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:

• 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.
  

• Predict equilibrium structure
• Predict relative stability
• Reproduce molecular motion

Tools:

• Energy minimization
• Molecular dynamics
• Vibrational; analysis
• Monte Carlo

Potential Energy Surface (PES):

• Born-Oppenheimer 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.

• Born-Oppenheimer 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)

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.

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: Hamiltonian matrix
• E: energy
• S: overlap matrix

• 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 halo-substituted 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.

• 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 Urey-Bradley terms (1,3-nonbonded interactions) or cross-interaction terms, electrostatic terms, H-bonding 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.
  
  

• Ethane molecule
• Vb(C1-C2)=1/2k(b-b0)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(b-b0)2
  
  

• 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.
  
  

1. General Harmonic Force Fields (GHFF)
   • All elements of the F matrix are considered in the analysis
   • Good for very small molecules with high symmetry
2. Constrained Force Fields (CFF)
   • More appropriate for larger molecules
   • Systematic simplifications are made of the complete force field
   • Basic assumption: off-diagonal 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.)
     1. Direct constraints Force Field
     • Fixes values of all interaction force constants from vibrations separated by at least two noncommon atoms.

• 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: 1,4-cyclohexadiene
  
  Although hand-held models can facilitate comp[rehension of the 3-D 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 hand-held models and as indicated by solid state structures that were influenced by crystal packing forces. Unlike the hand-held models, the computational force-field 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:
  1. Bond potentials
  2. Angle potentials
  3. Dihedral potentials
  4. 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:
  1. All of the above terms
  2. Urey-Bradley terms (1,3-nonbonded interactions)
  3. Cross terms
  4. Electrostatic terms
  5. H-bonding terms
• Many more parameters-->further optimization
• Systematic errors point to terms that need to be added. Each modification means a reoptimization.

• 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.

1. General Harmonic Force Fields (GHFF)
   • All elements of the F matrix are considered in the analysis
   • Good for very small molecules with high symmetry
2. Constrained Force Fields(CFF)
   • More appropriate for larger molecules
   • Systematic simplifications are made of the complete force field
   • Basic assumption: off-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.
   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.