Molecular force field, analytical energy function

Molecular mechanical force fields use the equations of classical mechanics to describe the potential energy surfaces and physical properties of molecules. A molecule is described as a collection of atoms that interact with each other by simple analytical functions. This description is called a force field. One component of a force field is the energy arising from compression and stretching a bond. 

In molecular-mechanics calculations, the atoms are considered to move in a force field defined by an energy function based on classical (rather than quantum) mechanics. Thus, the energy of a given molecular conformation is not calculated in an iterative SCF procedure, as in quantum-chemical approaches, but rather uses an analytical formula based on effective potentials. 

If we could analytically evaluate the action of the operator in Eq. [144] for any potential energy function or force field, there would be no need for molecular dynamics simulation. Since that is generally not possible, we need to devise a numerical scheme that will solve Eq. [144] to a desired accuracy, while preserving the symmetry of the equations of motion (e.g., time-reversal symmetry). 

Many molecular modelling techraques that use force-field models require the derivatives of the energy (i e the force) to be calculated with respect to the coordinates. It is preferable that analytical expressions for these derivatives are available because they are more accurate and faster than numerical derivatives. A molecular mechanics energy is usually expressed in terms of a combination of internal coordinates of the system (bonds, angles, torsions, etc.) and interatomic distances (for the non-bonded interactions). The atomic positions in molecular mechanics are invariably expressed in terms of Cartesian coordinates (unlike quantum mechanics, where internal coordinates are often used). The calculation of derivatives with respect to the atomic coordinates usually requires the chain rule to be applied. For example, for an energy function that depends upon the separation between two atoms (such as the Lennard-Jones potential. Coulomb electrostatic interaction or bond-stretching term) we can write ... 

The explicit expressions used for each of the terms in (16.94) define what is called a molecular-mechanics force field, since the derivatives of the potential-energy function determine the forces on the atoms. A force field contains analytical formulas for the terms in (16.94) and values for all the parameters that occur in these formulas. The MM method is sometimes called the empirical-force-field method. Empirical force fields are used not only for single-molecule molecular-mechanics calculations of energy differences, geometries, and vibrational frequencies, but also for molecular-dynamics simulations of liquids and solutions, where Newton s second law is integrated to follow the motions of atoms with time in systems containing hundreds of molecules. 

---