Force of atoms or molecules in relation to

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.

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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. 

 

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