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In your introduction, briefly describe what is meant by a minimum and transition state in the context of a potential energy surface. What is the gradient and the curvature at each of these points? (for thought later on, how would a frequency calculation confirm a structure is at either of these points?)

The potential energy surface (PES) represents the potential energy of the molecule visualised along two dimensions. The potential energy is a function of 3N-6 independent nuclear coordinates where <math>E=f(q_1, q_2,q_3, ... ,q_{3N-6})</math>. If the energy is plotted only along one dimension, an energy profile is obtained. This can also be obtained by taking a vertical slice of the PES. When the first derivative of the PES is zero <math>(\tfrac{\partial E(q_1)}{\partial q_1}=0)</math>, it corresponds to a minimum, maximum or saddle point. To better understand the nature of these points, the second derivative <math>(\tfrac{\partial E^2}{\partial q_1^2})</math>, corresponding to the frequency is calculated. If this value is positive, it is the minimum of the PES. This typically corresponds to reactant or products. If the value is negative, it could be a saddle point which corresponds to the transition state of a molecule. As the bonds can be approximated to a harmonic oscillator, it obeys Hooke's law where:

<math>F=kx</math>

<math>V = -\int F \mathrm{d}x</math>

<math>V= \tfrac{1}{2}kx^2</math>

<math> k = \tfrac{\partial E^2}{\partial q_1^2} </math>

We can thus solve for the frequency: <math>  \nu =   {1\over {2 \pi}} \sqrt{k \over \mu} </math> where: <math>\mu = \cfrac{m_1 m_2}{m_1 + m_2},\!\</math>

The above calculations are valid due to Born-Oppenheimer approximation, where the electronic distribution of a molecule adjust instantaneously to the movement of a nuclei and that the energy is a function of nuclei positions, where nuclear kinetic energy is not taken into account.

Through Gaussian calculations, the reactants, transition states and products were optimised. All calculations were done at PM6 level except exercise 2 where the optimised PM6 structures were further optimised at B3LYP 6-31G(d) level. From these optimisations, the molecular orbitals, bonds lengths and intrinsic reaction coordinates could be obtained and analysed. The thermodynamic data from Gaussian calculations were also used to support the analysis on competing reactions for Exercise 2, Exercise 3 and the extension.