Molecular Simulation/Intramolecular Forces

The representation of the potential energy of a torsional angle rotation using a cosine representation.

Bond Stretches

Bond stretches are the alteration of the bond length within a molecule. Using water as an example we will explain the different forms of stretches. The two modes of stretching are symmetric; to which in the case of water the two oxygen hydrogen bonds lengthen and shorten at the same rate and asymmetric; to which they lengthen and shorten at opposite times. The energy associated with this motion can be described using the following equation:

${\mathcal {V}}_{bond}(r)={\frac {1}{2}}k_{bond}\left(r-r_{e}\right)^{2}$

We see a Hooke's law relationship where k is a constant for the bond length and the x² term in the Hooke's law equation is replaced with the change in bond length subtracted from the equilibrium bond length. This will give an average change in bond length. This relationship will give us an accurate value for the energy of the bond length of the given molecule.

Bond Angle Bending

Bond angle energies are brought about from an alteration of the optimized bond angle to a less favorable conformation. This can be either a decrease in angle; the orbitals get closer to one another which will increase the potential energy, or increase in angle; the orbitals more away from the ideal overlap which give the molecule favorable orbital interactions to increase the stability. This will increase the potential energy by moving the molecule out of the low energy conformation. This energy relationship can be observed through the following equation:

${\mathcal {V}}_{angle}(\theta )={\frac {1}{2}}k_{angle}\left(\theta -\theta _{e}\right)^{2}$

We again see a hooks law relationship where k is a constant for the bond length and the x² term in the Hooks law equation is replaced with the change in bond angle subtracted from the equilibrium bond angle. This will give an average change in bond angle. This relationship will give us an accurate value for the energy of the bond length of the given molecule.

Torsional Rotations

The torsional potential energy surface of butane.

Torsional rotations, which can also be described as dihedral interactions come about when a system of 4 or more atoms form a molecule. This interaction relies upon the rotation central bond into a favorable orientation with the most stable overlap conformation. Some of these known conformations are called periplanar, anticlinal, gauche, and antiperiplaner, listed in order of highest to lowest energy. Observing Newman projections upon the energy we can see where the unfavorable confirmational interactions will lie. The methyl groups here will be considered large and have unfavorable overlap with one another. Mathematically this can be represented using the following equation:

${\mathcal {V}}_{dihedral}(\Phi )={k_{\Phi }}({1+\cos(n\Phi -\delta }))$

Where k is the rotational barrier of the system, n is the periodicity as the rotations repeat around 360 degrees, ɸ is the dihedral angle, and δ is the offset of the function

Improper Torsion

An improper torsion is a special type of torsional potential that is designed to enforce the planarity of a set of atoms (ijkl) where i is the index of the central atom. The torsional potential is defined as the angle between the planes defined by atoms ijk and jkl.

Internal Non-Bonded Interactions

The atoms of a molecule will also have intramolecular steric, dispersive, and electrostatic interactions with each other. These interactions within a molecule determine its conformation. The steric repulsion as well as Coulombic and dispersive interactions all together in the conformation of the molecule. Every atom in the system can be assigned a partial charge ($q_{i}$). These interactions are calculated using standard Lennard-Jones and Coulombic pairwise interaction terms,

${\mathcal {V}}_{NB}(r_{ij})={4\epsilon _{ij}}\left[\left({\frac {\sigma _{ij}}{r_{ij}}}\right)^{12}-\left({\frac {\sigma _{ij}}{r_{ij}}}\right)^{6}\right]+{\frac {q_{1}q_{2}}{4\pi \epsilon _{0}}}{\frac {1}{r_{ij}}}$

Here we see the combination of the Leonard Jones equation as well as Coulomb's law. With this relationship we can calculate the force resulting between the non bonding intermolecular forces. It should be noted at this point that interactions between bonded or atoms forming angles are excluded from this relationship. These interactions have been included though the calculations in the previous sections.