Potential energy functions interactions

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

Molecular dynamics conceptually involves two phases, namely, the force calculations and the numerical integration of the equations of motion. In the first phase, force interactions among particles based on the negative gradient of the potential energy function U,... 

Figure 7-8. Bonded (upper row) and non-bonded (lower row) contributions to a typioal molecular mechanics force field potential energy function. The latter two types of Interactions can also occur within the same molecule.

Molecular Mechanics uses an analytical, differentiable, and relatively simple potential energy function, V(R), for describing the interactions between a set of atoms specified by their Cartesian coordinates R. 

Fig. 1. Potential energies of interaction between two coUoidal particles as a function of their surface-surface separation, for electrical double layers due...

A potential energy function is a mathematical equation that allows for the potential energy, V, of a chemical system to be calculated as a function of its tliree-dimensional (3D) structure, R. The equation includes terms describing the various physical interactions that dictate the structure and properties of a chemical system. The total potential energy of a chemical system with a defined 3D strucmre, V(R)iai, can be separated into terms for the internal, V(/ )i,iBmai, and external, V(/ )extemai, potential energy as described in the following equations. 

Equations (l)-(3) in combination are a potential energy function that is representative of those commonly used in biomolecular simulations. As discussed above, the fonn of this equation is adequate to treat the physical interactions that occur in biological systems. The accuracy of that treatment, however, is dictated by the parameters used in the potential energy function, and it is the combination of the potential energy function and the parameters that comprises a force field. In the remainder of this chapter we describe various aspects of force fields including their derivation (i.e., optimization of the parameters), those widely available, and their applicability. 

According to the namre of the empirical potential energy function, described in Chapter 2, different motions can take place on different time scales, e.g., bond stretching and bond angle bending vs. dihedral angle librations and non-bond interactions. Multiple time step (MTS) methods [38-40,42] allow one to use different integration time steps in the same simulation so as to treat the time development of the slow and fast movements most effectively. 

The correlation of electron motion in molecular systems is responsible for many important effects, but its theoretical treatment has proved to be very difficult. Thus many quantum valence calculations use wave functions which are adjusted to optimize kinetic energy effects and the potential energy of interaction of nuclei and electrons but which do not adequately allow for electron correlation and hence yield excessive electron repulsion energy. This problem may be subdivided into cases of overlapping and nonoverlapping electron distributions. Both are very important but we shall concern ourselves here with only the nonoverlapping case. 

We now describe a relatively simple MD model of a low-index crystal surface, which was conceived for the purpose of studying the rate of mass transport (8). The effect of temperature on surface transport involves several competing processes. A rough surface structure complicates the trajectories somewhat, and the diffusion of clusters of atoms must be considered. In order to simplify the model as much as possible, but retain the essential dynamics of the mobile atoms, we will consider a model in which the atoms move on a "substrate" represented by an analytic potential energy function that is adjusted to match that of a surface of a (100) face-centered cubic crystal composed of atoms interacting with a Lennard-Jones... 

Muller et al. focused on polybead molecules in the united atom approximation as a test system these are chains formed by spherical methylene beads connected by rigid bonds of length 1.53 A. The angle between successive bonds of a chain is also fixed at 112°. The torsion angles around the chain backbone are restricted to three rotational isomeric states, the trans (t) and gauche states (g+ and g ). The three-fold torsional potential energy function introduced [142] in a study of butane was used to calculate the RIS correlation matrix. Second order interactions , reflected in the so-called pentane effect, which almost excludes the consecutive combination of g+g- states (and vice-versa) are taken into account. In analogy to the polyethylene molecule, a standard RIS-model [143] was used to account for the pentane effect. 

The local conformational preferences of a PE chain are described by more complicated torsion potential energy functions than those in a random walk. The simulation must not only establish the coordinates on the 2nnd lattice of every second carbon atom in the initial configurations of the PE chains, but must also describe the intramolecular short range interactions of these carbon atoms, as well as the contributions to the short-range interactions from that... 

The potential energy function prohibits double occupancy of any site on the 2nnd lattice. In the initial formulation, which was designed for the simulation of infinitely dilute chains in a structureless medium that behaves as a solvent, the remaining part of the potential energy function contains a finite repulsion for sites that are one lattice unit apart, and a finite attraction for sites that are two lattice units apart [153]. The finite interaction energies for these two types of sites were obtained by generalizing the lattice formulation of the second virial coefficient, B2, described by Post and Zimm as [159] ... 

The DLVO theory, with the addition of hydration forces, may be used as a first approximation to explain the preceding experimental results. The potential energy of interaction between spherical particles and a plane surface may be plotted as a function of particle-surface separation distance. The total potential energy, Vt, includes contributions from Van der Waals energy of interaction, the Born repulsion, the electrostatic potential, and the hydration force potential. [Israelachvili (13)]. 

Here, avaw is a positive constant, and and ctJ are the usual Lennard-Jones parameters found in macromolecular force fields. The role played by the term avdw (1 — A)2 in the denominator is to eliminate the singularity of the van der Waals interaction. Introduction of this soft-core potential results in bounded derivatives of the potential energy function when A tends towards 0. 

Similarly, we often try to interpret changes in the free energy in terms of contributions to the potential energy function. For instance, one might want to know whether AA is primarily driven by electrostatic or van der Waals interactions. Alternatively, one might be interested in finding out what are the contributions to AA arising from... 

Here we present and discuss an example calculation to make some of the concepts discussed above more definite. We treat a model for methane (CH4) solute at infinite dilution in liquid under conventional conditions. This model would be of interest to conceptual issues of hydrophobic effects, and general hydration effects in molecular biosciences [1,9], but the specific calculation here serves only as an illustration of these methods. An important element of this method is that nothing depends restric-tively on the representation of the mechanical potential energy function. In contrast, the problem of methane dissolved in liquid water would typically be treated from the perspective of the van der Waals model of liquids, adopting a reference system characterized by the pairwise-additive repulsive forces between the methane and water molecules, and then correcting for methane-water molecule attractive interactions. In the present circumstance this should be satisfactory in fact. Nevertheless, the question frequently arises whether the attractive interactions substantially affect the statistical problems [60-62], and the present methods avoid such a limitation. 

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