Molecular Mechanics Notes

(Leach - Ch 3; Grant & Richards, Ch. 3)

• empirical potentials fitted to experimental and calculational data
• composed of stretches, bend, torsion and non-bonded components (e.g. stretch component has a term for each bond in the molecule)
• potential energy curves for individual terms are approximately transferable (e.g. CH stretch in ethane almost the same as in octane)
• terms consist of functional forms and parameters
• parameters chosen to fit structures (in some cases also vibrational spectra, steric energies
• each atomic number divided into atom types, based on bonding and environment (e.g. carbon: sp³, sp², sp, aromatic, carbonyl, etc.)
• parameters assigned based on the atom types involved (e.g. different C-C bond length and force constant for sp³-sp³ vs sp²-sp²)
• force field comprised of functional forms, parameters and atom types
• examples: MM2, MM3, Amber, Sybyl, Dreiding, UFF, MMFF, etc.
• some force field use united atoms (i.e. H's condensed into the heavy atoms) to reduce the total number of atoms (but with a reduction in accuracy)
• do not mix and match - each developed to be internally self consistent
• MM force fields differ from force fields used for vibrational analysis, and analytical potential energy surfaces used for dynamics - these are custom fit for individual systems
• MM force fields are designed to be transferable, and can be used for broad classes of molecular systems (but stay within the scope of the original parameterization)

Bond Stretch term

• many use just quadratic term, but too large for very elongated bonds
• Morse (usually not used because of expense), cubic (wrong asymptotic form), quartic
• reference bond length not the same as the equilibrium bond length, because of non-bonded contributions

Angle Bend term

• usually quadratic is sufficient
• special atom type may be used for very strained atoms (e.g. cyclopropane)

Torsion term

• most use a single cosine with appropriate barrier multiplicity
• some use a sum of cosines for 1-fold (dipole), 2-fold (conjugation) and 3-fold (steric) contributions
• improper torsions used for out-of-plane bends (and chirality constraints in united atom force fields)

Non-bonded term

• van der Waals interactions
• repulsive part - overlap of electron distributions - Pauli exclusion
• attractive part - London or dispersion forces
• instantaneous dipole - induce dipole interaction r ^-6
• Lennard-Jones: 4 epsilon ( (sigma / r)¹² - (sigma / r)⁶) (easy to compute, but r ^-12 rises too rapidly)
• Buckingham: A exp(-Br) - C r^-6 (QM suggests exp repulsion better, but is harder to compute)
• tabulate s and e for each atom - mixed terms: sigma_AB = (sigma_AA + sigma_BB)/2;  epsilon_AB = (epsilon_AA epsilon_BB)^1/2

Electrostatics

• charges can be obtained from MO calculations (expensive, some difficulties)
• atom centered charges obtained by fit to ESP - may vary with conformation
• can also include atom centered multipoles for better fit to ESP
• possibly use off-center charges for better representation of ESP around lone pairs
• cheaper (but less accurate) method for calculating charges - electronegativity equalization
• polarization effects - compute iteratively (expensive and not that much of an improvement)
• include polarization effects in an average way with distance dependent dielectric constant

Hydrogen bonding

• some force fields add extra term: A/r¹² - C/r¹⁰
• others just use a balance between electrostatic and non-bonded terms

Parameterization

• difficult, computationally intensive, inexact
• fit to structures (and properties) for a set of molecules
• recent generation of force fields fit to ab initio data at minima and distorted geometries
• trial and error fit, or least squares fit
• different parameter sets and functional forms can give similar structures and energies
• don't mix and match

Energencies

• steric energy - energy relative to an artificial structure with no interactions - can be used to compare different conformers of same molecule
• strain energy - energy relative to a strainless molecule - e.g. all trans hydrocarbons
• heat of formation - average bond energies added to energy t get approximate atomization energy
• very dangerous to decompose steric energy into components
• different force fields can give similar energies and structures but quite different components

Applications

• geometry and relative energies of conformers of the same molecule
• effect of substituents on geometry and strain energy
• well parameterized for organics, less so for inorganics
• specialty force fields for proteins, DNA, for liquid simulation
• molecular mechanics cannot be used for reactions that break bonds
• simple organic problems: ring strain in cycloalkanes, conformational analysis, Bredt's rule
• high end biochemistry problems: docking of substrates into active sites, refining x-ray structures, determining structures from NMR data, free energy simulations