Intermolecular potentials Lennard-Jones

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. 

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... 

Fig. 5.1 A schematic projection of the 3n dimensional (per molecule) potential energy surface for intermolecular interaction. Lennard-Jones potential energy is plotted against molecule-molecule separation in one plane, the shifts in the position of the minimum and the curvature of an internal molecular vibration in the other. The heavy upper curve, a, represents the gas-gas pair interaction, the lower heavy curve, p, measures condensation. The lighter parabolic curves show the internal vibration in the dilute gas, the gas dimer, and the condensed phase. For the CH symmetric stretch of methane (3143.7 cm-1) at 300 K, RT corresponds to 8% of the oscillator zpe, and 210% of the LJ well depth for the gas-gas dimer (Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. /. Phys. Chem. A 105, 9284 (2001))...

All of the transport properties from the Chapman-Enskog theory depend on 2 collision integrals that describe the interactions between molecules. The values of the collision integrals themselves, discussed next, vary depending on the specified intermolecular potential (e.g., a hard-sphere potential or Lennard-Jones potential). However, the forms of the transport coefficients written in terms of the collision integrals, as in Eqs. 12.87 and 12.89, do not depend on the particular interaction potential function. 

The Lennard-Jones potential is a rather crude representation of an actual intermolecular potential, chosen more often for computational simplicity than chemical accuracy. More accurate for many chemical purposes is the Morse potential... 

One of the simplest orientational-dependent potentials that has been used for polar molecules is the Stockmayer potential.48 It consists of a spherically symmetric Lennard-Jones potential plus a term representing the interaction between two point dipoles. This latter term contains the orientational dependence. Carbon monoxide and nitrogen both have permanent quadrupole moments. Therefore, an obvious generalization of Stockmayer potential is a Lennard-Jones potential plus terms involving quadrupole-quadrupole, dipole-dipole interactions. That is, the orientational part of the potential is derived from a multipole expansion of the electrostatic interaction between the charge distributions on two different molecules and only permanent (not induced) multipoles are considered. Further, the expansion is truncated at the quadrupole-quadrupole term. In all of the simulations discussed here, we have used potentials of this type. The components of the intermolecular potentials we considered are given by ... 

Potential parameters (such as the Kihara core or Lennard-Jones potentials of the previous sections) can be calculated from a small set of fundamental, ab initio intermolecular energies, rather than fits of the potentials to phase equilibria and spectroscopic data. 

The multidimensional potential energy surface was written as the sum of a gas-phase (LEPS) energy surface incorporating the main features of the one-dimensional double-well potential in Example 10.1, solvent-solute interactions described by Lennard-Jones potentials with added (Coulomb) interactions corresponding to point charges, and solvent solvent interactions including intermolecular degrees of freedom. The solvent consisted of 64 water molecules. 

Common to these methods is the choice of the potential energies (1) intermolecular, (2) intramolecular, and (3) fluid-solid potential energy. The first one is the fluid-fluid potential and, for example, can be calculated from the 12-6 Lennard-Jones potential... 

The fundamental importance of bonding energies between bodies are traditionally divided into two broad classes chemical bond (short-range force), and physical or intermolecular bond (long-range force). The energies are largely dependent on the distance at which one body feels the presence of the other. Usually, they are called a Lennard-Jones potential [34] which has a minimum value at a certain distance. 

For the van der Waals interaction one is able to select a large number of different types of intermolecular potentials as seen in Ref. [53], Presently we have selected the 6-12 Lennard-Jones potential and we model the van der Waals contributions as... 

The temperature-dependent second and third virial coefficient describe the increasing two- and three-particle collisions between the gas molecules and their accompanying increase in gas density. The virial coefficients are calculated using a suitable intermolecular por-tential model (usually a 12-6 Lennard-Jones Potential) from rudimentary statistical thermodynamics. 

There are other empirical potential functions which are especially useful to describe intermolecular potentials, i.e., the potentials between atoms, which are not connected by chemical bonds. Two of these (the Buckingham and the Lennard-Jones potential) are discussed in Secs. 2.5.4 and 5.2. 

Because of their importance to nucleation kinetics, there have been a number of attempts to calculate free energies of formation of clusters theoretically. The most important approaches for the current discussion are harmonic models, " Monte Carlo studies, and molecular dynamics calcula-tions. In the harmonic model the cluster is assumed to be composed of constituent atoms with harmonic intermolecular forces. The most recent calculations, which use the harmonic model, have taken the geometries of the clusters to be those determined by the minimum in the two-body additive Lennard-Jones potential surface. The oscillator frequencies have been obtained by diagonalizing the Lennard-Jones force constant matrix. In the harmonic model the translational and rotational modes of the clusters are treated classically, and the vibrational modes are treated quantum mechanically. The harmonic models work best at low temjjeratures where anharmonic-ity effects are least important and the system is dominated by a single structure. 

In the dense phase the intermolecular potential consists mainly of a two-body term to which small three-body contributions should be added. This problem is poorly documented for molecular systems, and the classic example remains that of argon where an effective two-body Lennard-Jones potential accounts fairly well for the thermodynamic data simply as a result of cancellation of errors. For vibrational energy relaxation one is not directly concerned with the whole intermolecular potential, but rather by its vibrationally dependent part. As mentioned earlier, three-body effects are not usually observable and may be masked by inadequate knowledge of the true potential. Nevertheless one can expect some simply observable solvent effects describable by changes of either the intermolecular or the vibrational potentials. 

To provide a more quantitative explanation of the magnitudes of the properties of different materials, we must consider several types of intermolecular forces in greater detail than we gave to the Lennard-Jones model potential in Chapter 9. The Lennard-Jones potential describes net repulsive and attractive forces between molecules, but it does not show the origins of these forces. We discuss other intermolecular forces in the following paragraphs and show how they arise from molecular structure. Intermolecular forces are distinguished from intramolecular forces, which lead to the covalent chemical bonds discussed in Chapters 3 and 6. Intramolecular forces between atoms in the covalent bond establish and maintain... 

Corresponding States (CS) The principle of CS applies to conformal fluids [Lehmd, T. L., Jr., and P. S. Chappelear, Ind. Eng. Chem., 60 (1968) 15]. Two fluids are conformal if their intermolecu-lar interactions are equivalent when scaled in dimensionless form. For example, the Lennard-Jones (LJ) intermolecular pair potential energy U can be written in dimensionless form as... 

The forces of attraction and repulsion between molecules must be considered for a more accurate and rigorous representation of the gas flow. Chapman and Enskog proposed a well-known theory in which they use a distribution function, the Boltzmann equation, instead of the mean free path. Using this approach, for a pair of non-polar molecules, an intermolecular potential, V (r), is given in the potential function proposed by the Lennard-Jones potential ... 

The potential U(r ) is a sum over all intra- and intermolecular interactions in the fluid, and is assumed known. In most applications it is approximated as a sum of binary interactions, 17(r ) = IZ > w(rzj) where ry is the vector distance from particle i to particle j. Some generic models are often used. For atomic fluids the simplest of these is the hard sphere model, in which z/(r) = 0 for r > a and M(r) = c for r < a, where a is the hard sphere radius. A. more sophisticated model is the Lennard Jones potential... 

When one adds the attractive van der Waals potential terms to the repulsive term, one obtains the Lennard-Jones expression for the intermolecular potential energy for a simple fluid such as an inert gas like argon. On the basis of the above, the Lennard-Jones potential function may be written... 

A plot of the Lennard-Jones potential against distance between two molecules is shown in fig. 2.2. It is clear that the repulsive component dominates for values of r less than the molecular diameter, where it rises to values over 5eLj for r = 0.9a. The stable minimum occurs at r = 1.12a. By the time the separation between the two molecules is equal to three times their diameter, the intermolecular... 

Fig. 2.2 Plot of the Lennard-Jones potential u r) in units of the attractive potential energy Clj against intermolecular distance, r, in units of the molecular diameter a. The vertical and horizontal straight lines at r = a show the potential energy for a hard-sphere representation of the system.

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