In recent decades, the shell model has come into widespread use as a model Hamiltonian for use in molecular dynamics simulations it is these applications of the shell model that are of interest to us here. 

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... 

Born-Model Calculations.- A very much more complex model is that due to Catlow and co-workers who have developed a methodology whereby structure prediction takes place based upon Born s model for ionic solids. The interactions which are considered, are, for the most part, non-bonded interactions, and thus can be considered as a potential model and not a force constant model. 

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]. 

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