Lennard-Jones form

Extensive computer simulations have been caiTied out on the near-surface and surface behaviour of materials having a simple cubic lattice structure. The interaction potential between pairs of atoms which has frequently been used for inert gas solids, such as solid argon, takes die Lennard-Jones form where d is the inter-nuclear distance, is the potential interaction energy at the minimum conesponding to the point of... 

A complete set of intermolecular potential functions has been developed for use in computer simulations of proteins in their native environment. Parameters have been reported for 25 peptide residues as well as the common neutral and charged terminal groups. The potential functions have the simple Coulomb plus Lennard-Jones form and are compatible with the widely used models for water, TIP4P, TIP3P and SPC. The parameters were obtained and tested primarily in conjunction with Monte Carlo statistical mechanics simulations of 36 pure organic liquids and numerous aqueous solutions of organic ions representative of subunits in the side chains and backbones of proteins... 

In some force fields, especially those using the Lennard-Jones form in eq. (2.12), the /fg parameter is defined as the geometrical mean of atomic radii, implicitly via the geometrical mean rale used for the C and C2 constants. 

Vvdw may be represented by a potential of Lennard-Jones form (as in the last term of eq. (11.3)). A common alternative is the Buckingham form ... 

The nonbonded interactions commonly consist of a sum of two-body repulsion and dispersion energy terms between atoms that are often of the Lennard-Jones form in addition to the energy from the interactions between fixed partial atomic or ionic charges (Coulomb interaction)... 

The dispersion interactions are weak compared with repulsion, but they are longer range, which results in an attractive well with a depth e at an interatomic separation of am n = 21/6a. The interatomic distance at which the net potential is zero is often used to define the atomic diameter. In addition to the Lennard-Jones form, the exponential-6 form of the dispersion-repulsion interaction,... 

A simpler potential of the form of Eq. (10) has been used by Pearson et al. to model Si and SiC surfaces . The two-body term is of the familiar Lennard-Jones form while the three-body interaction is modeled by an Axilrod-Teller potential . The physical significance of this potential form is restricted to weakly bound systems, although it apparently can be extended to model covalent interactions. 

An important addition compared to previous models was the parameterization of the internucleosomal interaction potential in the form of an anisotropic attractive potential of the Lennard-Jones form, the so-called Gay-Berne potential [90]. Here, the depth and location of the potential minimum can be set independently for radial and axial interactions, effectively allowing the use of an ellipsoid as a good first-order approximation of the shape of the nucleosome. The potential had to be calibrated from independent experimental data, which exists, e.g., from the studies of mononucleosome liquid crystals by the Livolant group [44,46] (see above). The position of the potential minima in axial and radial direction were obtained from the periodicity of the liquid crystal in these directions, and the depth of the potential minimum was estimated from a simulation of liquid crystals using the same potential. 

There are presently two main difficulties which handicap attempts at exact calculation. The first concerns the intermolecular potential, and the hazards of extrapolation from models derived from viscosity measurements have been discussed. Furthermore, such a method is of dubious validity for polyatomic molecules, because the intermolecular repulsive potential will generally appear to become progressively shallower with increasing molecular dimensions if the viscosity data are cast, for example, in the Lennard-Jones form. Energy transfer depends... 

Several other functional forms, besides the Lennard-Jones and Buckingham potentials, have been used to describe the non-bonded interactions. For example, Kihara (1953) used a form in which the repulsive potential becomes infinite at very short distances, about -jfo- that at which the minimum of the function occurs if the distance, at which the potential becomes infinite, is reduced to zero, then the Lennard-Jones form results (Rowlinson, 1965). Kitaigorodskii (1961) derived another function from the Buckingham form. By defining r0 as the distance between the atoms at which U is a minimum, and letting z = r/r0, a = br0, and C72/3 be the value of 17 at r = 2r0/3 he obtained... 

Carbon tetrafluoride. Carbon tetra-fluoride, which undergoes a transition to a plastically crystalline (orientationally disordered) phase, has been investigated by the Parrinello-Rahman molecular dynamics method under constant-pressure conditions (6). A simple intermolecular potential model of the Lennard-Jones form was derived by taking into account the experimen-... 

Thus we again assume a Lennard-Jones form, where now the well depth and range parameters depend on the solute s internal vibrational coordinates. Without loss of generality we can define these coordinates so that q = Q = 0 corresponds to the minimum in the intramolecular potential. The solute-solvent potential in Hb above is actually then

(r, 0, 0), where clearly e = e(0, 0) and a = cr(0, 0). The oscillator-bath interaction term is... 

We have seen that the nearest-neighbor distance jn ionic solids is determined by the joint action of the Coulomb interaction and the predominantly repulsive overlap interaction that we approximate with the Lennard-Jones form. First, let us explore the properties of KCI, an ionic solid with ion charge Z = 1. We have seen that the Coulomb energy per ion pair in the face-centered cubic structure is... 

In some force fields, especially those using the Lennard-Jones form in eq. (2.12), tire... 

Because of the high symmetry of the main cavities of sieve A, it might seem reasonable, in the spirit of the cell theory of liquids (5), to assume that the various charged species of the zeolite are distributed with complete spherical symmetry about the center of each cavity. This, of course, leads to zero electric field throughout the cavity and hence to zero contribution from ion-induced dipole and similar forces. The remaining terms, dispersion and repulsion, are assumed to be of the Lennard-Jones form and are taken to be additive. This theory then becomes essentially the same as the cell theory of liquids, except that we permit several molecules per cell. The total P.E. is given as = 299(fi) +... 

Though simple analytic expressions such as the Lennard-Jones form adopted above (at least the attractive part) may be constructed on the basis of a sound quantum mechanical plausibility argument, the derivation of pair potentials for free electron systems is even more enlightening and strikes to the heart of bonding in metallic systems. An even more important reason for the current discussion is the explicit way in which the effective description of the total energy referred to in eqn (4.4) is effected. As will be shown below, the electronic degrees of freedom are literally integrated out with the result that... 

For the intermolecular potential, the Lennard-Jones form as modified by Rushbrooke,... 

After the pioneering quantum mechanical work not much new ground was broken until computers and software had matured enough to try fresh attacks. In the meantime the study of intermolecular forces was mainly pursued by thermodynamicists who fitted model potentials, often of the Lennard-Jones form [10] 4e[(cr/R) — (cr// ) ], to quantities like second virial coefficients, viscosity and diffusion coefficients, etc. Much of this work is described in the authoritative monograph of Hirschfelder et al. [11] who, incidentally, also gave a good account of the relationship of Drude s classical work to that of London. 

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