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In the following exercises, the reaction pathways of different pericyclic reactions were studied – Diels-Alder and cheletropic. A Diels-Alder reaction is a [4+2] cycloaddition which occurs between a conjugated diene and a substituted alkene, a dienophile to form a substituted cyclohexene system, while for a cheletropic reaction, both new bonds are being made to the same atom on one of the reagents. 

The stationary points of the potential energy surfaces (PES) of the reactions were identified and characterised through frequency calculations, and the determination of thermodynamically stability of the different isomers in certain reactions were carried out. 

'''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?'''

A PES is a multi-dimensional surface plot which shows the variation of potential energy across two or more
reaction coordinates. The potential energy is a function of the independent degrees of freedom of the molecule (for a non-linear molecule: 3N-6 and for a linear: 3N-5, where N = the number of atoms the molecule has), where each of the
different degrees of freedom is a normal mode. The normal modes are a linear combination of all the individual translation, rotation and vibration of the molecule and have individual force constants, corresponding to their own normal vibrational modes.

Each vibrational mode has a restoring force as defined by Hooke’s law, which states that:

F = -kx, where F is the restoring force, k the force constant, and x , the displacement of the spring/ bonds from its equilibrium
position. 

A minimum and transition state on a PES both correspond to a stationary point where the gradient is zero and the first derivative of the Potential Energy vs Bond Distances graph is ∂V(ri)/∂ri = 0. In a reaction pathway, the reactants and products are at minimum points whereas the transition state is the saddle point which is a local maximum on the minimum energy pathway linking the reactants and products. 

'''How would a frequency calculation confirm a structure is at either of these points?'''

By considering the vibrational frequencies of the different points, minima and transition state can be identified. 

In Gaussian, the vibrational frequencies are computed by determining the second derivatives of the energy with respect to the Cartesian nuclear coordinates and then transforming to mass-weighted coordinates. The following equation is used to calculate the force constants between the bonds.

[[File:At3815VibrationalFreq.PNG|centre|125px]]

, where k is the molecular force constant and μ, the reduced mass. 

As both reactants and products sit in a minimum well, the second derivative of the potential energy function is positive for each degree of freedom as all the coordinates are minimised. As a result, the minima will only have positive and real frequencies and will remain at a minimum unless sufficient energy is provided for them to react.  

Transition states however, are minimised along all other coordinates except along the reaction coordinate, resulting in a single imaginary frequency. 

In this lab, two main optimisation methods were used to calculate the optimised geometry of the transition states. The geometry optimizations proceed as a set of single point energy calculations, where an initial starting geometry calculation is performed. The geometry of the molecule is then varied until the one with lowest energy structure is obtained. 

Firstly, a semi-empirical method, PM6 was used where the use of severe approximations and parametrised integrals result in limited accuracy of calculations. However, as this method is fast, it is used for initial optimisation (for exercises 2 and 3). Subsequently, a more accurate (but at a much greater computational cost) Density Functional Theory (DFT) optimisation method B3LYP/6-31G(d) was used. 


=== Exercise 2 ===
{| class="wikitable"
|+Table 2.1: Energies of reactants, TS and products
|
  |
<nowiki>  </nowiki>Energy calculation from PM6 (Hatrees/ particle)
  
  |
<nowiki>  </nowiki>Energy calculation from PM6 (kJmol<sup>-1</sup>)
  
  |
<nowiki>  </nowiki>Energy calculation from B3LYP/6-31G(d) (Hatrees/ particle)
  
  |
<nowiki>  </nowiki>Energy calculated from B3LYP/6-31G(d) (kJmol<sup>-1</sup>)  
 
 |-
  |
<nowiki>  </nowiki>Cyclohexadiene
  
  |
<nowiki>  </nowiki>0.118063
  
  |
<nowiki>  </nowiki>309.9744
  
  |
<nowiki>  </nowiki>-233.324375
  
  |
<nowiki>  </nowiki>-612593.1466  
 
 |-
  |
<nowiki>  </nowiki>1,3-dioxole
  
  |
<nowiki>  </nowiki>-0.052279
  
  |
<nowiki>  </nowiki>-137.259
  
  |
<nowiki>  </nowiki>-267.068132
  
  |
<nowiki>  </nowiki>-701187.3806  
 
 |-
  |
<nowiki>  </nowiki>Sum of reactants
  
  |
<nowiki>  </nowiki>0.065784
  
  |
<nowiki>  </nowiki>172.7159
  
  |
<nowiki>  </nowiki>-500.392507
  
  |
<nowiki>  </nowiki>-1313780.527  
 
 |-
  |
<nowiki>  </nowiki>Endo Transition State
  
  |
<nowiki>  </nowiki>0.137941
  
  |
<nowiki>  </nowiki>362.1641
  
  |
<nowiki>  </nowiki>-500.332154
  
  |
<nowiki>  </nowiki>-1313622.07  
 
 |-
  |
<nowiki>  </nowiki>Endo Product
  
  |
<nowiki>  </nowiki>0.037804
  
  |
<nowiki>  </nowiki>99.2544
  
  |
<nowiki>  </nowiki>-500.418691
  
  |
<nowiki>  </nowiki>-1313849.273  
 
 |-
  |
<nowiki>  </nowiki>Exo Transition State
  
  |
<nowiki>  </nowiki>0.138904
  
  |
<nowiki>  </nowiki>364.6925
  
  |
<nowiki>  </nowiki>-500.329163
  
  |
<nowiki>  </nowiki>-1313614.217  
 
 |-
  |
<nowiki>  </nowiki>Exo Product
  
  |
<nowiki>  </nowiki>0.037975
  
  |
<nowiki>  </nowiki>99.70336
  
  |
<nowiki>  </nowiki>-500.417321
  
  |
<nowiki>  </nowiki>-1313845.676  
 
|}
{| class="wikitable"
|+Table 2.2: Activation Barriers and Reaction Energies
|  
  
  |
<nowiki>  </nowiki>PM6 level/ kJmol<sup>-1</sup>
  
  |
<nowiki>  </nowiki>B3LYP/6-31G(d)/ kJmol<sup>-1</sup>  
 
 |-
  |
<nowiki>  </nowiki>Activation Barrier endo
  
  |
<nowiki>  </nowiki>189.4482
  
  |
<nowiki>  </nowiki>158.457  
 
 |-
  |
<nowiki>  </nowiki>Reaction energy endo
  
  |
<nowiki>  </nowiki>-73.4615
  
  |
<nowiki>  </nowiki>-68.746  
 
 |-
  |
<nowiki>  </nowiki>Activation Barrier exo
  
  |
<nowiki>  </nowiki>191.9766
  
  |
<nowiki>  </nowiki>166.31  
 
 |-
  |
<nowiki>  </nowiki>Reaction energy exo
  
  |
<nowiki>  </nowiki>-73.01254
  
  |
<nowiki>  </nowiki>-65.149  
 
|}

===Exercise 3 calc===
{| class="wikitable"
 
  |  
  
  |
<nowiki>  </nowiki>Energy calculation (Hatrees/ particle)
  
  |
<nowiki>  </nowiki>Energy calculation (kJmol<sup>-1</sup>)  
 
 |-
  |
<nowiki>  </nowiki>Xylylene
  
  |
<nowiki>  </nowiki>0.178447
  
  |
<nowiki>  </nowiki>468.5125985  
 
 |-
  |
<nowiki>  </nowiki>Sulfur dioxide
  
  |
<nowiki>  </nowiki>-0.118614
  
  |
<nowiki>  </nowiki>-311.421057  
 
 |-
  |
<nowiki>  </nowiki>Sum of reactants
  
  |
<nowiki>  </nowiki>0.059833
  
  |
<nowiki>  </nowiki>157.0915415  
 
 |-
  |
<nowiki>  </nowiki>Endo Transition State
  
  |
<nowiki>  </nowiki>0.091344
  
  |
<nowiki>  </nowiki>239.8237  
 
 |-
  |
<nowiki>  </nowiki>Endo Product
  
  |
<nowiki>  </nowiki>0.021699
  
  |
<nowiki>  </nowiki>56.97072  
 
 |-
  |
<nowiki>  </nowiki>Exo Transition State
  
  |
<nowiki>  </nowiki>0.092467
  
  |
<nowiki>  </nowiki>242.7721  
 
 |-
  |
<nowiki>  </nowiki>Exo Product
  
  |
<nowiki>  </nowiki>0.021452
  
  |
<nowiki>  </nowiki>56.32223  
 
 |-
  |
<nowiki>  </nowiki>Chelotropic Transition State
  
  |
<nowiki>  </nowiki>0.071188
  
  |
<nowiki>  </nowiki>186.9041  
 
 |-
  |
<nowiki>  </nowiki>Chelotropic Product
  
  |
<nowiki>  </nowiki>-0.000002
  
  |
<nowiki>  </nowiki>-0.00525  
 
|}

==== IRC pathway ====
{| class="wikitable"
!Reaction
!IRC Pathway
|-
|Diels-Alder (Endo)
|[[File:At3815PM6IRCDATSendo.gif|thumb| 500px]]
|-
|Diels-Alder (Exo)
|
|-
|Cheletropic Reaction
|
|}

==== Optimised transition states and products ====
{| class="wikitable"
!Optimised structure
!Diels-Alder (endo)
!Diels-Alder (exo)
!Cheletropic Reaction
|-
|Transition State
|
|
|
|-
|Product
|
|
|
|}