Potential functions Lennard-Jones form

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... 

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. 

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... 

A novel hybrid molecular simulation technique was developed to simulate AFM over experimental timescales. This method combines a dynamic element model for the tip-cantilever system in AFM and an MD relaxation approach for the sample. The hybrid simulation technique was applied to investigate the atomic scale friction and adhesion properties of SAMs as a function of chain length [81], The Ryckaert-Bellmans potential, harmonic potential, and Lennard-Jones potential were used. The Ryckaert-Bellmans potential, which is for torsion, has the form... 

Generally the most important component of any molecular mechanics force field is the nonbonding potential function. The traditional form is the Lennard-Jones "6-12" potential (Eq. 2.47), where e and r are parameters that depend on the identities of the two interacting atoms and r is the distance between the atoms. 

The interactions of polyatomic molecules are frequently modeled by pair potentials, both Lennard-Jones and electrostatic, between all constituent atoms. The model potential used must also account for intramolecular geometries by including the bonded terms bond lengths, bond angles, and dihedral angles. The result is the molecular modeling potential function that generally is of the form... 

The remaining critical component in the simulations is the potential functions that describe the intra- and intermolecular energetics for the system. The intermolecular part is usually represented in a Coulomb plus Lennard-Jones form with the interactions occurring between sites located on the nuclei. Simple potential functions are now available that give excellent thermodynamic and structural results for many pure liquids including water,... 

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.

A better approach than non-linear scaling is to attempt to reduce and/or eliminate the singularity in the function that occurs on the step when a noninteracting group starts to interact. A clever approach has been described that reduces the problem by modifying the Lennard-Jones van der Waals term in the potential function.30,31 For a pair of atoms where one group vanishes at the X =1 endpoint, the modified Lennard-Jones 6-12 function takes the form ... 

In a statistical Monte Carlo simulation the pair potentials are introduced by means of analytical functions. In the election of that analytical form for the pair potential, it must be considered that when a Monte Carlo calculation is performed, the more time consuming step is the evaluation of the energy for the different configurations. Given that this calculation must be done millions of times, the chosen analytic functions must be of enough accuracy and flexibility but also they must be fastly computed. In this way it is wise to avoid exponential terms and to minimize the number of interatomic distances to be calculated at each configuration which depends on the quantity of interaction centers chosen for each molecule. A very commonly used function consists of a sum of rn terms, r being the distance between the different interaction centers, usually, situated at the nuclei. In particular, non-bonded interactions are usually represented by an atom-atom centered monopole expression (Coulomb term) plus a Lennard-Jones 6-12 term, as indicated in equation (51). 

Here the atoms in the system are numbered by i, j, k, l = 1,..., N. The distance between two atoms i, j is ry, q is the (partial) charge on an atom, 6 is the angle defined by the coordinates (i, j, k) of three consecutive atoms, and 4> is the dihedral angle defined by the positions of four consecutive atoms, e0 is the dielectric permittivity of vacuum, n is the dihedral multiplicity. The potential function, as given in equation (6), has many parameters that depend on the atoms involved. The first term accounts for Coulombic interactions. The second term is the Lennard-Jones interaction energy. It is composed of a strongly repulsive term and a van der Waals-like attractive term. The form of the repulsive term is chosen ad hoc and has the function of defining the size of the atom. The Ay coefficients are a function of the van der Waals radii of the... 

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). 

A considerable amount of work has been done on the development of water-ion potential energy functions." " Most of these functions are of the standard Lennard-Jones plus Coulomb form, with parameters selected to give the experimental free energy or enthalpy of solvation. ... 

Figure 4.31. The total rotational potential, in kcal/mol, as a function of the dihedral angle

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