Molecular Structure and Bonding

Molecular Mechanics

Background When it is of interest to study structure and bonding in very large molecular systems comprising hundreds or even thousands of atoms, many of the more sophisticated molecular modeling techniques are too demanding of computer time resources to be of general use. Molecular mechanics calculations, also known as force field calculations, can be of considerable use in the qualitative descriptions in these cases. In these cases, we concentrate on the structural aspect and not on the electronic and/or spectroscopic properties. In essence, we describe the potential energy surface without invoking any quantum mechanical calculations or descriptions. The Born-Oppenheimer approximation, fundamental to our molecular description, 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. This can be interpreted in one of two manners, one of which allows the study of the electronic structure, one of which allows the study of the molecular mechanics structure.

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.

The Born-Oppenheimer surface is then a multidimensional surface describing the energy of the molecule in terms of nuclear positions.

Energy = f(nuclear positions)

Based on our history of physical and chemical experiences, we can think of molecules as mechanical assemblies made up of simple elements like balls (atoms), rods or sticks (bonds), and flexible joints (bond angles and torsion angles). Present: Molecular mechanics tends to be one of the standard methods of structural chemistry.

Potential Function:

There are many variations in the functional forms of force fields.

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 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.

Example: Bond Potential

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

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: 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.

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: 1,4-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.

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: 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)
Modified Urey-Bradley 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.