Potential interatomic

The other feature, which warrants its own section, is the issue of interatomic potentials. 

1 Interatomic potentials. All molecular dynamics simulations and some MC simulations depend on the form of the interaction between pairs of particles (atoms 

Interatomic potentials began with empirical formulations (empirical in the sense that analytical calculations based on them... no computers were being used yet... gave reasonable agreement with experiments). The most famous of these was the Lennard-Jones (1924) potential for noble gas atoms these were essentially van der Waals interactions. Another is the Weber potential for covalent interactions between silicon atoms (Stillinger and Weber 1985) to take into account the directed covalent bonds, interactions between three atoms have to be considered. This potential is well-tested and provides a good description of both the crystalline and 

Most treatments, even when intended for materials scientists, of these competing forms of quantum-mechanical simplification are written in terms accessible only to mathematical physicists. Fortunately, a few translators , following in the tradition of William Hume-Rothery, have explained the essentials of the various approaches in simple terms, notably David Pettifor and Alan Cottrell (e.g., Cottrell 1998), from whom the formulation at the end of the preceding paragraph has been borrowed. 

It may be that in years to come, interatomic potentials can be estimated experimentally by the use of the atomic force microscope (Section 6.2.3). A first step in this direction has been taken by Jarvis et al. (1996), who used a force feedback loop in an AFM to prevent sudden springback when the probing silicon tip approaches the silicon specimen. The authors claim that their method means that force-distance spectroscopy of specific sites is possible - mechanical characterisation of the potentials of specific chemical bonds . 

Bartok-Ptoay, The Gaussian Approximation Potential, Springer Theses, 

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A related approach carries out lattice sums using a suitable interatomic potential, much as has been done for rare gas crystals [82]. One may also obtain the dispersion component to E by estimating the Hamaker constant A by means of the Lifshitz theory (Eq. VI-30), but again using lattice sums [83]. Thus for a FCC crystal the dispersion contributions are... 

A fundamental approach by Steele [8] treats monolayer adsorption in terms of interatomic potential functions, and includes pair and higher order interactions. Young and Crowell [11] and Honig [20] give additional details on the general subject a recent treatment is by Rybolt [21]. 

A more natural way to account for the anisotropy is to treat tire parairreters in an interatomic potential, such as equation (A 1.5.64). as fiurctioirs of the relative orientation of the interacting molecules. Comer [131] was perhaps the first to use such an approach. Pack [132] pointed out that Legendre expansions of the well depth e and equilibrium location of the interaction potential converge more rapidly tirair Legendre expansions of the potential itself... 

Aziz R A 1984 Interatomic potentials for rare-gases pure and mixed interactions Inert Gases Potentials, Dynamics and Energy Transfer in Doped Crystals ed M L Klein (Berlin Springer) oh 2, pp 5-86... 

Infomiation about interatomic potentials comes from scattering experiments as well as from model potentials fitted to the themiodynamic and transport properties of the system. We will confine our discussion mainly to... 

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. 

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. 

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

Jarvis S P, Yamada H, Yamamoto S-l, Tokumoto H and Pethica J B 1996 Direct mechanical measurement of interatomic potentials Nature 384 247... 

The Empirical Conformational Energy Program for Peptides, ECEPP [63, 64], is one of the first empirical interatomic potentials whose derivation is based both on gas-phase and X-ray crystal data [65], It was developed in 1975 and updated in 1983 and 1992. The actual distribution (dated May, 2000) can be downloaded without charge for academic use. 

Gale J D, C R A Catlow and W C Mackrodt 1992. Periodic Ab Initio Determination of Interatomic Potentials for Alumina. Modelling and Simulation in Materials Science and Engineering 1 73-81. 

Fig. 5.37 Comparison of the calculated phonon dispersion curve for Al with the experimental values measured using neutron diffraction. (Figure redrawn from Michin Y, D Farkas, M ] Mehl and D A Papaconstantopoulos 1999. Interatomic Potentials for Monomatomic Metals from Experimental Data and ab initio Calculations. Physical Review 359 3393-3407.)...

The primary requirement for carrying out MD simulations is a suitable interatomic potential for the description of forces acting between atoms in the cascade. A general discussion on MD can be found (47) and detailed summaries of the use of MD in ion—soHd interactions are also available (43,48). 

More refined estimates of cr are possible, using real interatomic potentials (Chapter 4) they give about E/15 instead of E/8. 

T. Mohri, Interatomic Potential and Phase Stability, Springer Series in Solid-State Sciences 114, ed. by K. Terakura and H. Akai, Springer-Verlag, (1993), 168-177. 

The rest of this paper will be organized as follows In the next section we describe a set of interatomic potentials which we have developed to model iron-nickel alloys and report the calculational techniques of molecular dynamics simulations which we... 

We have used the embedded-atom method (EAM) to develop interatomic potentials suitable for the simulation of iron-nickel alloys. In this method the total energy of the alloys can be written as... 

Table 1. Parameters of the interatomic potentials. Distances are given in as, densities in flg, charges in e and energies in Ry. ri4s and Vc have been set to 0.57 and 8.33 ag for iron. The corresponding values for nickel are 0.85 and 8.78 ag ao denotes the equilibrium lattice constant of the elements po is the electron density at equilibrium for the perfect lattices, i.e. 0.002776 ag and 0.003543 ag for iron and nickel respectively.

In view of the complications of the intermolecular potential (as compared to the interatomic potential of the rare gas atoms) the comparisons for molecules in Tables II, III, and IV should be judged with caution. The apparent discrepancies from the theories for single atoms can be misleading. An example is the calculation for CH4 on the Slater-Kirkwood theory where Table IV shows the absurd value of 24 for the effective number of electrons. Pitzer and Catalano32 have applied the Slater-Kirkwood equation to the intermolecular potential of CH4 by addition of all the individual atom interactions and, with N = 4 for carbon and 1 for hydrogen, obtained agreement within 5 per cent for the London energy at the potential minimum. 

The dynamics of any molecular system could in principle be calculated by considering all the acting intermolecular and intramolecular forces by means of interatomic potentials as a function of distance. Such precise modelling is, obviously, beyond our reach at present and different levels of approximation were imagined to mimic the behavior of a real macromolecule while restricting the number of degrees of freedom to within computational limits. One successful... 

TABLE 15. Parameters A, B, C for the intermolecular interatomic potentials 1 (energies in kcal mol 1) and van der Waals radii used in surface and volume calculations (A)23,24... 

Tersoff, J., Modeling Solid-State Chemistry Interatomic Potentials for Multicomponent Systems," Phys. Rev. B, Vol. 39, No. 8,1989, pp. 5566-5568. 

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