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 cm1 in calculated
spectroscopic frequencies.
Description:
 Simplest, mechanical method for molecular structure
 Also called empirical force field or Westheimer methods.
 Molecular Mechanics methods are based on the classical Newtonian
physics of balls and springs, but require quantum mechanical concepts like
force constants. However, instead of considering the electrons explicitly
and representing the potential energy function as a sum of the nuclear
repulsion energy and the electronic energy obtained from an approximate
solution to the electronic Schrodinger equation, an empirical potential
function is used.
 The methods, a natural outgrowth from older ideas of bonding between
atoms in molecules, and Van der Waals forces between nonbonded atoms.
Goals:
 Predict equilibrium structure
 Predict relative stability
 Reproduce molecular motion
Tools:
 Energy minimization
 Molecular dynamics
 Vibrational; analysis
 Monte Carlo
Potential Energy Surface (PES):

BornOppenheimer Approximation
States that the Schrodinger Equation for a molecule can be
separated into a part describing the motions of the electrons and a part
describing the motions of the nuclei and that these two motions can be
studied independently.
Electronic Structure: It is common practice to establish the
positions of the nuclei of the system by some method and then to study the
electronic structure using fixed nuclear positions.
Molecular Mechanics (Vibrational Spectroscopy): The motions of the
nuclei are studied and the electrons are not explicitly examined at all,
but are assumed to find an optimal distribution about the
nuclei.
 BornOppenheimer surface is then a multidimensional surface describing
the energy of the molecule in terms of nuclear positions.
Energy = f(nuclear positions)
 Total energy=Kinetic Energy + Potential Energy
 Kinetic energy = 1/2*m*v**2
 Potential energy = V(r)
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
 H: Hamiltonian matrix
 E: energy
 S: overlap matrix
FG  EL = 0
Vibrational Secular Determinant
 G: inverse kinetic energy matrix
 E: unit matrix
 L: eigenfrequencies, related to normal mode frequencies
 F: force constants
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:
 First development:
 Hill. T.L. J. Chem. Phys. 1946, 14, 465
Proposed that Van der Waals interactions together with stretching and
bending deformations should be used to minimize steric energies and that
this would lead to information regarding structure and energy in congested
systems.
 Second set of simultaneous developments:
 Dostrovsky, I.: Hughes,E.d.; Ingold,C.K. J.Chem. Soc. 1946, 173.
 de la Mare, P.; Fowden, L.; Hughes, E.D.; Ingold, C.K.; Mackie, J.
Chem. Soc. 1944. 3200.
Simultaneously and independently utilized the same basic schemes in an
effort to better understand the rates of which various halides underwent
SN2 reactions. The complexity of the problem was so great and the necessary
available information of such limited accuracy, that the results were not
very convincing at the time, but they foreshadowed events to some.
 Third set of developments:
 Westheimer, F.H.; Mayer, J.E. J. Chem. Phys. 1946, 14, 733.
 Westheimer, F.H.; Mayer, J.E. J. Chem. Phys.. 1947, 15, 252.
 Reiger, M.; Westheimer, F.H. J. Am. Chem. Soc. 1950, 72, 19.
 Westheimer, F.H. "Steric Effects in Organic Chemistry", 1956.
The most significant work was contributed in this set, on a less
complicated problem than studied earlier. This problem concerned the rates
of racemization of some optically active halosubstituted biphenyls, giving
some very impressive results. Westheimer's calculations were important in
showing that the method could be used to rationalize certain properties
involving geometries and energies of molecules.
General Time Line:
1940's: Westheimer method not widely used
1950's: Advent of computers caused methods to grow in popularity at a rapid
rate.
Present: Molecular mechanics tends to be one of the standard methods of structural chemistry.
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
 There are
many varieties of force fields in existence.
 This function is presumed to have a minimum corresponding to stable equilibrium
geometries.
 Treatment relies heavily on availability of exptal data for
parameterization; thus, EFF methods are interpolations and sometimes
extrapolations of existing data.
Assumptions:
Force Field (the set of potential functions):
 Bonds have 'Natural lengths and angles. All force fields consider a
molecule as a collection of [particles held together by some sort of
elastic forces. The atoms of a molecule may be thought of as joined
together by mutually independent springs, restoring "natural" values of
bond lengths and angles.
 As in a diagonal valence force field, one can then assume a harmonic
potential with Hooke's law functions for bond stretching and bending. All
forces are defined in terms of potential E Functions of the internal
coordinates of the molecules that constitute the molecular force field.
 Inthese expressions, the sums extend over all bonds, angles, torsions,
and nonbonded interactions between all atoms not bound to each other or to
a common atom.
 Simple force fields include bond stretching, angle bending, torsions,
and nonbonded interactions between all atoms not bound to each other or to
a common atom.
 More elaborate force fields may also include either UreyBradley terms
(1,3nonbonded interactions) or crossinteraction terms, electrostatic
terms, Hbonding terms, etc.)
 Molecules adjust geometrics according to these values.
 Steric interactions are included using V.d.W. potentials.
 Strained systems deform in predictable ways with "strain" energies that
can be calculated.
Example: Bond Potential
 Ethane molecule
 Vb(C1C2)=1/2k(bb0)2
 b0:Strain free bond length, 1.54A
 k: force constant
 Picture (parabola: E vs b)
 In general, for the entire molecule:
Vb(b)=E1/2Kb(bb0)2
Internal Energy:
 The sum of all these terms is called the steric energy of a molecule.
 In the simplest force fields, perhaps a dozen or so parameters are
necessary to describe the alkanes. Such a force field gives a reasonable
approximation to molecular structures and E differences, but the values
obtained from such a force field are clearly inferior to values measured
experimentally.
 To better reproduce the available exptal data, further optimization of
the parameters of the introduction of more of them by modifying the
equations that make up the force field might be considered.
Vibrational Spectroscopic Force Fields:
 General Harmonic Force Fields (GHFF)
 All elements of the F matrix are considered in the analysis
 Good for very small molecules with high symmetry
 Constrained Force Fields (CFF)
 More appropriate for larger molecules
 Systematic simplifications are made of the complete force field
 Basic assumption: offdiagonal elements of the F matrix are much
smaller than diagonal elements and can be set to zero.
(i.e., higher freq. stretch will not interact much with a low freq. bend.)
 Direct constraints Force Field
 Fixes values of all interaction force constants from vibrations
separated by at least two noncommon atoms.
Force Field:
 First used by spectroscopists to mean a set of equations designed
to reproduce or predict vibrational spectra.
 Adopted by Molecular Mechanics as a model for describing PES for all
degrees of freedom of a molecular system.
 Similar to the idea of hand held models, but more sophisticated in the
sense that they allow some internal motion:
Example: 1,4cyclohexadiene
Although handheld models can facilitate comp[rehension of the 3D
aspects of molecular structure (conformation), they can be misleading. The
fact that the models do also for internal motion (vibrations), they may not
correspond to a chemically real structural property.
In this molecule, all pieces of experimental evidence say that it is planar
with wide amplitude ring puckering. On the other hand, according to the
precise Dreiding molecular models, the planar form of this molecule is very
unstable, a slight jiggling of the model causing a sudden change to boat
conformation.
Unfortunately, many experimental NMR results of dihydrobenzene,
dihydronapthalene, and dihydroanthracens were interpreted on the
preconceived notion that planar forms are unstable as predeicted by the
handheld models and as indicated by solid state structures that were
influenced by crystal packing forces. Unlike the handheld models, the
computational forcefield model correctly predicts the conformation of
various dihydrobenzenes.
 However, even the computational model may be misused and so familiarity
with the methodology and the literature can help ensure proper
use
Mathematical expression for the MM force field:
V=Evb(b) + EVo(o) + EVr(r)+EVx(x)+ V related to k
bonds........angles....dihedrals..o.o.p. bends V=1/2r ki/mi
reduced mass
EVnb(r) +
nonbonded
Where the forces for the internal motions are described in terms of
potential energy functions which in turn sum to give the overall molecular
potential energy or steric energy,E, of the molecule:
Et=Eb+Eo (Bayer strain)+E(1) (Pitzer strain)+Enb "structural
deformation" description of strain
Each component representing a molecular deformation from an arbitrary
('natural') reference geometry.
However, no molecule is strain free. F nonbonded interaction
Works ok; Saturated hydrocarbons behave as though their e were
localized=>structural units and environment independent.
Varieties of Force Fields:
 Many modifications
 Presumed to have a minimum corresponding to stable equilibrium geometries.
 Treatment relies heavily on availability of experimental data for
parameterization.
 In the simplest force fields, perhaps a dozen or so parameters are
necessary to describe the alkanes. Such a force field gives a reasonable
approximation to molecular structures and E differences, but the values
obtained from such a force field are clearly inferior to values measured
experimentally
 To better reproduce the available exptal data, further optimization of
the parameters or the introduction of more of them by modifying the
equations that make up the force field might be considered.
 EFF methods are interpolations and sometimes extrapolations of existing
data.
 Pictorial diagram of various components.
 Simple force fields:
 Bond potentials
 Angle potentials
 Dihedral potentials
 nonbonded potentials
 About a dozen or so parameters
 Reasonable approximation to molecular structure and E differences \
Absolute values probably inferior to experimental values
 More elaborate force fields:
 All of the above terms
 UreyBradley terms (1,3nonbonded interactions)
 Cross terms
 Electrostatic terms
 Hbonding terms
 Many more parameters>further optimization
 Systematic errors point to terms that need to be added. Each
modification means a reoptimization.
Potential Functions:
 In general, there are no rules as to what functions are to be
chosen or what parameters are to be used.
 Some EFF's are designed to reproduce strictures and E's only.
 Other EFF's want to calculate strictures, energies, and vibrational
spectra (CFF's)
 All force fields consider a molecule as a collection of particles held
together by some sort of elastic forces.
 Each of these forces are defined in terms of potential energy functions
of the internal coordinated of the molecules that constitute the molecular
force field.
 The function is presumed to have a minimum corresponding to stable
equilibrium geometries.
Classification of Force Fields:
Vibrational Spectroscopic Force Fields: General Harmonic
Force Fields (GHFF)
 All elements of the F matrix are considered in
the analysis
 Good for very small molecules with high symmetry
 Constrained Force Fields(CFF)
 More appropriate for larger molecules
 Systematic simplifications are made of the complete force field
 Basic assumption: offdiagonal elements and can be set to zero.(i.e.,
higher freq. stretch will not interact much with a low freq.. bend.)
Direct constraints Force Field Fixes values of all
interaction force constants from vibrations separated by at least two
noncommon atoms.
 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 stretchstretch and stretchbend terms.
 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.
 UreyBradley
Force Field: Adds 1,3 interactions longer ranged nb terms
 In
addition to conventional stretching and bending force constants, all
interactions that are stretching/stretchbend coordinated are defined in
terms of interactions between nonbonded atoms.
Well established
shortcomings. Inadequacies are limited by allowing stretchstretch and
stretchbend interactions missing in the simple model (Modifies
UreyBradley 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 UreyBradley 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)
Modified UreyBradley Force Fields
 Central Force Fields
 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.