Interatomic potential energy function

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

RIS theory is used to study the unperturbed dimensions of PMPS chains as a function of their stereochemical structure. The required conformational energies are obtained from semi-empirical, interatomic potential energy functions and from known results on PDMS. 

Starting point is QM calculation within the framework of density-functional theory (DFT) (Hohenberg and Kohn, 1964 Kohn and Sham, 1965 Payne et al., 1992). DFT-based energy calculations can be used to evaluate the parameters of classical interatomic interaction potentials, which can be used to perform MS, MC, and MD simulations such ab initio potential parametrization is a key to improving the transferability of the classical force field. In Fig. 1, an interatomic potential energy function for Si-H interactions is given as an example of such a parametrization (Ohira et al., 1995). 

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. 

The substantial progress made in recent years by computer modeling may be ascribed to two reasons. Firstly, the great increase in computer power has made the necessary number crunching a lot easier more calculations, on larger systems can be completed in shorter periods of time. Secondly, the heart of the model, the interatomic potential energy functions, from which the interatomic forces are calculated, have improved considerably. One can now make detailed quantitative comparisons with experimental measurements, instead of the somewhat qualitative comparisons of a few years ago. The increased confidence in the potentials has... 

Figure 4-15 A van der Waals Potential Energy Function. The Energy minimum is shallow and the interatomic repulsion energy is steep near the van der Waals radius.

The concepts. All interatomic interactions are modeled with a set of mathematical functions which, when summed over all interactions, gives the potential energy of a molecule. The potential energy functions, the PEFs, contain adjustable parameters which, for a start, are taken from similar work or are merely guessed. 

The Molecular Origins of Elasticity. Recall from Section 1.0.4 that atoms are held together by interatomic bonds and that there are eqnations such as Eq. (1.13) that relate the interatomic force, F, to the potential energy function between the atoms, U, and the separation distance, r ... 

Solution According to Eq. (5.5), the elastic modulus, E, is proportional to the stiffness of the theoretical springs that model the bonds between atoms in the solid, Sq. According to Equation (5.1), So is in turn proportional to the second derivative of the potential energy function, U, with respect to interatomic separation distance, r. 

The atomistic cause of thermal expansion is often explained by the attractive and repulsive forces between atoms in a solid. The potential energy functions (force applied through a distance) for interatomic attraction, repulsion, and their sum are plotted in Figure 7.2. The base of the trough in the com-... 

The interatomic distances are primarily determined by the position of the minimum in the potential energy function describing the interactions between the atoms in the crystal. The question is then, what are the sizes of the atoms and ions The extension of electron density for an atom or an ion is not rigorously defined no exact size can be... 

Lennard-Jones potential As two atoms approach one another there is the attraction due to London dispersion forces and eventually a van der Waals repulsion as the interatomic distance r gets smaller than the equilibrium distance. A well-known potential energy function to describe this behavior is the Lennard-Jones (6-12) potential (LJ). The LJ (6-12) potential represents the attractive part as r-6-dependent whereas the repulsive part is represented by an r n term. Another often used nonbonded interaction potential is the Buckingham potential which uses a similar distance dependence for the attractive part as the LJ (6-12) potential but where the repulsive part is represented by an exponential function. 

Behavior remarkably similar to that revealed by the one-dimensional model crystals is generally observed for lattice vibrations in three dimensions. Here the dynamical matrix is constructed fundamentally in the same way, based on the model used for the interatomic forces, or derivatives of the crystal s potential energy function, and the equivalent of Eq. (7) is solved for the eigenvalues and eigenvectors [2-4, 29]. Naturally, the phonon wavevector in three dimensions is a vector with three components, q = (qx, qy, qz)> and both the fiequency of the wave, co(q), and its polarization, e q), are functions... 

Before proceeding, it is necessary to specify more clearly what kind of anharmonic behavior will be dealt with in the framework of rovibrational modes. Consequently, a brief but important digression concerning the conventional treatment of interatomic potential functions in molecular studies is presented. We are interested in having at our disposal a potential energy function (either in analytical form or expressed as a series in powers of proper analytical terms) for reproducing the sequence... 

Analytic potential energy functions are mathematical expressions that give the potential energy of a system of atoms as a function of relative atomic positions. Interatomic forces, which govern the motion of atoms in a typical dynamic simulation, are obtained by taking derivatives of the potential energy function with respect to nuclear coordinates. 

This chapter discusses analytic potential energy functions that have been developed for materials simulation. The emphasis is not on an exhaustive literature survey of interatomic potentials and their application rather, we provide an overview of how some of the more successful analytic functions summarized in Table 1 are related to quantum mechanical bonding. Concepts are emphasized over mathematical rigor, and equations are used primarily to illustrate derivations or to show relationships between different approaches. Atomic units are used for simplicity where appropriate. The discussion is restricted to metallic and covalent bonding. 

Pair-additive interactions continued to be used in most materials-related simulations for over 20 years after Vineyard s work despite well-known deficiencies in their ability to model surface and bulk properties of most materials. Quantitative simulation of materials properties was therefore very limited. A breakthrough in materials-related atomistic simulation occurred in the 1980s, however, with the development of several many-body analytic potential energy functions that allow accurate quantitative predictions of structures and dynamics of materials.These methods demonstrated that even relatively simple analytic interatomic potential functions can capture many of the details of chemical bonding, provided the functional form is carefully derived from sound physical principles. 

In Chapter 4, Professor Donald W. Brenner and his co-workers Olga A. Shenderova and Denis A. Areshkin explore density functional theory and quantum-based analytic interatomic forces as they pertain to simulations of materials. The study of interfaces, fracture, point defects, and the new area of nanotechnology can be aided by atomistic simulations. Atom-level simulations require the use of an appropriate force field model because quantum mechanical calculations, although useful, are too compute-intensive for handling large systems or long simulation times. For these cases, analytic potential energy functions can be used to provide detailed information. Use of reliable quantum mechanical models to derive the functions is explained in this chapter. 

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