Stillinger potential

Two cell models of water have been reported. Weissman and Blum 63> considered the motion of a water molecule in a cell generated by an expanded but perfect ice lattice. Weres and Rice 64> developed a much more detailed model, based on a more sophisticated description of the cell and a good, nonparametric water-water interaction namely the Ben-Naim-Stillinger potential 60>. The major features of the WR model are the following ... 

The function/c is a smoothing function with the value 1 up to some distance Yy (typically chosen to include just the first neighbour shell) and then smoothly tapers to zero at the cutoff distance, by is the bond-order term, which incorporates an angular term dependent upon the bond angle 6yk- The Tersoff pofenfial is more broadly applicable than the Stillinger-Weber potential, but does contain more parameters. 

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... 

Thermodynamic properties. A molecular-dynamics simulation method (using a steepest decent method) with Stillinger-Weber potential is employed to optimize structures and to obtain the cohesive en-... 

A full-scale treatment of crystal growth, however, requires methods adapted for larger scales on top of these quantum-mechanical methods, such as effective potential methods like the embedded atom method (EAM) [11] or Stillinger-Weber potentials [10] with three-body forces necessary. The potentials are obtained from quantum mechanical calculations and then used in Monte Carlo or molecular dynamics methods, to be discussed below. 

Different results were obtained in the case of the Stillinger-Weber (SW) potential for silicon (10). In this case, material deposited on the (ill) orientation was disordered, without distinct layers of atoms. Growth on the (100) face did produce about ten distinct layers at sufficiently high temperatures, although it is not clear whether thicker deposits would retain this order since some degeneration was observed as successive layers were deposited. 

One of the most important developments in the theory of water is the invention of a realistic effective intermolecular pair potential. Although others had made similar suggestions earlier 59>, it is Stillinger and co-workers 60> who have shown, with the aid of molecular dynamics calculations 3>, how satisfactory such a potential can be. The availability of a water-water potential drastically reduces the scope for parameterization in model based theories, and permits investigation of such concepts as "broken hydrogen bond. We shall frequently have occasion to call upon its properties in the following discussion. 

In a similar study, Khor and Das Sarma studied the diffusion of Si, Si2 and Si3 on the (001), (011), and (111) surfaces of silicon. In their study, the forces on all of the atoms were determined by the Stillinger-Weber potential. For single atoms on all surfaces, they report an upper bound for diffusion of 4.8 X 10 cm /sec at 1600K. This value is significantly less than both the range determined by NoorBatcha of 2.031 x 10" to 15.8 x 10" cm /sec ", and the experimental estimate of 10 cm /sec. They also report that diatomic Si2 molecules diffuse more readily than single atoms on the Si(l 11) surface. 

The potential model for water used in our work is taken from the work of Stillinger and David (8) (SD) as modified by Halley and co-workers (9) (HRR). Models for water that are capable of hetero-lytic dissociation into ions OH and H+ could take two different approaches. Probably the best approach, promising but not yet... 

I admire very much the X-ray diffraction pattern and the calculated fit you have shown, which are much better than the best agreements of Stillinger and Rahman (J. Chem. Phys. 60, 1545, 1974). I wonder what was the potential you were using, how was it determined, and what structure of liquid water was assumed ... 

Clementi, J. Chem. Phys., 64, 1351 (1976) the full details of the computed X-ray diffraction intensity are available in G. C. Lie, M. Yoshimine, and E. Clementi, J. Chem. Phys., 64, 2314 (1976). The previous computations by Stillinger and Rahman did not use a quantum-mechanically derived potential, but an empirical potential. Present quantum-mechanical techniques, if properly used, can yield remarkably accurate potentials. This fact is not fully appreciated by a large number of chemists, possibly discouraged by the rather large amount of poor theoretical chemistry computations currently in the literature. It is notable that the repulsive part of a potential can be inferred from experiments, in general, with poor accuracy. 

Notice that Rahman and Stillinger make use of the concept of an effective pair potential. The potential energy of any substance may always be resolved systematically into pair, triplet, quadruplet,. .. contributions. If we consider N particles each with position vector R, then formally we can write... 

For atoms with sp3 hybridization (e.g., Si in Si02), the parameter cosdg = —1/3 corresponds to the angle in the tetrahedron. The three-body term with an angular dependence is also used in the Stillinger-Weber potential [63] for systems with the diamond lattice... 

The characteristic time of this diffusion was estimated by carrying out the molecular dynamic relaxation of the film surface within the limits of the above model at 500°C. In MD calculations, the pair interaction energy between atoms is approximated by the Buckingham pair potential (Zr O, O-O) (see Table 9.4). To describe covalent bonds more correctly, a three-body O-Zr-O term in the Stillinger-Weber form was introduced in addition to the Coulomb term. 

Potential functions based on the sp3-hybrid tetrahedral structure model of water molecules in which two positive and two negative charges are placed at the tetrahedral positions in the molecule have been proposed by Bjerrum 1), Ben-Naim and Stillinger (BNS model) (2), and Stillinger-Rahman (ST2) (3). 

Stillinger FH, Rahman A (1978) Revised central force potentials for water. J Chem Phys 68(2) 666... 

A simple model, which has been quite successful in solids with the diamond or zinc-blende stmcrnre, was introduced by Stillinger and Weber (Stillinger and Weber, 1985). The first term in the potential is the product of a Lennard-Jones-like pair-wise interaction and a cut-off function smoothly terminating the potential at some distance r. The second term is a multi-variable three-body potential written as a separable product of two radial functions and an angular function ... 

As can be seen, there are eight parameters in the Stillinger-Weber potential A,B,p, q, /r, Tc, Z, and a. These parameters are fitted to experimental data, such as lattice constants and cohesive energy, for the diamond stmcture. The angle is the angle centered on atom i. If djik is 1010.47°, cos 6jii = the angular function has a minimum of zero, which... 

Eor simplicity, however, let s adopt an atomic-level description of the molecular solution. As with the central-force models (Lemberg and Stillinger, 1975), the interactions treated could be sufficiently complicated as to form complex molecules, but we will focus on the interaction contributions to the chemical potentials of atoms. In that case, all = 1, and any quantum mechanical effects on the internal structures of molecules that may be formed will have to be described by the quantum mechanical approximations that are the target here. 

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