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).

 

  The method then treats a molecule as a collection of particles held together by simple harmonic forces. These various types of forces are described in terms of individual potential functions which in sum constitute the overall molecular potential energy or steric energy of the molecule.

In its simplest representation, the molecular mechanics equation is

E = Es + Eb + Ew + Enb

  where Es is the energy involved in the deformation of a bond, either by stretching or compression, Eb is the energy involved in angle bending, Ew is the torsional angle energy, and Enb is the energy involved in interactions between atoms that are not directly bonded. We will take a look at each of these individually below.

 

In terms of a ball and spring description, a bond deformation will involve energy changes that can be approximated via Hooke's Law. If the strain free energy of a particular bond length is taken as lo, then any deviation from this value will lead to an increase in the potential energy. The mathematical form of this potential energy function describing the deformation is

Es = Sumbonds 1/2 kl (l - lo)2

where kl is the force constant of the particular bond (related to the strength of the bond), l is the bond length at the deformed structure, lo is the strain free bond length ('equilibrium' bond length), and the summation is over all the bonds in the molecule.

  Let's look at an example of this functional form for a couple of bonding situations. In the following exercise, we can choose values of the force constant as well as the equilibrium bond distance and plot the resulting potential energy description for deformations from equilibrium. For example, let us consider a Csp3 - Csp3o=1.54 angstroms, and the force constant for this bond is given as, kl = 65 kcal/mol. Insert these into the boxes and press run for the result.

 

 

 

  While this Hooke's Law description is fairly good at geometries just near the equilibrium bond length, it can be significantly improved by the use of a modified functional form that allows for more realistic deformations towards dissociation or compression. One choice of function that is very widely used is the Morse Potential Function.

 

Es = Sumbonds De (1 - exp(-a*(l - lo)))< sup>2

Again, let's take a look at this curve, and compare it to the other Hooke's Law description for the same Csp3 - Csp3 bond. What do you notice ?

  Now try the exercise with other types of bond situations and note the differences.

  Exercise:

Let's take a look at the ethylene molecule bond steric energy, Sumbonds1/2 kl (l - lo)2.

For this exercise,

  • kC=C=9.6 kcal/mol*A2
  • kH-C=4.6 kcal/mol*A2
  • bC=C=1.337 A2
  • bH-C=1.337 A2

  Now, evaluate the bond steric energy for two cases and compare:

 

    Near equilibrium case:
    • C=C = 1.3 A2
    • C-H = 1.0 A2

 

    Farther from equilibrium:
    • C=C = 1.34 A2
    • C-H = 1.09 A2

 

 

    Angle Potential Function, Es

In terms of a ball and spring description, angle deformations can also involve a description that can be approximated near equlibrium values using a Hookean Law formulat. Analogous to the above situation for bonds, if the strain free energy of a particular angle is given as bo, then any deviation from this value will lead to an increase in the potential energy. The mathematical form of this potential energy function describing the deformation is

Es = Sumangles 1/2 kb (b - bo)2

where kb is the force constant of the particular angle (related to the strength of the bonding situation), b is the angle at the deformed structure, bo is the strain free angle ('equilibrium' angle), and the summation is over all the angles in the molecule.

  As for bonds, we can take a look at the functional form for bond deformation. Although it will look very similar to the bond graphics, what you will notice is that it is much easier to deform angles than it is to deform bonds, and so the order of magnitude of energy involved has changed by about a factor of 10.

In the following exercise, we can choose values of the force constant as well as the equilibrium angle and plot the resulting potential energy description for deformations from equilibrium. The equilibrium value is taken as, bo=109 angstroms, and the force constant for this bond is given as, kl = 5 kcal/mol. Insert these into the boxes and press run for the result.

  For angle deformations up to about 10 degrees from the equilibrium situations, the quadratic description of angle bending seems to work out well. However, this angle deformation description can be improved by altering the functional form slightly, including more terms in the expansion. For example, the addition of a cubic term can enhance the description for angle deformations far from the equilibrium value.

Es = Sumbonds [1/2 kb (b - bo)2+k'b(b - bo)3]

Take a look at the visual comparison of the two functions by plotting the two of these functions together.

 

 

    Torsional Angle Potential, Ew

    It is well known that intramolecular rotations about bonds do not occur without energy costs. The minimum amount of energy needed to get a molecule to undergo one complete rotation around a certain bond can be referred to as the rotational barrier for that bond. The classic illustration of this idea is the rotation about a carbon-carbon bond in ethane. It was proposed over 100 years ago by Bischoff that ethane preferred a staggered conformation over an eclipsed conformation and that there is a restricted rotation, or barrier to rotation, between the carbon-carbon bond in ethanes that have the H's substituted for other functional groups. One must account for this type of energy perturbation within the molecular mechanics force field. The torsional energy term is typicall represented using a Fourier series expansion.

    Ew = V1/2(1 + cos(w)) + V2/2(1 + cos(2w)) + V3/2(1 + cos(3w))

    where the Vi terms (Fourier coefficients) have important physical interpretations. The V1 term has been attributed to residual dipole-dipole interactions, van der Waals interactions, or any other direct interactions between atoms not otherwise accounted for. The V2 term is attributed to conjugation or hyperconjugation, which is geometrically related to p orbitals. The V3 term is attributed to steric, or bonding/antibonding interactions.

       

    • Rotational Barrier in Ethane

      Let's look at the rotational barrier in ethane. This can be described in terms of the change in the potential energy of the molecule as a function of the change in the torsional angle, tau, of the molecular system.

      We can take a look at the energy profile for the carbon-carbon bond rotation in ethane in the following exercise. Rotation of the back carbon is described by the changing torsional angle between the two hydrogens that are shown in the diagram above. (For clarity, the remaining hydrogens are not shown in this diagram).

        [Luke's ethane torsional potential]

        As seen in the energy profile for the C-C bond rotation in ethane, a rotational barrier of only 3 kcal/mol separates one low-energy form of ethane from another. The low-energy conformations, whose torsional angles are 60, 180, and 300 degrees are called 'staggered' forms. The high-energy conformations, whose torsional angles are 0, 120, 240, and 360 degrees are called 'eclipsed' forms.

        Exercise:

      • Draw out the energy profile for yourself on paper and draw in the Newman project diagrams for each peak and each well for ethane.
      • Some collisions between molecules of ethane at room temperature may provide as much as 20 kcal. This means that the 3 kcal/mol rotational barrier is easily surmounted, and thus rotation in ethane is considered as essentially 'free' rotation. Where does the 3 kcal/mol of rotational energy come from in this case?

        It is not likely that the barrier arises because hydrogens which are vicinal (on adjacent carbons) bump into each other in the eclipsed conformation. This is because the vicinal hydrogen atoms are far apart relative to their van der Waals radii even in the eclipsed, high energy conformation.

        Instead, it is believed that the energy barrier comes from the electronic repulsions between carbon-hydrogen bonds as they pass by one another. Since the eclipsing of 3 paris of carbon-hydrogen bonds produces a rotational barrier of 3 kcal/mol, each eclipsing interaction raises the energy of ethane by about 1 kcal/mol.

       

    • Butane Torsional Potential

      As mentioned above, if we substitute one of the three hydrogens on each carbon atom in ethane, we no longer have the situation of an essentially 'free' rotation. Instead, a significant barrier to rotation is noticed. The energy profile looks significantly different in this case as well.

        Exercise:

      • As an example, let's take a look at the barrier to rotation in butane. First try and identify the possible conformational structures:
        • Syn-peripanar
        • Syn-clinal
        • Anti-clinal
        • Anti-periplanar
      • Try and guess what the torsional energy profile diagram will look like for this molecule. How much energy do you expect for the interaction between the two methyl groups? What is your expectation of the energy difference between the cis and trans case?

         

      Now let's take a look:

        [Luke's stuff]

       

       

    • Nonbonded Interaction Potential

     

     

      [the rest is for later .........]

    Full Force Field Description

      In it's entirety, the force field which represents the steric energy for all types of interactions in a molecule can get quite complicated. The following expression illustrates this. This functional form includes much more than the few terms described above. In general, one must also include such terms as cross terms which factor in how an angle will deform as a bond deforms, for example. Other important terms include out of plane angle bending terms, electrostatic terms and hydrogen bonding terms.

     

     

  • 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

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:
  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.
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:
  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.