Intermolecular interactions Lennard—Jones potential

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))...

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... 

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. 

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 ... 

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. 

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 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... 

Our Monte Carlo (MC) simulation uses the Metropolis sampling technique and periodic boundary conditions with image method in a cubic box(21). The NVT ensemble is favored when our interest is in solvent effects as in this paper. A total of 344 molecules are included in the simulation with one solute molecule and 343 solvent molecules. The volume of the cube is determined by the density of the solvent and in all cases used here the temperature is T = 298K. The molecules are rigid in the equilibrium structure and the intermolecular interaction is the Lennard-Jones potential plus the Coulombic term... 

The liquid structure factor of CCI4 and its derivatives with respect to temperature at fixed pressure or fixed volume, needed by eq. (2), were evaluated by Molecular Dynamics (MD) simulations. We have used the OPLS model for tetrachloromethane [9] In this model, the CCI4 molecules are described as rigid tetrahedra (dc-ci = 1 -769 A) and the intermolecular potentials are atom centered 6-12 Lennard-Jones potentials plus the coulombic interaction with partial charges on C and Cl. We performed NVT simulations with 512 molecules for about 1 ns each. The different x-ray structure factors were obtained from the accumulated partial radial distribution functions [10], using the atomic form-factors from the DABAX database [11]. In order to estimate the partial derivatives of the structure factor, we have used finite differences we considered two different temperatures, Ti = 300 K and T2 = 328 K, and two molar volumes, Vi = 97.3 cm mol and V2 = 100.65 cm mol which are the molar volumes along the liquid-vapor coexistence line for the two temperatures Tj and Tz respectively [12]. Three simulations were then run for the temperature and molar volume conditions (TiiVi), T2,V )... 

Extension of more advanced methods, in particular, density functional theory, to non-equilibrium phenomena is the principal aim of this survey. We shall consider a simple one-component fluid with van der Waals interactions as a suitable medium for exploration of basic theoretical problems of interfacial dynamics. In the case when intermolecular interactions are long-range, in particular, in the most important case of Lennard-Jones potential, the transformation from the nonlocal (density functional) to local (van der Waals-Landau-Cahn) equations fails due to divergences appearing in the commonly used expansion of the interaction term in the expression for free energy. Setting bound-... 

The intramolecular and intermolecular solvent forces can be obtained in several ways. For rare gas solvents, it is most commonly the case that Lennard— Jones potentials are used to represent the intermolecular interactions (although more sophisticated potential energy functions are available from atom—atom scattering data). Molecular solvents, such as water, require more complicated models. The construction of these models is an active field and well beyond the scope of this review. Many of these models aim to reproduce bulk properties of the solvent and are quite successful in doing that (although some properties. 

A force sensor in an AFM can only work if the probe interacts with the force field associated with a surface. The interaction force between the probe and the surface in ambient air is illustrated in Figure 9.2. The total intermolecular pair potential is obtained by assuming one attractive (-Ci/z ) and another repulsive potential (C2/z )-Superimposing the two gives an expression for the well-known Lennard-Jones potential, where Ci and C2 are the corresponding coefficients for the attractive and repulsive interactions, respectively, and z is the distance between the sample surface and rest position of the cantilever. 

The molecules interact according to a cut-and-shifted Lennard-Jones potential, i/s-Lj(Os written in terms of the intermolecular distance r between two beads on two different molecules as... 

Rather, flie assignment is more serious wifli intermolecular interaction potential used. For simple molecules, empirical model potential such as fliose based on Lennard-Jones potential and even hard-sphere potential can be used. But, for complex molecules, potential function and related parameter value should be determined by some theoretical calculations. For example, contribution of hydrogen-bond interaction is highly large to the total interaction for such molecules as HjO, alcohols etc., one can produce semi-empirical potential based on quantum-chemical molecular orbital calculation. Molecular ensemble design is now complex unified mefliod, which contains both quantum chemical and statistical mechanical calculations. 

From the definitions of Wa and Wq, it can be seen that is a function of surface energies. If the structure of a material and the molecular potential energy-separation relationships are known, the surface energy can be calculated by evaluating the work required to separate to infinity the material either side of a chosen plane. For a material in which the dominant intermolecular forces are dispersion force interactions, the Lennard-Jones potential (see Dispersion forces and Polar forces) will apply, and the calculation is relatively simple. It gives work of cohesion on phase 1... 

In the original version of this model an attractive term based on Lennard-Jones potential constants was included in order to account for intermolecular attraction between adsorbed molecules. However, it was found that the fit of the experimental isotherm was not significantly improved by the inclusion of these factors so they were omitted in later studies. This observation suggests that, at least for nonpolar sorbates, the adsorption equilibrium behavior is governed by the attractive potential of the framework, which determines the Henry constant, and the repulsive interaction between molecules due to their finite size. Attractive forces between adsorbed molecules appear to be of only secondary importance. 

Eigure 4.33 shows a plot of the Lennard-Jones potential between two atoms or molecules. The quantity s in Equation 4.12 is equal to the depth of the potential well and is a measure of the strength of the intermolecular force. The quantity a- is a measure of the size of the molecules and is defined as the separation at which 7=0. Table 4.7 lists the Lennard-Jones parameters, s and a, for a few atoms and molecules. The Lennard-Jones potential is a good representation of generic intermolecular interactions in nonionic systems and is used extensively in the modeling and simulation of chemical systems. 

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