It is difficult to accept that the behavior of atomistic systems, which behave according to quantum rather than classical laws, could be accurately described by the application of classical Newtonian mechanics. This approach can be justified, however, by considering the de Broglie expression for the thermal wavelength-Л,

where T is the temperature and M is the atomic mass. The approximation of classical behavior holds if Л = a, where a is the mean nearest neighbor separation. This holds for “heavy” liquid systems at all but the lowest temperatures at which quantum effects become important.

To describe the molecular dynamics of a system classically a function representing the potential energy of the system, together with the related parameters is required. Typically, the energy is calculated from the sum of bonded and non-bonded interactions,

E total Ebond + E angle + E dihedral + E non-bond

The exact form of the terms in the above potential function and the associated parameters varies across different molecular mechanical force fields. Some force fields also include a cross term representing coupling between the first three terms in Equation (8.2). Examples of commonly used force fields include the Allinger MM2 and MM3 series (3,4), CHARMM (5), AMBER (6), and GROMOS (7). Each force field has a slightly different ethos and is typically suited to the study of one class of molecules. Of those mentioned above CHARMM, AMBER, and GROMOS would be considered protein force fields suited to the simulation of proteins, nucleic acids, and with suitable parameterization, carbohydrates. The underlying equations for these three force fields are similar, although there are subtle differences. For the purposes of discussion here we will restrict ourselves to the AMBER force field equation, which is

V(rn) =J2 Kr (Г — req )2 +J2 Kв (Є — ®eq )2

bonds angles

+ ^ V [1+cos(^- 7)]+^ Rj

dihedrals i < j |_ i

where the potential energy V is a function of the positions r of n atoms. Kr, req, Kв, Qeq, Vn, n, ф, 7, Aij, Bij, £r, qi, and qj are all empirically defined parameters. The first three terms in the above equation correspond to the bond, angle, and dihedral terms, respectively, while the last term describes the non-bonded van der Waals and electrostatic interactions.