Lennard-Jones formula

Concrete calculations of Bc a are feasible only if the law according to which the forces of attraction and repulsion depend on intermolecular distance is known. For spherical molecules, this law is approximated by the Lennard-Jones formula ... 

A useful approximation to the intermolecular potential energy U(r) of a pair of spherically symmetric nonpolar molecules is given by the Lennard-Jones formula U(r) = Ar — Cr where A and C are constants and r is the internuclear separation. 

Jones, 1 Roy. Soo. Proc., A, vol. 105, p. 650 (1924). The first attempt to calculate the mole refraction from the quadratic Stark effect formula was made by Lennard-Jones (Jones), with the old quantum theory, f Z. f. Physik, vol. 38, p. 518 (1926). 

So far we have discussed various techniques for computing the PMF. The other type of free energy calculation commonly performed is alchemical transformation where two different systems are compared. Such calculations have many applications such as Lennard-Jones fluid with and without dipoles for each particles, comparison of ethanol (CH3CH2OH) and ethane thiol (CH3CH2SH), replacing one amino acid by another in a protein, changing the formula for a compound in drug discovery, etc. 

We assume for simplicity that the solvent is pure water, and that only the water-oxygen atoms have explicit Lennard-Jones interactions with the solute (this is typical of several common water models). We have seen that AWnp can be viewed as the free energy to change A from zero to one. Therefore, a well-known thermodynamic integration formula gives... 

A useful compendium of formulas for estimating the interaction parameters e and cr in the Lennard-Jones or Stockmayer potentials has been presented by Svehla [389]. His work... 

Figure 16. Phase diagram of the Lennard-Jones fluid with different approximations MSA (crosses) and DHH (dot line) (taken from Ref. [65]), DHH+DHHDS (solid line) (see Ref. [81]), BB approximation (black circles) (courtesy of the author), the chemical potential had been calculated by using Lee formula). Simulation data are from Lotfi et al. [102] (open circles) and from Panagiotopoulos [103] (solid triangles).

The dispersion and repulsion interactions form the Lennard-Jones (Barrer, 1978 Masel, 1996 Razmus and Hall, 1991 Gregg and Sing, 1982 Steele, 1974 Adamson, 1976 Rigby et al., 1986) potential, with an equilibrium distance (r0) where 4 d + 4 r = 0. This distance is taken as the mean of the van der Waals radii of the interacting pair. Once the attractive, dispersion constant, A, is known, B is readily obtained by setting at chp/dr = 0 at r0. Hence, B = Ar /2. The most commonly used expression for calculating A is the Kirkwood-Muller formula ... 

Kirkwood-Muller formula, 83 Lennard-Jones potential, 83 nonspecific contributions, 82 pillared clays, 120... 

The last term in the formula (1-196) describes electrostatic and Van der Waals interactions between atoms. In the Amber force field the Van der Waals interactions are approximated by the Lennard-Jones potential with appropriate Atj and force field parameters parametrized for monoatomic systems, i.e. i = j. Mixing rules are applied to obtain parameters for pairs of different atom types. Cornell et al.300 determined the parameters of various Lenard-Jones potentials by extensive Monte Carlo simulations for a number of simple liquids containing all necessary atom types in order to reproduce densities and enthalpies of vaporization of these liquids. Finally, the energy of electrostatic interactions between non-bonded atoms is calculated using a simple classical Coulomb potential with the partial atomic charges qt and q, obtained, e.g. by fitting them to reproduce the electrostatic potential around the molecule. 

To estimate the contribution from the Lennaid-Jones interaction gi we also utilized Pierotti s formula Because the solvent entropy change caused by the establishment of the Lennard-Jones type interaction is thought to be small, gi is approximately equal to the... 

Formula (4) was also used for Ar...Ar interaction, and respective parameters were adjusted to yield the same minimum distance and well depth as obtained from the 6-12 Lennard-Jones potential [9]. 

The general formula of the Lennard-Jones 6-12 potential function [Lennard-Jones, 1924 Lennard-Jones, 1929] is ... 

The success of the preceding scheme for diatomic molecules I7,i8,i , 20,21) ied Hund 22> and Mulliken 23> to apply the same theory to polyatomic molecules. In the beginning, there seemed to be no direct relation between molecular orbitals (MO s) and the bonds in a chemical formula, because MO s normally extend over the whole molecule and are not restricted to the region between two atoms. The difficulty was overcome by using equivalent localized MO s instead of the delocalized ones 24>25>. The mathematical definition of equivalent MO s was given only in 1949 by Lennard-Jones and his coworkers 26.27), but the concept of localization... 

It is valid for both endothermic and exothermic bimolecular reactions between simple radicals and molecules (mostly organic) in gaseous phase. Semenov realized that the coefficients of the correlation formula may differ somewhat for individual classes of the reaction partners. Ref. [15] reviews the empirical models of radical abstraction. A rationale for the above correlation was provided by Morse or Lennard-Jones who considered the potentials of overlap in the reacting partners [16]. 

By use of these formulas accurate calculations of the transport coefficients can thus be performed provided that the Lennard-Jones potential parameters like i and ci are known. Extensive lists of these parameters are given for many substances by Hirschfelder et al [39] and Bird et al [5], among others. 

The interface between the droplet and the gas is not discontinuous the average molecular density decreases over a narrow region from the liquid side to the vapor. When the size of the droplet becomes sufhctently small compared with the thickness of the transition layer, the use of classical thermodynamics and the bulk surface tension become inaccurate the Kelvin relation and Laplace formula no longer apply. This effect has been studied by molecular dynamics calculations of the behavior of liquid droplets composed of 41 to 2(X)4 molecules that interact through a Lennard-Jones (LI) intermolecular potential (Thomp.son et al., 1984). The results of this analysis are shown in Fig. 9.5, in which the nondimensional pressure difference between the drop interior and the surrounding vapor (Pd — p)rr / ij is... 

The van der Waals interaction for each atom-type pair (w, v) is calculated according to the general formula of Lennard-Jones 6-12 potential Junction as... 

The n= 12 soft sphere model is the high-temperature limit of the 12-6 Lennard-Jones (LJ) potential. Agrawal and Kofke [182] used this limit as the starting point for another Gibbs-Duhem integration, which proceeded to lower temperatures until reaching the solid-liquid-vapor triple point. The complete solid-fluid coexistence line, from infinite temperature to the triple point, can be conveniently represented by the empirical formula [182]... 

Let us plot the logarithmic values of the shear viscosities calculated by the PNM model, divided by the dilute gas values (which vary as the square root of temperature 7 as a function of ejkT, e being the depth of the Lennard-Jones potential well. The results are given in Fig. 1 for a reduced density of na = 0.818, a being the hard core diameter. The experimental points are well fitted by two approximately straight lines. We observe that Eyring s formula for shear viscosity at constant density... 

Both types of functions are commonly used. Several sets of a, b, c, and d, coefficients are available [1-3]. Equally good results can be obtained using Lennard-Jones-type functions alone or Buckingham-type functions alone or mixtures of Lennard-Jones and Buckingham functions [4]. The attraction coefficients in these expressions are generally but not always calculated with the formula of Slater and Kirkwood [5] ... 

More precise formulae appear to be necessary only if the anharmonicity effects of the molecule at high excitations are better known. Also, the exact interfragment potential and the corresponding centrifugal maxima have to be available to determine better rotational factors equations (24) and (25) have been based on an interfragment potential at the distances important for centrifugal maxima of the form V(g) oIl - 2( ,/ ) ], which corresponds to only the attractive part of a Lennard-Jones potential). 

Equation (46), usually attributed to Hudson and McCoubrey, was also proposed by Srivastava and Madan. It was obtained by equating the attractive portion of the Lennard-Jones potential to the London formula for Cj. The quantities f are the ionization energies of the pure components. A similar derivation based on the Kirkwood-Miiller treatment of dispersion forces leads to equation (47), which requires knowledge of the diamagnetic susceptibilities... 

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