Force fields are a computational chemistrymethod to describe the potential energy of a system of atoms or molecules inrelation to its particle coordinates.
With the aid of molecular modelling, thedifficulties of using quantum mechanical calculations, i.e. obtaining a clearpotential energy surface diagram from a nuclear configuration of electrons, canbe overlooked. This is a crucial technique, which allows for larger and/orslower calculations than what the tools of quantum mechanics allow. Molecularmodelling takes the atom to be the smallest aspect in further observations,rather than the electron, which is what quantum mechanics does.
This is done bydepicting the energy through use of a parametric function of 3D nuclearcoordinates of the atoms and/or molecules we are interested in. The sets ofparameters that are used to obtain this “larger scale” energy are force fields.This is done on the basis that molecules arecomposed of a set number of “building blocks”, which are structurally similarthroughout many if not all molecules. A clear example would be a carbonyl groupon an organic molecule. While these can have varying bond lengths (from ~1.20Åin carboxylic acids to a ~1.47 Å bond in epoxides) and can be found in the IRspectrum in a range rather than a very specific wavenumber (1630-1750cm-1),these are small discrepancies and taking a fixed value in the midpoint of theseranges are the C=O “standard”, this facilitates the study of moleculescontaining a carbonyl element.
Thus, the C=O force constants are comparable. The mathematical approach to force fields is theaddition of individual properties within molecules to give an overall pictureof molecular properties and behaviours. Thus, the parametric function modellingrequires an efficient, simple and fast approach to the calculation. The typicalmathematical (functional) form for an atomistic force field is a sum ofadditive terms: U(r) = U str + Ubend +U tor +U SR + U elect, where U(r) is an energy functiondependent on bond length, r, which is a summation of bonding and long rangeterms. In order to facilitate the calculation of thetotal energy, a number of assumptions and approximates have to be put in placewhen taking each individual term above into consideration. For example, Ustr,referencing the total energy of a bond stretching (str), we can look at thisbehaviour in terms of a known function. The most appropriate in this case isthe example of a Morse potential (Fig. 1 below).
This is due to the fact that the bond stretchingenergy can be assimilated to a simple harmonic oscillator very near itsequilibrium point, but as the distance increases, the bond will dissociate.This is not a perfect model for the breaking of a bond as that requires more indepth electronic arrangement analysis, but the anharmonicity of thisapproximation brings it closer. Thus, the stretching energy derived from aMorse potential is the following: , where Deis the dissociation energy of the bond, a is a force constant related to thebond’s vibrational frequency, and r0 isthe equilibrium bond length. Ethane and ethene are the two examples portrayingthe different additive terms. Ethane is comprised of two carbon sp3 atomsbonded together via one single bond, whereas ethene has two sp2atoms connected by a double bond.
The saturated hydrocarbon has a longerequilibrium bond length, 1.523 Å 2, than ethene, 1.337 Å 2.However, the values of the bond dissociation energies and of the force constantof these two molecules show an inverse relationship to that of the radii –ethane has lower values, 348 kJ mol-1 and 1.385 respectively, andethene has 614 kJ mol-1 and 1.583.
Due to the rigidity of the doublebond, which would increase with a higher bond order, the Morse potentialapproximation would get closer to simple harmonic behaviour due to itsinability to stretch. The higher order bonds approaching the simple harmonicapproximation can be seen below, in Figure 2, where the red line shows a singlebond between two sp3 carbons, and the green line shows a double bondbetween two sp2 carbons. This method is not quite the most efficientto depict this energy term as it quite expensive and time consuming and inorder to solve these problems, a summation of simple harmonic oscillator termsis ultimately adopted to give: , where kis a constant relating the stretching force of the bond. This cannot model thebreaking of a bond, but it does show a proportional relationship between theenergy and the deviation from the equilibrium radius, which is a reasonablemodel that can work. The following two terms, U bend and Utor, are both angle terms, referring to the bending and torsionalangle respectively. They differ in that the bending term requires athree-atom/two bond model to represent the angle in question, whereas thetorsional angle refers to a four-atom/three bond model. The bending energy canbe modelled by a summation of harmonic terms using the square of the deviationof equilibrium angle, depicted by the difference shown in the equation: , where kbendis a constant depending on the three atoms under observation and is the equilibrium angle.
The approximationand thus the energy of the bond will obviously differ depending on the atomsinvolved and the bonding (single bond, double bond, etc). This can be seen inFigure 2.The angle between C-C-H is 109.47°2 when theC-C bond has order one, and it is 121.40°2 when the bond order istwo. Since the three atoms in question are the same in both molecules, theforce constant is very similar (0.
04 and 0.05 respectively), thus can be takento be the same. By observing the figure above, and looking at their parametervalues, these two molecules behave quite similarly around their equilibriumbending angle point. This approximation cannot fully account for veryhigh-energy rotations without adding a number of higher order terms in thesummation, but, once again, the simple harmonic model is sufficient toconsider. The torsional angle consists of a Fourier seriesdepending on V, an energy constant, the dihedral angle, , angleshift and n, the phase factor: . The needfor a summation over a Fourier series is in order to account for the symmetryof the dihedral angle breaking down in a four-atom A-B-C-D system where the twoexternal atoms, i.e.
A and D, are two different groups. Ethane in this case has only one torsional termas, due to the symmetry of the three hydrogens on each carbon, eachconformation is equivalent. Ethene has a much higher energy barrier to overcomein order to freely rotate around the torsional angle, thus the potential energysurface for this molecule is quite high. The previous terms discussed and derived abovecan be grouped together and considered “bonding” terms, as they refer tointramolecular atoms and thus energies. In order to have a full picture of amolecule’s behaviour, we must also consider “non-bonding” terms, which includelong and short distance interactions between neighbouring molecules.
The maintwo interactions considered here are the short-range and electrostatic terms,depicted by USR and Uelect respectively. The short-range term is modelled by theLennard-Jones Potential, and is done so to mainly include the Van Der Waalsdispersion forces and the repulsion between neighbouring electron clouds: 3, where is the well depth and rm is theminimum energy interaction distance 3. The Lennard-Jones modelaccounts for the short-range repulsion terms with the more negative (r-12)term, and the other exponential refers to the attractive, longer-range termssuch as the London dispersion-attractive forces. This is a useful modellingtechnique, as it does not require a lot of parameters, and allows for bothlarge and small molecules to be calculated. The final of the additive energy terms is thesecond “non bonding” term – the term depicting the electrostatic potentialenergy, Uelect. This final term is a product of the partial chargeson two atoms that can be used to identify long-range Coulombic interactions: , where Dis a constant calculated using the dielectric constant, which depends on the solvent involved in thestudy. These are the most crucial force field terms inorder to model a molecule’s behaviour in a way that is time and cost efficient,with examinable results. There are more force fields that could be analysed,such as the behaviour of hydrogen bonding and certain 3D aspects.
In order tohave a perfect model, we must also look at cross-term interactions, for examplestretch-bend, torsion-bend, etc. However, for the two molecules in question,ethane and ethene the force fields mentioned above can prove to be sufficientanalysis, and this can potentially be verified using experimental methods suchas spectroscopy. In more complicated molecules, such as long-chain polymers andproteins, which are less readily available, these modelling methods might notsuffice in providing a full behavioural picture.