Potentials effective pair

A notable example of a potential that does include many-body terms is the Barker-Fisher-Watts potential for argon, which combines a pairwise potential with an Axilrod-Teller triple 

Fig 4 38. Comparison of the Lennard-]ones potential for argon with the Barker-Fisher-Watts pair potential is Boltzmann s constant 

This potential function contains eleven constants a, Ag... A5, Q, Cg, Qo and S. The function is expressed in terms of r which is given by r = r/r, where is the separation at the minimum in the potential. The true interaction energy as a function of the separation, r, is then obtained by multiplying v r ) by the depth of the potential well, e  

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Sprik M 1993 Effective pair potentials and beyond Computer Simulation in Chemical Physics vol 397 NATO ASI Series C ed M P Allen and D J Tildesley (Dordrecht Kluwer) pp 211-59... 

Figure 4.29 from Computer Simulation in Chemical Physics, edited by Allen M P and D J Tildesley, 1993. Effective Pair Potentials and Beyond, Sprik M, with kind permission from Kluwer Academic Publishers. 

Figure 2.3 Site-site interaetion for water 2.6.2 The Effective Pair Potential...

Notice that Rahman and Stiliinger 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... 

In the case of fluids which consist of simple non-polar particles, such as liquid argon, it is widely believed that Ui is nearly pairwise additive. In other words, the functions for n > 2 are small. Water fails to conform to this simplification, and if we truncate the series after the term, then we have to understand that the potential involved is an effective pair potential which takes into account the higher order-terms. 

For each atom type there are two parameters to be determined, the van der Waals radius and the atom softness, Rq and, It should be noted that since the van der Waals energy is calculated between pairs of atoms, but parameterized against experimental data, the derived parameters represent an effective pair potential, which at least partly includes many-body contributions. 

Table 1 Effective Pair Potentials in mRyd/atom for various distances between the pairs for a 50-50 PdV alloy. KKR-CPA-GPM values are taken from ref(10).

Berendsen HJC, Grigera JR, Straatsma TP (1987) The missing term in effective pair potentials. J Phys Chem 91(24) 6269-6271... 

Thirumalai, D. Hall, R. W. Berne, B. J., A path integral Monte Carlo study of liquid neon and the quantum effective pair potential, J. Chem. Phys. 1984, 81, 2523-2527... 

High-temperature Effective Pair Potentials for the System N2-N and 02-0. 

An interesting combined use of discrete molecular and continuum techniques was demonstrated by Floris et al.181,182 They used the PCM to develop effective pair potentials and then applied these to molecular dynamics simulations of metal ion hydration. Another approach to such systems is to do an ab initio cluster calculation for the first hydration shell, which would typically involve four to eight water molecules, and then to depict the remainder of the solvent as a continuum. This was done by Sanchez Marcos et al. for a group of five cations 183 the continuum model was that developed by Rivail, Rinaldi et al.14,108-112 (Section III.2.ii). Their results are compared in Table 14 with those of Floris et al.,139 who used a similar procedure but PCM-based. In... 

The pairwise nature of the bond-boost makes this task easier since such traps would show up as a non-convexity of some of the biased effective pair potentials, which in the canonical ensemble can be taken to be the pairwise potential of mean force (PMF, denoted as V). Thus, assuming that V is approximately quadratic for lei safety condition can be enforced by setting Sa[Pg.92]

Section 4 presents a variety of solid-gas surface processes adsorption, desorption, catalytic reaction, and surface diffusion. Non-ideal behavior of the systems is considered through the effective pair potentials of inter-molecular interactions. A wide circle of experimental data can be described on taking into account a non-ideal behavior of the surrounding medium. 

The form of the potential for the system under study was discussed in many publications [28,202,207,208]. Effective pair potentials are widely used in theoretical estimates and numerical calculations. When a many-particle interatomic potential is taken into account, the quantitative description of experimental data improves. For example, the consideration of three-body interactions along with two-particle interactions made it possible to quantitatively describe the stratification curve for interstitial hydrogen in palladium [209]. Let us describe the pair interaction of all the components (hydrogen and metal atoms in the a. and (j phases) by the Lennard Jones potential cpy(ry) = 4 zi [(ff )12- / )6], where Sy and ai are the parameters of the corresponding potentials. All the distances ry, are considered within c.s. of radius r (1 < r < R), where R is the largest radius of the radii of interaction Ry between atoms / and /). 

The correlation functions play an essential role in the static description of homogenous classical liquids whose particles are taken to interact through an effective pair potential. The starting point of the liquid-state theory, in terms of correlation functions, is the well-known Ornstein-Zernike equation [25]. The total correlation function h r) defined in Section II is actually a sum of two contributions that is illustrated by the following relationship... 

This section reports the results obtained from two self-consistent schemes that have been extended to deal with the triple-dipole interaction, which reduces to a state-dependent effective pair potential. The pair potential of Aziz and Slaman [106] (AS) is combined with the Axilrod and Teller [108] (AT) triple-dipole potential to describe the interactions in Xe and Kr fluids. Note that other investigations have been performed with ab initio potentials [113, 114]. 

Contrary to numerical simulation, taking the three-body contribution into account in IETs requires that we define an effective pair potential. Making use of the two- and three-body contributions allows one to write the state-dependent effective potential weff(V) under the standard form [10, 112, 117]. 

The integral equation theory consists in obtaining the pair correlation function g(r) by solving the set of equations formed by (1) the Omstein-Zernike equation (OZ) (21) and (2) a closure relation [76, 80] that involves the effective pair potential weff(r). Once the pair correlation function is obtained, some thermodynamic properties then may be calculated. When the three-body forces are explicitly taken into account, the excess internal energy and the virial pressure, previously defined by Eqs. (4) and (5) have to be, extended respectively [112, 119] so that... 

Since experiments for Kr have been performed at small angle neutron scattering for some low density states, we present the results of the Fourier transform of the direct correlation function, c(q) = (S(q) — 1 )/p.S (7/), rather than those of the structure factor S(q). Figure 20 shows the curves of c q). As it can be seen, the theoretical results, obtained by HMSA+WCA and MD with the AS plus AT potentials, are in excellent agreement with the experimental data [12]. While the AT contribution is included by means of an effective pair potential in the SCIET, it is used under its original form owing to Eqs.(l 19) and... 

As discussed earlier (see Secs. II and VI), for polystyrene spheres in water the DLVO pair potential provides an expression for the effective interparticle interaction that, with an appropriate renormalization of the charge, accounts for the main features of the structure of 3D homogeneous suspensions. One might think that the DLVO potential should be a good assumption under most circumstances. This, however, turns out to be the case at least for the systems being considered here. Then the question is, how to measure the effective pair potential One way to do it is described here in some detail. For sufficiently dilute suspensions, one can resort to the low concentration approximation to obtain the pair potential directly from the measured radial distribution functions, i.e.,... 

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