== A Study of Transition State and Reactivity using Computational Methods ==
=== Understanding Potential Energy Surfaces (PES) ===
The plotting of PES is one of the most effective methods to graphically conceptualise the variations in energy of the reaction system as the molecular structures change, generating crucial information on the molecular properties and the nature of the reaction.

The critical points on a PES are the '''minima''' and '''first-order saddle point'''. The minima corresponds to the molecular structure at equilibrium, while the first-order saddle points corresponds to the transition state structure. Both points are stationary points, where the first derivative of energy with respect to nuclear coordinates equals to zero. Due to the fact that the negative of the first energy derivative is equivalent to the force, there are no forces acting at these points as well.

By calculating the Hessian (i.e. second energy derivative), one can deduce the local curvature of the PES and use it to determine the nature of the stationary points. At the minimum, the second derivative will be more than zero, while at the transition state (first-order saddle point), the second derivative will be less than zero. From the proportional relationship between vibrational frequency and the the square root of the eigenvalues of the mass-weighted Hessian, a convenient means to check whether or not the optimised structure is a minimum or a transition state would be to run a frequency calculation: there will be no negative frequencies at minima points, while there will be exactly one negative frequency at the point corresponding to the transition state.

In a reaction, the chemical system transits from a minimum corresponding to the reactants, to another minimum corresponding to the products, passing through points of highest energy that corresponds to transition states. <ref name="TS and reactivity1" /><ref name="TS and reactivity2_supergoodlink" />

=== Analysis of Reactions using Computational Chemistry ===
Gaussian and GaussView are the computational Chemistry program and graphical user interface used respectively. The two electronic structure methods applied are:
 1) '''Semi-empirical method PM6:''' to obtain the initial geometry of the molecules.
 2) '''Density Functional Theory (DFT) method B3LYP:''' to further optimise the geometry obtained from the semi-empirical method.

The molecular structure of the product will first be minimised to find the equilibrium geometry, ensuring that there are no negative frequencies in vibration calculations. After which, the structure of the product will be modified to reflect the transition state by deleting the bonds that form during the reaction, increasing the bond lengths between them and freezing the atoms involved. The modified structure is optimised to a minimum again, before running transition state optimisation to locate the first-order saddle point in the PES. The vibration calculation of the transition state is verified to have only one negative frequency. The vibration of the molecular structure at this imaginary frequency corresponds to the movement of the reaction system through the highest point along the PES. From the optimisation calculations, the barrier heights of different reaction pathways and the relative energies of the formed products can be obtained to understand which reaction pathway is kinetically and/or thermodynamically favoured. An intrinsic reaction coordinate (IRC) calculation can also be set up to visualise the reaction progress.

By monitoring the reaction progress via the IRC and visualising the molecular orbitals involved to form a transition state, a better understanding can be gained on how factors such as stereochemistry and orbtial interactions can have a significant influence the kinetics and thermodynamics of reactions.

== Cycloaddition ==
Cycloaddition reaction involves two σ bonds forming at the same time to generate a ring structure. The Diels-Alder reaction is specifically [4+2] cycloaddition which involves a conjugated diene reacting with a alkene functionality to give cyclohexene. By Woodward-Hoffman rules, the Diels-Alder reaction is an "orbital-controlled, thermally permitted and [a] concerted pericyclic reaction". Using the frontier molecular orbital theory, we can compare the relative energies of the HOMO and LUMO in the reactants and thus determine the interaction between the frontier orbitals responsible for bond formation. Since the diene and dienophile each has one HOMO and one LUMO, there are two possible pairs of frontier orbital interactions in a Diels-Alder reaction. Based on the energy gap between each pair of HOMO and LUMO, the Diels-Alder reaction can be categorised into '''normal, inverse and neutral electron-demand''' (Fig 1)<ref name="electron demand" />:

{|class="wikitable" style="margin: auto;"
! style="background: #787171; color: black;" |Fig 1: Three possible types of Diels-Alder reaction
|-
|[[File:cycloaddition_electron demand_V2.png|800px]]
|}

== Exercise 1: Reaction of Butadiene and Ethylene ==
==== Reaction Scheme ====

[[File:rxn mechanism_ex 1.jpeg|250px]]

==== Molecular Orbital (MO) Diagrams ====

{| class="wikitable" style="margin: auto;"
!Fig 2: Schematic MO Diagram for Reaction between Butadiene and Ethylene
|-
|[[File: Ex 1_schematic MO diagram.png|800 px]]
|}

{| class="wikitable" style="margin: auto;"
!colspan="4" |'''Table 1''': Molecular Orbitals (jmol)
|-
|! style="background: #0D4F8B; color: white;"|'''MO'''
|! style="background: #0D4F8B; color: white;"|'''Butadiene'''
|! style="background: #0D4F8B; color: white;"|'''Ethylene'''
|! style="background: #0D4F8B; color: white;"|'''Transition State'''
|-
|! style="background: #0D4F8B; color: white;"|'''LUMO +1'''
! style="background: #b4afaf; color: white;" |
! style="background: #b4afaf; color: white;" |
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 1_transition_state_MO.log</uploadedFileContents><script> frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Antisymmetric''
|-
|! style="background: #0D4F8B; color: white;"|'''LUMO'''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 1_butadiene_MO.log</uploadedFileContents><script> frame 22; mo 12; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Symmetric''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 1_ethylene_MO.log</uploadedFileContents><script> frame 14; mo 7; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Antisymmetric''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 1_transition_state_MO.log</uploadedFileContents><script> frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Symmetric''
|-
|! style="background: #0D4F8B; color: white;"|'''HOMO'''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 1_butadiene_MO.log</uploadedFileContents><script> frame 22; mo 11; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Antisymmetric''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 1_ethylene_MO.log</uploadedFileContents><script> frame 14; mo 6; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Symmetric''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 1_transition_state_MO.log</uploadedFileContents><script> frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Symmetric''
|-
|! style="background: #0D4F8B; color: white;"|'''HOMO -1'''
! style="background: #b4afaf; color: white;" |
! style="background: #b4afaf; color: white;" |
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 1_transition_state_MO.log</uploadedFileContents><script> frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Antisymmetric''
|}

==== Symmetry Requirements ====
In Fig 2 and Table 1, MOs are determined to be symmetric or antisymmetric based on an imaginary mirror plane of reflection cutting through the central C-C bond of the molecule. From Fig 1, it can be seen that only MOs of the same symmetry can interact to generate a molecular orbital in the transition state. It can be inferred that for a reaction to be allowed, the interacting HOMO and LUMO from each reactant would need to have the same symmetry, i.e. the interaction has to be either symmetric-symmetric or asymmetric-asymmetric interaction, where orbital overlap would be non-zero. Conversely, a reaction would be forbidden if the HOMO and LUMO required for bond formation between the two species are of opposite symmetries, and orbital overlap would be zero.

==== Neutral Electron-Demand ====
This reaction can be categorised as a Diels-Alder reaction with neutral electron-demand, since the HOMO(ethylene)-LUMO(butadiene) energy gap is only slightly smaller than the HOMO(butadiene)-LUMO(ethylene) energy gap and both pairs of interaction contribute almost equally to the reactivity of the system.<ref name="electron demand" />

==== Bond length Variations during Reaction ====

{| class="wikitable" style="margin: auto;"
!Fig 3: carbon labels in butadiene and ethylene 
[[File:ex 1_C labels.png|200px]]
|}

{| class="wikitable" style="margin: auto;"
!colspan="7"|Table 2: C-C bond length variation during Diels-Alder Reaction 
|-
|
|! style="background: #0D4F8B; color: white;"|'''C1-C2/Å'''
|! style="background: #0D4F8B; color: white;"|'''C2-C3/Å'''
|! style="background: #0D4F8B; color: white;"|'''C3-C4/Å'''
|! style="background: #0D4F8B; color: white;"|'''C4-C5/Å'''
|! style="background: #0D4F8B; color: white;"|'''C5-C6/Å'''
|! style="background: #0D4F8B; color: white;"|'''C6-C1/Å'''
|-
|! style="background: #0D4F8B; color: white;"|'''Reactants'''
|1.33
|3.41
|1.34
|1.47
|1.34
|3.41
|-
|! style="background: #0D4F8B; color: white;"|'''Transition State'''
|1.38
|2.11
|1.38
|1.41
|1.38
|2.11
|-
|! style="background: #0D4F8B; color: white;"|'''Product'''
|1.54
|1.54
|1.50
|1.34
|1.50
|1.54
|-
|}

{| class="wikitable" style="margin: auto;"
!Graph 1: C-C bond length variation during Diels-Alder Reaction
|-
|[[File:ex 1_variation of CC bond lengths.png]]
|}

The bond formation occurs at C2-C3 and C6-C1, which is why the bond lengths of C2-C3 and C6-C1 show a steady decrease as the butadiene and ethylene molecules move towards each other.

The typical sp2 C-C bond length is 1.34 Å<ref name="c-c bond lengths" />, which is very similar to the bond lengths C1-C2, C3-43 and C5-61 in the reactants. In butadiene, C4-C5 is shorter than the typical sp3 C-C bond length of 1.54 Å, which demonstrates that the double bonds C3-43 and C5-61 are conjugated with each other.

In the transition state, the bond lengths of C2-C3 and C6-C1 are 2.11 Å each. This is less than two times the Van der Waals radius of carbon, which is equals to 1.70 Å <ref name="C VDW radius" />. In other words, C2-C3 and C6-C1 are in the midst of bond formation, where the bond lengths are in between the internuclear separation of two non-bonded carbons at 3.40 Å and a C-C single bond at 1.54 Å.

The C-C bond lengths in the final product are all longer relative to the reactants, demonstrating that the double bonds have transformed into single bonds, except at C4-C5 where the single bond is converted to a double bond after the reaction. The bond lengths in the product are also in close agreement with literature values<ref name="c-c bond lengths" />.

==== Vibration Corresponding to Bond Formation ====
[[File:Ex1_vibrations.gif|center|400 px]]

The bond formation is synchronous for this reaction. The imaginary frequency is at 948.52i cm-1.

== Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole ==
==== Reaction Scheme ====
[[File:rxn mechanism_ex 2.png|400px]]

==== MO Diagrams ====
{| class="wikitable" style="margin: auto;"
!Fig 4: Schematic MO Diagram for Transition State Formation
|-
!Exo Transition State
|-
|[[File: ex 2_exo MO diagram v2.png|800 px]]
|-
!Endo Transition State
|-
|[[File: ex 2_endo transition state.png|800 px]]
|}

{| class="wikitable" style="margin: auto;"
!colspan="5" |Table 3: Molecular Orbitals (jmol)
|-
|MO
|Cyclohexadiene
|1,3-Dioxole
|Exo Transition State
|Endo Transition State
|-
|LUMO +1
! style="background: #b4afaf; color: white;" |
! style="background: #b4afaf; color: white;" |
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 2 Exo TS jmol.log</uploadedFileContents><script> frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Antisymmetric''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 2 Endo TS jmol.log</uploadedFileContents><script> frame 28; mo 43; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Antisymmetric''
|-
|LUMO
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 2_cyclohexadiene MO.log</uploadedFileContents><script> frame 18; mo 23; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Symmetric''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 2_dioxole MO.log</uploadedFileContents><script> frame 12; mo 20; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Antisymmetric''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 2 Exo TS jmol.log</uploadedFileContents><script> frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Symmetric''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 2 Endo TS jmol.log</uploadedFileContents><script> frame 28; mo 42; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Symmetric''
|-
|HOMO
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 2_cyclohexadiene MO.log</uploadedFileContents><script> frame 18; mo 22; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Antisymmetric''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 2_dioxole MO.log</uploadedFileContents><script> frame 12; mo 19; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Symmetric''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 2 Exo TS jmol.log</uploadedFileContents><script> frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Symmetric''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 2 Endo TS jmol.log</uploadedFileContents><script> frame 28; mo 41; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Symmetric''
|-
|HOMO -1
! style="background: #b4afaf; color: white;" |
! style="background: #b4afaf; color: white;" |
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 2 Exo TS jmol.log</uploadedFileContents><script> frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Antisymmetric''
|<jmol><jmolApplet>
<color>white</color>
<size>200</size>
<uploadedFileContents>ex 2 Endo TS jmol.log</uploadedFileContents><script> frame 28; mo 40; mo nodots nomesh fill translucent; mo titleformat ""</script>
</jmolApplet></jmol>''Antisymmetric''
|}

==== Inverse Electron-Demand ====
Due to the electron donating effect from the -O(C=O)O functionality, 1,3-dioxole is electron rich and thus its HOMO and LUMO are both higher than those from cyclohexadiene. This results in HOMO(1,3-dioxole)-LUMO(cyclohexadiene) energy gap to be significantly smaller than the HOMO(cyclohexadiene)-LUMO(1,3-dioxole) energy gap, resulting in a Diels-Alder reaction with inverse electron-demand.<ref name="electron demand" />

The HOMO(1,3-dioxole)-LUMO(cyclohexadiene) symmetric-symmetric orbital overlap would be stronger than HOMO(cyclohexadiene)-LUMO(1,3-dioxole) antisymmetric-antisymmetric orbital overlap since the energies of the frontier orbitals are more similar in the former pair. However, the relative ordering of the MOs in the exo and endo transition states show that the HOMO-1 and LUMO+1 interactions still come from the antisymmetric-antisymmetric orbital overlap. This indicates that while the orbital overlap between HOMO(1,3-dioxole)-LUMO(cyclohexadiene) was quite large, it did not change the relative order of the MOs when compared to the transition state formed from the neutral Diels-Alder reaction seen in exercise 1.

==== Relevant Reaction Energies ====
Table 4 summarises the "Sum of Electronic and Thermal Free Energies" values under the "Thermochemistry" section in the .log file of the structures optimised using '''B3LYP calculations'''.

{| class="wikitable" style="margin: auto;"
!colspan="5"|Table 4: Reaction Energies (in kJ mol-1) at 298 K
|-
|Position in PES
|Compound
|Energy/(Hartree per particle)
|Total Energy/kJ mol-1
|Relative Energy (to 2 d.p.)/kJ mol-1 
|-
| rowspan="2" |Reactants
|Cyclohexadiene
|<nowiki>-233.324375</nowiki>
| rowspan="2" |-1313780.62721
| rowspan="2" |0.00
|-
|1,3-Dioxole
|<nowiki>-267.068132</nowiki>
|-
| rowspan="2" |Transition State
|Exo
|<nowiki>-500.329163</nowiki>
|<nowiki>-1313614.31752</nowiki>
|166.30
|-
|Endo
|<nowiki>-500.332152</nowiki>
|<nowiki>-1313622.16514</nowiki>
|158.46
|-
| rowspan="2" |Product
|Exo
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|<nowiki>-65.15</nowiki>
|-
|Endo
|<nowiki>-500.418692</nowiki>
|<nowiki>-1313849.37593</nowiki>
|<nowiki>-68.75</nowiki>
|}

The energy of the reactants is set to 0 kJ mol-1 under the 'Relative energy' column, so that energy of the transition state would be equivalent to the height of the activation barrier and the relative thermodynamic stabilities of the exo and endo products can be more easily compared.

The results show that endo transition state has a lower energy than the exo transition state, thus the activation barrier for the formation of endo product is smaller. The endo product also has a lower energy relative to the exo product. Therefore, the formation of the endo product is both kinetically and thermodynamically favoured.

==== Secondary Orbital Interactions ====
The endo pathway is more kinetically favourable due to the presence of secondary orbital interactions.

{| class="wikitable" style="margin: auto;"
!colspan="2"|Fig 5: HOMO in Exo and Endo Transition State
|-
!Exo
!Endo
|-
|[[File:ex 2_exo HOMO.png]]
|[[File:ex 2_endo HOMO.png]]
|-
|[[File:ex 2_exo HOMO primary interactions.png|200px]]
|[[File:ex 2_endo HOMO sec orb interactions.png|150px]]
|}

Fig 5 shows the snapshots of the HOMO exo and endo transition states, as well as the schematic HOMO diagram including the oxygen orbitals. It can be seen that in the exo transition state, there are only primary interactions for bond formation, while in the endo transition state, the orbitals from oxygen (highlighted in red) can significantly interact with the orbitals from the cyclohexadiene (highlighted in blue). The presence of secondary orbital interactions (highlighted as bold dotted curved lines in Fig 5) during the bond formation process stabilises the transition state, lowering the activation barrier for the formation of the endo product. This causes the endo pathway to be kinetically favoured.

==== Steric factors ====
Furthermore, the endo pathway is more thermodynamically favourable due to smaller steric repulsion in the final product.

{| class="wikitable" style="margin: auto;"
!colspan="2"|Fig 6: Structure of Exo and Endo Products
|-
!Exo
!Endo
|-
|[[File:ex 2_exo product.png]]
|[[File:ex 2_endo product.png]]
|}

As can be seen in Fig 6, the highly unfavourable steric repulsion between the methylene groups causes the exo product to be much higher in energy relative to the endo product, thus the endo product is more thermodynamically stable.

== Exercise 3: Diels-Alder vs Cheletropic ==
==== Reaction Scheme ====
[[File:Rxn_mechanism_ex_3_with arrows.png|570 px]]

==== Reaction Pathways ====
{| class="wikitable" style="margin: auto;"
!colspan="3"|Fig 7: Exo, Endo and Cheletropic Reaction Pathways
|-
!Exo
!Endo
!Cheletropic
|-
|[[File:ex 3_exo TS.gif]]
|[[File:ex 3_endo TS.gif]]
|[[File:ex 3_cheletropic TS.gif]]
|-
|[[File:ex 3_IRC exo.png|320px]]
|[[File:ex 3_IRC endo.png|320px]]
|[[File:ex 3_IRC cheletropic.png|320px]]
|}

==== Normal Electron-Demand ====
SO2 is electron poor, thus its HOMO and LUMO are both lower than those from o-xylylene. Therefore, the most significant frontier orbital interactions are between HOMO(o-xylylene) and LUMO(SO2) and the Diels-Alder reaction has a normal electron-demand.

O-xylylene is very unstable as it is anti-aromatic, having a planar structure with 8 π electrons. Due to this instability, the aromatic ring is generated before the bonds are formed with SO2, as visualised in Fig 7. The steepest part of the PES in all reaction pathways corresponds to the restoration of aromaticity in the o-xylylene molecule, and the PES tapers off at the point when the bond formation occurs.

By restoring the aromaticity first, the reaction mechanism for the Diels-Alder pathways would proceed via a charge transfer-like mechanism, in the direction from the HOMO(o-xylene) towards the LUMO(SO2). The SO2 molecule induces opposite charges on each end of the cis-butadiene fragment, allowing aromaticity to be restored before the bond formation process. As a result, the bond formation process is asynchronous for exo and endo pathways, where the C-O bond forms ahead of the the C-S bond. <ref name="comparison of DA and chel mechanisms" /> As for the cheletropic reaction pathway, in order to accommodate the restoration of aromaticity before bond formation, the HOMO(o-xylene) donates into LUMO(SO2) simultaneously as the HOMO(SO2) donates into LUMO(o-xylylene), in a manner analagous to σ bond donation/π back-donation in <ref name="TM analogy" />. Therefore, the bond formation is synchronous, where C-S bonds form at the same time.<ref name="comparison of DA and chel mechanisms" />

==== Reaction Profiles ====
Table 5 summarises the "Sum of Electronic and Thermal Free Energies" values under the "Thermochemistry" section in the .log file of the structures optimised using '''PM6 calculations'''. '' ''
{| class="wikitable" style="margin: auto;"
!colspan="5"|Table 5: Reaction Energies (in kJ mol-1) at 298 K
|-
|Position in PES
|Compound
|Energy/(Hartree per particle)
|Total Energy/kJ mol-1
|Relative Energy (to 2 d.p.)/kJ mol-1 
|-
| rowspan="2" |Reactant
|o-xylylene
|0.177828
| rowspan="2" |155.466369
| rowspan="2" |0.00
|-
|SO2
|<nowiki>-0.118614</nowiki>
|-
| rowspan="3" |Transition State
|Exo
|0.092077
|241.748182
|86.28
|-
|Endo
|0.090560
|237.765298
|82.30
|-
|Cheletropic
|0.099062
|260.087301
|104.62
|-
| rowspan="3" |Product
|Exo
|0.021454
|56.3274813
|<nowiki>-99.14</nowiki>
|-
|Endo
|0.021695
|56.9602268
|<nowiki>-98.51</nowiki>
|-
|Cheletropic
|0.000005
|0.013127501
|<nowiki>-155.45</nowiki>
|}

From the relative energy, we can plot a reaction profile of energy against reaction coordinates to graphically show the relative reaction barriers and the relative stabilities of the products formed from each pathway (Graph 2):

{| class="wikitable" style="margin: auto;"
!Graph 2: Reaction Profile for Exo, Endo and Cheletropic Reaction Pathways
|-
|[[file:Ex 3_Reaction Profile.png]]
|}

From Table 5 and Graph 2, we can conclude that the endo pathway is the most kinetically favoured, while the cheletropic pathway forms the most thermodynamically stable product.

==== Kinetic and Thermodynamic Product ====
{| class="wikitable" style="margin: auto;"
!colspan="3"|Fig 8: Exo, Endo and Cheletropic Transition State and Product
|-
!Exo
!Endo
!Cheletropic
|-
|[[File:ex 3_exo transition state.png]]
|[[File:ex 3_endo transition state.png]]
|[[File:ex 3_cheletropic transition state.png]]
|-
|[[File:ex 3_exo product.PNG]]
|[[File:ex 3_endo product.PNG]]
|[[File:ex 3_cheletropic product.PNG]]
|}

Fig 8 shows the HOMO of each of the transition state, and the structure of the products formed from each pathway. The endo pathway is most kinetically favoured as it is the only transition state where the oxygen orbital points to the orbitals in the carbons in the aromatic ring, for secondary orbital interaction. Although the secondary orbital interaction is not extensive, the endo transition state will be most stabilised out of the three pathways, causing the formation of endo product via Diels-Alder reaction to be kinetically favoured. The relative stability of the endo and exo product are similar, but the cheletropic product has a much lower energy. This shows that the five-membered ring is more thermodynamically stable than the Diels-Alder products. This conclusion agrees closely with literature, which had shown that the Diels-Alder products are unstable and readily undergo a retro-Diels-Alder reaction to yield the reactants. <ref name="comparison of DA and chel mechanisms" />

==== Alternative Reactive Site in O-Xylylene ====
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. Table 6 and Graph 3 show the results of the reaction energies for a Diels-Alder reaction at this alternative reactive site:

{| class="wikitable" style="margin: auto;"
!colspan="5"|Table 6: Reaction Energies at Alternative Site in O-Xylylene (in kJ mol-1) at 298 K
|-
|Position in PES
|Compound
|Energy/(Hartree per particle)
|Total Energy/kJ mol-1
|Relative Energy (to 2 d.p.)/kJ mol-1 
|-
| rowspan="2" |Reactants
|o-xylylene
|<nowiki>0.177828</nowiki>
| rowspan="2" |155.466369
| rowspan="2" |0.00
|-
|SO2
|<nowiki>-0.118614
</nowiki>
|-
| rowspan="2" |Transition State
|Exo
|<nowiki>0.105055</nowiki>
|<nowiki>275.821924</nowiki>
|120.36
|-
|Endo
|<nowiki>0.102070</nowiki>
|<nowiki>267.984805</nowiki>
|112.52
|-
| rowspan="2" |Product
|Exo
|<nowiki>0.067305</nowiki>
|<nowiki>176.709291</nowiki>
|<nowiki>+21.24</nowiki>
|-
|Endo
|<nowiki>0.065610</nowiki>
|<nowiki>172.259068</nowiki>
|<nowiki>+16.79</nowiki>
|}

{| class="wikitable" style="margin: auto;"
!Graph 3: Reaction Profiles at Alternative Site in O-Xylylene 
|-
|[[File: Ex 3_unstable site reaction profile.png]]
|}

Both the exo and endo pathways have higher activation barriers at this alternative reactive site, as compared to Diels-Alder reaction at the first cis-butadiene fragment. Additionally, for both the exo and endo products, the energy of the product is higher than the energy of the reactants. Therefore, the reaction at the second cis-butadiene fragment is both kinetically and thermodynamically unfavourable.

== Conclusion ==
Plotting of the PES is a powerful method to compare the reaction barriers and relative stability of products between different pathways. In exercise 1 to 3, the Diels-Alder reaction was used as an excellent example of how orbital symmetry and steric factors can significantly affect the reaction energies. Exercise 1 to 3 each illustrates neutral, inverse and normal electron-demand Diels-Alder reaction respectively, providing a good range of the possible frontier orbital interactions. Exercise 1 shows that reactions are allowed only if the symmetry of the interacting HOMO and LUMO from each species are the same. Exercise 2 demonstrates factors that can determine the kinetic and thermodynamic products. The presence of secondary orbital interactions can stabilise transition states, and lower the activation barrier to form the kinetic product, while steric repulsions can raise the energy of the product and cause it to be less thermodynamically favoured. Exercise 3 compares the exo and endo Diels-Alder and cheletropic reaction pathways, highlighting the important point that while the three possible mechanisms are concerted, the bond formation can be either synchronous or asynchronous depending on which product is formed. Exercise 3 also shows that not all reactive sites can give kinetically and thermodynamically favoured products.

== Calculation Files ==
=== Ex 1 ===
[[File:Ex 1_ethylene MO.log]]
 [[File:Ex 1_butadiene MO.log]]
 [[File:Ex 1_transition state MO.log]]
 [[File:ex1IRC.log]]

=== Ex 2 ===
[[File:ex 2_cyclohexadiene MO.log]]
 [[File:ex 2_dioxole MO.log]]
 [[File:ex 2_Exo TS jmol.log]]
 [[File:ex 2_Endo TS jmol.log]]
 [[File:ex 2_exoproduct.log]]
 [[File:ex 2_endoproduct.log]]
 [[File:ex 2_exoTS_631Gd_IRC.log]]
 [[File:ex 2_endoTS_631Gd_IRC.log]]

=== Ex 3 ===
[[File:ex 3_optmin_xylylene.log]]
 [[File:ex 3_optmin_SO2.log]]
 [[File:ex 3_Exo TS.log]]
 [[File:ex 3_Endo TS.log]]
 [[File:ex 3_Cheletropic TS.log]]
 [[File:ex 3_Exo Product.log]]
 [[File:ex 3_Endo Product.log]]
 [[File:ex 3_Cheletropic Product.log]]
 [[File:ex 3_Exo IRC.log]]
 [[File:ex 3_Endo IRC.log]]
 [[File:ex 3_Cheletropic IRC.log]]
 [[File:ex 3_Part 2_Exo TS.log]]
 [[File:ex 3_Part 2_Endo TS.log]]
 [[File:ex 3_Part 2_Exo Product.log]]
 [[File:ex 3_Part 2_Endo Product.log]]

== References ==
<ref name="TS and reactivity1" /> http://www.uni-heidelberg.de/institute/fak12/AC/hofmann/acf_theo/MinNoMin.pdf
 <ref name="TS and reactivity2_supergoodlink" /> http://www.pitt.edu/~jordan/chem3430/289.TACC.pdf
 <ref name="electron demand" /> http://onlinelibrary.wiley.com/doi/10.1002/jlac.199719971209/epdf
 <ref name="c-c bond lengths" />http://pubs.acs.org/doi/pdf/10.1021/jp0269367
 <ref name="C VDW radius" /> http://pubs.acs.org/doi/pdf/10.1021/jp953141%2B
 <<ref name="comparison of DA and chel mechanisms" /> http://pubs.acs.org/doi/pdf/10.1021/jo00114a039
 <ref name="TM analogy" /> http://onlinelibrary.wiley.com/book/10.1002/9781118558409