The Lennard-Jones potential

If we set n = 12 and m = 6 in Eq.7.6.1 we obtain the well known and extensively studied Lennard-Jones (LJ), or 12-6, potential. This potential is usually written in the reduced form  

To obtain numerical values for the parameters c and r the LJ potential must be related to some macroscopic property, typically vapor phase volumetric behavior, as expressed through the second virial coefficient and transport properties, such as viscosity and diffiisivity. Values for several compounds are given by Reid et al. 

Can we expect the LJ, or the more general Mie potential, to be applicable to all types of molecules  

Molecules that deviate from this behavior cannot be expected to follow Eq.7.6.1. For large and/or polar molecules, their relative orientation plays an important role in their intermolecular potential. 

The Lennard-Jones potential represents an approximation, even for simple molecules. As a result, different sets of parameter values are obtained from second virial coefficient data and from transport properties data and the discrepancy increases, for the aforementioned reasons, as the deviation from the simple molecule concept becomes larger (Hirsch-felder et al, 1954). 

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Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. 

The gradient model for interfacial tension described in Eqs. III-42 and III-43 is limited to interaction potentials that decay more rapidly than r. Thus it can be applied to the Lennard-Jones potential but not to a longer range interaction such as dipole-dipole interaction. Where does this limitation come from, and what does it imply for interfacial tensions of various liquids ... 

One fascinating feature of the physical chemistry of surfaces is the direct influence of intermolecular forces on interfacial phenomena. The calculation of surface tension in section III-2B, for example, is based on the Lennard-Jones potential function illustrated in Fig. III-6. The wide use of this model potential is based in physical analysis of intermolecular forces that we summarize in this chapter. In this chapter, we briefly discuss the fundamental electromagnetic forces. The electrostatic forces between charged species are covered in Chapter V. 

We have two interaction potential energies between uncharged molecules that vary with distance to the minus sixth power as found in the Lennard-Jones potential. Thus far, none of these interactions accounts for the general attraction between atoms and molecules that are neither charged nor possess a dipole moment. After all, CO and Nj are similarly sized, and have roughly comparable heats of vaporization and hence molecular attraction, although only the former has a dipole moment. 

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.

Figure 7-12. Plot of the van der Waals interaction energy according to the Lennard-Jones potential given in Eq. (27) (Sj, = 2.0 kcal mol , / (, = 1.5 A). The calculated collision diameter tr is 1.34 A.

The Lennard-Jones potential is characterised by an attractive part that varies as r ° and a repulsive part that varies as These two components are drawn in Figure 4.35. The r ° variation is of course the same power-law relationship foimd for the leading term in theoretical treatments of the dispersion energy such as the Drude model. There are no... 

The Lennard-Jones potential is constructed from a repulsive component (ar and an attractive nent (ar ). 

A comparison of the pairwise contribution to the Barker-Fisher-Watts potential with the Lennard-Jones potential for argon is shown in Figure 4.38. 

For the Lennard-Jones potential the long-range contribution can be determined analytically ... 

The shift makes the potential deviate from the true potential, and so any calculated thermodynamic properties will be changed. The true values can be retrieved but it is difficult to do so, and the shifted potential is thus rarely used in real simulations. Moreover, while it is relatively straightforward to implement for a homogeneous system under the influence of a simple potential such as the Lennard-jones potential, it is not easy for inhomogeneous systems containing rnany different types of atom. 

Forces Molecules are attracted to surfaces as the result of two types of forces dispersion-repulsion forces (also called London or van der Waals forces) such as described by the Lennard-Jones potential for molecule-molecule interactions and electrostatic forces, which exist as the result of a molecule or surface group having a permanent electric dipole or quadrupole moment or net electric charge. 

The classical kinetic theoty of gases treats a system of non-interacting particles, but in real gases there is a short-range interaction which has an effect on the physical properties of gases. The most simple description of this interaction uses the Lennard-Jones potential which postulates a central force between molecules, giving an energy of interaction as a function of the inter-nuclear distance, r. 

Figure 3.7 The Lennard-Jones potential of the interaction of gaseous atoms as a function of the internuclear distance...

It is interesting to note that all three mechanisms contributing to the attractive van der Waals interactions vary as the reciprocal of the separation distance to the sixth power. It is for this reason that the Lennard-Jones potential has been extensively used to model van der Waals forces. 

In the Yukawa potential, A is an inverse range parameter. The value A = 1.8 is appropriate for the inert gases. Each of the above potentials has a hard core. Real molecules are hard but not infinitely so. A slightly softer core is more desirable. The Lennard-Jones potential... 

The first step towards the development of appropriate expressions is the decomposition of the nonassociative pair potential into repulsive and attractive terms. In this work we apply the Weeks-Chandler-Andersen separation of interactions [117], according to which the attractive part of the Lennard-Jones potential is defined by... 

The simplest atomistic model for the formation of a crystal in continuous space requires the definition of some effective attractive potential between any two atoms, which is defined independently of the other atoms in the cluster or crystal. The most frequently studied potential is the Lennard-Jones potential... 

Here C and C2 are suitable constants. The Lennard-Jones potential can also be written as... 

The main difference between the three functions is in the repulsive part at short distances the Lennard-Jones potential is much too hard, and the Exp.-6 also tends to overestimate the repulsion. It furthermore has the problem of inverting at short distances. For chemical purposes these problems are irrelevant, energies in excess of lOOkcal/mol are sufficient to break most bonds, and will never be sampled in actual calculations. The behaviour in the attractive part of the potential, which is essential for intermolecular interactions, is very similar for the three functions, as shown in... 

Figure 2.10. Part of the better description of the Morse and Exp.-6 potentials may be due to the fact that they have three parameters, while the Lennard-Jones potential only employs two. Since the equilibrium distance and the well depth fix two constants, there is no additional flexibility in the Lennard-Jones function to fit the form of the repulsive interaction.

Most force fields employ the Lennard-Jones potential, despite the known inferiority to an exponential type function. Let us examine the reason for this in a little more detail. 

Kihara20 used a core model in which the Lennard-Jones potential is assumed to hold for the shortest distance between the molecular cores instead of molecular centers. By use of linear, tetrahedral, and other shapes of cores, various molecules can be approximated. Thomaes,41 Rowlinson,35 Hamann, McManamey, and Pearse,14 Atoji and Lipscomb,1 Pitzer,30 and Balescu,4 have used other models of attracting centers and other mathemtical methods, but obtain similar conclusions. The primary effect is to steepen the potential curve so that in terms of inverse powers of the inter-... 

In these equations, a and e are parameters in the Lennard-Jones potential function for interactions between unlike molecules, the customary mixing rules were used ... 

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