Lennard-Jones forces, electrostatic

But consistent with the overall theme of this chapter, we need to ask ourselves whether we really have to conclude that the mechanism of vibrational energy relaxation is fundamentally electrostatic just because we find the overall relaxation rate to be sensitive to Coulombic forces. Let us attempt to get at this question through another mechanistic analysis of the INM vibrational influence spectrum, this time looking at the respective contributions of the electrostatic part of the solvent force on our vibrating bond, the nonelectrostatic part (in most simulations, the Lennard-Jones forces), and whatever cross terms there may be. 

The results of such a calculation, shown in Fig. 8 (52), seem to tell a very different story from the earliest studies. With nondipolar I2 as a solute and C02 as a solvent, the complete domination of the solvent response by the Lennard-Jones forces is impressive, but perhaps not all that startling. One might surmise that the quadupole-quadrupole forces at work in this example are a bit too weak to accomplish much. Yet, when we have a dipolar solute dissolved in the strongly polar solvent CH3CN, we get almost the same kind of complete control by Lennard-Jones forces. Electrostatics now seems totally unimportant. 

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. 

Ihi.. same molecule but separated by at least three bonds (i.e. have a 1, h relationship where n > 4). In a simple force field the non-bonded term is usually modelled using a Coulomb piilential term for electrostatic interactions and a Lennard-Jones potential for van der IV.uls interactions. 

The range of systems that have been studied by force field methods is extremely varied. Some force fields liave been developed to study just one atomic or molecular sp>ecies under a wider range of conditions. For example, the chlorine model of Rodger, Stone and TUdesley [Rodger et al 1988] can be used to study the solid, liquid and gaseous phases. This is an anisotropic site model, in which the interaction between a pair of sites on two molecules dep>ends not only upon the separation between the sites (as in an isotropic model such as the Lennard-Jones model) but also upon the orientation of the site-site vector with resp>ect to the bond vectors of the two molecules. The model includes an electrostatic component which contciins dipwle-dipole, dipole-quadrupole and quadrupole-quadrupole terms, and the van der Waals contribution is modelled using a Buckingham-like function. 

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 first two terms on the right-hand side of Eq. (83) are usually assumed to be harmonic, as given for example by Eq. (6-74). The third term is often developed in a Fourier series, as given by Eq. (82). The potential function appropriate to the interaction between nonbonded atoms is taken to be of the Lennard-Jones type (Section 6.7.3). In all of these cases the necessary force constants are estimated by comparing the results obtained from a large number of similar molecules. If electrostatic interactions are to be considered, effective atomic charges must be suggested and Coulomb s law applied directly [see Eq. (6-81)]. 

Consider the case of an ion in water. The force field for the ion may be described using a Lennard-Jones and electrostatic potential with charge q. Assume that we... 

A key to both methods is the force field that is used,65 or more precisely, the inter- and possibly intramolecular potentials, from which can be obtained the forces acting upon the particles and the total energy of the system. An elementary level is to take only solute-solvent intermolecular interactions into account. These are typically viewed as being electrostatic and dispersion/exchange-repulsion (sometimes denoted van der Waals) they are represented by Coulombic and (frequently) Lennard-Jones expressions ... 

The first three terms, stretch, bend and torsion, are common to most force fields although their explicit form may vary. The nonbonded terms may be further divided into contributions from Van der Waals (VdW), electrostatic and hydrogen-bond interactions. Most force fields include potential functions for the first two interaction types (Lennard-Jones type or Buckingham type functions for VdW interactions and charge-charge or dipole-dipole terms for the electrostatic interactions). Explicit hydrogen-bond functions are less common and such interactions are often modeled by the VdW expression with special parameters for the atoms which participate in the hydrogen bond (see below). 

It has been traditional to define a van der Waals potential (which combines Coulomb s law and the Lennard-Jones 6-12 potential function) and thereby subsume electronic shell repulsion, London forces, and electrostatic interactions under the term van der Waals interaction. Unfortunately, the resulting expression is an oversimplified treatment of the electrostatic interactions, which are only calculated between close neighbors and are considered to be spatially isotropic. Both of these implicit assumptions are untrue and do not represent physically realistic approximations. We prefer to use the term van der Waals distance for the intemuclear separation at which the 6-12 potential function is a minimum (see Fig. 6), the van der Waals radius being one-half this value when the two interacting atoms are identical, and explicitly treat the Lennard-Jones and electrostatic terms separately. While the term van der Waals interaction may have some value as a shorthand in structure description, it should be avoided when energetics are treated quantitatively. 

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. 

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