treats molecules as arrays of hard, impenetrable balls connected by springs. The collection is governed by classical, mechanical principles, so that the energy may be represented a a sum of terms, each representing one possible mode of mechanical deformation of the molecule from an idealized geometry.

E_total = E_stretch + E_bend + E_torsion + E_vdW + E_dipole

A. E_stretch

• The energy of a pair of atoms as a function of the distance between them is given by a Morse curve:
  

• r_o is the equilibrium bond distance.

• Stretching or compressing the bond gives an increase in energy.

• The Morse curve is approximated with a mathematically simpler Hooke's Law function, which reproduces the bottom of the well pretty well, but fails at very long bond lengths:

  V = k (r - r_o)² / 2 + k' (r - r_o)³

  • V is the potential

  • k and k' are force constants appropriate for the particular atoms

  • r is the actual bond distance

• Typically the cubic term is used only in the later stages of computation because it becomes more attractive than is realistic at long bond lengths.

B. E_bend

• Also is based on a Hooke's Law type of potential

• Most modern computations include a higher order term as well:

  

  • V is the potential

  • q_o is the ideal value of the bond angle

  • q is the actual angle

  • k and k' are force constants

C. E_torsion

represents energy arising from bonds not fully staggered. It includes three-fold (like ethane) potentials, as well as two-fold (alkenes, carbonyls) and one-fold (alkynes) potentials.

• A typical form is:

  

  • q is the dihedral angle

  • the v's are force constants appropriate to the particular kinds of atoms and bonds involved.

D. E_vdW

refers to repulsions between non-bonded atoms.

• These potentials often are constructed of two terms:

  E_vdW = a/r¹² - b/r⁶

• This is known as a 6-12 potential (r is the atom separation),

• But the so-called "exponential minus six" is now generally regarded as more accurate:

  V = k e^-k'/p - k''p^-6

  • p = (r_i + r_j)/r_ij,

  • the k's are force constants,

  • r_i and r_j are the covalent radii of the atoms

  • r_ij is the separation between them.

E. E_dipole

is the energy of interaction of bond dipoles and any point charges,

• It can be derived from classical electrostatics:

  

  • the m are the bond dipoles

  • q is the angle between them

  • f are the angles made by the bond dipoles with a line joining their midpoints

  • r is the separation

  • e is the dielectric constant of the medium between the dipoles.

• Not all programs allow adjustment of e, and there is not general agreement that the variations in energy that result actually are representative of a real physical situation.

F. Evaluation of Force Constants

• In principle, the force constants, dipole moments, and standard structure parameters needed in the equations above all could be obtained expermentally.

• In practice, only the structure parameters are determined directly.

• All others are generated by ab initio calculations (see, for example, the MM3 references cited)

• The values so obtained are adjusted to give the best fit of structure and energy for a test suite of organic molecules.

In contrast to molecular orbital (MO) methods, molecular mechanics (MM) is bond-oriented.

• The placement of bonds must be defined in the input, whereas for MO calculations, we place the atoms and "let the bonds fall where they may".

• In fact, input to an MM program is a model corresponding to the classical valence bond picture of the molecule.

• Note that this approach assumes that the properties of bonds and atoms are independent of the molecule in which they are located.

• If the parameterization of the program is not exact for the kinds of bonds specified, the calculation will fail, which is one of the major defects in molecular mechanics.

G. Conjugated Molecules

Molecules containing multiple isolated double bonds are easily handled in MM by using the same force constants applied to simple alkenes like propene. In conjugated molecules, however, the geometry and extent of conjugation are interrelated.

One way to deal with this would be to define parameters specifically for the various kinds of atoms and bonds of each possible conjugated structural situation.

• This approach works well for structures like benzene rings, that do not vary significantly in geometry as substituents are changed.

• The MMX force field that PCModel employs therefore makes this option available for aromatics, by defining an aromatic carbon type.

For other kinds of structures, a better option is to do an MO calculation on the conjugated system, derive bond orders from it, and use the bond orders to modifiy the force field.

• Using the new parameters, a standard MM calculation then can be done, the MO calculation repeated with the new geometry, and the processes cycled to convergence.

• The eventual p-delocalization energy then can be included in the computation of the enthalpy of formation.

PCModel incorporates force field parameters that will reproduce the properties of simply conjugated functional groups like esters and amides without requiring a MO calculation on these groups.

• However, adjacent conjugated groups will not experience any interaction with these groups unless a p- calculation is done.

• The method employed is a standard VESCF procedure, and can be done as either a closed or open shell type, so that monoradicals are handled reasonably well.