Pair-potential

The third virial coefficient C(7) depends upon tliree-body interactions, both additive and non-additive. The relationship is well understood [106. 107. 111]. If the pair potential is known precisely, then C(7) ought to serve as a good probe of the non-additive, tliree-body interaction energy. The importance of the non-additive contribution has been confimied by C(7) measurements. Unfortunately, large experimental uncertainties in C (7) have precluded unequivocal tests of details of the non-additive, tliree-body interaction. 

The viscosity, themial conductivity and diffusion coefficient of a monatomic gas at low pressure depend only on the pair potential but through a more involved sequence of integrations than the second virial coefficient. The transport properties can be expressed in temis of collision integrals defined [111] by... 

Price S L and Stone A J 1980 Evaluation of anisotropic model intermolecular pair potentials using an ab initio SCF-CI surface Moi. Phys. 40 805... 

For systems in which the constituent particles interact via short-range pair potentials, W = (F. 

Figure A2.2.5 shows a sketch off(r) for Leimard-Jones pair potential. Now if AA is the excess Flelmholtz free energy relative to its ideal gas value, then (-pi4) = and AU/N= [5(pA /5V)/(5p)]. Then,...

Figure A2.2.5. Sketch off(r) for the Leimard-Jones pair potential u r) = 4E[(a/r) - (a/r) ] full curve -(Be =...

A few of the simpler pair potentials are listed below, (a) The potential for hard spheres of diameter a... 

It follows that atoms or molecules interacting with the same pair potential E (r.j/a), but with different e and a, have the same thennodynamic properties, derived from A INkT, at the same scaled temperature T and scaled... 

Combining tliis witli the Omstein-Zemike equation, we have two equations and tluee unknowns h(r),c(r) and B(r) for a given pair potential u r). The problem then is to calculate or approximate the bridge fiinctions for which there is no simple general relation, although some progress for particular classes of systems has been made recently. 

For a pairwise additive potential, each temi in the sum of pair potentials gives the same result in the above expression and there are N(N - l)/2 such temis. It follows that... 

We conclude this section by discussing an expression for the excess chemical potential in temrs of the pair correlation fimction and a parameter X, which couples the interactions of one particle with the rest. The idea of a coupling parameter was mtrodiiced by Onsager [20] and Kirkwood [Hj. The choice of X depends on the system considered. In an electrolyte solution it could be the charge, but in general it is some variable that characterizes the pair potential. The potential energy of the system... 

The principle ideas and main results of tlie theory at the level of the second virial coefficient are presented below. The Mayer/-function for the solute pair potential can be written as the sum of temis ... 

A very successfiil first-order perturbation theory is due to Weeks, Chandler and Andersen pair potential u r) is divided into a reference part u r) and a perturbation w r)... 

Figure A3.1.1. Typical pair potentials. Illustrated here are the Lennard-Jones potential, and the Weeks-Chandler- Anderson potential, which gives the same repulsive force as the Leimard-Jones potential.

Assuming that additive pair potentials are sufficient to describe the inter-particle interactions in solution, the local equilibrium solvent shell structure can be described using the pair correlation fiinction g r, r2). If the potential only depends on inter-particle distance, reduces to the radial distribution fiinction g(r) = g... 

At low solvent density, where isolated binary collisions prevail, the radial distribution fiinction g(r) is simply related to the pair potential u(r) via g ir) = exp[-n(r)//r7]. Correspondingly, at higher density one defines a fiinction w r) = -kT a[g r). It can be shown that the gradient of this fiinction is equivalent to the mean force between two particles obtamed by holding them at fixed distance r and averaging over the remaining N -2 particles of the system. Hence w r) is called the potential of mean force. Choosing the low-density system as a reference state one has the relation... 

Ghrayeb R, Purushotham M, Hou M and Bauer E 1987 Estimate of repulsive interatomic pair potentials by low-energy alkalimetal-ion scattering and computer simulation Phys. Rev. B 36 7364-70... 

Figure B3.3.4. Lennard-Jones pair potential showing the and r eontributions.

Figure B3.3.11. The classical ring polymer isomorphism, forA = 2 atoms, using/ = 5 beads. The wavy lines represent quantum spring bonds between different imaginary-time representations of the same atom. The dashed lines represent real pair-potential interactions, each diminished by a factor P, between the atoms, linking corresponding imaginary times.

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... 

Parrinello M and Rahman A 1980 Crystal structure and pair potentials a molecular dynamics study Phys. Rev.L 45 1196-9... 

Rare-gas clusters can be produced easily using supersonic expansion. They are attractive to study theoretically because the interaction potentials are relatively simple and dominated by the van der Waals interactions. The Lennard-Jones pair potential describes the stmctures of the rare-gas clusters well and predicts magic clusters with icosahedral stmctures [139, 140]. The first five icosahedral clusters occur at 13, 55, 147, 309 and 561 atoms and are observed in experiments of Ar, Kr and Xe clusters [1411. Small helium clusters are difficult to produce because of the extremely weak interactions between helium atoms. Due to the large zero-point energy, bulk helium is a quantum fluid and does not solidify under standard pressure. Large helium clusters, which are liquid-like, have been produced and studied by Toennies and coworkers [142]. Recent experiments have provided evidence of... 

The homonuclear rare gas pairs are of special interest as models for intennolecular forces, but they are quite difficult to study spectroscopically. They have no microwave or infrared spectmm. However, their vibration-rotation energy levels can be detennined from their electronic absorjDtion spectra, which he in the vacuum ultraviolet (VUV) region of the spectmm. In the most recent work, Hennan et al [24] have measured vibrational and rotational frequencies to great precision. In the case of Ar-Ar, the results have been incoriDorated into a multiproperty analysis by Aziz [25] to develop a highly accurate pair potential. 

The intennolecular forces between water molecules are strongly non-additive. It is not realistic to expect any pair potential to reproduce the properties of both the water dimer and the larger clusters, let alone liquid water. There has therefore been a great deal of work on developing potential models with explicit pairwise-additive and nonadditive parts [44, 50, 51]. It appears that, when this is done, the energy of the larger clusters and ice has a nonadditive contribution of about 30%. 

Fellers R S, Leforestier C, Braly L B, Brown M G and Saykally R J 1999 Speotrosoopio determination of the water pair potential Science 284 945-8... 

Abstract. A smooth empirical potential is constructed for use in off-lattice protein folding studies. Our potential is a function of the amino acid labels and of the distances between the Ca atoms of a protein. The potential is a sum of smooth surface potential terms that model solvent interactions and of pair potentials that are functions of a distance, with a smooth cutoff at 12 Angstrom. Techniques include the use of a fully automatic and reliable estimator for smooth densities, of cluster analysis to group together amino acid pairs with similar distance distributions, and of quadratic progrmnming to find appropriate weights with which the various terms enter the total potential. For nine small test proteins, the new potential has local minima within 1.3-4.7A of the PDB geometry, with one exception that has an error of S.SA. 

Our potential is a sum of smooth surface potentials that model amino acid-solvent interactions and of smooth pair potentials that model amino acid-amino acid interactions. As in [24], we take as essential only the Ca atoms. 

To find appropriate empirical pair potentials from the known protein structures in the Brookhaven Protein Data Bank, it is necessary to calculate densities for the distance distribution of Ga-atoms at given bond distance d and given residue assignments ai,a2- Up to a constant factor that is immaterial for subsequent structure determination by global optimization, the potentials then ciiiergo as the negative logarithm of the densities. Since... 

While smooth pair potentials are the rule in the literature, surface terms have traditionally been discontinuous the only potential using smooth surface terms seems to appear in Lund et al. [19], where the surface term is a function of a smooth approximation to the number of neighbors of a Ca atom.)... 

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