Force fields are a computational chemistry

method to describe the potential energy of a system of atoms or molecules in

relation to its particle coordinates. With the aid of molecular modelling, the

difficulties of using quantum mechanical calculations, i.e. obtaining a clear

potential energy surface diagram from a nuclear configuration of electrons, can

be overlooked. This is a crucial technique, which allows for larger and/or

slower calculations than what the tools of quantum mechanics allow. Molecular

modelling takes the atom to be the smallest aspect in further observations,

rather than the electron, which is what quantum mechanics does. This is done by

depicting the energy through use of a parametric function of 3D nuclear

coordinates of the atoms and/or molecules we are interested in. The sets of

parameters that are used to obtain this “larger scale” energy are force fields.

This is done on the basis that molecules are

composed of a set number of “building blocks”, which are structurally similar

throughout many if not all molecules. A clear example would be a carbonyl group

on an organic molecule. While these can have varying bond lengths (from ~1.20Å

in carboxylic acids to a ~1.47 Å bond in epoxides) and can be found in the IR

spectrum 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 these

ranges are the C=O “standard”, this facilitates the study of molecules

containing a carbonyl element. Thus, the C=O force constants are comparable. The mathematical approach to force fields is the

addition of individual properties within molecules to give an overall picture

of molecular properties and behaviours. Thus, the parametric function modelling

requires an efficient, simple and fast approach to the calculation. The typical

mathematical (functional) form for an atomistic force field is a sum of

additive terms: U(r) = U str + Ubend +U tor +

U SR + U elect, where U(r) is an energy function

dependent on bond length, r, which is a summation of bonding and long range

terms. In order to facilitate the calculation of the

total energy, a number of assumptions and approximates have to be put in place

when taking each individual term above into consideration. For example, Ustr,

referencing the total energy of a bond stretching (str), we can look at this

behaviour in terms of a known function. The most appropriate in this case is

the example of a Morse potential (Fig. 1 below).This is due to the fact that the bond stretching

energy can be assimilated to a simple harmonic oscillator very near its

equilibrium 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 in

depth electronic arrangement analysis, but the anharmonicity of this

approximation brings it closer. Thus, the stretching energy derived from a

Morse potential is the following: , where De

is the dissociation energy of the bond, a is a force constant related to the

bond’s vibrational frequency, and r0 is

the equilibrium bond length. Ethane and ethene are the two examples portraying

the different additive terms. Ethane is comprised of two carbon sp3 atoms

bonded together via one single bond, whereas ethene has two sp2

atoms connected by a double bond. The saturated hydrocarbon has a longer

equilibrium bond length, 1.523 Å 2, than ethene, 1.337 Å 2.

However, the values of the bond dissociation energies and of the force constant

of these two molecules show an inverse relationship to that of the radii –

ethane has lower values, 348 kJ mol-1 and 1.385 respectively, and

ethene has 614 kJ mol-1 and 1.583. Due to the rigidity of the double

bond, which would increase with a higher bond order, the Morse potential

approximation would get closer to simple harmonic behaviour due to its

inability to stretch. The higher order bonds approaching the simple harmonic

approximation can be seen below, in Figure 2, where the red line shows a single

bond between two sp3 carbons, and the green line shows a double bond

between two sp2 carbons. This method is not quite the most efficient

to depict this energy term as it quite expensive and time consuming and in

order to solve these problems, a summation of simple harmonic oscillator terms

is ultimately adopted to give: , where k

is a constant relating the stretching force of the bond. This cannot model the

breaking of a bond, but it does show a proportional relationship between the

energy and the deviation from the equilibrium radius, which is a reasonable

model that can work. The following two terms, U bend and U

tor, are both angle terms, referring to the bending and torsional

angle respectively. They differ in that the bending term requires a

three-atom/two bond model to represent the angle in question, whereas the

torsional angle refers to a four-atom/three bond model. The bending energy can

be modelled by a summation of harmonic terms using the square of the deviation

of equilibrium angle, depicted by the difference shown in the equation: , where kbend

is a constant depending on the three atoms under observation and is the equilibrium angle. The approximation

and thus the energy of the bond will obviously differ depending on the atoms

involved and the bonding (single bond, double bond, etc). This can be seen in

Figure 2.The angle between C-C-H is 109.47°2 when the

C-C bond has order one, and it is 121.40°2 when the bond order is

two. Since the three atoms in question are the same in both molecules, the

force constant is very similar (0.04 and 0.05 respectively), thus can be taken

to be the same. By observing the figure above, and looking at their parameter

values, these two molecules behave quite similarly around their equilibrium

bending angle point. This approximation cannot fully account for very

high-energy rotations without adding a number of higher order terms in the

summation, but, once again, the simple harmonic model is sufficient to

consider. The torsional angle consists of a Fourier series

depending on V, an energy constant, the dihedral angle, , angle

shift and n, the phase factor: . The need

for a summation over a Fourier series is in order to account for the symmetry

of the dihedral angle breaking down in a four-atom A-B-C-D system where the two

external atoms, i.e. A and D, are two different groups. Ethane in this case has only one torsional term

as, due to the symmetry of the three hydrogens on each carbon, each

conformation is equivalent. Ethene has a much higher energy barrier to overcome

in order to freely rotate around the torsional angle, thus the potential energy

surface for this molecule is quite high. The previous terms discussed and derived above

can be grouped together and considered “bonding” terms, as they refer to

intramolecular atoms and thus energies. In order to have a full picture of a

molecule’s behaviour, we must also consider “non-bonding” terms, which include

long and short distance interactions between neighbouring molecules. The main

two interactions considered here are the short-range and electrostatic terms,

depicted by USR and Uelect respectively. The short-range term is modelled by the

Lennard-Jones Potential, and is done so to mainly include the Van Der Waals

dispersion forces and the repulsion between neighbouring electron clouds: 3, where is the well depth and rm is the

minimum energy interaction distance 3. The Lennard-Jones model

accounts for the short-range repulsion terms with the more negative (r-12)

term, and the other exponential refers to the attractive, longer-range terms

such as the London dispersion-attractive forces. This is a useful modelling

technique, as it does not require a lot of parameters, and allows for both

large and small molecules to be calculated. The final of the additive energy terms is the

second “non bonding” term – the term depicting the electrostatic potential

energy, Uelect. This final term is a product of the partial charges

on two atoms that can be used to identify long-range Coulombic interactions: , where D

is a constant calculated using the dielectric constant, which depends on the solvent involved in the

study. These are the most crucial force field terms in

order 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 to

have a perfect model, we must also look at cross-term interactions, for example

stretch-bend, torsion-bend, etc. However, for the two molecules in question,

ethane and ethene the force fields mentioned above can prove to be sufficient

analysis, and this can potentially be verified using experimental methods such

as spectroscopy. In more complicated molecules, such as long-chain polymers and

proteins, which are less readily available, these modelling methods might not

suffice in providing a full behavioural picture.