Molecular Mechanics

Brief Review:

Molecular mechanics is an approach to modeling the behavior of matter. It begins with a fundamental assumption that matter consists of 'atoms' and the potential energy of a collection of atoms can be defined for every set of positions. The collection of atoms is treated as a mechanical system moving with this potential energy. Details about the system, such as the equilibrium structure, the vibrational spectrum, thermodynamic properties, equations of state, and reaction rates, can then be computed. Unlike quantum mechanical approaches, electrons are not explicitly included in these calculations. This is possible due to the BO approximation, which states that the electronic and nuclear motions can be uncoupled from one another and considered separately. MM assumes that the electrons in a system find their optimum distribution, and approaches chemical problems from the standpoint of the nuclear structure. A molecule from this perspective is considered to be a collection of masses that are interacting with each other via (almost) harmonic forces, and it is rather analogous to a system composed of weights joined together by springs ( a ball and spring model). Potential energy functions are used to describe these interactions between nuclei. With judicious parameterization, the electronic system is implicitly taken into account. Any deviation of the model from the "ideal" Molecular geometry will correspond to an increase in energy. The description of a molecule in terms of harmonic interactions is only a first approximation. As insights into molecular behavior have been discovered, and as more and better experimental data have become available and been utilized, it has become possible to use increasingly more sophisticated equations to reproduce molecular behavior.

The potential energy surfaces are also called "force fields" because the derivative of the potential energy determines the force acting on the atom when dynamics are considered. Spectroscopists frequently use the term "force field" to mean a similar (but different) set of equations designed to reproduce or predict vibrational spectra. A really accurate force field would give both structure, vibrational spectra, and related properties. however, current spectroscopic force fields cannot ordinarily be used to determine structure, and current molecular mechanics force fields usually give spectra that are only fair. Spectroscopic force fields does not include electrostatic or van der Waals interactions, and hence such force fields can be applied to a group of molecules only if those interactions remain constant ion the group. Because spectroscopic frequencies can be measured to an accuracy that is an order of magnitude greater than their structural significance, a force field that gives excellent structural results will still show errors of the order of 30 cm-1 in calculated spectroscopic frequencies.


Description:

Goals:

Tools:


Potential Energy Surface (PES):


Origins of Molecular Mechanics:

Many of the terms and ideas of MM have their origin in normal coordinate analysis. Vibrational Spectroscopists are interested in the forces that hold molecules together, and the approach they have taken is to do a normal coordinate analysis.

Andrews, D.H. Phys. Rev. 1930, 36, 544.


Vibrational Spectroscopic Force Fields vs Molecular Mechanics Force Fields


Spectroscopists: Determine forces holding molecules together from knowledge of structure and vibrational frequencies.

Vibrational energies and wavefunctions and force constants are related by a secular determinant, just like electronic energies and wavefunctions are related through the Schrodinger equation.

|H - ES| = 0

Electronic Secular Determinant

|FG - EL| = 0

Vibrational Secular Determinant


History of Molecular Mechanics:


Basic ideas of MM date back to the 1930's. Serious attempts to use the method were not forthcoming until 1946. In that year, three important developments occurred:

General Time Line:

Potential Function:

V=EVb(b) + EVo(o)+EVr(r) + EVx(x)+


EVnb(r) + EVbb'(bb') + EVbr(b,r) +....

nonbonded..bond/bond coupling..bond/angle coupling


Assumptions:

Force Field (the set of potential functions): Example: Bond Potential Internal Energy: Vibrational Spectroscopic Force Fields:
  1. General Harmonic Force Fields (GHFF)
  2. Constrained Force Fields (CFF)
Force Field:

Potential Functions: Classification of Force Fields:

Vibrational Spectroscopic Force Fields:
  1. General Harmonic Force Fields (GHFF)
  2. Constrained Force Fields(CFF)

    Direct constraints Force Field
    • Fixes values of all interaction force constants from vibrations separated by at least two noncommon atoms.
    1. Valence Force Field
      • All interaction force constants between stretching and bending of different bonds set equal to zero.
      • No cross terms included.

      Too simplistic for quantitative application. A slight relaxation is to allow some stretch-stretch and stretch-bend terms.
    2. Central Force Fields
      • Steric effects considered in a very limited way
      • Describes angle bending in terms of nonbonded interaction distances. Rarely used. Like VFF, this is too simplistic to explain molecular frequencies. Relaxation of some constraints improves the situation by allowing interactions between interbond parameters.
    3. Urey-Bradley Force Field: Adds 1,3 interactions longer ranged nb terms
      • In addition to conventional stretching and bending force constants, all interactions that are stretching/stretch-bend coordinated are defined in terms of interactions between nonbonded atoms.
        Well established shortcomings. Inadequacies are limited by allowing stretch-stretch and stretch-bend interactions missing in the simple model (Modifies Urey-Bradley Force Field)
      • Nonbonded terms can eliminate need for cross terms (eliminates force constants; therefore replaced w/ others ~ same # of parameters on Valence Force Field and Urey-Bradley Force Field.

      Orbital Valence Force Field
      • Like the UBFF in its treatment of interactions in terms of repulsions between nonbonded atoms.
      • Unlike UBFF, treats angle distortions in terms of maximizing orbital overlap.
    Molecular Mechanics Force Fields/Empirical Force Fields (EFF)
    1. Modified Urey-Bradley Force Fields
    2. Central Force Fields
    3. Valence Force Fields (most popular)

    Note:
    • The MM representation is a gross oversimplification, justified only by history of research.
    • Authors of various EFF's partition the component terms in vastly different ways:
      • Some have every conceivable kind of interaction treated.
      • Others have only 5 - 10 terms,>li>Missing terms accounted by judicious parameterization
    • The numerical value of V has no inherent physical meaning.
    • Directly dependent on potential functions and parameters.
    • Difference between energies of isomers is usually predicted well implying that MM is usually well suited for confrontational analysis.