Lecture Notes

Although our focus is on hybrid quantum mechanical and molecular mechanical potentials, we start by briefly reviewing empirical force fields or molecular mechanics that have been widely used in the study of liquids and proteins. Perhaps the most common goal in computer simulations of molecular systems is the prediction of equilibrium molecular structure, the relative stabilities of molecules at and away from equilibrium, and reproduction of the dynamics of molecular motions - in terms of fundamental inter and intramolecular forces. To achieve this, we need to know the potential energy of the system as a function of the molecular structure or coordinates. Chemical intuition and experience indicate that atoms, bonds, bond angles, etc. show regularity and have characteristic values and properties, reflected in equilibrium geometry, vibrational frequencies, and dissociation energies. Thus, molecular mechanics assumes that the potential energy of a molecule near its equilibrium configuration is a simple function of atomic distances, bond angles, etc., which is transferable among typical molecules.

(1)

where the terms in equation (1) are energy functions for, respectively, bond stretching, angle bending, torsional angles, out-of-plane bends, non-bonded interactions, and various cross terms.

A practical problem in molecular mechanics is the choice of "internal" coordinates to specify and define the molecular energy. The reason that there is a problem is because the total number of internal coordinates in a molecule is typically greater than the number of degrees of freedom of the system. In other words, many internal coordinates are linearly dependent, or redundant. For example, in formaldehyde, H₂C=O, only 2 out of the 3 bond angles are independent because the sum of the 3 is always 360^o for a planar molecule. Therefore, just 2 angles will be sufficient to describe the angle bending energies; however, there is no particular reason or advantage to choosing any pair out of the 3 possibilities. A direct consequence is that the force constants parameterized for this molecule are not unique, because the three force constants are linearly related. This artifact lead to nontransferability.

A well defined force field (and force constants) is the so-called spectroscopic force constants, obtained from a Taylor expansion of the potential energy function at equilibrium geometry, using a well-defined set of independent combination out of the redundant coordinates. However, this set itself is not unique. We only note that typically, spectroscopic force fields are not appropriate for simulation of molecular systems such as liquids and proteins.

Molecular mechanics attempts to incorporate the physical nature of interactions into the force field, rather than a mathematical formality. For a particularly type of force constant, for example, the H-C=O angle bending, by considering a large number of molecules containing this unit, one hopes that the correlation with other coordinates will be removed in the parameterization. Thus, it may be transferable.

Bond stretch. Often a simple harmonic, spring-like representation is used to describe bond stretch in force fields for biomolecules (e.g., CHARMM, AMBER):

(2)

A more general expression perhaps is the Morse function for diatomic molecules:

(3)

D_e is the experimental dissociation energy in the case of diatomic molecules. It will be an empirical parameter in force field parameterizations. Note that eq (2) is the first term of the Taylor expansion of eq (3) about the equilibrium point.

It should be pointed out that the value b₀ is not the equilibrium bond length in any particular molecule in general. It is the bond length of the bond in a hypothetical unperturbed state. Interactions with all other terms in the force field will lead to, hopefully, the correct equilibrium bond distance in a molecule up on minimization of the potential energy.

Angle Bending. It has an expression of a simple harmonic representation. In some cases, anharmonic terms may also be included.

(4)

Similarly, ?₀ should not be recognized as equilibrium bond angle in any molecules.

Torsional Potentials. A cosine series is traditionally used:

(5)

The number of terms N varies depending on the type of molecules and force fields.

Out-of-Plane Bend. For planar molecules, such as H₂C=O, there is the displacement of the trigonal atom above and below the molecular plane. Although this motion is a linear combination of torsional angles, it is sufficiently different and is often considered as an internal coordinate in the force field. Now, this introduces additional redundancy in coordinate, and defining such a motion. It is often treated as an "improper" dihedral angle shown below:

(6)

In AMBER, v_? is expressed as k_? (1 - cos 2?).

Nonbonded Interactions. Non-bonded interactions are generally considered the most difficult part in force field parameterizations because it deals short and long range electrostatic and dispersion interactions. In non-polarizable (or effective pair-wise potential) force fields, the typical expression is a Lennard-Jones term plus a Coulombic term, though other forms are also used.

  (7)

where {q_i}, {s_i}, and {e_i} are empirical parameters.

Call MacKerrel.

The determination of the force constants is not an easy job. One usually start by defining a set of molecules whose associated experimental properties are the target to be fitted by computations with an force field. Then the resulting parameters are used to predict properties of other molecules, outside of the training set, to verify the performance of the force field. The experimental (or sometimes ab initio results) properties include gas phase molecular structures, vibrational frequencies, torsional barriers, crystal lattice constants, sublimation energies, hydrogen-bonding energies and geometries, liquid properties (density, heats of vaporization, radial distribution functions), and free energies of solvation.