Two-body potential functions

Although the various types of bonds may differ very greatly from each other they are very similar in one respect. The bond reacts to an increase in distance with a moderate restoring force, while the repulsive forces, that come into effect when the distance is reduced and which act over a small distance, react very sensitively to reduction of the equilibrium position. Hence, the potential function is of the form shown for the bonding energy eigenvalue in Fig. 2.1, and can be approximated, for example, by a superimposition of an r and an r term with n m (Mie potential 

In the case of an ionic bond the first term is rather precise with m=l. Concerning the second term an exp-( const, r) law has a more fundamental basis as repulsion term (see Fig. 2.3). However, since the exponential term is also not strictly valid and the various models become equivalent for small displacements (cf. Morse potential 

The condition for the equilibrium distance (f), viz. de/dr = 0, yields as a correlation between the bonding parameters A and B 

2 Bonding aspects Prom atoms to solid state 

This makes it possible to express the potential in terms of the equilibrium parameters  

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The quantitative comparisons with the available experimental data are less favorable in this case. The transition state and product in water appear to be shifted up in energy by about 15 kcal/mol. The computed curves are more in line with experimental data for ketones, where formation of hydrates is far less favorable than for formaldehyde. The discrepancy likely comes from an overly exothermic hydration energy for the charge-localized hydroxide ion, which lowers the reactant end of the pmfs. This results from the use of two-body potential functions, that is, the TIP4P water molecules are not polarized by the ion, so water-water repulsions between molecules in the first solvent shell are underestimated. Until the polarization can be explicitly treated, ions that have attractive interactions with single water molecules greater than about 18-20 kcal/mol should probably be avoided. For example, CN would be less problematic since its single molecule hydration enthalpy is only 14 kcal/mol, versus 25 for... 

However, despite of the great importance of quantum mechanical potentials from the purely theoretical point of view, simple effective two-body potential functions for water seem at present to be preferable for the extensive simulations of complex aqueous systems of geochemical interest. A very promising and powerful method of Car-Parrinello ah initio molecular dynamics, which completely eliminates the need for a potential interaction model in MD simulations (e.g., Fois et al. 1994 Tukerman et al. 1995, 1997) still remains computationally extremely demanding and limited to relatively small systems N < 100 and a total simulation time of a few picoseconds), which also presently limits its application for complex geochemical fluids. On the other hand, it may soon become a method of choice, if the current exponential growth of supercomputing power will continue in the near future. 

Explanation of the stability of the cubic crystals on the basis of a two-body potential function was thus unsuccessful. As a next step three-body interactions were considered by several authors. Axilrod summed the triple-dipole interaction, a nonadditive part of the van der Waals attraction, in both the cubic and hexagonal close packings. He concluded The difference in the triple-dipole interaction for the cubic and hexagonal close-packed lattices, although favoring the former structure, is less... 

In his early survey of computer experiments in materials science , Beeler (1970), in the book chapter already cited, divides such experiments into four categories. One is the Monte Carlo approach. The second is the dynamic approach (today usually named molecular dynamics), in which a finite system of N particles (usually atoms) is treated by setting up 3A equations of motion which are coupled through an assumed two-body potential, and the set of 3A differential equations is then solved numerically on a computer to give the space trajectories and velocities of all particles as function of successive time steps. The third is what Beeler called the variational approach, used to establish equilibrium configurations of atoms in (for instance) a crystal dislocation and also to establish what happens to the atoms when the defect moves each atom is moved in turn, one at a time, in a self-consistent iterative process, until the total energy of the system is minimised. The fourth category of computer experiment is what Beeler called a pattern development... 

After this computer experiment, a great number of papers followed. Some of them attempted to simulate with the ab-initio data the properties of the ion in solution at room temperature [76,77], others [78] attempted to determine, via Monte Carlo simulations, the free energy, enthalpy and entropy for the reaction (24). The discrepancy between experimental and simulated data was rationalized in terms of the inadequacy of a two-body potential to represent correctly the n-body system. In addition, the radial distribution function for the Li+(H20)6 cluster showed [78] only one maximum, pointing out that the six water molecules are in the first hydration shell of the ion. The Monte Carlo simulation [77] for the system Li+(H20)2oo predicted five water molecules in the first hydration shell. A subsequent MD simulation [79] of a system composed of one Li+ ion and 343 water molecules at T=298 K, with periodic boundary conditions, yielded... 

Configuration energies are determined by optimizing different structures with respect to atomic positions using the above mentioned functional expressions. In Fig.5 we compare the two different potential functional forms with the MP4 ah initio results. Solid lines are for the sum of the three-body... 

One source of information on intermolecular potentials is gas phase virial coefficient and viscosity data. The usual procedure is to postulate some two-body potential involving 2 or 3 parameters and then to determine these parameters by fitting the experimental data. Unfortunately, this data for carbon monoxide and nitrogen can be adequately represented by spherically symmetric potentials such as the Lennard-Jones (6-12) potential.48 That is, this data is not very sensitive to the orientational-dependent forces between two carbon monoxide or nitrogen molecules. These forces actually exist, however, and are responsible for the behavior of the correlation functions and - In the gas phase, where orientational forces are relatively unimportant, these functions resemble those in Figure 6. On the other hand, in the liquid these functions behave quite differently and resemble those in Figures 7 and 8. 

The most straightforward way refers to the Bethe-Salpeter equation, i.e. an equation for the two-body Green function. It may be solved for the Coulomb potential and a two-body perturbative theory can be developed starting from this solution. This method was rarely used in the bound state QED calculations, being very complicated. 

Simulations of the liquid water properties have been the subject of many papers, see Ref. (374) for a review. Recently a two-body potential for the water dimer was computed by SAPT(DFT)375. Its accuracy was checked375 by comparison with the experimental second virial coefficients at various temperatures. As shown on Figure 1-16, the agreement between the theory and experiment is excellent. Given an accurate pair potential, and three-body terms computed by SAPT376, simulations of the radial 0-0, 0-H, and H-H distribution functions could be... 

Actually, computational convenience has almost always suggested using pairwise additive potentials for simulations of condensed phases also, though strictly two-body potentials are only acceptable for rarefied gases. The computational convenience of two-body potentials is maintained, however, if non-additive effects are included implicitly, i. e. with the so called two-body effective potentials. All empirical or semi empirical functions whose parameters have been optimized with respect to properties of the system in condensed phase belong to this class. As already observed, this makes these potentials state-dependent, with unpredictable performance under different thermodynamic conditions. 

Fig. 1.2 The Lennard-Jones 6-12 two-body potential [potential energy V(r)] and force F(r) for two interacting atoms ( point-like molecules) as a function of separation distance r. Reproduced with permission from [7] copyright Springer Verlag...

This so-called embedded atom potential consists of two terms. The first term is a two-body potential that represents the repulsion between the ion and the rest of the ions in the system. The second term is a many-body function that represents the energy to embed an atom in the position i, where there is an electron density p that comes from the linear superposition of spherically averaged atomic electron densities. 

Figure 24. (a) Same as Fig. 23a except the strength of the N-W(l 10) two-body potential parameter is varied (b) same as (a) except as a function of the two-body range parameter (c) same as (a) except as a function of the two-body interaction minimum. The figure is reprinted with permission from Kara and DePristo (1988a). 

Attempts to represent the three-body interactions for water in terms of an analytic function fitted to ab initio results date back to the work of dementi and Corongiu [191] and Niesar et al. [67]. These authors used about 200 three-body energies computed at the Hartree-Fock level and fitted them to parametrize a simple polarization model in which induced dipoles were generated on each molecule by the electrostatic field of other molecules. Thus, the induction effects were distorted in order to describe the exchange effects. The three-body potentials obtained in this way and their many-body polarization extensions have been used in simulations of liquid water. We know now that the two-body potentials used in that work were insufficiently accurate for a meaningful evaluation of the role of three-body effects. 

The thermodynamic properties of a fluid can be calculated from the two-, three- and higher-order correlation functions. Fortunately, only the two-body correlation functions are required for systems with pairwise additive potentials, which means that for such systems we need only a theory at the level of the two-particle correlations. The average value of the total energy... 

For simplicity we discuss a classical fluid system of N equal particles of mass m contained in a box of volume V and interacting by a two-body potential that is velocity independent (e.g., a 6-12 Lennard-Jones potential). The system is in equilibrium with a reservoir at temperature T [i.e., an (NVT) ensemble]. A configuration of the N particles is defined by their Cartesian coordinates and is denoted by the 3N vector x the ensemble of these vectors defines the configurational space ft of volume V. The momenta of the particles are denoted by the 3N vector p and the corresponding space by ft. Because the forces do not depend on the velocities, the contributions of the kinetic energy, pyim, and the interaction energy, (x), to the canonical partition function Q are separated,... 

The two-body potential V2 depends only on the distance between neighboring atoms and contains an exponentially-decaying function that terminates the potentid at a cutoff distance rg... 

In writing the final form for qi, the integrations over the angular variables (0 have been carried out. The function h l,2, E) is referred to as a two-body correlation function and carries the full effects of the two-body potential tpiij). The polarization is now given by the following expression ... 

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