ChemWiki - User contributions [en]

Rep:Mod:ts cyy113

2017-11-07T19:14:45Z

Cyy113: /* Extension */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are between a (C-C) single bond length and twice the Van Der Waals radius of C. This shows that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the table below:

{| class="wikitable" style="margin: auto;"
|+ Table 9: '''Reaction barrier and energy of the reaction between cyclohexadiene and 1,3-dioxole'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+160</nowiki>
|<nowiki>+168</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-67.4</nowiki>
|<nowiki>-63.8</nowiki>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 12: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 12 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 13. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 13: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 14: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 15: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+81.7</nowiki>
|<nowiki>+85.7</nowiki>
|<nowiki>+104.0</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-99.1</nowiki>
|<nowiki>-99.4</nowiki>
|<nowiki>-155.7</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram2 cyy113.PNG|centre|frame|Fig. 16: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 17: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 18: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 12: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 13: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+107.9</nowiki>
|<nowiki>+115.7</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+12.2</nowiki>
|<nowiki>+16.6</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.

==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:XYLENE3 cyy113.LOG]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-07T19:14:01Z

Cyy113: /* Exercise 3 */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are between a (C-C) single bond length and twice the Van Der Waals radius of C. This shows that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the table below:

{| class="wikitable" style="margin: auto;"
|+ Table 9: '''Reaction barrier and energy of the reaction between cyclohexadiene and 1,3-dioxole'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+160</nowiki>
|<nowiki>+168</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-67.4</nowiki>
|<nowiki>-63.8</nowiki>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 12: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 12 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 13. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 13: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 14: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 15: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+81.7</nowiki>
|<nowiki>+85.7</nowiki>
|<nowiki>+104.0</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-99.1</nowiki>
|<nowiki>-99.4</nowiki>
|<nowiki>-155.7</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram2 cyy113.PNG|centre|frame|Fig. 16: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 17: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+107.9</nowiki>
|<nowiki>+115.7</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+12.2</nowiki>
|<nowiki>+16.6</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.

==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:XYLENE3 cyy113.LOG]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-07T19:12:15Z

Cyy113: /* Reaction Barriers and Reaction Energies */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are between a (C-C) single bond length and twice the Van Der Waals radius of C. This shows that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the table below:

{| class="wikitable" style="margin: auto;"
|+ Table 9: '''Reaction barrier and energy of the reaction between cyclohexadiene and 1,3-dioxole'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+160</nowiki>
|<nowiki>+168</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-67.4</nowiki>
|<nowiki>-63.8</nowiki>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 12: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 12 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 13. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 13: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+81.7</nowiki>
|<nowiki>+85.7</nowiki>
|<nowiki>+104.0</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-99.1</nowiki>
|<nowiki>-99.4</nowiki>
|<nowiki>-155.7</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram2 cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+107.9</nowiki>
|<nowiki>+115.7</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+12.2</nowiki>
|<nowiki>+16.6</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.

==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:XYLENE3 cyy113.LOG]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-07T19:04:57Z

Cyy113: /* Reaction Barriers and Reaction Energies */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are between a (C-C) single bond length and twice the Van Der Waals radius of C. This shows that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+81.7</nowiki>
|<nowiki>+85.7</nowiki>
|<nowiki>+104.0</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-99.1</nowiki>
|<nowiki>-99.4</nowiki>
|<nowiki>-155.7</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram2 cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+107.9</nowiki>
|<nowiki>+115.7</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+12.2</nowiki>
|<nowiki>+16.6</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.

==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:XYLENE3 cyy113.LOG]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

File:Ex3combinedenergyprofilediagram2 cyy113.PNG

2017-11-07T19:04:19Z

Cyy113:

Rep:Mod:ts cyy113

2017-11-07T13:56:53Z

Cyy113: /* Bond Length Analysis */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are between a (C-C) single bond length and twice the Van Der Waals radius of C. This shows that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+81.7</nowiki>
|<nowiki>+85.7</nowiki>
|<nowiki>+104.0</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-99.1</nowiki>
|<nowiki>-99.4</nowiki>
|<nowiki>-155.7</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+107.9</nowiki>
|<nowiki>+115.7</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+12.2</nowiki>
|<nowiki>+16.6</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.

==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:XYLENE3 cyy113.LOG]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-06T15:18:10Z

Cyy113: /* Extension */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+81.7</nowiki>
|<nowiki>+85.7</nowiki>
|<nowiki>+104.0</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-99.1</nowiki>
|<nowiki>-99.4</nowiki>
|<nowiki>-155.7</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+107.9</nowiki>
|<nowiki>+115.7</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+12.2</nowiki>
|<nowiki>+16.6</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.

==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:XYLENE3 cyy113.LOG]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-06T15:14:16Z

Cyy113: /* Exercise 2 */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+81.7</nowiki>
|<nowiki>+85.7</nowiki>
|<nowiki>+104.0</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-99.1</nowiki>
|<nowiki>-99.4</nowiki>
|<nowiki>-155.7</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:XYLENE3 cyy113.LOG]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-06T12:06:52Z

Cyy113: /* Exercise 3 */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+81.7</nowiki>
|<nowiki>+85.7</nowiki>
|<nowiki>+104.0</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-99.1</nowiki>
|<nowiki>-99.4</nowiki>
|<nowiki>-155.7</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:XYLENE3 cyy113.LOG]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-06T12:06:17Z

Cyy113: /* Exercise 3 */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+81.7</nowiki>
|<nowiki>+85.7</nowiki>
|<nowiki>+104.0</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-99.1</nowiki>
|<nowiki>-99.4</nowiki>
|<nowiki>-155.7</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene3 cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

File:XYLENE3 cyy113.LOG

2017-11-06T12:05:26Z

Cyy113:

File:Xylene cyy113.log

2017-11-06T12:04:31Z

Cyy113: Cyy113 uploaded a new version of File:Xylene cyy113.log

Rep:Mod:ts cyy113

2017-11-06T12:02:49Z

Cyy113: /* Reaction Barriers and Reaction Energies */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+81.7</nowiki>
|<nowiki>+85.7</nowiki>
|<nowiki>+104.0</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-99.1</nowiki>
|<nowiki>-99.4</nowiki>
|<nowiki>-155.7</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-06T12:01:01Z

Cyy113: /* Reaction Barriers and Reaction Energies */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+81.7</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-99.1</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-06T11:58:18Z

Cyy113: /* Reaction Barriers and Reaction Energies */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178044
|467.454558
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

File:Xylene cyy113.log

2017-11-06T11:55:08Z

Cyy113: Cyy113 uploaded a new version of File:Xylene cyy113.log

File:TS ex2 exo cyy113.LOG

2017-11-06T11:35:29Z

Cyy113: Cyy113 uploaded a new version of File:TS ex2 exo cyy113.LOG

Rep:Mod:ts cyy113

2017-11-03T23:55:09Z

Cyy113: /* Exercise 3 */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-03T23:54:52Z

Cyy113: /* Exercise 3 */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frameless|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-03T21:50:51Z

Cyy113: /* Analysis of MO diagrams */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which cannot be the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-03T21:48:25Z

Cyy113: /* Reaction barriers and reaction energies */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which is not the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction Barriers and Reaction Energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

Rep:Mod:ts cyy113

2017-11-03T21:44:27Z

Cyy113: /* Extension */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which is not the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction barriers and reaction energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:Endotsextracyy113.log]]
|-
|Exo TS
|[[File:Exotsextra cyy113.log]]
|-
|Endo Product
|[[File:Endoproductmin4cyy113.log]]
|-
|Exo Product
|[[File:EXOPRODUCTMIN4cyy113.LOG]]
|}

== References ==

File:Endoproductmin4cyy113.log

2017-11-03T21:43:42Z

Cyy113:

File:Endotsextracyy113.log

2017-11-03T21:43:14Z

Cyy113:

File:EXOPRODUCTMIN4cyy113.LOG

2017-11-03T21:42:43Z

Cyy113:

File:Exotsextra cyy113.log

2017-11-03T21:42:12Z

Cyy113:

Rep:Mod:ts cyy113

2017-11-03T21:41:01Z

Cyy113: /* Exercise 3 */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which is not the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction barriers and reaction energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}

=== Extension ===

== References ==

Rep:Mod:ts cyy113 edit

2017-11-03T21:40:44Z

Cyy113:

{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO2
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}
{| class="wikitable"
|+'''Table X: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}
{| class="wikitable"
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
'''The relevant C-C bond lengths are tabulated below:'''
{| class="wikitable"
|+Table 2
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>G. S. Manku, ''Theoretical principles of inorganic chemistry'', Inter-India Publ., New Delhi, 1986, pp.97</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|sp2 = 1.31-1.34

sp3 = 1.53-1.55
|0.77
|}
Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>. It is given by the following equation.<ref>P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.</ref>

<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> Equation 2: Equation of the overlap integral </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable"
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of S
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

== intro ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point. Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\partial V/\partial q_i</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\partial^2 V/\partial q_i^2</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = 1/2 kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \partial^2 V/\partial x^2 = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3), reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {1/2\pi}\sqrt {k/m} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

{|
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|300x300px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|300x300px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |''Fig. Y: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction''
|}

== MO Analysis ==
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 

The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 9 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{|
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of exo TS</center>]]
|-
| colspan="2" | <center>''Fig. 9: MO diagrams for the formation of the (A) endo TS and (B) exo TS''</center>
|}
{| class="wikitable"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|
|
|-
|LUMO (MO 42)
|
|
|-
|HOMO (MO 41)
|
|
|-
|HOMO -1 (MO 40)
|
|
|}

Fig. 11 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 12. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 12: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref>J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, 2005, 85-90.</ref>
{| class="wikitable"
!Endo TS
!Exo TS
!Cheletropic
|-
|endo
|exo
|che
|}
{| class="wikitable"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}
{| class="wikitable"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

File:INDENEIRC CYY113.LOG

2017-11-03T21:40:30Z

Cyy113:

File:Xylene exo ts irccyy113.log

2017-11-03T21:39:56Z

Cyy113:

File:EX3ENDOTSIRC cyy113.LOG

2017-11-03T21:39:06Z

Cyy113:

File:INDENEPRODUCT CYY113.LOG

2017-11-03T21:38:10Z

Cyy113:

File:EXOPRODUCT OPT CYY113.LOG

2017-11-03T21:37:40Z

Cyy113:

File:ENDOPRODUCT cyy113ex3.LOG

2017-11-03T21:36:51Z

Cyy113:

File:SO2 CYY113.LOG

2017-11-03T21:35:07Z

Cyy113:

File:Xylene cyy113.log

2017-11-03T21:34:43Z

Cyy113:

Rep:Mod:ts cyy113

2017-11-03T21:33:18Z

Cyy113: /* Exercise 2 */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which is not the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction barriers and reaction energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:ENDOPRODUCT 631 CYY113.LOG]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
=== Extension ===

== References ==

File:ENDOPRODUCT 631 CYY113.LOG

2017-11-03T21:32:57Z

Cyy113:

Rep:Mod:ts cyy113 edit

2017-11-03T21:30:09Z

Cyy113:

{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:Endots631mo cyy113.log]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}
{| class="wikitable"
|+'''Table X: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}
{| class="wikitable"
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
'''The relevant C-C bond lengths are tabulated below:'''
{| class="wikitable"
|+Table 2
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>G. S. Manku, ''Theoretical principles of inorganic chemistry'', Inter-India Publ., New Delhi, 1986, pp.97</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|sp2 = 1.31-1.34

sp3 = 1.53-1.55
|0.77
|}
Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>. It is given by the following equation.<ref>P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.</ref>

<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> Equation 2: Equation of the overlap integral </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable"
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of S
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

== intro ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point. Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\partial V/\partial q_i</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\partial^2 V/\partial q_i^2</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = 1/2 kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \partial^2 V/\partial x^2 = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3), reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {1/2\pi}\sqrt {k/m} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

{|
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|300x300px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|300x300px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |''Fig. Y: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction''
|}

== MO Analysis ==
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 

The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 9 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{|
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of exo TS</center>]]
|-
| colspan="2" | <center>''Fig. 9: MO diagrams for the formation of the (A) endo TS and (B) exo TS''</center>
|}
{| class="wikitable"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|
|
|-
|LUMO (MO 42)
|
|
|-
|HOMO (MO 41)
|
|
|-
|HOMO -1 (MO 40)
|
|
|}

Fig. 11 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 12. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 12: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref>J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, 2005, 85-90.</ref>
{| class="wikitable"
!Endo TS
!Exo TS
!Cheletropic
|-
|endo
|exo
|che
|}
{| class="wikitable"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}
{| class="wikitable"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

Rep:Mod:ts cyy113

2017-11-03T21:30:04Z

Cyy113: /* Exercise 2 */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which is not the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction barriers and reaction energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|[[File:Endo PRODUCT MIN cyy113.LOG]]
|[[File:Endots631mo cyy113.log]]
|-
|Exo Product
|[[File:ExoPRODUCT MIN cyy113.LOG]]
|[[File:EXOPRODUCT631 CYY113.LOG]]
|}

=== Exercise 3 ===
=== Extension ===

== References ==

File:EXOPRODUCT631 CYY113.LOG

2017-11-03T21:29:22Z

Cyy113:

File:ExoPRODUCT MIN cyy113.LOG

2017-11-03T21:28:52Z

Cyy113:

File:Endots631mo cyy113.log

2017-11-03T21:27:58Z

Cyy113: Cyy113 uploaded a new version of File:Endots631mo cyy113.log

File:Endo PRODUCT MIN cyy113.LOG

2017-11-03T21:26:43Z

Cyy113:

Rep:Mod:ts cyy113 edit

2017-11-03T21:15:42Z

Cyy113:

{| class="wikitable"
! rowspan="2" |Species or IRC
! colspan="2" |Basis Set
|-
!PM6
!BY3LYP/6-31(d)
|-
|Cyclohexadiene
|[[File:CYCLOHEXADIENE MIN cyy113.LOG]]
|[[File:CYCLOHEXADIENE MIN 631 CYY113.LOG]]
|-
|1,3-dioxole
|[[File:DIOXOLE MIN cyy113.LOG]]
|[[File:DIOXOLE631 2 cyy113.LOG]]
|-
|Endo TS
|[[File:TS ex2 endo cyy113.LOG]]
|[[File:Endots631mo_cyy113.log]]
|-
|Exo TS
|[[File:TS ex2 exo cyy113.LOG]]
|[[File:Exots631mo2 cyy113.log]]
|-
|Endo Product
|
|
|-
|Exo Product
|
|
|}
{| class="wikitable"
|+'''Table X: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}
{| class="wikitable"
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
'''The relevant C-C bond lengths are tabulated below:'''
{| class="wikitable"
|+Table 2
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>G. S. Manku, ''Theoretical principles of inorganic chemistry'', Inter-India Publ., New Delhi, 1986, pp.97</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|sp2 = 1.31-1.34

sp3 = 1.53-1.55
|0.77
|}
Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>. It is given by the following equation.<ref>P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.</ref>

<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> Equation 2: Equation of the overlap integral </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable"
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of S
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

== intro ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point. Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\partial V/\partial q_i</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\partial^2 V/\partial q_i^2</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = 1/2 kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \partial^2 V/\partial x^2 = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3), reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {1/2\pi}\sqrt {k/m} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

{|
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|300x300px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|300x300px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |''Fig. Y: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction''
|}

== MO Analysis ==
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 

The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 9 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{|
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of exo TS</center>]]
|-
| colspan="2" | <center>''Fig. 9: MO diagrams for the formation of the (A) endo TS and (B) exo TS''</center>
|}
{| class="wikitable"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|
|
|-
|LUMO (MO 42)
|
|
|-
|HOMO (MO 41)
|
|
|-
|HOMO -1 (MO 40)
|
|
|}

Fig. 11 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 12. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 12: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref>J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, 2005, 85-90.</ref>
{| class="wikitable"
!Endo TS
!Exo TS
!Cheletropic
|-
|endo
|exo
|che
|}
{| class="wikitable"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}
{| class="wikitable"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

Rep:Mod:ts cyy113 edit

2017-11-03T21:09:37Z

Cyy113:

{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}
{| class="wikitable"
|+'''Table X: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}
{| class="wikitable"
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
'''The relevant C-C bond lengths are tabulated below:'''
{| class="wikitable"
|+Table 2
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>G. S. Manku, ''Theoretical principles of inorganic chemistry'', Inter-India Publ., New Delhi, 1986, pp.97</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|sp2 = 1.31-1.34

sp3 = 1.53-1.55
|0.77
|}
Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>. It is given by the following equation.<ref>P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.</ref>

<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> Equation 2: Equation of the overlap integral </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable"
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of S
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

== intro ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point. Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\partial V/\partial q_i</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\partial^2 V/\partial q_i^2</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = 1/2 kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \partial^2 V/\partial x^2 = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3), reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {1/2\pi}\sqrt {k/m} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

{|
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|300x300px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|300x300px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |''Fig. Y: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction''
|}

== MO Analysis ==
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 

The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 9 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{|
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of exo TS</center>]]
|-
| colspan="2" | <center>''Fig. 9: MO diagrams for the formation of the (A) endo TS and (B) exo TS''</center>
|}
{| class="wikitable"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|
|
|-
|LUMO (MO 42)
|
|
|-
|HOMO (MO 41)
|
|
|-
|HOMO -1 (MO 40)
|
|
|}

Fig. 11 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 12. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 12: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref>J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, 2005, 85-90.</ref>
{| class="wikitable"
!Endo TS
!Exo TS
!Cheletropic
|-
|endo
|exo
|che
|}
{| class="wikitable"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}
{| class="wikitable"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

Rep:Mod:ts cyy113

2017-11-03T21:08:40Z

Cyy113: /* Exercise 1 */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which is not the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction barriers and reaction energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|Butadiene
|[[File:BUTADIENEWITHMO CYY113 2.LOG]]
|-
|Ethene
|[[File:ETHENE_WITHMO_CYY113.LOG]]
|-
|Butadiene/Ethene TS
|[[File:TS WITHMO CYY113.LOG]]
|-
|Cyclohexene
|[[File:Cyclohexeneproduct cyy113.log]]
|-
|IRC
|[[File:CYCLOHEXENE TS IRC3 CYY113.LOG]]
|}

=== Exercise 2 ===
=== Exercise 3 ===
=== Extension ===

== References ==

Rep:Mod:ts cyy113

2017-11-03T20:57:58Z

Cyy113: /* Conclusion */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which is not the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction barriers and reaction energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

==Log Files==
=== Exercise 1 ===
=== Exercise 2 ===
=== Exercise 3 ===
=== Extension ===

== References ==

Rep:Mod:ts cyy113

2017-11-03T20:57:01Z

Cyy113: /* Conclusion */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which is not the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction barriers and reaction energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==
The reaction dynamics of several Diels-Alder and a cheletropic reaction were analysed. From an optimisation of the atom coordinates in the reactants, transition state and products using Gaussian, various physical parameters (C-C bond lengths, free energies, MO energies) were extracted. In addition, an internal reaction coordinate (IRC) calculation was also run to visualise the trajectory of the reactants towards each other and how these physical parameters change along the reaction coordinate. The results obtained agreed with theory, in particular the Frontier Molecular Orbital theory, concerted mechanism of Diels-Alder reactions and the Endo Rule arising from secondary orbital interactions. The results also demonstrated that when SO2 is used as a dienophile, the Diels-Alder pathway is kinetically controlled while the cheletropic pathway is thermodynamically controlled.

== References ==

Rep:Mod:ts cyy113

2017-11-03T20:43:12Z

Cyy113: /* Extension */

== Introduction ==
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 dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point.<ref name="atkins">P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.,pp 470, 908-909</ref> Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\frac{\partial V}{\partial q_i}</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\frac{\partial^2 V}{\partial q_i^2}</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring<ref name="atkins" />, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = \frac{1}{2} kx^2 </math></center>
<center> ''Equation 1: Hooke's Law'' </center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \frac{\partial^2 V}{\partial x^2} = k </math></center>
<center> ''Equation 2: Second derivative of Hooke's Law'' </center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3)<ref name="atkins" />, reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {\frac{1}{2\pi}}\sqrt {\frac{k}{m}} </math></center>
<center> ''Equation 3: Relation between frequency and <math>k</math>'' </center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions.

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.

== Exercise 1 ==
The following Diels-Alder cycloaddition was investigated. Butadiene was the diene while ethene was the dienophile. Butadiene had to be in the correct s-cis conformation so that there is effective spatial overlap with ethene. Reactants, TS and products were optimised at the PM6 level.
<center>[[File:Q1 rxnscheme cyy113.PNG]]</center>
<center> Fig 1: Scheme of reaction between butadiene and ethene. Scheme was generated via ChemDraw </center>
=== Optimisation Results ===

1) Optimise the reactants and TS at the '''PM6 level'''.

{| class="wikitable" style="margin: auto;"
!Butadiene (s-cis)
!Ethene
!TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>BUTADIENEWITHMO CYY113 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
<center> Fig. 2: Optimised JMol files reactants and TS </center> 

2) Confirm that you have the correct TS with a '''frequency calculation''' and '''IRC'''.
 '''Frequency Calculation'''

{|
|[[File:Cyclohexene ts freq cyy113_2.PNG|frame|none|alt=Fig. 3: Frequency values of butadiene/ethene TS|Fig. 3: Frequency values of butadiene/ethene TS]]
|To confirm that the TS is correctly optimised, its frequencies must be calculated and checked that there is only 1 imaginary vibration. This is represented by GaussView as a negative vibration. Since the TS occurs on a maximum point on the Free Energy surface, the second derivative of such a point is negative. Frequency values are in fact the second derivatives of energy, hence a single negative frequency suggests that there is only 1 TS on the reaction coordinate chosen. This would be indicative of a good reaction coordinate, as additional negative frequencies suggests the presence of more than 1 stationary point, which could be a maximum in the order parameter chosen but a minimum in another order parameter. 
The transition state has been computed correctly as it only shows 1 negative frequency. (Fig. 3)
|}
'''IRC''' 
A well-defined, asymmetric Free Energy Surface was obtained, further confirming that the reaction coordinate was well chosen. (Fig. 4) A predicted trajectory was computed using a Intrinsic Reaction Coordinate function on Gaussian. (Fig. 5)
{| style="margin: auto;"
|-
|[[File:Energy profile ex 1 cyy113.PNG|center|thumb|1500x1500px|alt=Fig. 4: Free Energy Profile of reaction|Fig. 4: Free Energy Profile of reaction]] || [[File:Cyclohexene ts irc3 cyy113.gif|center|frame|none|alt=Fig. 5: Trajectory of Butadiene/Ethene molecules|Fig. 5: Trajectory of Butadiene/Ethene molecules]]
|} 

3) Optimise the products at the PM6 level.
{| class="wikitable" style="margin: auto;"
!Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Cyclohexeneproduct cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 6 Optimised JMol file of cyclohexene </center> 

=== Analysis of MO diagrams ===
Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).
 Based on the values obtained from a Gaussian calculation using the PM6 basis set, secondary orbital mixing between the MOs of the transition state is expected, which stabilises LUMO+1 and destabilises the HOMO, as illustrated with the red arrows and energy levels.
[[File:MO q1 cyy113 2.PNG|centre|frame|Fig 7: MO diagram showing the formation of the butadiene/ethene TS]]
The HOMO-LUMO energy gap in butadiene is smaller than ethene due to increased conjugation. In fact, this reaction is reported to proceed inefficiently.<ref name="phychem">E. Anslyn and D. Dougherty, ''Modern physical chemistry'', University Science, Sausalito, Calif., 2004.,pp. 896 </ref> The reaction will proceed more efficiently if butadiene is made more electron rich by adding electron donating groups, or ethene is made more electron poor by adding electron withdrawing groups. Adding electron donating groups raises the energy levels, while adding electron withdrawing groups lower the energy levels. This effectively reduces the HOMO-LUMO energy gap allowing for more efficiency overlap.
 
For each of the reactants and the TS, open the .chk (checkpoint) file. Under the Edit menu, choose MOs and visualise the MOs. Include images (or Jmol objects) for each of the HOMO and LUMO of butadiene and ethene, and the four MOs these produce for the TS. Correlate these MOs with the ones in your MO diagram to show which orbitals interact. 
{| style="margin: auto;" cellpadding=5
|+ '''Table 1: Selected JMol images showing MOs of Butadiene, Ethene and Butadiene/Ethene TS'''
|
{| class="wikitable" style="margin: auto;"
!
!Butadiene
!Ethene
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 16; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENEWITHMO_CYY113_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_WITHMO_CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
||
{| class="wikitable"
!Butadiene/Ethene TS
|-
|LUMO+1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO-1
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
|}

What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 

The MOs of the transition state are all formed by overlapping and mixing reactant MOs of the same symmetry (Fig. 5), suggesting that a reaction is only allowed when reactant MOs of the same symmetry overlap. This can be justified mathematically. As the coordinates of the atoms change along a reaction coordinate, the wavefunctions of the reactant MOs (denoted as <math>\psi_a</math> and <math>\psi_b</math>) can no longer describe the electron density of the transition state<ref name="pearson">R. Pearson, ''Accounts of Chemical Research'', 1971, '''4''', 152-160.</ref>. A transition state wavefunction (denoted as <math>\psi</math>) will better describe this electron density (as <math>\psi^2</math>). The energy of the transition state wavefunction <math>E_\psi</math> is given via the Hamiltonian <math> \hat{H} </math> as follows:

<center><math> E_\psi = \frac{<\psi_a | \hat{H} | \psi_a> \pm 2 <\psi_a | \hat{H} | \psi_b> + <\psi_b | \hat{H} | \psi_b>}{2(1 \pm <\psi_a | \psi_b>)} </math></center>
<center>''Equation 4: Energy of transition state wavefunction''<ref name="phychem" /> </center>

The Hamiltonian is a totally symmetric operator, meaning that <math>\hat{H} | \psi ></math> has the same symmetry as <math>\psi </math>. If the symmetries of <math>\psi_a</math> and <math>\psi_b </math> were different, <math>E_\psi</math> would be the sum of <math>E_{\psi_a}</math> and <math>E_{\psi_b}</math>, which is not the case since the reactants are not at infinite separation.

Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>). It is given by the following equation.<ref name="pearson" />
<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center> ''Equation 5: Equation of the overlap integral'' </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 2: Summary of the possible values of <math>S</math>'''
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of <math>S</math>
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

=== Bond Length Analysis ===
Include measurements of the 4 C-C bond lengths of the reactants and the 6 C-C bond lengths of the TS and products. How do the bond lengths change as the reaction progresses? What are typical sp3 and sp2 C-C bond lengths? What is the Van der Waals radius of the C atom? How does this compare with the length of the partly formed C-C bonds in the TS.

The relevant C-C bond lengths are tabulated below:
{| class="wikitable" style="margin: auto;"
|+ ''' Table 3: Summary of C-C bond lengths and Van Der Waals radius '''
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp2 and sp3 bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp2 and

sp3 bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius
of C atom<ref>A. Bondi, ''The Journal of Physical Chemistry'', 1964, '''68''', 441-451.</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|<center>sp2-sp2 C-C = 1.31-1.34
sp2-sp3 C-C = 1.50 
sp3-sp3 C-C = 1.53-1.55</center>

|<center>1.70</center>
|}

The partly formed C-C bond lengths are shorter than twice of the Carbon Van Der Waals radius, which is evidence that a bond is indeed being formed. In addition, the C-C bond lengths in the cyclohexene product agreed well with literature values of cyclohexene and average sp2 C sp2 C, sp2 C sp3 C and sp3 C sp3 C bond lengths as well. This is further support that the cyclohexene product and TS has been optimised well.
 
Along the reaction coordinate, graphs showing the change in various C-C bond lengths are shown in the table below. The atoms are labelled based on Fig. 8.

[[File:Labelled bondlengths cy113.PNG|center|thumb|Fig 8: Labelled atoms of reactants]]
{| class="wikitable" style="margin: auto;"
|+ ''' Table 4: Change in bond lengths with reaction coordinate '''
!Bond length between
!C2 and C3
!C3 and C4
!C4 and C1
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength23 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength34 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength41 cyy113.PNG|centre|thumb]]Bond length increases
|-
!Bond length between
!C11 and C14
!C1 and C11
!C2 and C14
|-
|Graph showing the change in Bond Length with Reaction Coordinate
|[[File:Bondlength1411 cyy113.PNG|centre|thumb]]Bond length increases
|[[File:Bondlength111 cyy113.PNG|centre|thumb]]Bond length decreases
|[[File:Bondlength214 cyy113.PNG|centre|thumb]] Bond length decreases
|}

=== Vibrational Analysis ===
Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?
 The vibration that corresponds to the reaction path occurs at <math>-949 cm^{-1}</math>. The formation is synchronous as expected of a typical Diels Alder cycloaddition.
{|style="margin: auto;"
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script> frame 7; vibration 2; </script>
<uploadedFileContents>TS WITHMO CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Fig. 9: Reaction path vibration for Butadiene/Ethene TS
|}

== Exercise 2 ==
In this exercise, the following reaction was investigated. Cyclohexadiene was the diene while 1,3-dioxole was the dienophile. Reactants, TS and products were optimised at both the PM6 and BY3LYP/6-31(d) levels.
<center>[[File:Q2 rxnscheme cyy113.PNG]]</center>
<center> Fig. 10: Scheme of reaction between cyclohexadiene and 1,3-dioxole generated by ChemDraw </center>
=== Optimisation Results ===
Using any of the methods in the tutorial, locate both the endo and exo TSs at the B3LYP/6-31G(d) level (Note that it is always fastest to optimise with PM6 first and then reoptimise with B3LYP). 
Method 3 was used and the following optimisations were obtained: 
{| class="wikitable" style="margin: auto;"
|+ Table 5: ''' JMol log files of reactants and TS at PM6 and BY3LP 6-31(d) level '''
!
!Endo TS
!Exo TS
!Cyclohexadiene
!1,3-dioxole
|-
|PM6
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 endo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>TS ex2 exo cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE MIN cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|B3LYP/6-31G(d)
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Endots631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>Tsexo631 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>CYCLOHEXADIENE MIN 631 CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<uploadedFileContents>DIOXOLE631 2 cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Frequency Analysis ===
Confirm that you have a TS for each case using a frequency calculation. 
The frequencies of each TS are shown in the table below. In each case, there is only 1 negative frequency which confirms that a TS has been located on a suitable reaction coordinate.
{| class="wikitable" style="margin: auto;"
|+ Table 6: '''Summary of frequency values'''
!
! colspan="2" | Endo TS
! colspan="2" | Exo TS
|-
|Basis Set
!PM6
!BY3LYP/6-31G(d)
!PM6
!BY3LYP/6-31G(d)
|-
|Frequencies
|[[File:Endotsfreq cyy113.PNG|centre|frameless]]
|[[File:Endo631freq cyy113.PNG|centre|frameless]]
|[[File:Exofreq cyy113.PNG|centre|frameless]]
|[[File:Exo631freq cyy113.PNG|centre|frameless]]
|}

=== MO Analysis ===
Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. 
 The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 11 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{| style="margin: auto;"
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of Exo TS</center>]]
|-
| colspan="2" | <center>Fig. 11: MO diagrams for the formation of the (A) endo TS and (B) exo TS</center>
|}
{| class="wikitable" style="margin: auto;"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO (MO 42)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO (MO 41)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|HOMO -1 (MO 40)
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

Is this a normal or inverse demand DA reaction? 
This is an inverse demand DA reaction. In a normal demand DA reaction, the HOMO of the diene reacts with the LUMO of the dienophile but in an inverse demand DA reaction, the HOMO of the dienophile reacts with the LUMO of the diene.<ref name="bio">B. Oliveira, Z. Guo and G. Bernardes, ''Chem. Soc. Rev.'', 2017, '''46''', 4895-4950.</ref> Thus, in this reaction, the HOMO of the dienophile 1,3-dioxole reacted with the LUMO of the diene, hexadiene. This can be rationalised by Molecular Orbital (FMO) theory, which states that the rate of a Diels-Alder reaction is faster if the energy gap between the overlapping HOMO and LUMO orbitals is smaller.<ref name="bio" /> Based on the energy calculations of cyclohexadiene and 1,3-dioxole at the BY3LYP/6-31(d) level, it is found that the energy gap between LUMO (hexadiene) and HOMO (1,3-dioxole) is smaller than that between HOMO (hexadiene) and LUMO (1,3-dioxole).

In addition, the symmetries of the LUMO +1, LUMO, HOMO and HOMO +1 molecular orbitals in the transition state are all reversed in an inverse demand Diels-Alder reaction. This is reflected in the A-S-S-A ordering of these transition state MOs, which should have been S-A-A-S in a normal demand Diels-Alder reaction.

In fact, it is expected that this Diels Alder cycloaddition proceeds with inverse demand. Electrons can be donated into the dienophilic double bond via resonance effect from the adjacent oxygen atoms. Thus, the energies of the dienophile MOs will increase in energy<ref name="bio" /> such that the HOMO of the dienophile will be closer in energy to the LUMO of the diene.

=== Reaction Barriers and Reaction Energies ===
In the .log files for each calculation, find a section named "Thermochemistry". Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature (the corrected energies are labelled "Sum of electronic and thermal Free Energies", corresponding to the Gibbs free energy). 
 The energies of reactants, TS and products are summarised in the table below.
{| class="wikitable" style="margin: auto;"
|+'''Table 8: Summary of the energies of reactants, TS and products'''
! colspan="2" rowspan="2" |
! colspan="2" |Cyclohexadiene
! colspan="2" |1,3-dioxole
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.116877
|306.860587
|0.052276
|137.250648
|0.137939
|362.158872
|-
|BY3LYP/6-31(d)
|<nowiki>-233.324375</nowiki>
|<nowiki>-612593.19323</nowiki>
|<nowiki>-267.068650</nowiki>
|<nowiki>-701188.79399</nowiki>
|<nowiki>-500.332148</nowiki>
|<nowiki>-1313622.1546</nowiki>
|-
! colspan="2" rowspan="2" |
! colspan="2" |Exo TS
! colspan="2" |Endo Product
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
| rowspan="2" |Basis Set
|PM6
|0.138903
|364.689854
|0.037804
|99.2544096
|0.037977
|99.7086211
|-
|BY3LYP/6-31(d)
|<nowiki>-500.329168</nowiki>
|<nowiki>-1313614.3306</nowiki>
|<nowiki>-500.418691</nowiki>
|<nowiki>-1313849.3733</nowiki>
|<nowiki>-500.417320</nowiki>
|<nowiki>-1313845.77374</nowiki>
|}

The reaction barriers (i.e. activation energy) and reaction energies (i.e. <math> \Delta G </math>) were then calculated using energy values obtained from the BY3LYP/6-31G(d) basis set, as they are more accurate. The reaction barrier is the difference between the energies of the TS and the reactants, while the reaction energy is the difference between the energies of the reactants and products. They are summarised in the reaction profile diagrams below. (Fig. 12)

{| style="margin: auto;"
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|400x400px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<center>Fig. 12: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction</center>
|}

 
Which are the kinetically and thermodynamically favourable products? 
 
In this reaction, the endo product has a lower activation energy and hence the kinetically favourable product. This is in agreement with the fact that secondary orbital interactions exist only for the endo product in any Diels-Alder cycloaddition involving substituted dienophiles.<ref name ="endo" /> In this example, it is illustrated in Fig. 13. 

In addition, in this reaction the endo product has a more negative reaction energy and is hence the thermodynamically favourable product. This deviates from the fact that the endo product suffers from steric clash. However, since a significant extent of distortion is required from cyclohexadiene to form the transition state<ref> B. Levandowski and K. Houk, ''The Journal of Organic Chemistry'', 2015, '''80''', 3530-3537.</ref>, it is likely that the reaction proceeds with a late transition state. By Hammond's Postulate, this means that the transition state resembles the products<ref>R. Macomber, ''Organic chemistry'', University Science Books, Sausalito, 1st edn., 1996.,pp. 248 </ref>. As such, the same secondary orbital interactions that stabilise the endo transition state will also stabilise the endo product, leading to a more negative reaction energy.
 
Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy (Hint: in GaussView, set the isovalue to 0.01. In Jmol, change the mo cutoff to 0.01)? 
{| class="wikitable" style="margin: auto;"
!HOMO of Endo TS
!HOMO of Exo TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 6; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Endots631mo_cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script> frame 8; mo 43; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Exots631mo2 cyy113.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| colspan="2" |<center>Fig. 13: Analysis of secondary orbital overlaps in the HOMO of endo and exo TS</center>
|}
Fig. 13 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 14. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 14: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref name="endo">J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, '''2005''', 85-90.</ref>

==Exercise 3==
The Diels-Alder reaction between o-xylylene and SO2 is investigated in this exercise. o-xylylene is the diene while SO2 is the dienophile. 
[[File:Ex3rxnscheme cyy113.PNG|centre|frame|Fig. 15: Reaction scheme between o-xylylene and SO2]]
=== Optimisation Results ===
1) Optimise the TSs for the endo- and exo- Diels-Alder and the Cheletropic reactions at the '''PM6 level'''. 
The endo and exo TS each have a pair of enantiomers. In each case, only 1 enantiomer is displayed in the JMol files in the figure below.
{| class="wikitable" style="margin: auto;"
!Exo TS
!Endo TS
!Cheletropic TS
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>XYLYLENE exoex3 TS cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>EX3ENDOTS CYY113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>INDENE_TS_cyy113.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
 <center> Fig. 16: Optimised JMol files of the endo, exo and cheletropic TS </center> 

=== Reaction barriers and reaction energies ===
3) Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.
 The energies of the reactants, TS and products for the endo, exo and cheletropic reactions are summarised below:
{| class="wikitable" style="margin: auto;"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |o-xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}

Taking the same differences in energies as in Exercise 2, the reaction barrier (activation energy) and reaction energies are as follows:
{| class="wikitable" style="margin: auto;"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO2'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}

4) Using Excel or Chemdraw, draw a reaction profile that contains relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the cheletropic reaction. You can set the 0 energy level to the reactants at infinite separation. 
The relative energy levels are shown in the figure below, generated on ''ChemDraw''.
[[File:Ex3combinedenergyprofilediagram cyy113.PNG|centre|frame|Fig. 17: Energy profile diagrams of the endo, exo and cheletropic reactions]]
Between the endo/exo pathways, the endo product is kinetically favoured due to secondary orbital overlap. The exo product is thermodynamically favoured as it has less steric hindrance. These factors were previously discussed in Exercise 2 under section 3.4. Considering the cheletropic reaction and both Diels Alder reactions, the cheletropic TS has a higher energy than the Diels Alder TS, as it is more planar and there are strong eclipsing interactions developing between the aromatic 6-membered ring and the sulphone oxygen atoms which destabilises the TS.<ref> N. Isaacs and A. Laila, ''Tetrahedron Letters'', 1976, '''17''', 715-716. </ref> However, the cheletropic product is more stable than both Diels-Alder products. Although a 5-membered ring typically suffers from more torsional strain than 6-membered rings<ref>G. Odian, ''Principles of polymerization'', Wiley-Interscience, Hoboken, NJ, 2004. </ref>, in this case the presence of a large sulfur atom would cause more distortions in the 6-membered ring. Thus, it can be concluded that the cheletropic pathway is under thermodynamic control while the Diels-Alder pathway is under kinetic control. The cheletropic product will be formed in equillibrating conditions.

=== IRC Analysis ===
2) Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.
 The following figures illustrate the approach trajectory between o-xylylene and SO2 for the endo, exo and cheletropic reactions:
<center>[[File:Ex3endoirc cyy113.gif]]</center>
<center>(A): Approach trajectory for the formation of the endo product </center>
<center>[[File:Exoex3irc cyy113.gif]]</center>
<center>(B): Approach trajectory for the formation of the exo product </center>
<center>[[File:Indeneirc cyy113.gif]]</center>
<center>(C): Approach trajectory for the formation of the cheletropic product </center>
<center> Fig. 18: IRC visualisations of the approach trajectory between o-xylylene and SO2</center>

Xylylene is highly unstable. Look at the IRCs for the reactions - what happens to the bonding of the 6-membered ring during the course of the reaction?
 o-xylylene is unstable due to the presence of 2 dienes locked in an s-cis conformation, which can act as good dienes for Diels-Alder reactions. All the carbons are sp2 hybridised, thus the molecule is planar. As SO2 approaches o-xylylene from either the top or bottom face, the 6-membered ring becomes aromatic due to it having 6 <math>\pi</math> electrons. It is further evidenced by measuring the C-C bond lengths, which lie between an sp2C-sp2C and sp3C-sp3C bond at 1.40 Angstroms. This has a stabilising effect.

== Extension ==
There is a second cis-butadiene fragment in o-xylylene that can undergo a Diels-Alder reaction. If you have time, prove that the endo and exo Diels-Alder reactions are very thermodynamically and kinetically unfavourable at this site.
 The reaction between SO2 and the second cis-butadiene fragment is described in the scheme below. Similarly, endo and exo products are possible.
[[File:Ex4rxnscheme cyy113.PNG|centre|frame|Fig. 19: Reaction scheme for an alternative reaction between o-xylylene and SO2]]

In a similar way as Exercise 3, the reactants, TS and products are optimised with Gaussian at PM6 energy level and the table below summarises their energies.
{| class="wikitable" style="margin: auto;"
|+'''Table 11: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO2'''
! colspan="2" |Xylylene
! colspan="2" |SO2
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
!Energy (Hartrees)
!Energy (kJ mol-1)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}

Accordingly, the reaction barriers and reaction energies were calculated as follows:
{| class="wikitable" style="margin: auto;"
|+Table 12: Reaction barrier and reaction energy for the alternative reaction between o-xylylene and SO2
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol-1
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol-1
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
Both Diels-Alder pathways on this cis-butadiene fragment proceed with an activation energy barrier that is much larger than if it proceeded with the former cis-butadiene fragment in Exercise 3. Moreover, <math>\Delta G</math> is positive. This suggests that both reaction pathways at this cis-butadiene fragment are not feasible.
==Conclusion==

== References ==