The Problem: how to describe pair-wise interacting particles? One approach is the Lennard-Jones (L-J) potential. This is written as:
For most classical modeling, the equations are expressed in terms of reduced units. In reduced units, the distances, time and energy values are scaled by the energy function parameters.
The Lennard-Jones potential function reproduces the atomic interactions very well. A comparison among accurate potentials for various noble gases and the L-J potential function is shown below:
The forces on the atoms in a system, either a collection of atoms such as a gas, a molecule, a solid, or a system, are derived from a potential energy function. This general potential energy function (force field function) describes the energy of the molecule. The force field parameters, which are the interaction strengths, are derived from experiment or other calculations. This is contained in a potential energy function whose gradient is the force.
A description of the mathematical form of the potential is important in order to understand the nature of the forces being modeled. The origins of the method can be traced back to the valence force fields of vibrational spectroscopy. As the name implies, valence force fields incorporate only terms expressing the covalent bonding pattern of the molecule.
For most molecular systems, such as proteins, polymers, and other classes of macromolecules, the potential energy can also be described pictorially as;
where
The first set of terms (1 through 9) in the above expression measures the degree of deformation of bond lengths, bond angles, and torsional angles from equilibrium values. For bonds and angles these equilibrium values are denoted by O. The terms in the potential interaction are usually based a harmonic approximation and the interaction is expressed as a Hookes law function. The harmonic approximation used here for bond and angle strain energies is quite reasonable in most cases; that is, provided the temperature is not too extreme and chemical reactions are not being modeled. The steepness of the quadratic potential curve is determined by the magnitude of the force constant. Larger values for the force constants denote steeper, narrower parabolas, reflecting more restricted motion in the internal coordinate. Anharmonic corrections, additional terms past the linear restoring force are sometimes included.
The last two terms determine the long-range, or nonbonded, interactions of the system. It is important to note that for a system of N atoms each of these terms requires a double summation involving, at the most, N(N-1)/2 separate pairwise interactions (an order N2 computation). All other terms in the energy expression require sums only of order N. This order of magnitude difference is the primary reason for the vastly greater computational effort required to evaluate the nonbonded energy as compared to the rest of the potential energy calculation. A brief review on the derivation of the term N(N-1)/2 can be instructive. This term is derived from simple probability theory for the number of different sets of m objects taken from a system of N total objects, where the order in which the m objects are acquired is unimportant. The number of combinations of N objects is simply N!. If we choose m objects from this set, the number of combinations among the m objects does not matter (of which there are m!) and the arrangement of the remaining N-m objects is insignificant (a total of (N-m) possibilities. The total number of unique combinations is therefore,
The van der Waals energy is determined through the use of a 6-12 Lennard-Jones potential. Since individual interactions are described in terms of well-depth, and position of the minimum, it is important to note the form of the Lennard-Jones potential using these parameters:
The electrostatic energy between two atoms is based on a simple Coulombic interaction of two point charges Qi and Qj. The charges are expressed in terms of electrostatic units (eu's) with 1 eu = 1.6 10-19 coulombs. The constant, k, in the expression has a value of 332.5 (Å kcal eu2 mole-1) in order to express energies in terms of kcal/mole. The proper value of the dielectric constant is a matter of some contention. The most common practice is to set the dielectric constant = 1.0 for systems which explicitly include solvent, and to use a distance-dependent dielectric, = Rij, to model the solvent screening effect in the absence of explicit solvent. The use of the distance-dependent dielectric constant is also referred to as the application of a "solvent-averaged" potential. Of course, in vacuo studies set the dielectric constant to unity.
The potential interaction parameters have various methods of determination: