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Rep:Zw4415:TSexercise

2025-09-01T09:51:12Z

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=Introduction=

===The Transition State===

According to '''Computational Quantum Chemistry''' the reaction coordinate was functionalised by all 3N<sub>atoms</sub>-6 internal coordinates.

Potenyial energy surface describes the energy of a specific system where energy was tabulated by a set of reaction coordinates. [1]

The reactant and product with stationary structure are the minimum points on the potential energy surface.It could be calculated by :

[[File:Zinan_Wang_eq1.png|thumb|center|Equation. 1 [1] ]]

where '''R''' is a set of all internal coordinates and q is normal coordinates which is a linear combination of all internal coordinates.

The transition state is the point with the highest energy along the reaction pathway. The second derivative should be all positive except the one along the reaction coordinate.
Thus it could be calculated as below.
[[File:Zinan_Wang_eq2.png|thumb|center|Equation. 2 [1] ]]
[[File:Zinan_Wang_eq3.png|thumb|center|Equation. 3 [1] ]]

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 08:53, 23 March 2018 (UTC) You get this info by diagonalising the hessian matrix which is second derivatives in the basis of the degrees of freedom. the diagonalisation give the normal modes as eigenvectors and the force constants as eigenvalues.

===The computational method===
In this exercise, PM6 and B3LYP/6-31G(d) methods have been used to calculate the optimized structure of the reactants, products and the transition state. PM6 method is generally a faster method to give a rough approximation of the structure. A more precise optimization could be carried out by B3LYP with basis set of 6-31G(d) while it often takes quite a long time.

===Freqrency check===
All the normal modes for stationary structure ,thus the reactants and products, should have positive values as they represent the minimum energy points on the PES.
For all the transition state structures, only 1 negative value should be seen. It is because when calculating from a harmonic oscillator with negative force constant (The negative value of the second derivative along the reaction coordinate), frequency was obtained as a imaginary number. The negative value in the frequency table thus illustrate the imaginary number (e.g. 526.7i).

===IRC check===
IRC was obtained with the same basis set used to calculate the transition structrue. The gradient of the energy graph (thus the first derivative along the reaction coordinate) shows 0 at the Transition state, reactants and product, which confirms the success of obtaining transition state.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 08:54, 23 March 2018 (UTC) This section was ok you could have done abit more reading and gone into more detail.

=Exercise 1: Reaction of Butadiene with Ethylene=
In exercise 1, the simplest Diels-Alder reaction between butadiene and ethene was calculated.

The net reaction was visualized by the bond length measurment which represent the change of bond order.

===TS analysis===
All reactants and product were optimized at PM6 level.
Transition state were obtained at PM6 level with frequency checked to only have one negative value at around -930.

[[File: Zinan_Wang_EX1_IRC.PNG|thumb|The IRC for optimized transition state|300px]]

===MO diagram for the formation of the butadiene/ethene transition state===
[[File:Zinan_Wang_MO_TS_T.jpg|thumb|x800px|center|MO diagram for the formation of the cyclohexene]]

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) You did a good work in this exercise overall but you got a bit confused here in reporting the energies of your TS LUMOs. Both the LUMO and LUMO+1 are higher in energy than what you have used here in the diagram.)

The MO energy level of both reactant was obtained by Energy calculation using PM6 method. For this specific Diels-Alder reaction between butadiene and ethene with no substituents, the electrons are in normal demand (Diene being electron rich and ethene being electron poor).

The Transition state MO generated by both reactant has higher energy then the HOMO of ethene. That is due to the fact that it is MO for transition state (not the product) which is the highest energy point along the reaction pathway.

===Visualised MOs for HOMO and LUMO===
{| class="wikitable" style="border: none; background: none;"
|+|Table 1: Visualised MOs
! colspan="2"| cis-Butadiene !! colspan="2"| Ethene !! colspan="4"| Transition state
|-
| HOMO || LUMO || HOMO || LUMO || HOMO-1 || HOMO || LUMO || LUMO+1
|-
|<jmol>
<jmolApplet>
<title>HOMO diene</title>
<color>white</color>
<size>180</size>
<script>frame 24; mo 11; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZW4415_EX1_DIENE_OPT_PM6_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO diene</title>
<color>white</color>
<size>180</size>
<script>frame 24; mo 12; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZW4415_EX1_DIENE_OPT_PM6_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>HOMO ethene</title>
<color>white</color>
<size>180</size>
<script>frame 14; mo 6; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZW4415_EX1_ETHENE_OPT_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO ethene</title>
<color>white</color>
<size>180</size>
<script>frame 14; mo 7; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZW4415_EX1_ETHENE_OPT_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>HOMO-1 TS</title>
<color>white</color>
<size>180</size>
<script>frame 60; mo 16; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZW4415_EX1_TS_OPT_PM6_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>HOMO TS</title>
<color>white</color>
<size>180</size>
<script>frame 60; mo 17; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZW4415_EX1_TS_OPT_PM6_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<title>LUMO TS</title>
<color>white</color>
<size>180</size>
<script>frame 60; mo 18; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZW4415_EX1_TS_OPT_PM6_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO+1 TS</title>
<color>white</color>
<size>180</size>
<script>frame 60; mo 19; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZW4415_EX1_TS_OPT_PM6_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
! colspan="8"| Correlation with MO diagram
|-
|[[File:Zinan_Wang_MO_DI_HOMO.jpg|center]]
|[[File:Zinan_Wang_MO_DI_LUMO.jpg|center]]
|[[File:Zinan_Wang_MO_ENE_HOMO.jpg|center]]
|[[File:Zinan_Wang_MO_ENE_LUMO.jpg|center]]
|[[File:Zinan_Wang_MO_TS_HOMO1.jpg|center]]
|[[File:Zinan_Wang_MO_TS_HOMO.jpg|center]]
|[[File:Zinan_Wang_MO_TS_LUMO.jpg|center]]
|[[File:Zinan_Wang_MO_TS_LUMO1.jpg|center]]
|}

For a reaction to take place the interacting orbitals must have the same symmetry (a-a/s-s), also both orbitals should be in the similar energy level.
For both symmetric-symmetric interaction and antisymmetric-antisymmetric interaction, the overlap integral is non zero. For symmetric-antisymmetric interactions the overlap integral is zero.

===Bond length measurement===
The Van der Waals radii of carbon is around 1.7 Å.
A typical sp3-sp3 carbon bond length is about 1.54 Å, and a typical sp2-sp2 carbon bond length is about 1.33 Å.

{| class="wikitable"
!
!cis-Butadiene
!Ethene
!Transition State
!Product
!Bond length
!Bond order
|-
|
|[[File:Zinan_Wang_EX1_di_BOND.PNG|x200px]]
|[[File:Zinan_Wang_EX1_ene_BOND.PNG|x200px]]
|[[File:Zinan_Wang_EX1_TS_BOND.PNG|x200px]]
|[[File:Zinan_Wang_EX1_P_BOND.PNG|x200px]]
|
|
|-
|C1-C2
|<nowiki>-</nowiki>
|<nowiki>-</nowiki>
|2.11484
|1.53580
|decrease
|forming
|-
|C2-C3
|<nowiki>-</nowiki>
|1.32731
|1.38205
|1.53764
|increase
|breaking
|-
|C3-C4
|<nowiki>-</nowiki>
|<nowiki>-</nowiki>
|2.11485
|1.53579
|decrease
|forming
|-
|C4-C5
|1.33529
|<nowiki>-</nowiki>
|1.37977
|1.49261
|increase
|breaking
|-
|C5-C6
|1.46826
|<nowiki>-</nowiki>
|1.41113
|1.33306
|decrease
|forming
|-
|C6-C1
|1.33537
|<nowiki>-</nowiki>
|1.37972
|1.49261
|increase
|breaking
|}

At transition state, the partially formed bond length is about 2.11 Å, which is smaller than 2 times the van der Waals radii of carbon atom while still longer than a typical sp3-sp3 C-C single bond. That means the electrons from both atoms have been interacting (but yet to form a bond) at the transition state, after which the distance decreased to 1.54 Å indicating the formation of a sp3-sp3 carbon bond in the product.

===Bond vibration at reaction path===

As seen in this bond formation vibration, two bonds are forming synchronously.

[[File:Zinan_Wang_EX1_TS_vibration.gif|center|thumb|800px|The bond vibration at reaction path.]]

===.log Files for Exercise 1===
PM6 optimized s-cis-Butadiene: [[File:ZW4415_EX1_DIENE_OPT_PM6_4.LOG]]

PM6 optimized ethene: [[File:ZW4415_EX1_ETHENE_OPT_PM6.LOG]]

PM6 optimized transition state: [[File:ZW4415_EX1_TS_OPT_PM6_2.LOG]]

PM6 optimized product: [[File:ZW4415_EX1_PRODUCT_OPT_PM6.LOG]]

PM6 IRC: [[File:ZW4415_EX1_TS_IRC_PM6_2_2.LOG]]

=Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole=

This is a reaction between cyclohexadiene and 1,3-Dioxole. There are 2 pathways of Diels-Alder reaction, Endo and Exo. For both pathway, reactants, TS and products were approximated using PM6 method then reoptimized by B3LYP method with 6-31G(d) basis set. Frequencies were checked, Gibbs free energies were extracted and MO diagrams were constructed and adjusted.

===Visualized HOMOs and LUMOs,frequency check and Gibbs free energies===
([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) You have compared PM6 optimised reactants with a B3LYP optimised transition state, so your MO diagrams are incorrect.)

{| class="wikitable"
!
!HOMO
!LUMO
!frequency check
!Gibbs Free Energy (Hatree)
|-
|Cyclohexadiene
|<jmol>
<jmolApplet>
<title>HOMO Cyclohexadiene</title>
<color>white</color>
<size>280</size>
<script>frame 12; mo 16; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZINAN_WANG_EX2_REACT1_OPT_PM6_JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO Cyclohexadiene</title>
<color>white</color>
<size>280</size>
<script>frame 12; mo 17; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZINAN_WANG_EX2_REACT1_OPT_PM6_JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:ZINAN_WANG_EX2_REACT1.PNG]]
|<nowiki>-233.324375</nowiki>
|-
|1,3-Dioxole
|<jmol>
<jmolApplet>
<title>HOMO 1,3-Dioxole</title>
<color>white</color>
<size>280</size>
<script>frame 6; mo 14; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZINAN_WANG_EX2_REACT2_OPT_PM6_MO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO 1,3-Dioxole</title>
<color>white</color>
<size>280</size>
<script>frame 6; mo 15; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZINAN_WANG_EX2_REACT2_OPT_PM6_MO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:ZINAN_WANG_EX2_REACT2.PNG]]
|<nowiki>-267.068132</nowiki>
|-
|Exo transition state
|<jmol>
<jmolApplet>
<title>HOMO Exo TS</title>
<color>white</color>
<size>280</size>
<script>frame 8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZINAN_WANG_EX2_TS_EXO_OPT_B3LYP_MO.log</uploadedFileContents>
</jmolApplet>
</jmol>

<jmol>
<jmolApplet>
<title>HOMO-1 Exo TS</title>
<color>white</color>
<size>280</size>
<script>frame 8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZINAN_WANG_EX2_TS_EXO_OPT_B3LYP_MO.log</uploadedFileContents>
</jmolApplet>
</jmol>

|<jmol>
<jmolApplet>
<title>LUMO Exo TS</title>
<color>white</color>
<size>280</size>
<script>frame 8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZINAN_WANG_EX2_TS_EXO_OPT_B3LYP_MO.log</uploadedFileContents>
</jmolApplet>
</jmol>

<jmol>
<jmolApplet>
<title>LUMO+1 Exo TS</title>
<color>white</color>
<size>280</size>
<script>frame 8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>ZINAN_WANG_EX2_TS_EXO_OPT_B3LYP_MO.log</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Zinan_Wang_EX2_EXO_TS_FRQ_B3LYP.PNG]]
|<nowiki>-500.329169</nowiki>
|-
|Exo product
|
|
|[[File:Zinan_Wang_EX2_EXO_P_FRQ.PNG]]
|<nowiki>-500.417322</nowiki>
|-
|Endo transition state
|<jmol>
<jmolApplet>
<title>HOMO Endo TS</title>
<color>white</color>
<size>280</size>
<script>frame 6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>Zinan_Wang_EX2_TS_ENDO_OPT_B3LYP_MO.log</uploadedFileContents>
</jmolApplet>
</jmol>

<jmol>
<jmolApplet>
<title>HOMO-1 Endo TS</title>
<color>white</color>
<size>280</size>
<script>frame 6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>Zinan_Wang_EX2_TS_ENDO_OPT_B3LYP_MO.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO Endo TS</title>
<color>white</color>
<size>280</size>
<script>frame 6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>Zinan_Wang_EX2_TS_ENDO_OPT_B3LYP_MO.log</uploadedFileContents>
</jmolApplet>
</jmol>

<jmol>
<jmolApplet>
<title>LUMO+1 Endo TS</title>
<color>white</color>
<size>280</size>
<script>frame 6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""</script>
<uploadedFileContents>Zinan_Wang_EX2_TS_ENDO_OPT_B3LYP_MO.log</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Zinan_Wang_EX2_ENDO_TS_FRQ_B3LYP.PNG]]
|<nowiki>-500.332149</nowiki>
|-
|Endo product
|
|
|[[File:Zinan_Wang_EX2_ENDO_P_FRQ.PNG]]
|<nowiki>-500.418692</nowiki>
|}

According to the frequency calculation, all transition states are showing only 1 negative value and all the other species are showing all positive frequencies which confirms the success in obtaining the optimized structures.

===MO diagram===

Firstly the a single point energy calculation was carried out for two reactants being separated for 20 a.u.(i.e. assume no interaction)
The MO diagram for the resulting file was shown below
[[File:Zinan_Wang_EX2_REACT1and2.PNG|thumb|center|800px|The MOs from single point energy calculation ]]
The value of the each energy level was used to draw the corresponded MO diagram.
The energy levels for the MO of TS are higher than the final product as expected, as the TS represent the highest energy along the reaction coordination.

{| class="wikitable"
!
!EXO
!ENDO
|-
|The MO diagram for TS formation
|[[File:Zinan_Wang_EX2_EXO_TS_MO.jpg|500px]]
|[[File:Zinan_Wang_EX2_ENDO_TS_MO.jpg|500px]]
|-
|Reaction barriers (at room temp) / Hartree
|0.062817 (164.926034 kJ/mol)
|0.060358 (158.469929 kJ/mol)
|-
|Reaction energies (at room temp) /Hartree
|<nowiki>-0.024815</nowiki> (<nowiki>-65.1517825</nowiki> kJ/mol)
|<nowiki>-0.026185</nowiki> (<nowiki>-68.7487175</nowiki> kJ/mol)
|}

The reaction barrier and reaction energy are both lower for the Endo-pathway.
Therefore the endo pathway is both kinetic and thermodynamic favored.

From the single point calculation, the energy of the HOMO of the dieneophile (1,3-Dioxole) is higher than that of the diene (Cyclohexadiene) which shows that the dieneophile is more electron rich than the diene, thus inverse electron demand.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 08:58, 23 March 2018 (UTC) Well done as long as these are optimised this should give you the right answer

===Secondary orbital interactions===

In the endo position, there are 2 oxygen p orbitals from oxygen lone pair which has the silimar energy and same symmetry of the cyclohexadiene LUMO, therefore they would have non-zero overlap integral. This interaction would lower the energy of LUMO producing a stabalising effect for the whole structrue. Thus more thermodynamically favored.

In the exo position, the two orbitals mentioned above were too far apart, thus almost no overlap integral. Moreover, there is a steric clash between the ring structure of 1,3-Dioxole and the half of the ring structure of cyclohexadiene, which rise up the energy of the net structure, thus less stable.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 09:02, 23 March 2018 (UTC) This was a good section your study of the MO energies was particularly good.

===.log Files for Exercise 2===

B3LYP/6-31G(d) optimized cyclohexadiene:[[File:ZINAN_WANG_EX2_REACT1_OPT_B3LYP.LOG]]

B3LYP/6-31G(d) optimized 1,3-dioxole:[[File:Zinan_Wang_EX2_REACT2_OPT_B3LYP.log]]

B3LYP/6-31G(d) optimized Exo TS:[[File:ZINAN_WANG_EX2_TS_EXO_OPT_B3LYP_MO.log]]

B3LYP/6-31G(d) optimized Endo TS:[[File:ZINAN_WANG_EX2_TS_ENDO_OPT_B3LYP_CHRIS.LOG]]

B3LYP/6-31G(d) optimized Exo product:[[File:Zinan_Wang_EX2_PRODUCT_EXO_OPT_B3LYP.log]]

B3LYP/6-31G(d) optimized Endo product:[[File:ZINAN_WANG_EX2_PRODUCT_ENDO_OPT_B3LYP.LOG]]

=Exercise 3: Diels-Alder vs Cheletropic=

In exercise 3, xylylene and SO<sub>2</sub> are used as the reactant.
They could react through 3 pathways: Exo and Endo hetero-Diels Alder reaction or the Cheletropic reaction.
Energy barrier and reaction energy for each pathway has been calculated and plotted.

[[File:Zinan_Wang_EX3_Scheme.jpg|thumb|center|600px|The reaction scheme for Exercise 3]]

===Reaction energy calculation===
The Gibbs free energy for each reactants, transition states and products were extracted form the .log file at 'Thermochemistry' section and converted to kJ/mol.

{| class="wikitable"
!
!Gibbs free energy (Hatree)
!Gibbs free energy (kJ/mol)
|-
|Xylylene
|0.178134
|467.6908
|-
|SO<sub>2</sub>
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4211</nowiki>
|-
|'''EXO TS'''
|0.092078
|241.7508
|-
|'''ENDO TS'''
|0.090558
|237.76
|-
|'''Cheletropic TS'''
|0.099062
|260.0873
|-
|EXO product
|0.021456
|56.33273
|-
|ENDO product
|0.021696
|56.96285
|-
|Cheletropic product
|<nowiki>-0.000002</nowiki>
|<nowiki>-0.00525</nowiki>
|}

===Reaction coordinations===
{| class="wikitable"
!
!Exo DA reaction
!Endo DA reaction
!Cheletropic reaction
|-
|Reaction coordinate from IRC
(click to see animation)
|[[File:Zinan_Wang_EX3_irc_EXO_PM6.gif|400px]]
|[[File:Zinan_Wang_EX3_irc_ENDO_PM6.gif|400px]]
|[[File:Zinan_Wang_EX3_irc_Chele_PM6.gif|400px]]
|-
|IRC energy calculation
|[[File:Zinan_Wang_EX3_irc_EXO_PM6.PNG|400px]]
|[[File:Zinan_Wang_EX3_irc_ENDO_PM6.PNG|400px]]
|[[File:Zinan_Wang_EX3_irc_Chele_PM6.PNG|400px]]
|-
|Reaction barriers (kJ/mol)
|85.4811
|81.4903
|103.8176
|-
|Reaction energy (kJ/mol)
|<nowiki>-99.9370</nowiki>
|<nowiki>-99.3069</nowiki>
|<nowiki>-156.275</nowiki>
|}

From the reaction barriers calculated, the Endo path has slightly lower reaction barrier than the Exo path. These two DA reaction path also has the silimar reaction energy. Although the most kinetic path is the Endo, the Exo is a strong competing pathway.

The Cheletropic reaction has the highest reaction barrier while the lowest reaction energy, thus it is most thermodynamic favored.

[[File:Zinan_Wang_EX3_Reactionenergy.jpg|thumb|center|600px|The reaction profile]]

(Too many significant figures. You need to think about the errors related to to the convergence criteria you are using [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 15:03, 16 March 2018 (UTC))

===.log Files for Exercise 3===

PM6 Optimized Xylylene:[[File:ZINAN_WANG_EX3_REACT1_OPT_PM6.LOG]]

PM6 Optimized SO2:[[File:ZINAN_WANG_EX3_SO2_OPT_PM6.LOG]]

PM6 Optimized EXO product:[[File:ZINAN_WANG_EX3_P_EXO_OPT_PM6.LOG]]

PM6 Optimized EXO TS:[[File:ZINAN_WANG_EX3_TS_EXO_PM6.LOG]]

PM6 Optimized ENDO product:[[File:ZINAN_WANG_EX3_P_ENDO_OPT_PM6.LOG]]

PM6 Optimized ENDO TS:[[File:ZINAN_WANG_EX3_TS_ENDO_PM6.LOG]]

PM6 Optimized Cheletropic product:[[File:ZINAN_WANG_EX3_P_CHELE_OPT_PM6.LOG]]

PM6 Optimized Cheletropic TS:[[File:ZINAN_WANG_EX3_TS_CHELE_PM6.LOG]]

=Conclusion=

In this computational lab, 3 pericyclic reactions were investigated. Gaussian was used to optimized the structure of the reactants, products and transition states using either PM6 or B3LYP method with IRC and frequency checked to confirm the transition states. Gibbs free energy was extracted from the log file for each species and the reaction barriers and reaction energies were calculated from them. The kinetic favored pathway were determined by the lowest reaction barrier and thermodynamic favored pathway was determined by the lowest reaction energy.

This computational method could be carried out for more reactions to determine the reaction route under different conditions.

=Reference=

[1] J. J. W. McDouall, in Computational Quantum Chemistry: Molecular Structure and Properties in Silico, The Royal Society of Chemistry, London,
2013, ch. 1, pp. 1-62.

Rep:Zw4415:TS

2025-09-01T09:51:12Z

Move page script: Move page script moved page Zw4415:TS to Rep:Zw4415:TS: Move to report namespace

{| class="wikitable"
!
!Gibbs free energy (Hatree)
!Gibbs free energy (kJ/mol)
|-
|Xylylene
|0.178134
|467.6908
|-
|SO<sub>2</sub>
|0.118614
|311.4211
|-
|'''EXO TS'''
|0.092078
|241.7508
|-
|'''ENDO TS'''
|0.090558
|237.76
|-
|'''Cheletropic TS'''
|0.099062
|260.0873
|-
|EXO product
|0.021456
|56.33273
|-
|ENDO product
|0.021696
|56.96285
|-
|Cheletropic product
|<nowiki>-0.000002</nowiki>
|<nowiki>-0.00525</nowiki>
|}
{| class="wikitable"
!
!Exo DA reaction
!Endo DA reaction
!Cheletropic reaction
|-
|Reaction coordinate from IRC
|
|
|
|-
|IRC energy calculation
|
|
|
|}
{| class="wikitable"
!
!cis-Butadiene
!Ethene
!Transition State
!Product
!Bond length
!Bond order
|-
|
|image di
|image ene
|image ts
|img product
|
|
|-
|C1-C2
|<nowiki>-</nowiki>
|<nowiki>-</nowiki>
|2.11484
|1.53580
|decrease
|forming
|-
|C2-C3
|<nowiki>-</nowiki>
|1.32731
|1.38205
|1.53764
|increase
|breaking
|-
|C3-C4
|<nowiki>-</nowiki>
|<nowiki>-</nowiki>
|2.11485
|1.53579
|decrease
|forming
|-
|C4-C5
|1.33529
|<nowiki>-</nowiki>
|1.37977
|1.49261
|increase
|breaking
|-
|C5-C6
|1.46826
|<nowiki>-</nowiki>
|1.41113
|1.33306
|decrease
|forming
|-
|C6-C1
|1.33537
|<nowiki>-</nowiki>
|1.37972
|1.49261
|increase
|breaking
|}

Rep:ZZY15 TS

2025-09-01T09:51:12Z

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Rep:ZY3915liqsimu

2025-09-01T09:51:09Z

Move page script: Move page script moved page ZY3915liqsimu to Rep:ZY3915liqsimu: Move to report namespace

<span style=color:red> Overall feedback: The tasks were completed with quite a few mistakes. The report was very short and failed to convey a clear goal and motivation, but instead included vague statements about MD. Grammar and spelling were an issue. Please edit your work!. </span>

= Third year simulation experiment =

=== Liquid simulation and the diffusion coefficient ===
Zhuohao You

==== Abstract ====
Diffusion behaviour of water was modeled and investigated by molecular dynamic simulation with the assistant of high performance computing power. The connection of diffusion coefficient to the mean square displacement was exploited to calculated the diffusion coefficient base on the performed MSD for liquid, solid and vapour. A further experiment on diffusion coefficient of solid was carried to exam its relationship with temperature.<span style=color:red> The abstract of a scientific paper is meant to briefly convey what you have done and your main results and conclusions, perhaps with a very short motivation. While you have briefly touched upon what you have done, your abstract lacks specifics. What exactly were your main results and conclusions? Also spelling and grammar! </span>

=== Introduction ===
With the development of high performance computing system, the accuracy of molecular dynamic simulation (MSD) <span style=color:red> molecular dynamics is usually represented by the acronym "MD", "MDS" for molecular dynamics simulation(s) would be acceptable if specified. However "MSD" has the letters in the wrong order, and is a bit confusing given that MSD is also common for "mean squared displacement" </span> was brought to a new level <span style=color:red> Arguably, yes. However, you have performed relatively small simulations using cheap and cheerful LJ potentials, so perhaps this comment is not very relevant to what you have done. </span>. MSD is a useful tool that gives rise to calculation of macroscopic properties from microscopic scale systems. By considering the interaction for a single particle with a limited amount of nearby particles, 'exact' prediction of thermo and physical properties are possible depending in the scale of calculation. <span style=color:red> This point is arguable, since there a lot of technical subtleties, certainly an elaboration would be necessary after making such a bold claim with the use of "exact". </span>[1]

Using the college's high performance computing facilities <span style=color:red> simply "the college's" is not an adequate accreditation of the hpc resources you have used. </span>, simulation of simple liquid <span style=color:red> what about the other phases you have simulated? </span>was performed and an important property of diffusion coefficient was computed from the simulation with a method manipulating its relationship with the mean squared displacement of ensemble particles.

==== Aims and Objectives ====
In this experiment, simulation using Lennard-Jones potential was applied on a simple liquid system. (e.g. Argon) <span style=color:red> why single out argon? have you used LJ parameters for argon? </span>And investigation of the diffusion coefficient property of the system in liquid, solid and vapour phase was carried to give comparisons between the three states. Furtherly, a variation in temperature for the solid state was investigated to exploit the relationship between temperate and diffusion coefficient.

==== Methods ====
The input script was base on the given npt file with 8000 atoms and the molecular dynamic was calculated by the velocity Verlet algorithm with based on Lennard-Jones potential. All the simulation was completed on the college HPC system with the parallel computational pacakge LAMMPS. The diffusion coefficient was computed by the given method:
The easiest way to measure <math>D</math> is by exploiting its connection to the [http://en.wikipedia.org/wiki/Mean_squared_displacement mean squared displacement].

<math>D = \frac{1}{6}\frac{\partial\left\langle r^2\left(t\right)\right\rangle}{\partial t}</math>

<span style=color:red> This is not sufficient information for another scientist to reproduce your results. What LJ parameters have you used, what cutoff? You mention the NPT ensemble, what pressure and temperature? </span>

==== Results and discussion ====
The mean squared displacement (MSD), effectively measures how much the particles deviate from their equilibrium positions <span style=color:red> a more clear explanation would be valuable here </span> . The value of MSD represents the extent of random motion in the system, and it can be calculated with the equation:

[[File:Zyup001803.jpg]]

In this experiment, calculation of MSD was all completed by HPC and was given in the results.

[[File:Zyup0018701.jpg]] [[File:Zyup001802.jpg]]

<span style=color:red> No x axis label for the second graph. </span>

As shown in two graphs, the simulation for liquid, solid and vapour gives the evolution of mean squared displacement over ti,me for both cases. (8000 atoms and a million atoms respectively) The first thing to see on the graphs was the abnormal position for liquid state and gas state in the first figure, as the liquid phase gave a larger MSD as time goes, which on the other hand, for the second figure did have the gas curve laying above the liquid curve.

In a realistic sense, as the MSD measured the random of particles, the displacement for liquid molecules should be much smaller than the vapour counterpart, since the gas particles was supposed to be about 10 times more distant than liquid molecules in the space.

Therefore, it turn out that the simulation for vapour phase with this MSD method was inaccurate, or a much longer period of time was required for the system to reach the equilibrium.

<nowiki> </nowiki>As mentioned above, the diffusion coefficient was calculated by the relationship:

<math>D = \frac{1}{6}\frac{\partial\left\langle r^2\left(t\right)\right\rangle}{\partial t}</math> so one sixth of the gradient of the MSD graph was the diffusion coeffient:

D(liq)= 0.000171 cm2/s; D(sol)= 1.92x10-6 cm2/s; D(vap)= 0.000106 cm2/s (8000atoms)

D(liq)= 0.000177cm2/s; D(sol)= 0; D(vap)= 0.00627cm2/s (a million atoms)

The result was quite close to each other apart from the vapour case, and the data confirmed that for the 8000 atoms system, an equilibrium was not reach therefore the inaccuracy was due to a lack of simulation steps as the gradient was only valid in the diffusion region of the graph (i.e. the linear part). In the case of solid the diffusion coefficient was to low to be calculated.

===== Extension =====

<span style=color:red> why have you included an extension in the middle of your results section? </span>
As the simulation for solid was quite stable in the last section, further interest of examine the temperate-diffusion coefficient connection was developed from the literature[2]. Five additional simulation with different temperature for the solid system was carried to investigate if the MDS simulation could give a similar trend.
{| class="wikitable"
!T (reduced temperature)
!Diffusion coefficient cm2/s
|-
|0.6
|7.48E-07
|-
|0.7
|7.85E-07
|-
|0.8
|1.26E-06
|-
|0.9
|1.47E-06
|-
|1.0
|2.5E-06
|}

<nowiki> </nowiki>The results of simulation was given in the table, and a clear trend of D increasing with temperature was illustrated.
[[File:Zyup001805.jpg]][[File:Zyup001804.jpg]]

In general, the simulation gave the same relationship with the literature graph <span style=color:red> citation? </span>, though the fluctuation in the computed curve was greater due to the weakness in size and timesteps. This was saying the error in the simulation can be averaged out with large scale simulation andFurther investigate of this relation could be carried with a greater size (e.g. a million atoms) and more steps to provide more reliable data for the different states.

=== Conclusion ===
The MD simulation provides a powerful and relatively reliable tool for investigation of the simple systems as shown in the experiment, this provides an alternative method to gather thermo and physical data from Lab experiment. To ensure the accuracy of the simulated data, a large size of model to mimic the interaction and long time of random motion to reach equillibrium was required.

<span style=color:red> These are some very vague conclusions. The conclusion of a scientific paper is meant to summarise the main results and conclusions, and perhaps offer a brief outlook. </span>

===== Reference hav =====
# Computational Soft Matter: From Synthetic Polymers to Proteins, Lecture Notes, Norbert Attig, Kurt Binder, Helmut Grubmuller ¨ , Kurt Kremer (Eds.), John von Neumann Institute for Computing, Julich, ¨ NIC Series, Vol. 23, ISBN 3-00-012641-4, pp. 1-28, 2004.
#Molecular and condition parameters dependent diffusion coefficient of water in poly(vinyl alcohol): a molecular dynamics simulation study,Colloid and Polymer Science, 2017, 295(5),859-868

= TASK: =
'''<big>TASK</big>: Open the file HO.xls. In it, the velocity-Verlet algorithm is used to model the behaviour of a classical harmonic oscillator. Complete the three columns "ANALYTICAL", "ERROR", and "ENERGY": "ANALYTICAL" should contain the value of the classical solution for the position at time , "ERROR" should contain the ''absolute'' difference between "ANALYTICAL" and the velocity-Verlet solution (i.e. ERROR should always be positive -- make sure you leave the half step rows blank!), and "ENERGY" should contain the total energy of the oscillator for the velocity-Verlet solution. Remember that the position of a classical harmonic oscillator is given by (the values of , , and are worked out for you in the sheet).'''

[[File:Zyup00181.jpg]]

[[File:Zyup00182.jpg]]

[[File:Zyup00183.jpg]]

'''<big>TASK</big>: For the default timestep value, 0.1, estimate the positions of the maxima in the ERROR column as a function of time. Make a plot showing these values as a function of time, and fit an appropriate function to the data.'''

Error= C*t*sin( ωt + φ ) C is a constant that equals approx. 0.000417 in the case of timestep=0.1 ω=1.00 and φ=1.00

'''<big>TASK</big>: Experiment with different values of the timestep. What sort of a timestep do you need to use to ensure that the total energy does not change by more than 1% over the course of your "simulation"? Why do you think it is important to monitor the total energy of a physical system when modelling its behaviour numerically?'''

Timesteps below 0.63s would be valid in this case <span style=color:red> way too large </span>. Ideally the total energy is conserved in a closed system, so it is better to monitor the total energy of a system to ensure the simulation was not collapsed in terms of a strong fluctuation in total energy.

[[File:Zyup00184.jpg|800x263px]]
[[File:Zyup00185.jpg|714x300px]]

<span style=color:red> force is +ve, r_eq is 2^(1/6)*sigma. Numerical answers stated to way too many decimal places. </span>

'''<big>TASK</big>: Estimate the number of water molecules in 1ml of water under standard conditions. 55.5*N<sub>A</sub>/1000= 3.34*10<sup>22</sup>'''

'''Estimate the volume of 10000 water molecules under standard conditions. 10000/3.34*10<sup>22</sup>=2.99*10<sup>-19</sup>mL'''
[[File:Zyup00186.jpg|800x156px]]
[[File:Zyup00187.jpg|1000x200px]]

<span style=color:red> Atom positions not after PBC not correct. Well depth off by factor of 1000, temperature not correct. </span>

'''<big>TASK</big>: Why do you think giving atoms random starting coordinates causes problems in simulations? Hint: what happens if two atoms happen to be generated close together?'''

In case of two atoms generated on top of each other，the force between them will be very large and therefore leads to unwanted large acceleration to the system, cause a sudden blow up'''.'''

'''<big>TASK</big>: Satisfy yourself that this lattice spacing corresponds to a number density of lattice points of 0.8. Consider instead a face-centred cubic lattice with a lattice point number density of 1.2. What is the side length of the cubic unit cell?'''
1/(1.07722)3 = 0.800
4 atoms in one lattice, so 4/a3 = 1.2, a = 1.49380, side length is 1.49380.

'''<big>TASK</big>: Consider again the face-centred cubic lattice from the previous task. How many atoms would be created by the create_atoms command if you had defined that lattice instead?''' 4000

'''<big>TASK</big>: Using the [http://lammps.sandia.gov/doc/Section_commands.html#cmd_5 LAMMPS manual], find the purpose of the following commands in the input script:'''

<pre>
mass 1 1.0 for every atom in type 1 mass = 1.0 (reduced unit)
pair_style lj/cut 3.0 cutoff Lennard-Jones potential with no Coulomb at 3.0 potential with no Coulomb at 3.0
pair_coeff * * 1.0 1.0 for all the pairs coefficient 1.0 1.0 was applied
</pre>

'''<big>TASK</big>: Given that we are specifying <math>\mathbf{x}_i\left(0\right)</math> and <math>\mathbf{v}_i\left(0\right)</math>, which integration algorithm are we going to use?'''

velocity Verlet algorithm.

'''<big>TASK</big>: Look at the lines below.'''
<pre>
### SPECIFY TIMESTEP ###
variable timestep equal 0.001
variable n_steps equal floor(100/${timestep})
timestep ${timestep}

### RUN SIMULATION ###
run ${n_steps}
run 100000
</pre>
'''The second line (starting "variable timestep...") tells LAMMPS that if it encounters the text ${timestep} on a subsequent line, it should replace it by the value given. In this case, the value ${timestep} is always replaced by 0.001. In light of this, what do you think the purpose of these lines is? Why not just write:'''
<pre>
timestep 0.001
run 100000
</pre>
'''Ask the demonstrator if you need help.'''

Allows easy variation of timesteps without worrying about forgetting to change the relevant steps to run. As the change in steps will be made by the codes as soon as the value of timesteps was changed. Instantaneous change of two related value.

'''<big>TASK</big>: make plots of the energy, temperature, and pressure, against time for the 0.001 timestep experiment (attach a picture to your report). '''[[File:Zyup00188.jpg|800x426px]]

'''Does the simulation reach equilibrium? '''Yes

'''How long does this take? '''0.3 reduced time

'''When you have done this, make a single plot which shows the energy versus time for all of the timesteps (again, attach a picture to your report). '''
[[File:Zyup00189.jpg|800x446px]]

'''Choosing a timestep is a balancing act: the shorter the timestep, the more accurately the results of your simulation will reflect the physical reality; short timesteps, however, mean that the same number of simulation steps cover a shorter amount of actual time, and this is very unhelpful if the process you want to study requires observation over a long time. Of the five timesteps that you used, which is the largest to give acceptable results? '''0.0025

Fluctuating in the region that covers the most accurate value from 0.0001

'''Which one of the five is a ''particularly'' bad choice? Why?''' 0.015 it does not converge.

'''<big>TASK</big>: We need to choose <math>\gamma</math> so that the temperature is correct <math>T = \mathfrak{T}</math> if we multiply every velocity <math>\gamma</math>. We can write two equations:'''

<math>\frac{1}{2}\sum_i m_i v_i^2 = \frac{3}{2} N k_B T</math>

<math>\frac{1}{2}\sum_i m_i \left(\gamma v_i\right)^2 = \frac{3}{2} N k_B \mathfrak{T}</math>

'''Solve these to determine <math>\gamma</math>.'''

γ = ( <math>\mathfrak{T}</math> /T )0.5

<span style=color:red> I think you meant to the power of 0.5 here, but it is typed as multiply as 0.5. Be more careful! Also, if you showed any working out, then I could safely say this was a typo, but since you have not, I cannot justify treating it as to the power of 0.5 </span>

'''<big>TASK</big>: Use the [http://lammps.sandia.gov/doc/fix_ave_time.html manual page] to find out the importance of the three numbers ''100 1000 100000''. '''

• Nevery = 100 use input values every 100 timesteps

• Nrepeat = 1000 1000 of times to use input values for calculating averages

• Nfreq =10000 calculate averages every 10000 timesteps

'''How often will values of the temperature, etc., be sampled for the average? '''every 10000 timesteps

'''How many measurements contribute to the average? '''1000

'''Looking to the following line, how much time will you simulate? '''100000 unit time
'''<big>TASK</big>: When your simulations have finished, download the log files as before. At the end of the log file, LAMMPS will output the values and errors for the pressure, temperature, and density <math>\left(\frac{N}{V}\right)</math>. Use software of your choice to plot the density as a function of temperature for both of the pressures that you simulated.'''

[[File:Zyup001810.jpg|800x488px]]

'''Your graph(s) should include error bars in both the x and y directions. You should also include a line corresponding to the density predicted by the ideal gas law at that pressure. Is your simulated density lower or higher? Justify this. Does the discrepancy increase or decrease with pressure?'''

'''<nowiki/>'''Lower, as ideal gas law ignores any interactions between particles apart from collisions while the L-J system takes the potential energy into account so that results in a lower density.
discrepancy increase with pressure.

'''<big>TASK</big>: As in the last section, you need to run simulations at ten phase points. In this section, we will be in density-temperature <math>\left(\rho^*, T^*\right)</math> phase space, rather than pressure-temperature phase space. The two densities required at <math>0.2</math> and <math>0.8</math>, and the temperature range is <math>2.0, 2.2, 2.4, 2.6, 2.8</math>. Plot <math>C_V/V</math> as a function of temperature, where <math>V</math> is the volume of the simulation cell, for both of your densities (on the same graph). Is the trend the one you would expect? Attach an example of one of your input scripts to your report.'''

[[File:Zyup001811.jpg|800x420px]]

Supposed to be constant for liquid but the fluctuation was within an acceptable range

====== Scripts: ======
<nowiki>###</nowiki> SPECIFY THE REQUIRED THERMODYNAMIC STATE ###

variable D equal 0.2

variable T equal 2.0

variable timestep equal 0.0025

<nowiki>###</nowiki> DEFINE SIMULATION BOX GEOMETRY ###

lattice sc ${D}

region box block 0 15 0 15 0 15

create_box 1 box

create_atoms 1 box

<nowiki>###</nowiki> DEFINE PHYSICAL PROPERTIES OF ATOMS ###

mass 1 1.0

pair_style lj/cut/opt 3.0

pair_coeff 1 1 1.0 1.0

neighbor 2.0 bin

<nowiki>###</nowiki> ASSIGN ATOMIC VELOCITIES ###

velocity all create ${T} 12345 dist gaussian rot yes mom yes

<nowiki>###</nowiki> SPECIFY ENSEMBLE ###

timestep ${timestep}

fix nve all nve

<nowiki>###</nowiki> THERMODYNAMIC OUTPUT CONTROL ###

thermo_style custom time etotal temp press

thermo 10

<nowiki>###</nowiki> RECORD TRAJECTORY ###

dump traj all custom 1000 output-1 id x y z

<nowiki>###</nowiki> RUN SIMULATION TO MELT CRYSTAL ###

run 10000

unfix nve

reset_timestep 0

<nowiki>###</nowiki> BRING SYSTEM TO REQUIRED STATE ###

variable tdamp equal ${timestep}*100

variable pdamp equal ${timestep}*1000

fix nvt all nvt temp ${T} ${T} ${tdamp}

run 10000

reset_timestep 0

unfix nvt

fix nve all nve

<nowiki>###</nowiki> MEASURE SYSTEM STATE ###

thermo_style custom step etotal temp vol density

variable dens equal density

variable temp equal temp

variable volu equal vol

variable ener equal etotal

variable ener2 equal etotal*etotal

fix aves all ave/time 100 1000 100000 v_dens v_temp v_vol v_ener v_ener2 v_press2

run 100000

variable avedens equal f_aves[1]

variable avetemp equal f_aves[2]

variable avevolu equal f_aves[3]

variable heatc equal 3375*3375*(f_aves[5]-f_aves[4]*f_aves[4])/(f_aves[2]*f_aves[2])

print "Averages"

print "--------"

print "Density: ${avedens}"

print "Volume: ${avevolu}"

print "Temperature: ${avetemp}"

print "Cv/V: ${heatc}/${avevolu}"

'''<big>TASK</big>: perform simulations of the Lennard-Jones system in the three phases. When each is complete, download the trajectory and calculate <math>g(r)</math> and <math>\int g(r)\mathrm{d}r</math>. Plot the RDFs for the three systems on the same axes, and attach a copy to your report. '''

[[File:Zyup001812.jpg|800x457px]]

'''Discuss qualitatively the differences between the three RDFs, and what this tells you about the structure of the system in each phase. '''

Liquid and vapour drop constantly due to the evenly distributing simple cubic structure while solid has fluctuation because of the Fcc structure.

<span style=color:red> This does not really make sense, are you saying that the gas and liquid phase form a cubic lattice? - Then they would be the solid phase! We were looking for a discussion of the short range vs. long range order for each of the phases, relating this to the features of the RDF. </span>

'''In the solid case, illustrate which lattice sites the first three peaks correspond to.'''
''' What is the lattice spacing? '''

'''What is the coordination number for each of the first three peaks?'''

Lattice spacing around 1.45 reduced unit.

[0.5,0.5,0] corners; [1.0,0,0] centre of face; [1.0,0.5,0] centre of a different face

<span style=color:red> Illustration? What are the coordination numbers? </span>

'''<big>TASK</big>: make a plot for each of your simulations (solid, liquid, and gas), showing the mean squared displacement (the "total" MSD) as a function of timestep. Are these as you would expect? Estimate in each case. Be careful with the units! Repeat this procedure for the MSD data that you were given from the one million atom simulations.'''
[[File:Zyup001813.jpg]]
[[File:Zyup001814.jpg]]

<span style=color:red> Yes diffusion coefficient should be higher for the gas than liquid - this data is not so good. </span>

'''<big>TASK</big>: In the theoretical section at the beginning, the equation for the evolution of the position of a 1D harmonic oscillator as a function of time was given. Using this, evaluate the normalised velocity autocorrelation function for a 1D harmonic oscillator (it is analytic!):'''

<math>C\left(\tau\right) = \frac{\int_{-\infty}^{\infty} v\left(t\right)v\left(t + \tau\right)\mathrm{d}t}{\int_{-\infty}^{\infty} v^2\left(t\right)\mathrm{d}t}</math>

'''Be sure to show your working in your writeup. '''
[[File:Zyup001815.jpg]]

'''On the same graph, with x range 0 to 500, plot <math>C\left(\tau\right)</math> with <math>\omega = 1/2\pi</math> and the VACFs from your liquid and solid simulations. What do the minima in the VACFs for the liquid and solid system represent?'''

The minima give the location of the maximum difference for the liquid and solid system.

<span style=color:red> This is not what we were looking for. Think about was the VACF actually corresponds to, and how the velocity changes after initiating the simulation. What happens when they particles begin to collide? </span>

''' Discuss the origin of the differences between the liquid and solid VACFs. The harmonic oscillator VACF is very different to the Lennard Jones solid and liquid. Why is this? '''

Because the HO model has a periodic motion while the Lennard Jones solid and liquid move randomly there for there is no pattern in this kind of motion. i.e. the dependence on previous velocity is rather low.<span style=color:red> Could be better explained - I get what you are trying to say. First of all LJ particles do not move randomly... or what would be the point of the simulation? But unlike the LJ system, the velocity of a simple HO in a closed system is exactly periodic. </span>

Attach a copy of your plot to your writeup.

'''<nowiki/>'''[[File:Zyup001816.jpg|800x387]]
[[File:Zyup001817.jpg]]
[[File:Zyup001818.jpg]]

Rep:Yrt13 TS

2025-09-01T09:51:08Z

Move page script: Move page script moved page Yrt13 TS to Rep:Yrt13 TS: Move to report namespace

== 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?)''''' <sup>1</sup>

In the following exercises, the reaction pathways of several Diels-Alder reactions and their transitions states will be studied. A Diels-Alder reaction is a pericyclic reaction between a conjugated diene and a dienophile. Specifically, it is a [4+2] cycloaddition involving rearrangement of π bonds. Reaction pathways can be studied by investigating a potential energy surface. <sup>1</sup>

On a potential energy surface, the reactants and products correspond to the minimum points. A transition state is then identified as the maximum on the minimum energy pathway linking the reactants and the products. This is also known as a saddle point. Since the reactants, products and transition states lie on stationary points, the total gradient of the potential energy surface will be zero at these points. Therefore, the first derivative of the Potential Energy vs Bond Distances graph (gradient), ∂V(ri)/∂ri = 0

To differentiate between minimum points (reactants and products) and saddle points (transition states), we need to look further and consider their vibrational frequencies. Linear molecules have 3N - 5 normal vibrational modes / degrees of freedom while non-linear molecules have 3N - 6 normal vibrational modes, where N = the number of atoms the molecule has. Under the harmonic oscillator model, each vibrational mode has a restoring force F. This restoring force F is defined by Hooke's Law, '''F = - kx''', where ''''k'''' is the force constant and ''''x'''' is the displacement from equilibrium position / extension from equilibrium length.

Since the force is the negative derivative of the potential energy function, F = - ∂V(ri)/∂ri = 0. Also, '''k''' = - ∂F(ri)/∂ri = (∂2V(ri)/∂ri2), the second derivative of the potential energy function. The value and sign of '''k''' reveals the nature of the stationary point. For example, '''k''' is positive at a minimum point while '''k''' is negative at a maximum point. The value and sign of '''k''' is also significant as frequency calculations involve taking the square root of '''k'''.

Since the reactants and products sit in a minimum well, the second derivative of the potential energy function ('''k''') is positive in every degree of freedom as all degrees of freedom / coordinates are minimised. Therefore, reactants and products only have positive and real frequencies and will remain at this minimum 'forever' unless sufficient energy is provided for them to overcome reaction barriers.

The transition state, however, has a different mode of vibration along the reaction coordinate. Since the transition state lies on a saddle point (maximum on the minimum energy pathway), the second derivative of the potential energy function ('''k''') is positive in every degree of freedom except one. This is because all other coordinates are minimised '''except along the reaction coordinate'''. Therefore, the transition state will have (only) one imaginary frequency, resulting from the one negative '''k'''. <sup>1</sup>

Thus, the calculation of the second derivative of the potential energy surface ('''k''') can be used to distinguish the reactants/products from the transition states.

Two main optimisation methods were used in the following exercises. Firstly, the semi-empirical method PM6 was first used to optimise all reactants, products and transition states. PM6 was used as a first optimisation step as it is a fitted method that uses experimental data in order to save time and resources during calculation, hence it is not as accurate.

Some of the reactants, products and transition states were then further optimised using the Density Functional Theory (DFT) method B3LYP/6-31G(d). The B3LYP method is reasonably fast and is capable of reproducing chemical data. Thus, it produces more accurate results, at the cost of computational effort.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 00:12, 8 November 2017 (UTC) Excellent intro. Everything explained well. A few points you could have elaborated on how each degree of freedom is a normal mode, which is a linear combination of all the individual bends/rotation/bond lengths. hence why when you look at the vibration it looks like the whole molecule. You could have also gone into mmore detail about how the Quantum chemical methods work.

== Exercise 1 ==

In this exercise, the [4+2] cycloaddition reaction of butadiene with ethylene, to yield cyclohexane as a product, was studied. In this reaction, butadiene acts as the diene while ethylene acts as the dienophile. The reactants, transition state and product were optimised at the PM6 level. IRC and frequency calculations were also done with the transition state.

[[File:Yrt13 E1reactionscheme.PNG|300px]]

''Figure 1: Reaction scheme of Butadiene with Ethylene to form Cyclohexane. Adapted from https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_exercise#Exercise_1:_Reaction_of_Butadiene_with_Ethylene''

=== MO Diagram ===
'''''Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).'''''

[[File:Yrt13_Cyclohexene_MO_diagram2.PNG|500px]]

''Figure 2: MO Diagram for the formation of the butadiene/ethene TS. Note: This is the estimated MO diagram and energy calculations done at the PM6 Level were not used.''

The symmetry labels 'S' (symmetric) and 'A' (anti-symmetric) correspond to the symmetry of the molecular orbitals (MO) with respect to the mirror plane perpendicular to the sigma bond at the middle of the MO.

=== HOMO/LUMO of Butadiene and Ethylene ===

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |LUMO
of Butadiene
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13 BUTADIENE PM6 MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
! style="background: #0D4F8B; color: white; text-align: center;" |LUMO
of Ethylene
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 14; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13 ETHYLENE PM6 MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
|-
! style="background: #0D4F8B; color: white; text-align: center;" |HOMO
of Butadiene
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13 BUTADIENE PM6 MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
! style="background: #0D4F8B; color: white; text-align: center;" |HOMO
of Ethylene
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 14; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13 ETHYLENE PM6 MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
|}
''Figure 3: Jmols of the HOMOs and LUMOs of Butadiene and Ethylene.''

=== Four MOs produced at the Transition State ===

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;"|MO16
of TS (A)
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13_CYCLOHEXENE_TS_REOPT_MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
! style="background: #0D4F8B; color: white; text-align: center;"|MO17
of TS (S)
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13_CYCLOHEXENE_TS_REOPT_MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
|-
! style="background: #0D4F8B; color: white; text-align: center;"|MO18
of TS (S)
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 6; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13_CYCLOHEXENE_TS_REOPT_MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
! style="background: #0D4F8B; color: white; text-align: center;"|MO19
of TS (A)
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 6; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13_CYCLOHEXENE_TS_REOPT_MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
|}
''Figure 4: Jmols of the 4 relevant Transition State MOs.''

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) 11:20, 2 November 2017 (UTC) Your MO is very good, although you could have pointed out which orbitals are the HOMO and LUMO of the Transition State.)
=== Questions ===
'''''1. What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')?'''''

For a reaction to be allowed, the symmetry of the orbitals must be the same (i.e. both must be symmetric or anti-symmetric). If one of the orbitals is symmetric while the other is anti-symmetric, the reaction is 'forbidden'.

The extent of orbital overlap between two orbitals (A and B) can be quantified by the overlap integral (S<sub>AB</sub>) <sup>2</sup>. It is defined as:

[[File:Yrt13 Sab equation.PNG]]

If the product of both wavefunctions is symmetric, the overlap integral S<sub>AB</sub> is non-zero and the reaction is allowed. If the product of both wavefunctions is antisymmetric, S<sub>AB</sub> is zero and the reaction is forbidden.

'''''2. 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.'''''

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white;"|'''Interaction'''
! style="background: #0D4F8B; color: white;"|'''Orbital Overlap Integral'''
|-
|style="text-align: center;"| Symmetric-Antisymmetric
|style="text-align: center;"| Zero
|-
|style="text-align: center;"|Symmetric-Symmetric
|style="text-align: center;"|Non-Zero
|-
|style="text-align: center;"|Antisymmetric-Antisymmetric
|style="text-align: center;"|Non-Zero
|}
''Table 1: Summary of Orbital Overlap Integral with respect to Interaction type.''

=== Measurements ===
'''C-C Bond Lengths of Reactants'''
{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |Butadiene
! style="background: #0D4F8B; color: white; text-align: center;" |Bond
! style="background: #0D4F8B; color: white; text-align: center;" |C1-C4
! style="background: #0D4F8B; color: white; text-align: center;" |C4-C6
! style="background: #0D4F8B; color: white; text-align: center;" |C6-C8
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>select atomno=1, atomno=4, atomno=6, atomno=8; label display </script>
<uploadedFileContents>Yrt13 BUTADIENE PM6 MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
|style="text-align: center;" |Bond Distance
|1.33530 Å
|1.46835 Å
|1.33530 Å
|}
''Table 2: Summary of Bond distances in Butadiene.''

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |Ethylene
! style="background: #0D4F8B; color: white; text-align: center;" |Bond
! style="background: #0D4F8B; color: white; text-align: center;" |C1-C4
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>select atomno=1, atomno=4; label display </script>
<uploadedFileContents>Yrt13 ETHYLENE PM6 MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
|Bond Distance
|1.32731 Å
|}
''Table 3: Summary of Bond distances in Ethylene.''

'''C-C Bond Lengths of Transition State'''
{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |Transition State
! style="background: #0D4F8B; color: white; text-align: center;" |Bond
! style="background: #0D4F8B; color: white; text-align: center;" |C1-C4
! style="background: #0D4F8B; color: white; text-align: center;" |C4-C6
! style="background: #0D4F8B; color: white; text-align: center;" |C6-C8
! style="background: #0D4F8B; color: white; text-align: center;" |C8-C11
! style="background: #0D4F8B; color: white; text-align: center;" |C11-C14
! style="background: #0D4F8B; color: white; text-align: center;" |C14-C1
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>select atomno=1, atomno=4, atomno=6, atomno=8, atomno=11, atomno=14; label display </script>
<uploadedFileContents>Yrt13 CYCLOHEXENE TS REOPT MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
| style="text-align: center;" |Bond Distance
|1.37977 Å
|1.41111 Å
|1.37977 Å
|2.11473 Å
|1.38176 Å
|2.11473 Å
|}
''Table 4: Summary of Bond distances in the Cyclohexene Transition State.''

'''C-C Bond Lengths of Product, Cyclohexene'''
{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |Cyclohexene
! style="background: #0D4F8B; color: white; text-align: center;" |Bond
! style="background: #0D4F8B; color: white; text-align: center;" |C1-C4
! style="background: #0D4F8B; color: white; text-align: center;" |C4-C6
! style="background: #0D4F8B; color: white; text-align: center;" |C6-C8
! style="background: #0D4F8B; color: white; text-align: center;" |C8-C11
! style="background: #0D4F8B; color: white; text-align: center;" |C11-C14
! style="background: #0D4F8B; color: white; text-align: center;" |C14-C1
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>select atomno=1, atomno=4, atomno=6, atomno=8, atomno=11, atomno=14; label display </script>
<uploadedFileContents>Yrt13_CYCLOHEXENE_FINALOPT.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
| style="text-align: center;" |Bond Distance
|1.50084 Å
|1.33696 Å
|1.50081 Å
|1.53712 Å
|1.53459 Å
|1.53714 Å
|}
''Table 5: Summary of Bond distances in the Cyclohexene Product.''

=== Questions (continued) ===
'''''3. How do the bond lengths change as the reaction progresses?'''''

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |Bond
! style="background: #0D4F8B; color: white; text-align: center;" |C1-C4
! style="background: #0D4F8B; color: white; text-align: center;" |C4-C6
! style="background: #0D4F8B; color: white; text-align: center;"|C6-C8
! style="background: #0D4F8B; color: white; text-align: center;"|C8-C11
! style="background: #0D4F8B; color: white; text-align: center;" |C11-C14
! style="background: #0D4F8B; color: white; text-align: center;" |C14-C1
|-
|Bond Distance Changes
|Increases
|Decreases
|Increases
|Decreases
|Increases
|Decreases
|}
''Table 6: Summary of Changes in Bond distances (as reaction progressed).''

In particular, the bond length between C1-C14 (and between C8-C11) changes the most due to the formation of the new C-C bonds. The bond length is originally 3.40 Å (twice the Van der Waals radius of C) <sup>3</sup>

[[File:Yrt13_Bond141.PNG|500px]]

''Figure 5: Plot of the Bond Distance (between C1 and C14) against Intrinsic Reaction Coordinate.''

'''''4. What are typical sp3 and sp2 C-C bond lengths?''''' <sup>4</sup>

Typical sp<sup>3</sup>-sp<sup>3</sup> C-C bond length: '''1.54 Å'''

Typical sp<sup>3</sup>-sp<sup>2</sup> C-C bond length: '''1.50 Å'''

Typical sp<sup>2</sup>-sp<sup>2</sup> C-C bond length: '''1.34 Å'''

'''''5. 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.'''''

Van der Waals radius of the C atom: '''1.70 Å'''.<sup>3</sup>
Twice the Van der Waals radius = '''3.40 Å'''.

The length of the '''partly formed C-C bonds''' in the TS ('''C8-C11''' and '''C14-C1''') is about '''2.11 Å'''. The length of the partly formed C-C bonds is '''shorter''' than twice the length of the Van der Waals radius (3.40 Å). This suggests that the C-C bonds are indeed forming.

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) 11:20, 2 November 2017 (UTC) The reasoning is incomplete here: what tells you that the bonds are forming is that the TS bond length is in between twice the Van der Waals radius and the length of a single C-C bond.)

The length of the other 4 C-C bonds (C1-C4, C4-C6, C6-C8 and C11-C14) are between 1.34 Å (typical sp<sup>2</sup>-sp<sup>2</sup> C-C bond length) and 1.54 Å (typical sp<sup>3</sup>-sp<sup>3</sup> C-C bond length). This is because the nature of the C-C bonds are changing (from C-C single bond to C=C double bond, or vice versa).

'''''6. Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?'''''

<jmol> //Group all the HTML within "jmol"
<jmolApplet> //Initialise the applet
<uploadedFileContents>Yrt13 CYCLOHEXENE TS REOPT MO.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 7; rotate x -20; frank off</script> //Set the variables (vibrating and spinning) as the applet initialises. Also switched off the frank (JSmol that normally appears on the bottom right)
<name>AnthMal</name> //The name of the applet must be set. This is the name that the controls refer to (the target)
</jmolApplet>
<jmolbutton> //Adding a jmol button, which executes code on the target
<script>if(vibrating==0) vibrating=1; vibration 2; else; vibrating=0; vibration off; endif</script> //Using IF functions to make a toggle. If it's not vibrating, set vibration period to 2 and change "vibrating" variable to 1, else switch off vibration and change "vibrating" to 0
<text>Vibration</text>
<target>AnthMal</target>
</jmolbutton>

<jmolmenu> //The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>frame 7; if(vibrating==0) vibration off; else; vibration 2; endif</script> //Adding vibration code as a safety net. It might not be necessary but it ensures the applet behaves properly
<text>i948/cm</text>
<target>AnthMal</target>
</item>
<item>
<script>frame 8; if(vibrating==0) vibration off; else; vibration 2; endif</script>
<text>145/cm</text>
<target>AnthMal</target>
</item>
</jmolmenu>
</jmol>
''Figure 6: Illustration of the vibration that corresponds to the reaction path at the transition state.''

A concerted process<sup>5</sup> (such as the Diels-Alder reaction) is synchronous when the changes concerned (in this case, bond formation) progresses to the same extent at the transition state. As seen from the Jmol, both bonds are formed at the same rate. Hence, this is synchronous.

== Exercise 2 ==
In this exercise, the Diels-Alder reactions of Cyclohexadiene and 1,3-Dioxole to form the Endo and Exo product were studied. Cyclohexadiene acts as the diene while 1,3-Dioxole acts as the dienophile. The reactants, transition states and products were first optimised at the PM6 level. The initial geometries generated at the PM6 level were then further optimised at the B3LYP/6-31G(d) level. Frequency calculations were executed on all reactants, transition states and products. The reactants (cyclohexadiene and 1,3-dioxole) and the exo and endo products had no imaginary frequencies. Both the exo and endo transition states had 1 imaginary frequency each. IRC calculations were done on both transition states as well.

[[File:Yrt13 E2reactionscheme.PNG|400px]]

''Figure 7: Reaction scheme of Cyclohexadiene and 1,3-Dioxole. Adapted from https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_exercise#Exercise_2:_Reaction_of_Cyclohexadiene_and_1.2C3-Dioxole''

'''''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.'''''

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |LUMO of
Cyclohexadiene
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 6; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13 CYCLOHEXADIENE PM6_MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
! style="background: #0D4F8B; color: white; text-align: center;"|LUMO of
1,3-Dioxole
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 6; mo 15; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13_DIOXOLE_PM6_MO.log</uploadedFileContents>

</jmolApplet>

</jmol>
|-
! style="background: #0D4F8B; color: white; text-align: center;"|HOMO of
Cyclohexadiene
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 6; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13 CYCLOHEXADIENE PM6_MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
! style="background: #0D4F8B; color: white; text-align: center;"|HOMO of
1,3-Dioxole
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 6; mo 14; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13_DIOXOLE_PM6_MO.log</uploadedFileContents>

</jmolApplet>

</jmol>
|}
''Figure 8: Jmols of the HOMOs and LUMOs of Cyclohexadiene and 1,3-Dioxole.''

=== Endo ===

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;"|MO40 (HOMO -1)
of Endo TS (A)

|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 36; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13_E2ENDO_TS631GD_MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
! style="background: #0D4F8B; color: white; text-align: center;"|MO41 (HOMO)
of Endo TS (S)

|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 36; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13_E2ENDO_TS631GD_MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
|-
! style="background: #0D4F8B; color: white; text-align: center;"|MO42 (LUMO)
of Endo TS (S)

|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 36; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13_E2ENDO_TS631GD_MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
! style="background: #0D4F8B; color: white; text-align: center;"|MO43 (LUMO +1)
of Endo TS (A)

|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 36; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13_E2ENDO_TS631GD_MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
|}
''Figure 9: Jmols of the relevant Endo Transition State MOs.''

[[File:Yrt13_EndoTS_MO_diagram3.PNG|500px]]

''Figure 10: MO Diagram for the formation of the Cyclohexadiene/1,3-Dioxole Endo TS. Note: This is an estimated MO diagram. The ordering of the MOs are accurate but the exact energy levels obtained from calculations done at the PM6 and B3LYP/6-31G(d) Levels were not used.''

(It's quite difficult to see the phases in your TS MOs [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:36, 30 October 2017 (UTC))

=== Exo ===

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;"|MO40 (HOMO -1)
of Exo TS (A)

|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13 E2EXO TS631GD MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
! style="background: #0D4F8B; color: white; text-align: center;"|MO41 (HOMO)
of Exo TS (S)

|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13 E2EXO TS631GD MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
|-
! style="background: #0D4F8B; color: white; text-align: center;"|MO42 (LUMO)
of Exo TS (S)

|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13 E2EXO TS631GD MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
! style="background: #0D4F8B; color: white; text-align: center;"|MO43 (LUMO +1)
of Exo TS (A)

|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13 E2EXO TS631GD MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
|}
''Figure 11: Jmols of the relevant Exo Transition State MOs.''

[[File:Yrt13_ExoTS_MO_diagram3.PNG|500px]]

''Figure 12: MO Diagram for the formation of the Cyclohexadiene/1,3-Dioxole Exo TS. Note: This is an estimated MO diagram. The ordering of the MOs are accurate but the exact energy levels obtained from calculations done at the PM6 and B3LYP/6-31G(d) Levels were not used.''

=== Questions ===

'''''1. Is this a normal or inverse demand DA reaction?''''' <sup>6</sup>

As seen from the MO diagram, this is an Inverse Demand DA reaction. This is due to the (A-S-S-A) symmetry order of the Transition State MOs. <sup>6</sup>

In a normal demand Diels-Alder reaction, the HOMO of the diene and the LUMO of the dienophile are more similar in energy (compared to the LUMO of the diene and the HOMO of the dienophile). This means that the strongest orbital interaction is between the most similar frontier molecular orbitals - HOMO of diene and LUMO of dienophile. In this case, the symmetry order of the Transition State MOs will be 'S-A-A-S' as illustrated in the MO diagram of Figure 13.

[[File:Yrt13_NormaldemandDA_diagram.PNG|500px]]

''Figure 13: Expected MO diagram of a normal demand Diels-Alder reaction.''

However, in an inverse demand Diels-Alder reaction, the LUMO of the diene and the HOMO of the dienophile are more similar in energy. This could be due to an electron-rich dienophile or an electron-poor diene. This means that the strongest orbital interaction is between the LUMO of the diene and the HOMO of the dienophile. As a result, the symmetry order of the Transition State MOs will be 'A-S-S-A', as illustrated in the MO diagram of Figure 12.

Hence, the Diels-Alder reaction between Cyclohexadiene and 1,3-Dioxole is an Inverse Demand one. This is not surprising as 1,3-Dioxole is an electron-rich dienophile (due to the oxygen atoms acting as electron-donating groups). The electron lone pairs in the p orbitals of the Oxygen atoms are able to delocalise into the π electron system of the C=C double bond, making the dienophile more electron rich.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 00:21, 8 November 2017 (UTC) Excellent use of the MO diagram to show the ordering of the orbitals. This was presented very clearly was east to understand

'''''2. Tabulate the energies and determine the reaction barriers and reaction energies (in kJ/mol) at room temperature'''''

At the '''PM6 level''',

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |Compound
! style="background: #0D4F8B; color: white; text-align: center;" |Sum of electronic and thermal Free Energies
(Hartrees / particle)
! style="background: #0D4F8B; color: white; text-align: center;" |Sum of electronic and thermal Free Energies
(kJ / mol)
|-
|style="text-align: center;" |'''Cyclohexadiene'''
|style="text-align: center;" |0.116874
|style="text-align: center;" |306.85271
|-
|style="text-align: center;" |'''1.3-Dioxole'''
|style="text-align: center;" |0.052278
|style="text-align: center;" |137.255899
|-
|style="text-align: center;" |'''Endo TS'''
|style="text-align: center;" |0.137942
|style="text-align: center;" |362.166749
|-
|style="text-align: center;" |'''Endo Product'''
|style="text-align: center;" |0.037807
|style="text-align: center;" |99.2622861
|-
|style="text-align: center;" |'''Exo TS'''
|style="text-align: center;" |0.138902
|style="text-align: center;" |364.687229
|-
|style="text-align: center;" |'''Exo Product'''
|style="text-align: center;" |0.037977
|style="text-align: center;" |99.7086211
|}
''Table 7: Summary of the 'Sum of electronic and thermal Free Energies' of the various reactants, transition states and products obtained at the PM6 Level.''

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |Endo Reaction Barrier
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Endo Reaction Energy
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Exo Reaction Barrier
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Exo Reaction Energy
(kJ / mol)
|-
|style="text-align: center;" |362.166749 - (306.85271+137.255899) = '''-81.94186'''
|style="text-align: center;" |99.2622861 - (306.85271+137.255899) = '''-344.8463229'''
|style="text-align: center;" |364.687229 - (306.85271+137.255899) = '''-79.42138'''
|style="text-align: center;" |99.7086211 - (306.85271+137.255899) = '''-344.3999879'''
|-
|}
''Table 8: Summary of the Endo/Exo reaction barriers and energies (in kJ/mol) obtained at the PM6 Level at room temperature''

However, calculations done at the PM6 level are inaccurate as PM6 is a semi-empirical fitted method that uses experimental data to save time and resources during calculations. This can be seen from the negative reaction barrier values obtained for both the endo and exo cases. It is unlikely that the values of the reaction barriers are indeed negative, thus, calculations at the B3LYP/6-31G(d) level were done.

Also, instead of optimising the first point of the IRCs, reactants were taken at infinite separation to ensure fair comparison.

At the '''B3LYP/6-31G(d) level''',

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |Compound
! style="background: #0D4F8B; color: white; text-align: center;" |Sum of electronic and thermal Free Energies
(Hartrees / particle)
! style="background: #0D4F8B; color: white; text-align: center;" |Sum of electronic and thermal Free Energies
(kJ / mol)
|-
|style="text-align: center;" |'''Cyclohexadiene'''
|style="text-align: center;" | -233.324375
|style="text-align: center;" | -612593.193227
|-
|style="text-align: center;" |'''1.3-Dioxole'''
|style="text-align: center;" | -267.068132
|style="text-align: center;" | -701187.43398
|-
|style="text-align: center;" |'''Endo TS'''
|style="text-align: center;" | -500.332150
|style="text-align: center;" | -1313622.1599
|-
|style="text-align: center;" |'''Endo Product'''
|style="text-align: center;" | -500.418691
|style="text-align: center;" | -1313849.3733
|-
|style="text-align: center;" |'''Exo TS'''
|style="text-align: center;" | -500.329165
|style="text-align: center;" | -1313614.3228
|-
|style="text-align: center;" |'''Exo Product'''
|style="text-align: center;" | -500.417321
|style="text-align: center;" | -1313845.7764
|}
''Table 9: Summary of the 'Sum of electronic and thermal Free Energies' of the various reactants, transition states and products obtained at the B3LYP/6-31G(d) Level.''

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;"|Endo Reaction Barrier
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Endo Reaction Energy
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Exo Reaction Barrier
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Exo Reaction Energy
(kJ / mol)
|-
|style="text-align: center;" |(-1313622.1599) - ((-612593.193227)+(-701187.43398)) = '''158.467307'''
|style="text-align: center;" |(-1313849.3733) - ((-612593.193227)+(-701187.43398)) = '''-68.746093'''
|style="text-align: center;" |(-1313614.3228) - ((-612593.193227)+(-701187.43398)) = '''166.304407'''
|style="text-align: center;" |(-1313845.7764) - ((-612593.193227)+(-701187.43398)) = '''-65.149193'''
|-
|}
''Table 10: Summary of the Endo/Exo reaction barriers and energies (in kJ/mol) obtained at the B3LYP/6-31G(d) Level at room temperature''

'''''3. Which are the kinetically and thermodynamically favourable products?'''''

As seen from Table 10, the kinetically favourable product is the Endo product as it has the smaller reaction barrier.

The thermodynamically favourable product is also the Endo product as it has the more exothermic reaction energy.

Usually, the kinetically favourable product is the Endo product while the Thermodynamically favourable product is the Exo product.<sup>7</sup> The Endo product is usually the kinetically favourable product due to favourable and stabilising orbital interactions, lowering the energy of the Endo transition state. The Exo product is usually the thermodynamically favourable product as it experiences less steric hindrance. Thus, it is the more (thermodynamically) stable product.

However, in this case, the Endo product is both the kinetically and thermodynamically favourable product. This could be due to factors such as secondary orbital interactions. The Endo product has stabilising secondary orbital interactions between the p-Orbitals of the Oxygen atoms and the back lobes of the diene, lowering the energy of the Endo product. Hence, the Endo product is (also) the thermodynamically favourable product (instead of the expected Exo product).

'''''4. Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy'''''

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;"|HOMO of Exo TS
! style="background: #0D4F8B; color: white; text-align: center;"|HOMO of Endo TS
|-
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat""; mo cutoff 0.01; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13 E2EXO TS631GD MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
|<jmol>

<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 36; mo 41; mo nodots nomesh fill translucent; mo titleformat""; mo cutoff 0.01; set antialiasdisplay on</script>
<uploadedFileContents>Yrt13 E2ENDO TS631GD MO.LOG</uploadedFileContents>

</jmolApplet>

</jmol>
|}
''Figure 14: Jmols of the Exo and Endo Transition State HOMOs, showing secondary orbital interactions.''

There are no significant secondary orbital interactions in the Exo Transition State. However, in the Endo Transition State, there is a stabilising secondary orbital interaction between the p-Orbitals of the Oxygen atoms (on the 1,3-Dioxole) and the LUMO of Cyclohexadiene. This stabilising interaction lowers the energy of the Endo Transition State, hence lowering the reaction barrier. Even though the Endo Transition State also experiences unfavourable steric interactions, this steric effect is minor and also helps in orientating the reactant FMOs for better overlap. Thus, the electronic factor dominates and this explains why the kinetically favourable product is the Endo Product.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 00:26, 8 November 2017 (UTC) You have come to the correct conclusion. However Your arguments in steric are not quite correct and you can see this in the Jmols you have uploaded. Excellent script. You could have made it even better by including some info on the quantum methods that are used.

== Exercise 3 ==
In this exercise, the Diels-Alder and Cheletropic reactions between o-Xylylene and SO2 were studied. For the Diels-Alder reactions, o-Xylylene acts as the diene while SO2 acts as the dienophile. The reactants, transition states and products were optimised at the PM6 level. IRC and frequency calculations were also done with the transition states.

[[File:Yrt13 E3reactionscheme.PNG|500px]]

''Figure 15: Reaction scheme of o-Xylylene and SO2. Adapted from https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_exercise#Exercise_3:_Diels-Alder_vs_Cheletropic''

=== IRC Gifs ===
'''''Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.'''''

[[File:Yrt13 Endo Product IRC.gif]]

''Figure 16: Animation Gif of the IRC of the '''Endo''' Product''

[[File:Yrt13 Exo Product IRC.gif]]

''Figure 17: Animation Gif of the IRC of the '''Exo''' Product''

[[File:Yrt13 Cheletropic Product IRC.gif]]

''Figure 18: Animation Gif of the IRC of the '''Cheletropic''' Product''

=== Calculations ===
'''''Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred.'''''

At the PM6 Level,

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |Compound
! style="background: #0D4F8B; color: white; text-align: center;" |Sum of electronic and thermal Free Energies
(Hartrees / particle)
! style="background: #0D4F8B; color: white; text-align: center;" |Sum of electronic and thermal Free Energies
(kJ / mol)
|-
|style="text-align: center;" |'''Xylylene'''
|style="text-align: center;" |0.178045
|style="text-align: center;" |467.457183
|-
|style="text-align: center;" |'''SO2'''
|style="text-align: center;" | -0.118614
|style="text-align: center;" | -311.421081
|-
|style="text-align: center;" |'''Endo TS'''
|style="text-align: center;" |0.090562
|style="text-align: center;" |237.770549
|-
|style="text-align: center;" |'''Endo Product'''
|style="text-align: center;" |0.021702
|style="text-align: center;" |56.9786053
|-
|style="text-align: center;" |'''Exo TS'''
|style="text-align: center;" |0.092078
|style="text-align: center;" |241.750807
|-
|style="text-align: center;" |'''Exo Product'''
|style="text-align: center;" |0.021462
|style="text-align: center;" |56.3484853
|-
|style="text-align: center;" |'''Cheletropic TS'''
|style="text-align: center;" |0.099067
|style="text-align: center;" |260.100428
|-
|style="text-align: center;" |'''Cheletropic Product'''
|style="text-align: center;" |0.000002
|style="text-align: center;" |0.0052510004
|}
''Table 11: Summary of the 'Sum of electronic and thermal Free Energies' of the various reactants, transition states and products obtained at the PM6 Level.''

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |Endo Reaction Barrier
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Endo Reaction Energy
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Exo Reaction Barrier
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Exo Reaction Energy
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Cheletropic Reaction Barrier
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Cheletropic Reaction Energy
(kJ / mol)
|-
|style="text-align: center;" |237.770549 - (467.457183 - 311.421081) = '''81.734447'''
|style="text-align: center;" |56.9786053 - (467.457183 - 311.421081) = '''-99.0574967'''
|style="text-align: center;" |241.750807 - (467.457183 - 311.421081) = '''85.714705'''
|style="text-align: center;" |56.3484853 - (467.457183 - 311.421081) = '''-99.6876167'''
|style="text-align: center;" |260.100428 - (467.457183 - 311.421081) = '''104.064326'''
|style="text-align: center;" |0.0052510004 - (467.457183 - 311.421081) = '''-156.0308509996'''
|-
|}
''Table 12: Summary of the Endo/Exo/Cheletropic reaction barriers and energies (in kJ/mol) obtained at the PM6 Level at room temperature''

The most kinetically favourable reaction would be the reaction with the smallest reaction barrier / lowest energy transition state. Therefore, as seen from Table 12, the most kinetically favourable reaction is the Endo Diels-Alder reaction, followed by the Exo Diels-Alder reaction and finally, the Cheletropic reaction.

The most thermodynamically favourable reaction would be the reaction with the most exothermic reaction energy / lowest energy product. Therefore, as seen from Table 12, the most thermodynamically favourable reaction is the Cheletropic reaction, followed by the Exo Diels-Alder reaction and finally, the Endo Diels-Alder reaction.

=== Reaction Profile ===
'''''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.'''''

[[File:Yrt13_Reaction_Profile.PNG]]

(Two many decimal places - these decimal places indicate trust in the precision of the calculation. When you subtract absolute energies, the results typically become very sensitive to geometry and even the architecture of the computer you're running on. [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:42, 30 October 2017 (UTC))

''Figure 19: Reaction Profile containing the relative heights of the energy levels of the reactants, TSs and products from the endo- and exo- Diels-Alder reactions and the Cheletropic Reaction. Reaction Coordinates 1, 2 and 3 refers to the reactants, TSs and products respectively. ''

As concluded from the previous section, it is expected that the Endo Diels-Alder reaction is the more kinetically favourable reaction while the Exo Diels-Alder reaction is the more thermodynamically favourable reaction.<sup>7</sup> This is explained above in ''Exercise 2 Question 3'' with stereoelectronic factors (i.e. steric hindrance and secondary orbital interactions).

Now considering the Cheletropic reaction, it is the most kinetically unfavourable reaction because of steric effects. For the Cheletropic reaction to occur, the diene has to adopt the cisoid conformation. Bulky groups/substituents tend to favour this cisoid conformation and hence, due to the lack of such bulky groups, the Cheletropic reaction is not kinetically favourable.<sup>8</sup>

The Cheletropic reaction is also the most thermodynamically favourable reaction because it results in a more stable five-membered ring product being formed. Sulfur is a relatively larger atom (than Carbon) and hence, a five-membered ring is more stable than a six-membered ring in this case.<sup>8</sup>

(Good additional analysis [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:42, 30 October 2017 (UTC))

=== Questions ===
'''''1. 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?'''''

Xylylene is highly unstable and reactive. This is because it has 2 cis-butadiene fragments. At each of the cis-butadiene fragments, xylylene can undergo reactions such as the Diels-Alder reaction. Furthermore, it is a planar molecule with 2 open faces, thus there are no significant steric hindrances either.

During the course of the reaction, the 6-membered ring becomes aromatic and the product becomes extremely stable. Each of the 6 C-C bonds are now equal in length, with a length value of '''1.40 Å'''. This is an intermediate value between the average C-C (1.54 Å) and C=C (1.34 Å) bond length. This suggests that the 6 π electrons are delocalised over the planar ring system, hence making it aromatic according to Huckel's rule. The stability of the product drives the reaction forward, explaining the reactivity of Xylylene.

(A cyclic planar ring molecule is aromatic and follows Huckel's rule when the number of its π electrons equals to 4n + 2 where n = 0 or any positive integer.)

'''''2. 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.'''''

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |Compound
! style="background: #0D4F8B; color: white; text-align: center;" |Sum of electronic and thermal Free Energies
(Hartrees / particle)
! style="background: #0D4F8B; color: white; text-align: center;" |Sum of electronic and thermal Free Energies
(kJ / mol)
|-
|style="text-align: center;" |'''Xylylene'''
|style="text-align: center;" |0.178045
|style="text-align: center;" |467.457183
|-
|style="text-align: center;" |'''SO2'''
|style="text-align: center;" | -0.118614
|style="text-align: center;" | -311.421081
|-
|style="text-align: center;" |'''Endo TS'''
|style="text-align: center;" |0.102072
|style="text-align: center;" |267.990056
|-
|style="text-align: center;" |'''Endo Product'''
|style="text-align: center;" |0.065610
|style="text-align: center;" |172.259068
|-
|style="text-align: center;" |'''Exo TS'''
|style="text-align: center;" |0.105055
|style="text-align: center;" |275.821924
|-
|style="text-align: center;" |'''Exo Product'''
|style="text-align: center;" |0.067304
|style="text-align: center;" |176.706665
|}
''Table 13: Summary of the 'Sum of electronic and thermal Free Energies' of the various reactants, transition states and products obtained at the PM6 Level.''

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white; text-align: center;" |Endo Reaction Barrier
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Endo Reaction Energy
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Exo Reaction Barrier
(kJ / mol)
! style="background: #0D4F8B; color: white; text-align: center;" |Exo Reaction Energy
(kJ / mol)
|-
|style="text-align: center;" |267.990056 - (467.457183 - 311.421081) = '''111.953954'''
|style="text-align: center;" |172.259068 - (467.457183 - 311.421081) = '''16.222966'''
|style="text-align: center;" |275.821924 - (467.457183 - 311.421081) = '''119.785822'''
|style="text-align: center;" |176.706665 - (467.457183 - 311.421081) = '''20.670563'''
|-
|}
''Table 14: Summary of the Endo/Exo reaction barriers and energies (in kJ/mol) obtained at the PM6 Level at room temperature''

The Diels-Alder reactions at the second cis-butadiene fragment in o-xylylene are very thermodynamically and kinetically unfavourable.

Firstly, it is thermodynamically unfavourable due to the positive/endothermic reaction energies. As seen in Table 14, in both the endo and exo cases, the reaction energies are positive. On the other hand, the reaction energies of the Diels-Alder reactions at the first cis-butadiene fragment are negative/exothermic (Table 12) and hence more thermodynamically favourable.

Secondly, it is kinetically unfavourable due to the significant reaction barrier. Comparing the reaction barriers of the Diels-Alder reactions at the first and second cis-butadiene fragment (Table 12 and Table 14), the Endo reaction barrier has increased from +81.73 kJ/mol to +111.95 kJ/mol, while the Exo reaction barrier has increased from +85.71 kJ/mol to +119.79 kJ/mol. This suggests that Diels-Alder reactions at the second cis-butadiene fragment are more kinetically unfavourable.

Thus, Diels-Alder reactions at the second cis-butadiene fragment are more thermodynamically and kinetically unfavourable.

== Conclusion ==

Several Diels-Alder reaction pathways were studied in the previous exercises. The way that the reactants approached each other had a significant impact on the transition state structure and product. For example, the amount of steric hindrance present depended on the approach of the reactants. Also, depending on the overlap of the reactants in the transition state, secondary orbital interactions may be present. Steric hindrances and secondary orbital interactions are examples of stereoelectronic factors that could affect the energies of the transition state/product and hence alter the predicted outcome of a reaction. An example can be found in Exercise 2, where the Endo product is (also) the thermodynamically favourable product due to stabilising secondary orbital interactions.

== Log Files ==

'''''For each of your calculations, upload the log file and include a link in the wiki'''''

=== Exercise 1 ===

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white;"|'''Compound/IRC'''
! style="background: #0D4F8B; color: white;"|'''PM6'''
|-
|style="text-align: center;"| Butadiene
|style="text-align: center;"| [[File:Yrt13 BUTADIENE PM6 MO.LOG]]
|-
|style="text-align: center;"| Ethylene
|style="text-align: center;"| [[File:Yrt13 ETHYLENE PM6 MO.LOG]]
|-
|style="text-align: center;"| Transition State
|style="text-align: center;"| [[File:Yrt13 CYCLOHEXENE TS REOPT MO.LOG]]
|-
|style="text-align: center;"| Cyclohexene
|style="text-align: center;"| [[File:Yrt13 CYCLOHEXENE PRODUCT REOPT PM6.LOG]]
|-
|style="text-align: center;"| IRC
|style="text-align: center;"| [[File:Yrt13_IRC_CYCLOHEXENE.LOG]]
|}
''Table 15: List of log files obtained in Exercise 1.''

=== Exercise 2 ===

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white;"|'''Compound/IRC'''
! style="background: #0D4F8B; color: white;"|'''PM6'''
! style="background: #0D4F8B; color: white;"|'''B3LYP/6-31G(d)'''
|-
|style="text-align: center;"| Cyclohexadiene
|style="text-align: center;"| [[File:Yrt13 CYCLOHEXADIENE PM6 MO.LOG]]
|style="text-align: center;"| [[File:Yrt13_CYCLOHEXADIENE_631Gd.log]]
|-
|style="text-align: center;"| 1,3-Dioxole
|style="text-align: center;"| [[File:Yrt13 DIOXOLE PM6 MO.log]]
|style="text-align: center;"| [[File:Yrt13_DIOXOLE_631Gd.log]]
|-
|style="text-align: center;"| Endo Transition State
|style="text-align: center;"| [[File:Yrt13 E2ENDO TSPM6.LOG]]
|style="text-align: center;"| [[File:Yrt13 E2ENDO TS631GD MO.LOG]]
|-
|style="text-align: center;"| Endo Product
|style="text-align: center;"| [[File:Yrt13_E2_ENDO_FINALPRODUCT.LOG]]
|style="text-align: center;"| [[File:Yrt13_E2_ENDO_FINALPRODUCT_631Gd.log]]
|-
|style="text-align: center;"| Endo IRC
|style="text-align: center;"| [[File:Yrt13_IRC_E2_ENDO_PM6.LOG]]
|style="text-align: center;"|
|-
|style="text-align: center;"| Exo Transition State
|style="text-align: center;"| [[File:Yrt13 E2EXO TSPM6 MO.LOG]]
|style="text-align: center;"| [[File:Yrt13 E2EXO TS631GD MO.LOG]]
|-
|style="text-align: center;"| Exo Product
|style="text-align: center;"| [[File:Yrt13_E2_EXO_FINALPRODUCT.LOG]]
|style="text-align: center;"| [[File:Yrt13_E2_EXO_FINALPRODUCT_631Gd.log]]
|-
|style="text-align: center;"| Exo IRC
|style="text-align: center;"| [[File:Yrt13_IRC_E2_EXO_PM6.LOG]]
|style="text-align: center;"|
|}
''Table 16: List of log files obtained in Exercise 2.''

=== Exercise 3 ===

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white;"|'''Compound/IRC'''
! style="background: #0D4F8B; color: white;"|'''PM6'''
|-
|style="text-align: center;"| Xylylene
|style="text-align: center;"| [[File:Yrt13 E3 XYLYLENERXT2.LOG]]

|-
|style="text-align: center;"| SO2
|style="text-align: center;"| [[File:Yrt13 E3 SO2 MIN PM6.LOG]]

|-
|style="text-align: center;"| Endo Transition State
|style="text-align: center;"| [[File:Yrt13 E3 ENDO TS PM6.LOG]]

|-
|style="text-align: center;"| Endo Product
|style="text-align: center;"| [[File:Yrt13 E3 ENDOPRODUCTIRC PM6.LOG]]

|-
|style="text-align: center;"| Endo IRC
|style="text-align: center;"| [[File:Yrt13 IRC Endo TS PM6.log]]

|-
|style="text-align: center;"| Exo Transition State
|style="text-align: center;"| [[File:Yrt13 XYLYLENE TS.LOG]]

|-
|style="text-align: center;"| Exo Product
|style="text-align: center;"| [[File:Yrt13 XYLYLENE FROZENREOPT2.LOG]]

|-
|style="text-align: center;"| Exo IRC
|style="text-align: center;"| [[File:Yrt13 IRCXYLYLENE.LOG]]

|-
|style="text-align: center;"| Cheletropic Transition State
|style="text-align: center;"| [[File:Yrt13 E3 CHELETROPIC TS PM6.LOG]]

|-
|style="text-align: center;"| Cheletropic Product
|style="text-align: center;"| [[File:Yrt13 E3 Cheletropic2 PM6.log]]

|-
|style="text-align: center;"| Cheletropic IRC
|style="text-align: center;"| [[File:Yrt13 IRC CHELETROPIC.log]]
|}
''Table 17: List of log files obtained in Exercise 3.''

'''Log files from the 'Extra' section regarding the second cis-butadiene fragment can be found here.'''

{| class="wikitable"
|-
! style="background: #0D4F8B; color: white;"|'''Compound/IRC'''
! style="background: #0D4F8B; color: white;"|'''PM6'''
|-
|style="text-align: center;"| Endo Transition State
|style="text-align: center;"| [[File:Yrt13 CISBUTADIENEENDOTS.LOG]]

|-
|style="text-align: center;"| Endo Product
|style="text-align: center;"| [[File:Yrt13 ENDOPDTFROMIRC.LOG]]

|-
|style="text-align: center;"| Endo IRC
|style="text-align: center;"| [[File:Yrt13 IRC cisbutadieneENDOTS.log]]

|-
|style="text-align: center;"| Exo Transition State
|style="text-align: center;"| [[File:Yrt13_CISBUTADIENEEXOTS_REOPT.LOG]]

|-
|style="text-align: center;"| Exo Product
|style="text-align: center;"| [[File:Yrt13 EXOPDTFROMIRC.LOG]]

|-
|style="text-align: center;"| Exo IRC
|style="text-align: center;"| [[File:Yrt13 IRC CISBUTADIENEEXOTS.LOG]]
|}
''Table 18: List of log files obtained in Exercise 3 'Extra' section.''

== References ==

1. P. Deglmann and F. Furche, J. Chem. Phys., 2002, 117, 9535–9538.

2. R. M. Pitzer, J. Chem. Phys., 1973, 58, 3111.

3. S. S. Batsanov, Van der Waals Radii Elem., 2001, 37, 871–885.

4. A. G. Orpen, L. Brammer, F. H. Allen, O. Kennard, D. G. Watson and R. Taylor, J. Chem. Soc. Dalt. Trans., 1987, S1–S83.

5. K. N. Houk, J. GonzAlez and Y. Li, Acc. Chem. Res., 1995, 28, 81–90.

6. F. Liu, R. S. Paton, S. Kim, Y. Liang and K. N. Houk, J. Am. Chem. Soc., 2013, 135, 15642–15649.

7. J. H. Cooley and R. V. Williams, J. Chem. Educ., 1997, 74, 582–585.

8. D. Suárez, T. L. Sordo and J. A. Sordo, J. Org. Chem., 1995, 60, 2848–2852.

Rep:Yrt13TS

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Rep:Yg5515ts

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==3rd Year Computational Lab---Transition states and Reactivity==
In this Lab, the transition states (TS) of three Diels-Alder (DA) reactions were located and characterized by Gaussian.DA reactions play an important role in numerous new materials and natural products
Method 3 was adopted to locate the transition state. The procedure started from the optimized structure of either the reactant or the product, then the lengths of bonds which were involved in the reaction were adjusted and freezed to resemble that of the transition structure. The guessed transition state was optimized and checked with frequency calculations and intrinsic reaction coordinate(IRC).
===Introduction===
Computational quantum chemistry has provided an efficient way to compute experimental complex and expensive experiments. Computational method is able to offer useful information of molecular geometries and properties, and reaction kinetics and energetic, and orbitals as well.
In this lab, the reactivities of three DA reactions were explored by locating and characterizing the transition states from the PES. PES characterizes how the total potential energy varied as a function of the positions of nuclei geometry. Gaussian operates based on the Born-Oppenheimer approximation,which assumed that the nuclei are fixed in positions and electrons adjusts instantaneously to any movement of the nuclei. Gaussian investigates the electron distributions in accordance to the changes of nuclei geometries. For a molecule with N atoms, there are (3N-6) independent geometry variables which equal to the number of internal motions a molecule may have. <ref> Ot, W. J. (1990). Computational quantum chemistry. Journal of Molecular Structure: THEOCHEM (Vol. 207).</ref>

Minimum energy point is the stationary point on the PES and it is defined by the zero value of the first derivative. At the stationary point, the change of potential energy with respect to all 3N-6 coordinates is zero.

<math>\frac{\partial E_r}{\partial r}</math>= = -F(R) = 0

[[File:TS diagram.png|frame|center|400px|Figure 1 illustrates the reaction pathway from one energy minima to another via going through a transition state. The energy minima is a stationary point at which energy goes up in any directions on the PES, whereas transition state is a saddle point at which energy goes up in any directions apart from the coordinate aligned with the reaction pathway shown as the blue arrow in the figure ]]

The "opt" in job type served as molecular geometry optimization in Gaussian, which can be applied to search the local minimum on the PES by employing mathematical algorithms.

Although the optimization can help us to find stationary point, but it cannot provide further information to identify whether the stationary point is a energy minima or a stationary saddle point. A stationary saddle point lies at the maximum for only one coordinate which is the reaction coordinate and all the others lie at the minimum.To distinguish whether the stationary point is a energy minima or a transition state,the curvature of the PES around the stationary point need to be determined and therefore frequency calculations are required. It can be done by explore the second derivatives, which is the force constant.

<math>\frac{\partial^2 E_r}{\partial^2 r}</math>= = -F(R) = 0

The transition state is defined to satisfy the following relationship:

<math>\frac{\partial^2 E_r}{\partial r^2}</math> > 0 with one coordinate of <math>\frac{\partial^2 E_r}{\partial r^2}</math> < 0
This is manifested by the vibrational frequency analysis. There should be only one imaginary frequency for a transition state and zero for that of an energy minima.

The computational methods used in this lab were semi-empirical method PM6 and the Density Functional Theory (DFT) method B3LYP. PM6 was fast and relatively inaccurate compared with B3LYP. To carry out B3LYP, chemical species were optimized on PM6 level first.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 09:07, 17 April 2018 (BST) You have shown a good understanding here. We actaully put the second derivatives that you have said above into a Hessian matrix. and diagonalisation gives all the force constants and the eigenvectors are the normal modes.

===Exercise 1 Reaction of Butadiene with Ethylene===

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) You are missing the discussion on the mechanism of bond formation, and are not showing the vibration corresponding to the bond forming mode. Additionally, your MO diagram is wrong: you are not at all taking into account the computed relative energies of your MOs.)
====reaction scheme====

[[File:ex1 mechanism.png|frame|center|400px|Figure.2 The reaction scheme of the [4+2] cycloaddition of butadiene with ethylene ]]

Current mechanism studies showed controversies in determining whether the reaction was proceeded via a one step concerted or a stepwise mechanism. In this exercise, the concerted TS was located by employing the method 3 with a frequency calculation. Reactants, the product and the transition state were all optimized by PM6.

====Optimization at PM6 level====
{| class="wikitable"
| colspan="4" style="text-align: center; background: #f0ddf0" | '''Table 1. Optimisation of Reactants, Product and transition state(PM6)'''
|-
| style="text-align: center;" |Butadiene
| style="text-align: center;" |Ethlyene
| style="text-align: center;" |transition state
| style="text-align: center;" |Cyclohexene
|-
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====Frequency calculation and IRC====
{| class="wikitable"
| colspan="4" style="text-align: center; background: #f0ddf0" | '''Table 2. Frequency calculation and IRC for TS'''
|-
| style="text-align: center;" |Frequency calculation
| style="text-align: center;" |IRC(Total Energy)
| style="text-align: center;" |IRC(RMS Gradient Norm)
| style="text-align: center;" |reaction progress
|-
| [[File:Yihan ex1 TS freq.PNG]]
| [[File:Yihan ex1 IRC1.PNG]]
| [[File:yihan ex1 IRC2.PNG]]
| [[File:Yihan ex1 IRC movie.gif]]
|}
The frequency calculation showed that the PM6 optimized transition state has one imaginary frequency at -949.cm<sup>-1</sup>. An IRC analysis was done for confirmation. IRC followed a minimum energy pathway on the PES from the transition state to either the product or the reactant. In this experiment, an IRC path starting from the transition state to reactant was simulated. TS was successfully located and optimized because the energy gradients were at zero at the reactants , product and the TS.

====MO analysis====
[[File:Yihanex1MO.PNG|frame|center|100px|Figure 5, MO diagram of 1 reacting with 2. A represents asymmetric orbital label and S represents symmetric orbital label. HOMO is the highest occupied nolecular orbital and LUMO is the lowest occupied molecular orbital]]

{| class="wikitable"
| colspan="5" style="text-align: center; background: #f0ddf0" | '''Table 3. MO display of key orbitals (HOMO and LUMO) '''
|-
| style="text-align: center;" |Butadiene
| style="text-align: center;" |Ethlyene
| style="text-align: center;" |TS occupied orbitals (in phase interaction)
| style="text-align: center;" |TS unoccupied orbitals (out of phase interaction)
|-
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Generally, molecular orbitals interact according to a set of rules.Firstly, only the MO which have the correct symmetry and close in energy tend to have large interactions. For example, the symmetric orbitals will interact with symmetric orbitals with non zero orbital overlap. This is also true for antisymmetric orbitals. However, orbital overlap is zero if symmetric orbital is combined with antisymmetric orbitals. In addition, HOMO and LUMO is a pair of occupied and unoccupied orbitals which closet in energy, therefore, the antisymmetric HOMO of butadiene (MO2) interacted strongly with the antisymmetric LUMO of enthylene, generating antisymmetric MO5 (bonding) and MO8 (antibonding). Also, the symmetric LUMO of butadiene (MO2) interacted strongly with the symmetric HOMO of enthylene, generating antisymmetric MO6 (bonding) and MO7 (antibonding). The occupied MO5 and MO6 of TS were resulted from in phase combination of the corresponding reactants' MO, whereas the occupied MO are generated by out of phase interactions.

{| class="wikitable"
| colspan="3" style="text-align: center; background: #f0ddf0" | '''Table 4. summary of orbital integrals'''
|-
|
| style="text-align: center;" |antisymmetric orbital
| style="text-align: center;" |symmetric orbital
|-
| style="text-align: center;" |antisymmetric orbital
| style="text-align: center;" |non zero
| style="text-align: center;" |zero
|-
| style="text-align: center;" |symmetric orbital
| style="text-align: center;" |zero
| style="text-align: center;" |non zero
|}
The reaction is a [4+2] cycloaddition. Woodward-Hoffmann rule is the commonly applied to analyze the orbital symmetry requirement for DA reactions.
In a thermal pericyclic reactions, the total number of (4q+2)s and (4r)a components need to be odd. s stands for suprafacial and a stands for antarafacial. As for suprafacial components, new bonds form on the same face at both ends, whereas an antarafacial component form new bonds at opposite faces.
[[File:Yihan woodward.PNG|frame|center|400px|Figure 3 butadiene is a suprafacial component with four π electrons and the ethylene is also a suprafacial component with two π electrons]]

There is totally one component fit the (4q+2)s and (4r)a , therefore the reaction is allowed.

====Carbon bond lengths====
Keeping track of the changes of carbon bond lengths as the reaction progressed can provide useful information of the structure of the transition states , at which bonds are partially broken and formed.
The hybridizations of the carbon atoms involved in the reaction have changed during the reaction. Therefore, the bond length also changed because it depended on the type of hybridization. The more s character the bond has, the stronger the electrons are hold to the nuclei thus the bond becomes shorter. sp2 hybridization has more s character than sp3 hybridization, so sp2-sp2 C-C is expected to be shorter than that of sp3-sp3 C-C. In addition, double bonds are shorter and stronger than single bonds, because the additional bonding attracted the nuclei stronger, pulling them closer.

{| class="wikitable"
| colspan="2" style="text-align: center; background: #f0ddf0" | '''Table 5. summary of carbon bond lengths'''
|-
| style="text-align: center;" |bond type
| style="text-align: center;" |bond length/Å
|-
| style="text-align: center;" |sp3-sp3 C-C
| style="text-align: center;" |1.54
|-
| style="text-align: center;" |sp2-sp2 C-C
| style="text-align: center;" |1.47
|-
| style="text-align: center;" |sp2-sp3 C-C
| style="text-align: center;" |1.5
|-
| style="text-align: center;" |sp2-sp2 C=C
| style="text-align: center;" |1.34
|-
| style="text-align: center;" |vand der wal
| style="text-align: center;" |1.7
|}
{| class="wikitable"
| colspan="4" style="text-align: center; background: #f0ddf0" | '''Table 6. experimental bond length of reactants, product and transition state'''
|-
| style="text-align: center;" |Butadiene
| style="text-align: center;" |Ethylene
| style="text-align: center;" |transition state
| style="text-align: center;" |cyclohexene
|-
|[[File:Yihan diene labell.PNG|400px]]
|[[File:Yihan alkene label.PNG|350px]]
|[[File:YihanTS label.PNG]]
|[[File:Yihan Pdt label.PNG]]
|-
| style="text-align: center;" |C1-C2:1.47
C1-C4:1.34

C2-C3:1.34

| style="text-align: center;" |C1-C2:1.33
| style="text-align: center;" |C1-C2:1.38
C2-C3:1.41

C3-C4:1.38

C5-C6:1.38

C1-C6:2.11

C4-C5:2.11
| style="text-align: center;" |C1-C2:1.49
C1-C6:1.54

C6-C5:1.54

C4-C5:1.54
|}

{| class="wikitable"
| colspan="4" style="text-align: center; background: #f0ddf0" |'''Table 7. explanation of bond lengths of the transition state'''
|-
| style="text-align: center;" |bond number
| style="text-align: center;" |bond type changes
| style="text-align: center;" |explanation
|-
|C1-C2/C3-C4
|sp2-sp2 C=C to sp3-sp2 C-C
| 1.38 lies in the middle of the sp2-sp2 C=C (1.34) and sp3-sp2 C-C(1.5). The bond was lengthened to form a sp3-sp2 C-C bond
|-
|C2-C3
|sp2-sp2 C-C to sp2-sp2 C=C
|the bond was shortened to form a sp2-sp2 C=C
|-
|C1-C6
|non-bonding to sp3-sp3 C-C
|The sum of two carbon atoms' van der wal radii is 3.4Å, within which electrostatic attraction started to form, or in other words, a partial bond formed. The bond length is approximately the mean of the sum of two carbon atom's van der wal radii and the sp3-sp3 C-C.
|}

===Exercise 2 Reaction of Cyclohexadiene and 1,3-Dioxole===
====reaction scheme====

[[File:Exercise 2Reaction scheme.PNG|frame|center|400px|]]

As shown in the reaction scheme, this DA reaction can proceed via two pathways ,leading to either endo or exo products depending on the orientation of 1,3-Dioxole in the TS. In this exercise, the reactants, products and both the endo and exo TS were optimized on a B3LYP/6-31G(d) level. A MO and thermochemical analysis were also conducted.

The TS were located and characterized by employing Method 3 in the Tutorial. Firstly, the product was optimized. The bonds which were involved in the reaction were broken, altered and freezed to make the structure resemble the TS. Then the coordinates were unfreezed and the TS were optimized by B3LYP/6-31G(d). All the structures were firstly optimized on a fast and rough PM6 level then optimized by the slow and more accurate B3LYP/6-31G(d) method.

====Optimization of reactants and products====
{| class="wikitable"
| colspan="4" style="text-align: center; background: #f0ddf0" | '''Table 8. B3LYP/6-31G(d) Optimisation of reactant and products'''
|-
| style="text-align: center;" |Cyclohexadiene
| style="text-align: center;" |1,3-Dioxole
| style="text-align: center;" |product(endo)
| style="text-align: center;" |product (exo)
|-
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====TS optimization and characterization====
{| class="wikitable"
| colspan="4" style="text-align: center; background: #f0ddf0" | '''Table 9. B3LYP/6-31G(d)Optimisation of TS, frequency analysis and IRC'''
|-
|
| style="text-align: center;" |TS (endo)
| style="text-align: center;" |TS (exo)
|-
|optimized structure
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|-
|frequency analysis
|[[File:Yihan ex2Endo TS freq.PNG]]
|[[File:Yihan ex2Exo TS freq.PNG]]
|-
|IRC(total energy)
|[[File:YihanEndo IRC total E.PNG|400px]]
|[[File:YihanExo IRC total E.PNG|400px]]
|-
|IRC( RMS gradient norm)
|[[File:YihanEndo IRC g.PNG|400px]]
|[[File:YihanExo IRC g.PNG|400px]]
|-
|reaction progress
|[[File:Yihan ex2Endo movie.gif]]
|[[File:Yihan ex2Exo movie.gif]]
|}

Both endo and exo TS have only one imaginary frequencies. The convergences in the log file were checked. The geometries of the optimized TS were consistent with the structures of their corresponding products. IRC analysis showed that the TS were successfully located because the energy gradients were zero.

====MO analysis====

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) As before, your MO diagrams are wrong.)

|[[File:YihanMO 1endo.PNG]]
|[[File:YihanMO exo.PNG]]

{| class="wikitable"
|colspan="5" style="text-align: center; background: #f0ddf0" | '''Table 10. HOMO and LUMO of reactants, products and TS'''
|-
|
|Cyclohexadiene
|1,3-Dioxole
|TS(endo)
|TS(exo)
|-
|HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<title>A, MO1</title>
<script>frame 18; mo 22; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yihan reactant 1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<title>S, MO7</title>
<script>frame 44; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>YihanREACANT 2 OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<title>S, MO4</title>
<script>frame 30; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>YihanENDO TS OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<title>S,MO12</title>
<script>frame 16; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>YihanTS EXO OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<title>S,MO2</title>
<script>frame 18; mo 23; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yihan reactant 1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<title>A,MO8</title>
<script>frame 44; mo 20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>YihanREACANT 2 OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<title>S,MO5</title>
<script>frame 30; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>YihanENDO TS OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<title>S,MO13</title>
<size>250</size>
<script>frame 16; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>YihanTS EXO OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
{| class="wikitable"
|colspan="5" style="text-align: center; background: #f0ddf0" | '''Table 11.other key orbitals of TS '''
|-
|
|TS(endo)
|TS(exo)
|-
| HOMO-1
| <jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<title>A,MO3</title>
<script>frame 30; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>YihanENDO TS OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<title>A,MO11</title>
<size>250</size>
<script>frame 16; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>YihanTS EXO OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|LUMO+1
| <jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<title>A, MO6</title>
<script>frame 30; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>YihanENDO TS OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<title>A, MO14</title>
<script>frame 16; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>YihanTS EXO OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
The MO interactions occurred in the region of HOMO and LUMO.

{| class="wikitable"
| colspan="4" style="text-align: center; background: #f0ddf0" | '''Table 12. energy comparison of endo and exo products and TS by B3LYP/6-31G(d))'''
|-
|style="text-align: center;" |chemical species
|style="text-align: center;" |Sum of electronic and thermal Free Energiesy,Hartree
|style="text-align: center;" |kJ mol<sup>-1<sup>
|-
|style="text-align: center;" |Cyclohexadiene
|style="text-align: center;" |-233.3243
|style="text-align: center;" |-612593.14656
|-
|style="text-align: center;" |1,3-Dioxole
|style="text-align: center;" |-267.0686
|style="text-align: center;" |-701188.7195
|-
|style="text-align: center;" |TS(endo)
|style="text-align: center;" |-500.3321
|style="text-align: center;" |-1313622.0545
|-
|style="text-align: center;" |TS (exo)
|style="text-align: center;" |-500.3291
|style="text-align: center;" |-1313614.2305
|-
|style="text-align: center;" |product(endo)
|style="text-align: center;" |-500.4186
|style="text-align: center;" |-1313849.2732
|-
|style="text-align: center;" |product(exo)
|style="text-align: center;" |-500.4173
|style="text-align: center;" |-1313845.6730
|}

The MO energies were checked with single point energies. It was observed that both the endo TS and endo product were more energetically stable than their exo counterpart. This is due to the extra stabilization resulted from the secondary orbital interactions.

{| class="wikitable"
| colspan="2" style="text-align: center; background: #f0ddf0" | '''Table 13. comparison of orbital interactions of the HOMO of endo and exo TS '''
|-
|style="text-align: center;" |TS endo
|style="text-align: center;" |TS exo
|-
|[[File:YihanEndo orbital.PNG|200px]]
|[[File:YihanExo orbital.PNG|200px]]
|}
The reaction barriers were calculated as the energy differences between the total energy of reactants and the TS. Reaction energies, i.e. ΔG, was the energy differences between the reactants and products.
{| class="wikitable"
| colspan="3" style="text-align: center; background: #f0ddf0" | '''Table 14. Activation energy and ΔG for endo and exo reaction pathways'''
|-
|
|style="text-align: center;" |Activation energy,kJ mol<sup>-1<sup>
|style="text-align: center;" |ΔG,kJ mol<sup>-1<sup>
|-
|style="text-align: center;" |endo pathway
|style="text-align: center;" |159
|style="text-align: center;" |-68
|-
|style="text-align: center;" |exo pathway
|style="text-align: center;" |167
|style="text-align: center;" |-64
|}

The four red p orbitals indicated in the diagram are involved in the secondary orbital interaction. There are significant interactions between the non bonding orbitals of the oxygen atoms and the p orbitals of the diene component.The four p orbitals combined in phase. As the endo transition state was stabilized, the energy barrier for endo pathway was expected to be smaller than that of the exo pathway. Therefore, the activation energy of endo pathway was lower and ΔG was more negative. Overall, in combination of the single energy analysis, the endo products are both the kinetically stable and thermaldynamically stable products.

====Normal or Inverse electron demand====
According to the energy differences between the LUMO and HOMO pairs of the reactants and products, DA reactions are classified as two types:
Normal electron demand: electron deficient dienophile with low energy LUMO and the electron rich diene with high energy HOMO
Inverse electron demand: electron rich dienophile with high energy LUMO and the electron difficient diene with low energy HOMO

{| class="wikitable"
| colspan="3" style="text-align: center; background: #f0ddf0" | '''Table 15 Energy difference of different LUMO and HOMO pairs'''
|-
|
|style="text-align: center;" |HOMO energy/a.u.
|style="text-align: center;" |LUMO energy/a.u
|-
|style="text-align: center;" |Cyclohexadiene
|style="text-align: center;" |-0.20554
|style="text-align: center;" |-0.01711
|-
|style="text-align: center;" |1,3-Dioxole
|style="text-align: center;" |-0.19594
|style="text-align: center;" |+0.03795
|}

ΔE of HOMO (Cyclohexadiene) and LUMO (1,3-Dioxole):0.243
ΔE of HOMO (1,3-Dioxole) and LUMO (Cyclohexadiene):0.179

The closer the energies of two molecular orbitals, the larger the interactions. The energy difference between the HOMO of cyclohexadiene and LUMO of 1,3-Dioxole, therefore, this DA reaction has an inverse electron demand. 1,3-Dioxole has high energy LUMO because the lone pairs of the two oxygen atoms have the ability to donate electrons into the π cloud, raising the orbital energies.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 09:18, 17 April 2018 (BST) Your analysis of the MOs was good. and you have correctly deduced the electron demand of the reaction. well done. Your energies were correct and there for you have come to correct conclusions. This was your best section.

===Exercise 3 o-Xylylene-SO2 Cycloaddition===
====reaction scheme====
[[File:Ex3 Reaction scheme.png|700px]]

The reaction of o-Xylylene and SO2 can proceed via two pathways, DA and cheletropic. The DA reactions can go through both endo and exo transition state to the sultine product. The TS is a six membered heteroaromatic ring with 6 π electrons involved. The cheletropic reaction is a separate class of pericyclic reactions, they must also obey the Woodward Hoffmann rules. According to the selection rules for cheletropic reactions, o-Xylylene and SO2 reacted through a disrotatary fashion which the HOMO of the S atom pointed directly to the π system of the o-Xylylene, because the π system has 4n+2 π electrons. The TS is a five membered heteroaromatic ring with also 6 π electrons.<ref>Woodward, R.B.; Hoffman, R. Angew. Chem. Int. Ed. Engl. 1969, 8, 781–853.</ref>

====PM6 optimization====
The reactants, products and TS were optimized on PM6 level and the TS s were located by Method 3. Products were firstly optimized. Then, the bonds involved in the reaction were broken ,frozen and optimized to obtain a guessed TS stucture. The coordinated were unfrozed and optimized again to obtain the accurate TS structure.
{| class="wikitable"
| colspan="2" style="text-align: center; background: #f0ddf0" | '''Table 16 PM6 optimization of reactants'''
|-
|style="text-align: center;" |o-Xylylene
|style="text-align: center;" |SO2
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 26</script>
<uploadedFileContents>Yihan o-Xylylene.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 18</script>
<uploadedFileContents>Yihan REACTANT SO2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

{| class="wikitable"
|colspan="3" style="text-align: center; background: #f0ddf0" | '''Table 17 PM6 optimization of endo and exo products and TS'''
|-
|
|style="text-align: center;" |products
|style="text-align: center;" |TS
|-
|style="text-align: center;" |endo
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 76</script>
<uploadedFileContents>ex3ENDO PRODUCT.LOG ‎</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 26</script>
<uploadedFileContents>Yihan ex3 ENDO TS UNFREEZ.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|style="text-align: center;" |exo
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 70</script>
<uploadedFileContents>YihanEXOPRODUCT PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 12</script>
<uploadedFileContents>Yihan ex3EXO TS3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|style="text-align: center;" |Cheletropic
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 58</script>
<uploadedFileContents>Yihan chele PRODUCT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 60</script>
<uploadedFileContents>Yihan chele TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

(Your cheletropic TS and product geometries are wrong. The oxygen atoms are too close together and have bonded. Your exo geometries are actually endo [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 15:11, 4 April 2018 (BST))

====TS characterization====
IRC and frequency analysis were carried out to confirm a TS has been successfully located. All the IRC showed that the energy gradients were zero at reactants, transition states and products. All the reactions were proceeded via a concerted fashion. It was observed that the energies involved in the reaction were quite small compared with the reactions in exercise 1 and exercise 2 which was due to the high energy o-Xylylene. For both the endo and exo DA pathways, two single bonds were formed, i.e. C-S and C-O, meanwhile, two C=C were reduced to one C=C as well as the S=O. In cheletropic pathway, two C-S were formed and two C=C were reduced to one.<ref>Woodward, R.B.; Hoffman, R. Angew. Chem. Int. Ed. Engl. 1969, 8, 781–853.</ref>

{| class="wikitable"
| colspan="6" style="text-align: center; background: #f0ddf0" | '''Table 18 TS characterization of endo and exo pathways at PM6 level'''
|-
|
|style="text-align: center;" |reaction progress
|style="text-align: center;" |IRC(total energy)
|style="text-align: center;" |IRC(energy norm gradient)
|-
|style="text-align: center;" |endo
|[[File:Yihan ex3 DA endo.gif]]
|[[File:Yihan ex3Endo total E.PNG]]
|[[File:YihanEndo E gradient.PNG]]
|-
|exo
|[[File:Yihan ex3 DA exo.gif ]]
|[[File:Yihan Exo total E.PNG]]
|[[File:Yihan Exo E gradient.PNG]]
|-
|style="text-align: center;" |Cheletropic
|[[File:Yihan Chele IRC.gif]]
|[[File:YihanChele total E.PNG ]]
|[[File:YihanChele E gradient.PNG]]
|}
For all the chemical species, convergences were checked. As for the reactants and products, there were no imaginary frequencies present and for TS, there was only one imaginary frequency.

====Thermochemical analysis====

The energies of reactants, products and TS were obtained from the log files. Activation barriers and ΔG were calculated.
{| class="wikitable"
| colspan="4" style="text-align: center; background: #f0ddf0" | '''Table 19. energy comparison of endo and exo products and TS by B3LYP/6-31G(d))'''
|-
|style="text-align: center;" |chemical species
|style="text-align: center;" |Sum of Electronic and Thermal Free Energies/Hartree
|style="text-align: center;" |kJ mol<sup>-1<sup>
|-
! o-Xylylene
| style="text-align: center;" | +0.179059
| style="text-align: center;" | +470.119
|-
! SO<sub>2</sub>
| style="text-align: center;" | -0.119268
| style="text-align: center;" | -313.496
|-
! Diels-Alder Exo TS
| style="text-align: center;" | +0.090559
| style="text-align: center;" | +240.286
|-
! Diels-Alder Endo TS
| style="text-align: center;" | +0.090559
| style="text-align: center;" | +237.762
|-
! Cheletropic TS
| style="text-align: center;" | +0.099377
| style="text-align: center;" | +260.914
|-
! Diels-Alder Exo Product
| style="text-align: center;" | +0.021700
| style="text-align: center;" | +56.973
|-
! Diels-Alder Endo Product
| style="text-align: center;" | +0.021696
| style="text-align: center;" | +56.963
|-
! Cheletropic Product
| style="text-align: center;" | +0.000006
| style="text-align: center;" | +0.0158
|}

(Your cheletropic energy is inconsistent with the Jmol and log file you've produced [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 15:11, 4 April 2018 (BST))

{| class="wikitable"
| colspan="4" style="text-align: center; background: #f0ddf0" | '''Table 20. Activation energies and reaction energies'''
|-
|
|style="text-align: center;" |Activation energy,kJ mol<sup>-1<sup>
|style="text-align: center;" |ΔG,kJ mol<sup>-1<sup>
|-
! endo pathway
| style="text-align: center;" | 81.139
| style="text-align: center;" | 99.66
|-
! exo pathway
| style="text-align: center;" | 83.663
| style="text-align: center;" | 99.65
|-
! cheletropic reaction
| style="text-align: center;" | 104.291
| style="text-align: center;" | 156.60
|}
Activation barriers were calculated by the differences between the sum of energies of two reactants and the TS. ΔG was obtained by calculating the differences between the free energies of reactants and products.
====reaction profile====
[[File:Yihan reaction profile.png|600px]]

It was assumed that the reactants have zero energies with infinite separations. The reaction profile showed relative height of the TS and products of the three reaction pathways. Cheletropic products were the thermodynamic products because they were most energetically stable. There were extremely small energy differences between the exo and endo transition states as well as the products. However, the endo products and TS were slightly more stable than that of the exo. Therefore, the kinetic and thermodynamic products were generated from the endo pathway.Apart from the steric interactions, favorouble orbital interactions also play a role in energy stabilization. The non bonding p orbitals of S=O interacts with the π system.

Files:
Ex1:

butadiene : [[File:Yihan butadiene.LOG]]

transition state: [[File:Yihan TS PM6.LOG]]

cyclohexene: [[File:Yihan cyclohexene.LOG]]

Ex2

Cyclohexadiene: [[File:Yihan reactant 1.LOG]]

1,3-Dioxole : [[File:YihanREACANT 2 OPT.LOG]]

endo product:[[File:YihanPRODUCT ENDO OPT.LOG]]

exo product : [[File:YihanPRODUCT EXO OPTIMIZED.LOG]]

Ex3

o-Xylylene: [[File:Yihan o-Xylylene.LOG]]

SO2: [[File:Yihan REACTANT SO2.LOG]]

Endo TS: [[File:Yihan ex3 ENDO TS UNFREEZ.LOG]]

Exo TS: [[File:Yihan ex3EXO TS3.LOG]]

Cheletropic product: [[File:Yihan chele PRODUCT.LOG]]

Cheletropic TS: [[File:Yihan chele TS.LOG]]

Exo product: [[File:YihanEXOPRODUCT PM6.LOG]]

Endo product: [[File:ex3ENDO PRODUCT.LOG]]

References
<references/>

Rep:Year 3 computational lab physical module 3: Transition structures (ec1412)

2025-09-01T09:51:02Z

Move page script: Move page script moved page Year 3 computational lab physical module 3: Transition structures (ec1412) to Rep:Year 3 computational lab physical module 3: Transition structures (ec1412): Move to report namespace

== The Cope Rearrangement Tutorial ==

In this tutorial, we are making use of the Cope rearrangement of 1,5-hexadiene to study its chemical reactivity.

=== Objectives: ===

i) Optimisation of the structure of 1,5-hexadiene

ii) Optimisation of the "Chair" and "Boat" transition structures

iii) Determine the preferred reaction mechanism of 1,5-hexadiene using the optimisation data

== Optimisation of the structure of 1,5-hexadiene ==

=== i) Anti-periplanar conformation ===

<jmol>
<jmolApplet>
<title>1,5-hexadiene app</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>elvin1.mol</uploadedFileContents>
</jmolApplet>
</jmol>

'''Total energy''': -231.69260235 a.u

'''RMS Gradient Norm''': 0.00001824 a.u

'''Dipole moment''': 0.2021 Debye

'''Point Group''': C2

=== ii) Gauche conformation ===

<jmol>
<jmolApplet>
<title>1,5-hexadiene app</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>elvin2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

'''Total energy''': -231.68961574 a.u

'''RMS Gradient Norm''': 0.00001401 a.u

'''Dipole moment''': 0.4439 Debye

'''Point Group''': C1

Rep:Year 3 computational lab physical module 3: Transition structures

2025-09-01T09:51:02Z

Move page script: Move page script moved page Year 3 computational lab physical module 3: Transition structures to Rep:Year 3 computational lab physical module 3: Transition structures: Move to report namespace

== The Cope Rearrangement Tutorial ==

Rep:Year3 MgO fd915

2025-09-01T09:51:01Z

Move page script: Move page script moved page Year3 MgO fd915 to Rep:Year3 MgO fd915: Move to report namespace

Felix de Courcy-Ireland - fd915 - 01062960 - Third Year 'Thermal Expansion of MgO' Computational Laboratory

= Introduction =

The aim of the third year 'Thermal Expansion of MgO' computation laboratory was to use two different forms of computational simulation (one based upon the quasi-harmonic approximation and the other a molecular dynamics calculation) to study the thermal expansion of magnesium oxide (MgO), to compute free energies of MgO, and calculate the volumetric thermal expansion coefficient of this system using Eq. 1.

<math> Equation 1: \alpha_V = \frac{1}{V_0} \left( \frac{ \delta V}{ \delta T} \right)_P </math>

Where <math> \alpha_V </math> is the volumetric thermal expansion coefficient, <math> V_0 </math> is the initial volume of the system, and <math> \frac{ \delta V}{ \delta T} </math> is the rate of change in volume with temperature.
A good understanding of thermal expansion has many useful applications. Civil engineers need to be able to predict how building materials expand as they are heated - this has lead to the incorporation of an expansion joint into bridges. Research is also done specifically on the computer-simulated thermal expansion of MgO, as the earth's mantle<ref name="quantum corrections" /> contains a considerable amount of MgO, and so geophysicists are interested in fine-tuning these calculations.

== Quasi-Harmonic Simulation ==

A brief discussion of the underlying theory is required to facilitate proper discussion of the quasi-harmonic approximation, and the results from the calculations carried out.

The quasi-harmonic approximation is based on quantum mechanics, and using it, the vibrations in a crystal lattice can be treated as waves or as quantised particles - these particles are referred to as phonons. A phonon is a discrete quantum of vibration, just as a photon is a discrete quantum of light<ref name="OSSB" />. Due to the large amount of atoms in these crystal systems, they can be treated as an infinite lattice of atoms, with each atom having associated with it a set of orbitals. As the orbitals on these atoms overlap, there is effectively a continuum of energies formed through the crystal lattice, ranging from the lowest energy, most bonding configuration to the highest energy, most anti-bonding configuration. This effective continuum of orbitals can be shown graphically, whereby it becomes known as a band structure, see Fig 1.

[[File:E(k)_vs_k.PNG|400px|thumb|Figure 1: Plot of E(k) against k<ref name="Hoffmann"/>. The reason that the curve is sigmoidal is due to the fact that there are more degenerate levels as you get more bonding or anti-bonding, a consequence of Peierls Distortion <ref name="Hoffmann" />.]]

Fig. 1 shows a plot of the wavevector, k, and the energy levels corresponding energy to these values of k. The plot is for values of k between <math> \frac{-\pi}{a} </math> and <math> \frac{\pi}{a} </math>, as it is between this limit that unique values for the energy levels are obtained. This limit defines what is known as the first Brillouin Zone<ref name="lattice dynamics 2" />. Moving outside of this limit can be thought of in terms of translation through over the infinite lattice - if one translates by <math> \frac{2\pi}{a} </math> in any direction, one simply returns to the point where one started, and so the same energy value will be obtained.

Using the quasi-harmonic approximation, values of k are sampled to return their corresponding energy values, and from these energy values, phonon dispersion curves are obtained, see Fig. 2. These phonon dispersion curves show the energy levels associated with a specific path through k-space - the path taken is given as the x-axis.

[[File: First_Phonon_Disperson_Curves_Invert.png|400px|thumb|right|Figure 2: First Phonon Dispersion Curves: Notice that there are six bands, due to the fact that there are two atoms (one Magnesium and one Oxygen) per unit cell, and the unit cell is in three dimensions, giving a total of six bands.]]

Ideally, one would cycle through every value of k and obtain the corresponding molecular energy levels, and then use this information to calculate properties of interest in the material. However, as the crystal structure contains atoms on the order of Avogadro's constant, there are a similarly vast amount of k values, which would take a very long time to analyse. The solution to this is to sample the Brillouin Zone at regular intervals, using a grid of sufficient density so that enough energy levels are sampled, yet not looking at so many k-values that the computation takes too long. If more k-values in the first Brillouin Zone are sampled, more energy levels are used in the calculations of density of state, and free energy, making the results from these more accurate.

Fig 1 also illustrates a second important concept required in the analysis the quasi-harmonic approximation, and that is the concept of reciprocal space.

As a face-centred cubic structure, the crystal structure of MgO can be described with a single lattice parameter, a. As can be seen from Fig 1, the repeat period of the sinusoidal curve from the Brillouin Zone is <math> \frac{2\pi}{a} </math>. This value, <math> \frac{2\pi}{a} </math> is defined as the cell length, a*, in the reciprocal lattice<ref name="lattice dynamics 2" />.

<math> Equation 2: a* = \frac{2\pi}{a} </math> <ref name="lattice dynamics 2" />

Where <math> a </math> is the lattice parameter in direct space and <math> a* </math> is the lattice parameter in reciprocal space.

The first calculation method utilised the quasi-harmonic approximation, whereby shrinking factors where selected to choose a suitable grid density for sampling the phonon frequencies of the simulated MgO crystal at temperatures ranging from 0 - 1000 K.
The quasi-harmonic approximation expands upon the harmonic approximation to better simulate the real behaviour of a material - this progression from the harmonic to the quasi-harmonic approximation is a good example of the perturbation method, often used in physics<ref name="lattice dynamics 2" />. In the quasi-harmonic approximation, the equilibrium distance between atoms is permitted to change with temperature, therefore allowing the system to expand upon heating - this change in equilibrium bond distance is not permitted in the harmonic approximation<ref name="lattice dynamics 8" />.

== Molecular Dynamics Simulation ==

The second method used molecular dynamics to simulate the expansion of the MgO crystal system. Molecular dynamics calculations rely on solving Newtons laws of motion, and are therefore classically mechanic <ref name="lattice dynamics 8" />. At each fixed temperature, the initial system, with constant pressure and number of particles, was allowed to vary in volume. The simulated particles moved around randomly, as they would do in reality - and as the system equilibrated, the properties of the systems were computed from the average movement of the atoms.

= Experimental =

Two methods were used to calculate the volumetric expansion coefficient of MgO - making use of lattice dynamics and molecular dynamics. In a virtual Linux environment, DL Visualise<ref name="DLV" /> was used as the interface with the General Utility Lattice Program (GULP<ref name="GULP" />) to simulate the expansion of MgO from 0 - 1000 K. Data for optimised energies and volumes were plotted and analysed using Python on a Jupyter Notebook.

First, a phonon dispersion curves for the MgO system were calculated. 50 points along a path W-L-G-X-W-K in the reciprocal space were computed - and the phonon dispersion curves in Fig 2 were returned.

The shrinking factors were varied along the three axis of the reciprocal lattice, and the phonon density of states were calculated. Starting with a grid density of 1x1x1, the shrinking factors were increased up to 64x64x64, with the GULP program returning values for the minimised zero point energy for each calculation, as well as the phonon density of states graphs (see Fig's 3 - 8 and Table 1).

Next, the Helmholtz Free Energy of the MgO lattice was calculated, using the quasi-harmonic approximation. Shrinking factors from 1x1x1 to 64x64x64 were used to minimise the Helmholtz Free Energy at 300 K (see Table 3).

The Helmholtz Free Energy of the crystal was then optimised at 100 K intervals, between 0 and 1000 K, using a grid density of 32x32x32 - this was deemed to give the best balance between calculation accuracy and computational intensity. From the GULP output, plots were obtained for the Free Energy against temperature, and lattice constants from the optimised MgO system against temperature.

A further graph of optimised primitive cell area against temperature was plotted, and from the fitted gradient of this graph, the thermal expansion coefficient of MgO was obtained using eq. 1.

Finally, molecular dynamics calculations were carried out using a cell containing 32 MgO units. The number of particles, temperature and pressure were again kept constant, with the volume allowed to vary. Readings were taken at 100 K intervals, between 100 K - 1000 K. The values for the optimised unit cell volume were converted to corresponding values for the primitive unit cell. A plot of optimised primitive unit cell volume against temperature was then used to determine the volumetric thermal expansion coefficient.

= Results and Discussion =

== Phonon Modes of MgO ==

[[File: 1x1x1 shrinking factor invert.png|400px|thumb|Figure 3: The density of states graph for a 1x1x1 grid. The k-point at which the density of states was measured is point L. This was confirmed by examination of the log file from the GULP output, and by analysis of the graph of Phonon Dispersion Curves. At point L, there are four frequencies, with the two frequencies at 290 cm-1 and 350 cm-1 having twice the density as the frequencies at 680 and 805 cm<sup id="cite_ref-9">-1</sup>; this is due to the overlap of two pairs of the six bands.]]
[[File:5x5x5 shrinking factor invert.png|400px|thumb|right|Figure 4: 5x5x5]]

{{multiple image
| direction = vertical
| width = 300
| image1 = 3x3x3 shrinking factor invert.png
| alt1 = 3x3x3
| caption1 = Figure 5: Density of states plot, grid density = 3x3x3
| image2 = 8x8x8 shrinking factor invert.png
| alt2 = 8x8x8
| caption2 = Figure 6: Density of states plot, grid density = 8x8x8
}}

{{multiple image
| direction = vertical
| width = 300
| image1 = 32x32x32 shrinking factor invert.png
| alt1 = Yellow cartouche
| caption1 = Figure 7: Density of states plot, grid density = 32x32x32
| image2 = 64x64x64 shrinking factor invert.png
| alt2 = Red cartouche
| caption2 = Figure 8: Density of states plot, grid density = 64x64x64
}}

As shrinking factor increases and the grid is divided into smaller and smaller increments, the density of states distributes out across the entire frequency range. As the shrinking factor increases, the number of k values sampled increases, and so a fuller picture of the occupancy of energy states is obtained. As the Brillouin zone is divided up into smaller and smaller segments (by increasing the shrinking factor), more and more k values are measured, until effectively the plot of DOS versus energy appears to show a continuum.<ref name="Hoffmann" />. With a grid size of 5x5x5, the energy of system converges to a value in eV accurate to 6 dp (Fig 4), this is the limiting accuracy of the calculation. Graphically, the grid size needs to be increased closer to values of 32x32x32 (Fig 7) before there appears to be no change in the graph with increasing shrinking factor. However, although the graphs for the density of states do become smoother as the grid size increases, the main features of the graphs, such as the magnitude and positions of the peaks, are clear well before this.

{| class="wikitable float right"
|+ style="text-align: left;" | Table 1: Shows values for zero point energy of MgO as the grid density was changed.
! Grid Density
! Energy (eV)
|-
| 1x1x1
| 0.172063
|-
| 2x2x2
| 0.174209
|-
| 3x3x3
| 0.174331
|-
| 4x4x4
| 0.174339
|-
| 5x5x5
| 0.174340
|-
| 6x6x6
| 0.174340
|-
| 8x8x8
| 0.174340
|-
| 16x16x16
| 0.174340
|-
| 32x32x32
| 0.174340
|-
| 64x64x64
| 0.174340
|}

Hoffmann describes the density of states as a return from k-space to real space, with the density of states curve being related to the gradient of the dispersion curve<ref name="Hoffmann" />. The smaller the gradient of the dispersion curve in a given energy range, the greater the density of states in this energy range. In plain English, the flatter the dispersion curve of a given energy range, the more energy levels there are in this energy range.<ref name="Hoffmann" />

With the density of states calculated for MgO, one can consider how these calculations might be altered if different materials were to be analysed. The consideration of three compounds with varying lattice parameters - Calcium oxide, Faujasite and lithium metal - provide this opportunity (Table 2).

{| class="wikitable"
|+ style="text-align: left;" | Table 2: Values for lattice parameters in direct and reciprocal space of MgO, CaO, Faujasite and Lithium metal.
|-
! Compound
! a (Å)
! a* (Å)
! Grid Size
|-
| MgO<ref name="MgO" />
| 4.21
| 1.49
| 32x32x32
|-
| CaO<ref name="CaO lattice" />
| 4.81
| 1.31
| 28x28x28
|-
| Faujasite<ref name="faujasite" />
| 24.57
| 0.26
| 6x6x6
|-
| Lithium<ref name="lithium" />
| 3.5
| 1.79
| 38x38x38
|}

As the unit cell of the material under study becomes larger, the size of the reciprocal lattice shrinks. This means for a given grid size, more values of the k-space are sampled if the unit cell of a material is large, compared to whether it is small. Therefore, a smaller grid size can be used to sample the reciprocal space of a material with a larger unit cell, and still obtain as much information on phonon frequency.

In this instance, MgO and Cao have very similar lattice parameter sizes, however the lattice parameter for CaO is slightly bigger than that of MgO. This leads to a smaller value for the unit cell of the reciprocal lattice of CaO, a*. This smaller value for a* means that a slightly smaller grid size could be used in the calculation on CaO, however the differences in the value for a* of MgO and CaO are small (see Table 2) and so a change in grid size would only cause minor changes to the calculation.

The lattice parameter a for the perovskite is far larger than that of MgO, by a factor of 6. Due to the inverse relationship between a and a*, the unit cell of the reciprocal lattice of Faujasite is 6 times small than that of MgO. Consequently, a calculation on Faujasite requires a far less dense grid compared to that of a calculation on MgO.

Finally, the body-centred cubic lithium structure has a lattice parameter of a = 3.5 A, and so consequently will have a slightly larger value of a* versus MgO. Therefore calculations with lithium require a slightly more dense grid size than that of MgO.

== Calculating the Helmholtz Free Energy of MgO within the harmonic approximation ==

{| class="wikitable"
|+ style="text-align: left;" | Table 3: How optimised Helmholtz Free Energy of MgO varies with grid density
|-
! Grid Density
! Helmholtz Free Energy (eV)
! Helmholtz Free Energy (kJ/mol)
|-
| 1x1x1
|<nowiki>-40.930301</nowiki>
|<nowiki>-3949.148006</nowiki>
|-
| 2x2x2
|<nowiki>-40.926609</nowiki>
|<nowiki>-3948.791810</nowiki>
|-
| 3x3x3
| -40.926432
| -3948.774713
|-
| 4x4x4
|<nowiki>-40.926450</nowiki>
|<nowiki>-3948.776419</nowiki>
|-
| 8x8x8
|<nowiki>-40.926478</nowiki>
|<nowiki>-3948.779123</nowiki>
|-
| 16x16x16
|<nowiki>-40.926482</nowiki>
|<nowiki>-3948.779580</nowiki>
|-
| 20x20x20
| -40.926483
| -3948.779614
|-
| 24x24x24
| -40.926483
| -3948.779629
|-
| 32x32x32
|<nowiki>-40.926483</nowiki>
|<nowiki>-3948.779641</nowiki>
|-
| 64x64x64
|<nowiki>-40.926483</nowiki>
|<nowiki>-3948.779648</nowiki>
|}

The free energy decreases with grid density, until a value of around 20x20x20 - at this point, the programme has minimised the free energy of the system accurate to 0.01 meV. A grid density of 3x3x3 was found to be appropriate for calculations accurate to 1 meV, and a grid density of 4x4x4 was found to be accurate for calculations accurate to 0.5 meV and 0.1 meV.

{| class="wikitable"
|+ style="text-align: left;" | Table 4
|-
! Compound
! a (Å)
! a* (Å)
! Grid Size
|-
| MgO
| 4.21
| 1.49
| 20x20x20
|-
| CaO
| 4.81
| 1.31
| 18x18x18
|-
| Faujasite
| 24.57
| 0.26
| 3x3x3/4x4x4
|-
| Lithium
| 3.5
| 1.79
| 24x24x24
|}

Using the previously mentioned arguments, one can speculate on the possible changes to the calculation that would be required if different materials, with different lattice parameters were analysed, using the ratios of the a* values to approximate changes to grid density (Table 4).

== Thermal Expansion of MgO ==

Taking a value for the grid density of 32x32x32, graphs of Helmholtz Free Energy and temperature, and lattice parameter against temperature were plotted (Fig 9 and 10).

[[File: Free energy (eV) vs temp.png|450px| thumb| alt = Figure 9: Helmholtz Free Energy (eV) versus Temperature| Figure 9: Plot of Optimised Helmholtz Free Energy of MgO against Temperature using Quasi-Harmonic Approximation]]
[[File: Lattice constant vs. temp.png|450px|thumb| Figure 10: Plot of Optimised Lattice Constant of MgO vs. Temperature using Quasi-Harmonic Approximation]]
[[File: QH_plot_of_prim_cell_vol_vs_temp_0_-_1000K_-_new.png|400px|thumb|right| Figure 11: Determination of Volumetric Expansion Coefficient using Quasi-Harmonic Approximation]]

Both these curves show a linear region at the first two data points at 0 K and 100 K. Both curves then trace a line of fairly steady gradient for the final data points between 200 K and 1000 K. This linear portion can be attributed to quantum mechanical effects inherent in the use of the quasi-harmonic approximation, which contribute considerably at lower temperatures<ref name="lattice dynamics 8" />.

The coefficient for thermal expansion was calculated after omitting the first two data points at 0 K and 100 K due to these quantum mechanical. The value obtained was 2.711 x 10<sup>-5</sup>.

{| class="wikitable"
|+ style="text-align: left;" | Table 5: Literature values for the volumetric expansion coefficient at different temperatures
|-
! Temperature (K)
! <math> \alpha </math> (K<sup>-1</sup> x 10 <sup>-5 </sup>)<ref name="lit values coeff" />
|-
| 100
| 0.63
|-
| 200
| 2.24
|-
| 300
| 3.11
|-
| 400
| 3.57
|-
| 500
| 3.84
|-
| 600
| 4.04
|-
| 700
| 4.17
|-
| 800
| 4.29
|-
| 900
| 4.41
|-
| 1000
| 4.50
|}

The value calculated in the lab was an average of the thermal expansion coefficient over a large temperature range (800 K).

The main approximations in these calculations are inherent in the difference between using the harmonic approximation versus the quasi-harmonic approximation. In the harmonic approximation, the bond equilibrium distance does not change, whereas in the quasiharmonic approximation, the equilibrium bond distance does change with temperature. This leads to a change in volume as the temperature is raised. It is this change in volume with temperature that allows calculation of the volumetric expansion coefficient.

The value obtained for the thermal expansion coefficient varied across the temperature range that was used in the calculations.

If the results from 200-600 K are used, then the value for alpha obtained is 2.334 x 10<sup>-5</sup> - corresponding literature value is 3.84 x 10<sup>-5</sup>
If the results from 600-1000 K are used, the value obtained is 3.007 x 10<sup>-5</sup>, corresponding literature value is 4.39 x 10<sup>-5</sup>.

The calculation consistently underestimates the value for the thermal expansion coefficient.

The physical origin of thermal expansion is that as a sample is heated, the molecules in the given sample gain more kinetic energy, and the molecules vibrate more, with each molecule taking up a great amount of space. This leads to the thermal expansion of the material.

At the melting point of MgO, the sample undergoes a phase transition from a solid to a liquid. Phonons do not represent well the motion of ions in a liquid, as in a liquid, the atoms are free-flowing. Phonons require a regular arrangement of atoms through which to propagate, this regular arrangement of atoms breaks down in a liquid.

In the harmonic approximation, the length of the bond fluctuates between greater extremes as the temperature increases, however the equilibrium bond distance remains unchanged. In the quasi-harmonic approximation, the equilibrium bond distance is allowed to change as the temperature increases - this is a far more realistic representation. In the harmonic approximation, the temperature can increase infinitely, and the bond between the atoms will not break, however in the quasi-harmonic approximation, the equilibrium bond distance increases with temperature.

== Molecular Dynamics ==

Fig 12. shows the fitted plot used to calculate the volumetric thermal expansion coefficient using a molecular dynamics simulation.

[[File: MD_plot_of_prim_cell_vol_vs_temp_100_-_1000K_-_new.png|400px|thumb|right| Figure 12: Determination of Volumetric Expansion Coefficient using Molecular Dynamics]]

2.994205 x 10<sup>-5</sup> K<sup>-1</sup> predicted by the molecular dynamics simulations over the values 100 - 1000 K is closer to the experimental values obtained by Suzuki et al.<ref name="lit values coeff" /> compared to the value of 2.711 x 10<sup>-5</sup> obtained using the quasi-harmonic approximation over the temperature range 200 - 1000 K.

The two simulations are underpinned by contrasting models of motion, the quasi-harmonic approximation being based in quantum mechanics, and the molecular dynamics relying on classical, Newtonian mechanics.

It has been noted in the literature that at high temperatures<ref name="quasi-harmonic error" />, the quasi-harmonic approximation overestimates the molar volume of perovskite crystals of MgSiO3, as the approximation does not take into account higher-order anharmonic effects.

At lower temperatures, the molecular dynamics calculation is subject to quantum effects that can significantly alter the value for the volumetric thermal expansion coefficient<ref name="quantum corrections" /> . In the literature, similar molecular dynamics calculations have required quantum corrections of 37 % of the classically derived value at temperatures of 300 K. In the calculation carried out in this laboratory, there was no amendment to the molecular dynamics simulation to atone for these quantum mechanical effects, such as the use of a Wigner-Kirkwood expansion<ref name="quantum corrections" />.

[[File: Both_MD_and_QH_plots.png|600px|thumb|right|Figure 13: Comparison of Curves obtained using the Quasi-Harmonic Approximation versus Molecular Dynamics]]

In theory, the larger the cell used, the more reliable the value obtained from the molecular dynamics calculation, although using a larger cell would be more computationally expensive. The molecular dynamics calculations gave a value for the volumetric thermal expansion coefficient of 2.994 x 10<sup>-5</sup> K<sup>-1</sup> over the temperature range 100 - 1000 K. Experimentally determined values are listed in Table X, and these values are of the same order of magnitude as that calculated by MD. The MD values are also closer to those calculated using the quasi-harmonic approximation.

At high temperature, the quasi-harmonic approximation tends to infinity, as the bonds between the atoms are broken. With the molecular dynamics.

Harmonic approximations cannot account for the thermal dependence of equilibrium properties such as the volumetric thermal expansion coefficient, or the occurrence of phase transitions<ref name="lattice dynamics 8" />. At higher temperatures, the quasi-harmonic approximation suffers from some of these higher order anharmonic effects.

= Conclusion =

A lattice dynamics calculation using the quasi-harmonic approximation and molecular dynamics calculation were used in the calculation of the volumetric expansion coefficient. Both had with them errors associated with their relative basis in quantum mechanics and classical Newtonian mechanics. Both methods yielded values for the volumetric expansion coefficient to the correct order of magnitude, however both underestimated the experimentally determined values for the coefficient, and so further optimisation of the calculations is required, such as potentially modifying the molecular dynamics calculation with a Wigner-Kirkwood expansion.

= References =

<references>
<ref name = "GULP"> GULP - a computer program for the symmetry adapted simulation of solids, J.D. Gale, JCS Faraday Trans., 1997, 93, 629 </ref>
<ref name = "DLV"> B.G. Searle, Computer Physics Communications, 2001, 137, p. 25 </ref>
<ref name = "Hoffmann"> R. Hoffmann, Angew. Chemie-International Ed. English, 1987, 26, 846–878 </ref>
<ref name = "lattice dynamics 2"> M. T. Dove, Introduction to Lattice Dynamics, CUP, 2016, ch. 2, pp. 18–35 </ref>
<ref name = "lattice dynamics 8 "> M. T. Dove, Introduction to Lattice Dynamics, CUP, 2016, ch. 8, pp. 101-131 </ref>
<ref name = "quasi-harmonic error"> G. D. Price and A. Patel, Geophysical Research Letters, 1994, 21, 1659–1662. </ref>
<ref name = "quantum corrections"> M. Matsui, J. Chem. Phys., 1989, 91, 489. </ref>
<ref name = "CaO lattice"> D.K. Smith, H.R. Leider, J. Appl. Cryst., 1998, 1, 246. </ref>
<ref name = "MgO"> R.W.G Wyckoff, AJS Online, 1925, 9, 54, 448-459 </ref>
<ref name= "lit values coeff"> Y. Sumino, O. L. Anderson and I. Suzuki, Phys. Chem. Miner., 1983, 9, 38–47</ref>
<ref name = "OSSB"> S.H. Simon, The Oxford Solid State Basics, OUP Oxford, 2013, https://ebookcentral.proquest.com/lib/imperial/detail.action?docID=1336493</ref>
<ref name = "faujasite"> H. D. Simpson, H. Steinfink, Jour. Am. Chem. Soc., 1969 91 (23), 6225-6229 </ref>
<ref name = "lithium"> M.R. Nadler and C.P. Kempfer, Anal. Chem., 1959, 31, 2109. </ref>
</references>

Rep:Yc9014-transition

2025-09-01T09:51:00Z

Move page script: Move page script moved page Yc9014-transition to Rep:Yc9014-transition: Move to report namespace

== Introduction ==

== Exercise 1:Reaction of Butadiene with Ethylene ==

<u>molecular orbital structure</u>

HOMO and LUMO of both reactants can be visualized by GaussiView and shown in '''table 1''' as following.

{| class="wikitable"
! colspan="3" |Table1. HOMO and LUMO of reagents butadiene and ethene
|-
!
!HOMO
!LUMO
|-
|''cis''-Butadiene
|[[File:Diene_HOMO_cyy.jpg|250px]]
|[[File:Diene_LUMO_cyy.jpg|250px]]
|-
|
|Anti-symmetric (AS)
|Symmetric (S)
|-
|Ethylene
|[[File:Ethene_HOMO_cyy.jpg|250px]]
|[[File:Ethene_LUMO_cyy.jpg|250px]]
|-
|
|Symmetric (S)
|Anti-symmetric (AS)
|}

After the transition state was optimized and its identity proved by IRC, graph of the energy levels proceed from HOMO and LUMO of the reactants was visualized and shown in '''table 2'''.

{| class="wikitable"
! colspan="4" |Table 2. energy levels for transition states of the reaction of butadiene and ethylene
|-
!HOMO-1
!HOMO
!LUMO
!LUMO+1
|-
|[[image:Level_16_as.jpg|344x344px]]
|[[image:Level_17_s.jpg|344x344px]]
|[[image:Level_18_s.jpg|344x344px]]
|[[image:Level_19_as.jpg|344x344px]]
|-
|Anti-symmetric (AS)
|Symmetric (S)
|Symmetric (S)
|Anti-symmetric (AS)
|}

From the graphs in '''table 2'''

HOMO-1 is a in-phase combination of butadiene HOMO(AS) and ethylene LUMO(AS),

HOMO is a in-phase combination of butadiene LUMO(S) and ethylene HOMO(S),

LUMO is a anti-phase combination of butadiene LUMO(S) and ethylene HOMOMO(S),

LUMO+1 is a anti-phase combination of butadiene HOMO(AS) and ethylene LUMO(AS)

<u>MO daigram</u>

With these combination relationship and relative energy levels above, a MO diagram can be drawn as '''graph 1'''.

[[image:MO-1.jpg|thumb|center|Graph 1. MO diagram of transition state for reaction of butadiene and ethylene.]]

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) 16:06, 5 April 2017 (BST) MO diagram is correct but very hard to read do to its scale. Additionally you didn't label the TS MOs.)

As indicated from '''graph 1''',the symmetry of two potential reacting orbitals must match with each other. ie. symmetric orbital interacts with symmetric orbitals, asymmetric orbital interacts with asymmetric orbitals.
The orbital overlap can only be none-zero when the two orbitals have the same symmetry.The relationship of symmetry interaction and orbital overlap shown in '''table 3'''. For symmetrically mismatched orbitals(symmetric with asymmetric), no overlap means no interaction, therefore, no reaction happen.

{| class="wikitable" border="1"
|+ table 3
! symmetry interaction !! Orbital overlap integral
|-
| AS-AS || None-zero
|-
| AS-S || zero
|-
| S-S || none-zero
|}

<u>bond length</u>

[[image:Internuclear_distance_new.png|thumb|left|graph 2. Inter-nuclear distances of butadiene react with ethylene.|683x683px]]
[[image:Bond_distances_indicator.jpg|thumb|Graph 3. Carbon positions.|590x590px|none]]

{| class="wikitable" border="1"
|+ table 4 typical bond length
! bond !! bond length(Å)
|-
| sp3-sp3 || 1.54
|-
| sp3–sp2 || 1.50
|-
| sp2–sp2 || 1.47
|-
| benzene || 1.40
|-
| alkene || 1.34
|}

As can see from '''graph 2''' , the bond length of the double bond in butadiene and ethylene decreases and the single bond in butadiene experiences a increase in bond length while two new bonds forms between two molecules.
The Van der Waals radius of the C atom is 1.70.
the partly form C-C bond has a bond length(2.14 Å) longer than normal sp3-sp3 single bond(1.54 Å).

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) 16:06, 5 April 2017 (BST) Even though you did compute the bond lenghts, you didn't report which length corresponds to which bond in your Figure 3. This led you to failing to appreciate that the bond lenght of the C2-C3 and C4-C5 bonds at the transition state is at an intermediate value between sp3 bonding length and 2 x the Van der Waals radius of Carbon atoms.)

<u>reaction path at the transition state</u>

<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script>frame 15; vibration 2;rotate x -20; </script>
<uploadedFileContents>TS_MP6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

Both reactants and the transition state are symmetric, two bonds are identical and form synchronously.

== Exercise 2:Reaction of Cyclohexadiene and 1,3-Dioxole ==

<u>Molecular orbitals</u>

Following molecules are optimised at PM6 level. Cailulation of transition state at B3LYP/6-31G(d) level was conducted but failed due to unsolved software reason. Transition state of endo producted was reached and proved by IRC .Transition state of exo product met a "Maximum number of corrector steps exceeded" issue. Although additional keyword 'IRC=MaxCycle=n' was used, the IRC calculation still failed. If more time was given, this problem can be potentially fixed by setting the 'correction steps' to 'never and optimising again. MO graph shown as follow. HOMO and LUMO for reactants '''table 5'''.HOMO and LUMO for endo transition state '''table 6''' and exo transition state '''table 7'''.

(What was the software issue? Was it raised with one of the demonstrators? [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 18:21, 3 April 2017 (BST))

{| class="wikitable"
! colspan="3" |Table5. HOMO and LUMO of reagents butadiene and ethene
|-
!
!HOMO
!LUMO
|-
|''cyclohexdiene
|[[File:Cyclohexdiene_HOMO_as.png|250px]]
|[[File:Cyclohexdiene_LUMO_s.png|250px]]
|-
|
|Anti-symmetric (AS)
|Symmetric (S)
|-
|dioxole
|[[File:Dioxle_HOMO_s.png|250px]]
|[[File:Dioxole_LUMO_as.png|250px]]
|-
|
|Symmetric (S)
|Anti-symmetric (AS)
|}

{| class="wikitable"
! colspan="4" |Table 6. energy levels for transition states of endo DA reaction of Cyclohexadiene and 1,3-Dioxole
|-
!HOMO-1
!HOMO
!LUMO
!LUMO+1
|-
|[[image:HOMO-1_as.jpg|344x344px]]
|[[image:HOMO_s.jpg|344x344px]]
|[[image:LUMO_s.jpg|344x344px]]
|[[image:LUMO+1_as.jpg|344x344px]]
|-
|Anti-symmetric (AS)
|Symmetric (S)
|Symmetric (S)
|Anti-symmetric (AS)
|}

{| class="wikitable"
! colspan="4" |Table 7. energy levels for transition states of the exo DA reaction of Cyclohexadiene and 1,3-Dioxole
|-
!HOMO-1
!HOMO
!LUMO
!LUMO+1
|-
|[[image:Level_29_as.jpg|344x344px]]
|[[image:Level_30_s.jpg|344x344px]]
|[[image:Level_31_as.jpg|344x344px]]
|[[image:Level_32_s.jpg|344x344px]]
|-
|Anti-symmetric (AS)
|Symmetric (S)
|Anti-ymmetric (AS)
|Symmetric (S)
|}

It can been seen from the '''graph 6''' that endo product has the same orbital symmetry order (AS/S/S/AS from LUMO-1 to HOMO +1) with the the cyclohexene formation in exercise one, so it has a similar MO diagram with as graph**. However, the exo transition state has a different orbital symmetry order(AS /S/AS/S from LUMO-1 to HOMO) '''table 7'''. So the MO diagram is adjusted as following graphs.

[[image:Exo_MO_diagram.jpg|thumb|center|Graph 4. MO diagram of transition stateog exo reaction.]]
[[image:Endo_MO_diagram.jpg|thumb|center|Graph 4. MO diagram of transition stateog endo reaction.]]

It is an inverse DA reactions. A normal DA reaction happen between a electron-poor dienophile and an electron rich diene. An inverse DA happen between an electron-rich dienophile and an electron-poor diene. In the case, the diene is not very electron poor nor electron rich, but dienophile 1,3-Dioxole is very electron rich due to direct attach to two electron donating oxygen atom. The orbital energy rises in dienophile and HOMO of dienolphile interact with LUMO of diene and form most energetically favored new orbital

<u>Energy barrier and reaction energy</u>

At room temperature,1 Hartree= 627.509 kcal mol-1

(Why are you using kcal/mol? Below you write kJ/mol when they are kcal/mol values [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 18:21, 3 April 2017 (BST))

energy for Cyclohexadiene,0.118067. energy for 1,3-Dioxole -0.052286. Energy for reatant=(0.118067-0.052286)<math>\times</math>627.509kcal mol-1=41.27 kJ mol-1

energy for endo transition state, 0.137943<math>\times</math>627.509kcal mol-1=86.56 kJ mol-1
energy for endo product,0.037803<math>\times</math>627.509kcal mol-1=23.72 kJ mol-1

energy for exo transition state, 0.138903<math>\times</math>627.509kcal mol-1=87.16 kJ mol-1
energy for exo product,0.037975<math>\times</math>627.509kcal mol-1=23.83 kJ mol-1

{|class="wikitable" border="1"
|+ table 8. Activation energy and reaction energy for each route
|-
| || activation energy || reaction energy
|-
|exo
|45.89
|<math>-17.44</math>
|-
|endo
|45.29
|<math>-17.55</math>
|}

[[image:Exercise_2_reaction_coordinate.jpg|thumb|center|Graph 5. reaction coordinate of endo and exo DA reaction.]]
The calculation shows that endo product are both kinetic and thermo product. Endo product is the kinetic product because of the secondary effect. as can be seen from the graph below, the two middle orbitals on diene LUMO has a favorable interaction with the orbital from oxygen, which lower the energy of the transition state and facilitates the endo reaction.
[[image:Secondary_effect.jpg|thumb|center|Graph 6. reaction coordinate of endo and exo DA reaction.]]

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 13:54, 16 April 2017 (BST) You have got the correct conculsion and you have explained the SOO. However you should have used b3LYP and told the demonstators that your Guassian was not working as this could have been fixed.

== Exercise 3:Diels-Alder vs Cheletropic ==

<u>reaction coordinate with IRC calculation</u>

{| class="wikitable"
! colspan="3" |Table 9. reaction coordinate for three routes
|-
!cheletropic product
!endo
!exo
|-
|[[File:5mr_IRC.png|450px]]
|[[File:Endo_IRC_cyy.png|300px]]
|[[File:Exo_IRC_cyy.png|350px]]
|}

<i>NOTE: the exo TS is optimized in a different route so its IRC are from the product to reactants.The reactants to product graph is its inverse version from y axis.</i>

{| class="wikitable"
! colspan="3" |Table 10. IRC path for three routes
|-
!cheletropic product
!endo
!exo
|-
|[[File:Exercise_3_cheletropic.gif|550px]]
|[[File:Exercise_3_endo.gif|550px]]
|[[File:Exercise_3_exo.gif|550px]]
|}

<u>Activation energy and reaction energy</u>

At room temperature

The energy measurement in GaussView is in Hartree,
1 Hartree= 627.509 kcal mol-1

energy for so2, -0.118614.energy for xylyene,0.178554. Energy of the reactants=(-0.118614+0.178554)<math>\times</math>627.509kcal mol-1=37.61 kJ mol-1

energy for exo 6-membered-ring TS, 0.092079<math>\times</math>627.509kcal mol-1=57.78 kJ mol-1

energy for exo 6-membered-ring product, 0.056109<math>\times</math>627.509kcal mol-1=35.21 kJ mol-1

energy for endo 6-membered-ring TS, 0.090559<math>\times</math>627.509kcal mol-1=56.83 kJ mol-1

energy for endo 6-memberd-ring product, 0.021700<math>\times</math>627.509kcal mol-1=13 kJ mol-1

energy for 5-memberd-ring TS, 0.099060<math>\times</math>627.509kcal mol-1=62.16 kJ mol-1

energy for 5-memberd-ring product, -0.000002<math>\times</math>627.509kcal mol-1=-0.0012 kJ mol-1

{|class="wikitable" border="1"
|+ Table 11. Activation energy and reaction energy for each route
|-
| || activation energy || reaction energy
|-
|exo 6-membered-ring
|20.17
|<math>-2.4</math>
|-
|endo 6-membered-ring
|19.22
|<math>-24.61</math>
|-
|5-memberd-ring
|24.55
|<math>-37.61</math>
|}

(exo reaction energy should be more exothermic [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 18:21, 3 April 2017 (BST))

The endo Diels-Alder product is kinetically preferred as it has lowest activation energy.
The cheletropic product is aerodynamically preferred as it has lowest reaction energy.

(Check your work for autocorrections and typos like this! [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 18:21, 3 April 2017 (BST))

Draw the reaction profile for all three routes in one graph ('''graph 7''').

[[image:New_MO_coordinate.jpg|thumb|center|Graph 7. Reaction coordinate of three product.|344x344px]]
As can be seen in the graph. cheletropic product has the lowest energy so it is thermodynamic product. Endo product is the kinetic product.

<u>bond length of the 6-membered ring in xylyene</u>

[[image:IRC_cheletropic_bond.png|530x530px]]
[[image:IRC_endo_bond.png|530x530px]]
[[image:IRC_exo_bond.png|530x530px]]

(This is a nice use of the Python code. You should probably show explicitly which bonds these are, but it's clear that there is some aromaticity being developed in this just after the TS [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 18:21, 3 April 2017 (BST))

As can be seen from the graph, all nbond lengths changed. Two double bond on the ring extends and sing bonds shortens and finally all of they reaches a similar distances as the electron density delocalise in the 6 membered ring. The graph of endo and exo product are similar as they share the same structure. Cheletropic product has one bond slightly long than other. This is because the bond is shared with the neighboring 5 membered ring and experience a additional ring strain.

== Conclusion ==

In this computational lab, transition states of three Diels-Alder reactions are constructed and analysed by GaussView. For a Diels-Alder reaction starts from a substituted dienophile, the endo route is preferred because of the additional secondary interaction which lowers the activation energy. Exercise 1 and Exercise 2 researches the molecular orbital interaction and formation of HOMO and LUMO for the products. Exercise 2 and Exercise 3 focus on the relative energy barrier and reaction energy for different reaction routes and determine the kinetic and thermo product with the aid of these energies.

GaussiView is used to optimise the molecules and visualise the molecular orbitals. In this experiment, PM6 is most commonly used to achieve a reasonable optimization for small molecule. B3LYP/6-31G(d) level calculation is more time consuming but will give a more accurate result.

Rep:Yc9014-liquid

2025-09-01T09:50:56Z

Move page script: Move page script moved page Yc9014-liquid to Rep:Yc9014-liquid: Move to report namespace

== part 2:Introduction to molecular dynamics simulation ==

'''TASK: For the default timestep value, 0.1, estimate the positions of the maxima in the ERROR column as a function of time. Make a plot showing these values as a function of time, and fit an appropriate function to the data.'''

[[image:Energy_error.png|left|800x800px]]
[[image:Maxima_in_error.png]]

'''TASK: Experiment with different values of the timestep. What sort of a timestep do you need to use to ensure that the total energy does not change by more than 1% over the course of your "simulation"? ''Why do you think it is important to monitor the total energy of a physical system when modelling its behaviour numerically''?'''

[[image:Energy_timestep0.0007s.png|center|frameless|831x831px]]

To ensure total energy change within 1%, we need at most 0.0007s as our time step.
We assume that the total energy is a constant for this physical model, however it is not steady in reality, so me need to monitor the total energy to make sure the difference in total energy is within a acceptable range.

'''TASK: Lennard-Jones interaction'''

'''Find the separation, and at which the potential energy is zero. What is the force at this separation?'''

For a single Lennard-Jones interaction,<math>\phi\left(r\right) = 4\epsilon \left( \frac{\sigma^{12}}{r^{12}} - \frac{\sigma^6}{r^6} \right)</math>, when potential energy is zero,

<math> 0= 4\epsilon \left( \frac{\sigma^{12}}{r^{12}} - \frac{\sigma^6}{r^6} \right)</math>

<math> \frac{\sigma^{12}}{r^{12}} = \frac{\sigma^6}{r^6}</math>

Distance r is a positive value , so r=σ,

Force at this separation, <math>F = - \frac{\mathrm{d}U\left(\mathbf{r}^N\right)}{\mathrm{d}\mathbf{r}_i}= -4\epsilon \left( \frac{-12\sigma^{12}}{r^{13}} + 6\frac{\sigma^6}{r^7} \right)=-4\epsilon (\frac{-12 }{\sigma}+ \frac{6 }{\sigma}) = \frac{-24 \epsilon}{\sigma}</math>

'''Find the equilibrium separation, and work out the well depth <math>\phi\left(r_{eq}\right)</math>'''
At equilibrium separation, F=0

<math>F = -4\epsilon \left( \frac{-12\sigma^{12}}{r^{13}} + 6\frac{\sigma^6}{r^7} \right)=0</math>

<math>\frac{2\sigma^{6}}{r^{6}}=1 </math>

<math>r= 2^{1/6} \sigma</math>,

<math>\phi\left(r_{eq}\right) = 4\epsilon \left( \frac{\sigma^{12}}{r^{12}} - \frac{\sigma^6}{r^6} \right)=4\epsilon \left( \frac{\sigma^{12}}{(2^{\frac{1}{6}}r)^{12}} - \frac{\sigma^6}{(2^{\frac{1}{6}}r)^6} \right)=4\epsilon \left( \frac{\sigma^{12}}{4r^{12}} - \frac{\sigma^6}{2r^6} \right)=4\epsilon(\frac{1}{4}-\frac{1}{2})=-\epsilon</math>

'''Evaluate the integrals <math>\int_{2\sigma}^\infty \phi\left(r\right)\mathrm{d}r</math>, <math>\int_{2.5\sigma}^\infty \phi\left(r\right)\mathrm{d}r</math>, and <math>\int_{3\sigma}^\infty \phi\left(r\right)\mathrm{d}r</math> when <math>\sigma = \epsilon = 1.0</math>.'''

<math> \int_{2\sigma}^\infty \phi\left(r\right)\mathrm{d}r = 4\epsilon \int_{2\sigma}^\infty \left( \frac{-\sigma^{12}}{11r^{11}} + \frac{\sigma^6}{5r^5} \right)=0-4\epsilon(\frac {-\sigma^{12}}{11\times 2.5^{11}\times \sigma^{11}} + \frac{\sigma^{6}}{5\times 2.5^{5}\times \sigma^{5}})=-2.48\times 10^{-2}</math>

<math> \int_{2.5\sigma}^\infty \phi\left(r\right)\mathrm{d}r = 4\epsilon \int_{2.5\sigma}^\infty \left( \frac{-\sigma^{12}}{11r^{11}} + \frac{\sigma^6}{5r^5} \right)=0-4\epsilon(\frac {-\sigma^{12}}{11\times 2^{11}\times \sigma^{11}} + \frac{\sigma^{6}}{5\times 2^{5}\times \sigma^{5}})=-8.18\times 10^{-3}</math>

<math> \int_{3\sigma}^\infty \phi\left(r\right)\mathrm{d}r = 4\epsilon \int_{3\sigma}^\infty \left( \frac{-\sigma^{12}}{11r^{11}} + \frac{\sigma^6}{5r^5} \right)=0-4\epsilon(\frac {-\sigma^{12}}{11\times 3^{11}\times \sigma^{11}} + \frac{\sigma^{6}}{5\times 3^{5}\times \sigma^{5}})=-3.29\times 10^{-3}</math>

'''TASK: Estimate the number of water molecules in 1ml of water under standard conditions. Estimate the volume of 10000 water molecules under standard conditions.
'''

'''Estimate the number of water molecules in 1ml of water under standard conditions.'''

<math>n=m/M=\frac{1g}{18g/mol}=\frac{1}{18} mol</math>

<math>N=A \times n=6.02*10^{23}/mol \times \frac{1}{18} mol=3.34 \times 10^{-19} </math>

'''Estimate the volume of 10000 water molecules under standard conditions'''

<math>\frac{18mL}{A}\times 10000=2.99\times 10^{-19} mL</math>

'''TASK: Consider an atom at position <math>\left(0.5, 0.5, 0.5\right)</math> in a cubic simulation box which runs from <math>\left(0, 0, 0\right)</math> to <math>\left(1, 1, 1\right)</math>. In a single timestep, it moves along the vector <math>\left(0.7, 0.6, 0.2\right)</math>. At what point does it end up, after the periodic boundary conditions have been applied?'''

(0.2,0.1,0.7)

'''TASK: The Lennard-Jones parameters for argon are <math>\sigma = 0.34\mathrm{nm}</math>, <math>\epsilon\ /\ k_B= 120 \mathrm{K}</math>. If the LJ cutoff is <math>r^* = 3.2</math>, what is it in real units? What is the well depth in <math>\mathrm{kJ\ mol}^{-1}</math>? What is the reduced temperature <math>T^* = 1.5 </math>''' '''in real units?'''

r=r*<math>\times</math>σ=3.2 <math>\times</math> 0.34nm=1.088 nm

ε=120k<math>\times</math> kb=<math>120K \times 1.38 \times 10^{-23} J/K</math>=<math>1.65\times 10^{-21} J=1.65\times 10^{-24} kJ \times 6.02 \times 10^{23} /mol =0.9974 kJ/mol</math>

T=T*<math>\frac{\epsilon}{kb}=1.5\times 120K=180K</math>

== part 3 : Equilibrium ==

'''TASK: Why do you think giving atoms random starting coordinates causes problems in simulations? Hint: what happens if two atoms happen to be generated close together?'''

They overlap in space, which cannot happen in real life.

'''TASK: Satisfy yourself that this lattice spacing corresponds to a number density of lattice points of 0.8. Consider instead a face-centred cubic lattice with a lattice point number density of 1.2. What is the side length of the cubic unit cell?'''

Side length=<math>(\frac {4}{1.2})^{\frac{1}{3}}=1.4938</math>

'''TASK: Consider again the face-centred cubic lattice from the previous task. How many atoms would be created by the create_atoms command if you had defined that lattice instead?'''

4000

Simple cubic lattice has 1 atom/lattice point per unit cell, face centered cubic lattice has 4 atoms/lattice points per unit cell, our box contains 10*10*10=1000 unit cells, therefore 4000 atoms are generated.

'''TASK: Using the LAMMPS manual, find the purpose of the following commands in the input script:

mass 1 1.0
pair_style lj/cut 3.0
pair_coeff * * 1.0 1.0'''

For type I atom, mass =1.0
.The lj/cut 3.0 compute the standard 12/6 Lennard-Jones potential
<nowiki> </nowiki>while r=3 is the cutoff. Pair_coeff defines the coefficient, in this case, both <math>\epsilon</math> and <math>\sigma</math> are 1.

'''TASK: Given that we are specifying <math>\mathbf{x}_i\left(0\right)</math> and <math>\mathbf{v}_i\left(0\right)</math>, which integration algorithm are we going to use?'''

Velocity Verlet algorithm. Classical verlet algorithm cannot make use of <math>\mathbf{v}_i\left(0\right)</math> and is less accurate.

'''what do you think the purpose of these lines is? Why not just write:

timestep 0.001
run 100000'''

‘timestep 0.001’ only defines a time step at beginning, we need timestep as a varible so that every timestep appears later can be substitute by 0.001.

'''TASK: make plots of the energy, temperature, and pressure, against time for the 0.001 timestep experiment (attach a picture to your report). Does the simulation reach equilibrium? How long does this take? When you have done this, make a single plot which shows the energy versus time for all of the timesteps (again, attach a picture to your report). Choosing a timestep is a balancing act: the shorter the timestep, the more accurately the results of your simulation will reflect the physical reality; short timesteps, however, mean that the same number of simulation steps cover a shorter amount of actual time, and this is very unhelpful if the process you want to study requires observation over a long time. Of the five timesteps that you used, which is the largest to give acceptable results? Which one of the five is a particularly bad choice? Why?'''

[[image:E-3.png|546x546px]][[image:T-3.png]][[image:P-3.png]]

stimulation reaches equilibrium, taking 0.33s.

[[image:Energy-3.png]]

T=0.01 is the largest acceptable time step, t=0.015 is a bad choice because it does not reach equilibrium.

== part 4:Running simulations under specific conditions ==

'''TASK: We need to choose <math>\gamma</math> so that the temperature is correct <math>T = \mathfrak{T}</math> if we multiply every velocity <math>\gamma </math>. Determine <math>\gamma </math>.'''

<math>\frac{1}{2}\sum_i m_i v_i^2 = \frac{3}{2} N k_B T </math> equation (1)

<math>\frac{1}{2}\sum_i m_i \left(\gamma v_i\right)^2 = \frac{3}{2} N k_B \mathfrak{T}</math> equation (2)

divide two equations (2)/(1)

<math> \gamma^2 = \frac{T}{ \mathfrak{T} }</math>

<math> \gamma = (\frac{T}{ \mathfrak{T} })^{\frac{1}{2}}</math>

'''TASK: Use the manual page to find out the importance of the three numbers 100 1000 100000. How often will values of the temperature, etc., be sampled for the average? How many measurements contribute to the average? Looking to the following line, how much time will you simulate?
'''

1 in every 100 time steps is sample for average. 1000 measurements contribute to average. Only simulate once.

'''TASK: When your simulations have finished, download the log files as before. At the end of the log file, LAMMPS will output the values and errors for the pressure, temperature, and density \left(\frac{N}{V}\right). Use software of your choice to plot the density as a function of temperature for both of the pressures that you simulated. Your graph(s) should include error bars in both the x and y directions. You should also include a line corresponding to the density predicted by the ideal gas law at that pressure. Is your simulated density lower or higher? Justify this. Does the discrepancy increase or decrease with pressure'''

[[File:25.png|frameless|600x600px]][[File:30.png|frameless|600x600px]]

Estimated densities are lower than theoretic densities.

The discrepancy increases with the pressure.

== paet 5:Calculating heat capacities using statistical physics ==

'''TASK: As in the last section, you need to run simulations at ten phase points. In this section, we will be in density-temperature <math>\left(\rho^*, T^*\right)</math> phase space, rather than pressure-temperature phase space. The two densities required at 0.2 and 0.8, and the temperature range is 2.0, 2.2, 2.4, 2.6, 2.8. Plot C_V/V as a function of temperature, where V is the volume of the simulation cell, for both of your densities (on the same graph). Is the trend the one you would expect? Attach an example of one of your input scripts to your report.'''

Script for density= 0.2, temperature=2.0 :[[file:P=0.2_t=2.txt]]

[[image:Part_5.png|centre|480x480px|Caption]]

Not expected trend.Heat capacity expected to increase with temperature .

== part 6:Structural properties and the radial distribution function ==

'''TASK: perform simulations of the Lennard-Jones system in the three phases. When each is complete, download the trajectory and calculate g(r) and <math>\int g(r)\mathrm{d}r</math>. Plot the RDFs for the three systems on the same axes, and attach a copy to your report. Discuss qualitatively the differences between the three RDFs, and what this tells you about the structure of the system in each phase. In the solid case, illustrate which lattice sites the first three peaks correspond to. What is the lattice spacing? What is the coordination number for each of the first three peaks?

[[File:Gr.png|543x543px]][[File:Int(gr).png|543x543px]]

In graph g(r), we see that gas phase has a very smooth line with only one peak, liquid phase three observable peaks, and solid phase has countless peaks. In lattice model, solid has large density (<math>N/V)</math>, so the possibility of finding a neighbor atom(Int(gr)) is larger than liquid and gas. On the contrary, the density of gas is far smaller than both liquid and solid so it has smallest Int(gr). In g(r) graph, solid has many peaks due to its high repeat (lattice) in space, liquid has approximately three smooth observable peaks due to the hydration shells around the particle, gas has only little repeat in long range.

For the nearest neighbors:position would be (0.5,0.5,0)(0,0.5,0.5)(0.5,0,0.5)...12 atoms in total

For the second nearest neighbors:position would be (1,0,0)(-1,0,)(0,1,0)...6 atoms in total

For the third nearest neighbors:position would be (1,1,0)(-1,-1,)(0,1,1)...24 atoms in total

<math>1.875\sigma=(0.5)^2r</math>

<math>r=2.65 \sigma</math>

== part 7:Dynamical properties and the diffusion coefficient ==

'''TASK: make a plot for each of your simulations (solid, liquid, and gas), showing the mean squared displacement (the "total" MSD) as a function of timestep. Are these as you would expect? Estimate D in each case. Be careful with the units! Repeat this procedure for the MSD data that you were given from the one million atom simulations.'''

[[image:S-msd-gas.png]]
[[image:S-msd-liquid.png]]
[[image:S-msd-solid.png|475x475px]]

<math>D=6 \times MSD</math>

When time step=5000

<math>D_(gas) =204 \times 6=1224</math>

<math>D_(liquid) =2.44 \times 6=14.64</math>

<math>D_(solid) =0.0196 \times 6=0.1176</math>

No unit in the case as all the parameter we use are in their reduced units, however in reality the unit of D would be <math>distance^2/time</math>, for example, <math>cm^2/s</math>

One million stimulation graphs

[[image:B-msd-gas.png]]
[[image:B-msd-liquid.png]]
[[image:B-msd-solid.png|487x487px]]

'''TASK: In the theoretical section at the beginning, the equation for the evolution of the position of a 1D harmonic oscillator as a function of time was given. Using this, evaluate the normalised velocity autocorrelation function for a 1D harmonic oscillator
'''

<math>C\left(\tau\right) = \frac{\int_{-\infty}^{\infty} v\left(t\right)v\left(t + \tau\right)\mathrm{d}t}{\int_{-\infty}^{\infty} v^2\left(t\right)\mathrm{d}t}</math>=
<math> = \frac{\int_{-\infty}^{\infty}\sin\left(\omega t + \phi\right) sin\left(\omega (t+\tau) + \phi\right)\mathrm{d}t}{\int_{-\infty}^{\infty} \sin^2\left(\omega t + \phi\right) \mathrm{d}t}</math>

For denominator:

Given that <math>\cos(2x)=1-2\sin^2(x)</math>

<math>\int_{-\infty}^{\infty} \sin^2\left(\omega t + \phi\right) \mathrm{d}t=\frac{1}{2}\int_{-\infty}^{\infty} 1-\cos\left(2\omega t + 2\phi\right) \mathrm{d}t=\frac{1}{2}[ 1-\frac{1}{2\omega}\sin\left(2\omega t + 2\phi\right)]_{-\infty}^{\infty}</math>

For numerator:

Given that <math>\sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)</math>

<math>\int_{-\infty}^{\infty}\sin\left(\omega t + \phi\right) sin\left(\omega (t+\tau) + \phi\right)\mathrm{d}t=\int_{-\infty}^{\infty}\sin\left(\omega t + \phi\right) sin\left(\omega t + \phi\right)\cos(\omega\tau)+
\sin\left(\omega t + \phi\right) cos\left(\omega t + \phi\right)\sin(\omega\tau) \mathrm{d}t=[\frac{\cos(\omega\tau)}{2}(t-\frac{1}{2\omega}\sin(2\omega t+2\phi)) +\frac{\sin(\omega\tau)}{2\omega}\sin^2(\omega t + \phi)]_{-\infty}^{\infty}</math>

We know that t is infinity, sin and cos items are negligible compared to t, so

<math>C\left(\tau\right) =[\frac{\frac{\cos(\omega\tau)}{2}t}{\frac{1}{2}t}]_{-\infty}^{\infty}=\cos(\omega\tau)</math>

'''Task: Be sure to show your working in your writeup. On the same graph, with x range 0 to 500, plot <math> C\left(\tau\right)</math> with <math> \omega = 1/2\pi </math>and the VACFs from your liquid and solid simulations. What do the minima in the VACFs for the liquid and solid system represent? Discuss the origin of the differences between the liquid and solid VACFs. The harmonic oscillator VACF is very different to the Lennard Jones solid and liquid. Why is this? Attach a copy of your plot to your writeup.
'''

[[image:VACF-1.png]]

Minima in VACF means the state when it has the maximal velocity in the opposite direction, which is when the particle passes the original position. Liquid and solid VACF are different due to different lattice model. Solid has tight lattic point and can only vibrate in a small space and liquid has more degree of free down.

At long range Lennard Jones VACF is approaching 0, but harmonic oscillator VACF still repeats the trigonometric pattern. At long range in Lennard Jones model, the particle does not feel any interaction and so no force apply to the particle therefor the acceleration is 0. However, in harmonic oscillator, the particle repeat simple harmonic oscillation within a certain range, and the VACF is periodic.

'''TASK: Use the trapezium rule to approximate the integral under the velocity autocorrelation function for the solid, liquid, and gas, and use these values to estimate D in each case. You should make a plot of the running integral in each case. Are they as you expect? Repeat this procedure for the VACF data that you were given from the one million atom simulations. What do you think is the largest source of error in your estimates of D from the VACF?'''

[[image:VACF-G.png]]
[[image:VACF-L.png|494x494px]]
[[image:VACF-S.png|530x530px]]

As expected, gas has the largest diffusion coefficient and the solid has smallest coefficient.
Largest source of error: we assume t is much larger than cos(t) and sin(t) and ignore the trigonometric part, however when t is small, the assumption is no longer doable. There might be a a larger error when t is small.

Rep:YZ20215TS

2025-09-01T09:50:52Z

Move page script: Move page script moved page YZ20215TS to Rep:YZ20215TS: Move to report namespace

== Introduction ==

===Transition State ===
To investigate a reaction, it is crucial to firstly locate the Transition State of the reaction. The transition state of a reaction, on a 2D reaction coordinate, will be the highest point of energy connecting reactants and the products. In addition, it should also have the characteristics as shown below:

<math>\frac{\partial V}{\partial q_i}</math>= 0

<math>\frac{\partial^2 V}{\partial q_i^2}</math> < 0

Where as being a stationary point, its first derivative is equal to 0, while its second derivative should be negative as being the highest point on the reaction profile.

On a 3D PES (Potential Energy Surface), it is more difficult to determine the transition state, as it will be one saddle point among many other existing saddle points, however, being the maximum point on the minimum energy path, its first derivative and second derivative will both be equal to zero.

<math>\frac{\partial V}{\partial q_i}</math>= 0

<math>\frac{\partial^2 V}{\partial q_i^2}</math> = 0

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 11:35, 7 March 2018 (UTC) Guassian works on a 3N-6 Potential energy surface. You get the force constants by diagonalising the Hessian matrix

=== Computational Methods ===

In this report, three different pericylic reactions were investigated using computational methods, including the PM6 and the B3LYP/6-31G(d) Methods.

PM6, with its full name being Parameterization Method 6, is a semi-empirical method. This method is based on the Hartree-Fock Model, and the model works by minimising the total molecular potential energy by varying the expansion coefficients, c<sub>μi</sub>, which is the coefficient in the equation of LCAO (Linear Combination of Atomic Orbitals).

This method is not perfect as it is based on the wrong assumption of accounting electrons as being largely independent of each other.<ref>Ot, W. J. (1990). Computational quantum chemistry. Journal of Molecular Structure: THEOCHEM (Vol. 207). </ref>

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 11:39, 7 March 2018 (UTC) What you mean is that PM6 ignores the 2 electron integrals

In reality, this is not true and the electrons will repulse each other due to their negative charge. Therefore, this method needs to be parameterised, which means the results fitted by a set of parameters, to product results that agree the most with experimental data.

While the other method, B3LYP, representing Becke, three-parameter, Lee-Yang-Parr, is based on Density Fucntional Theory (DFT), which is an incorporation of partly exact exchange from Hartree–Fock theory as well as exchange-correlation energy from other sources. It has an exchange-correlation functional as shown below:

<math>E_{\rm xc}^{\rm B3LYP} = E_{\rm x}^{\rm LDA} + a_0 (E_{\rm x}^{\rm HF} - E_{\rm x}^{\rm LDA}) + a_{\rm x} (E_{\rm x}^{\rm GGA} - E_{\rm x}^{\rm LDA}) + E_{\rm c}^{\rm LDA} + a_{\rm c} (E_{\rm c}^{\rm GGA} - E_{\rm c}^{\rm LDA}),</math>

where <math>a_0=0.20 \,\;</math>, <math>a_{\rm x}=0.72\,\;</math>, and <math>a_{\rm c}=0.81\,\;</math>. <ref>{{ cite journal |author1=K. Kim |author2=K. D. Jordan | title = Comparison of Density Functional and MP2 Calculations on the Water Monomer and Dimer | journal = J. Phys. Chem. | volume = 98 | issue = 40 | pages = 10089–10094 | year = 1994 | doi = 10.1021/j100091a024 }}</ref><ref>{{ cite journal |author1=P.J. Stephens |author2=F. J. Devlin |author3=C. F. Chabalowski |author4=M. J. Frisch | title = ''Ab Initio'' Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields | journal = J. Phys. Chem. | volume = 98 | pages = 11623–11627 | year = 1994 | doi = 10.1021/j100096a001 | issue = 45 }}</ref>

6-31G is a basis set of basis function among many others including 3-21G, etc.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 11:39, 7 March 2018 (UTC) Well done. You have clearly red beyonf the script here and included the equations. Good work.

=== Optimisation Methods ===

In this lab, three methods were used to optimise the TSs, with difficulty increasing from Method 1 to Method 3. Method 1 is the easiest and fastest one, but it is based on existing knowledge of the Transition State. Method 2, compared with Method 1, is more reliable as well as relatively fast, but it also has the limitation of requirement on knowledge of TS. Method 3 takes the most time to run, however, it does not have the limitation of the first two methods.

In this report, three pericylic reactions were investigated with all the Transition States being run with Method 3 and will be shown below.

== Excercise 1- Diels-Alder reaction of butadiene with ethylene ==

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) Good job on the whole exercise. Well done!)

[[File:Yz20215 E1 scheme.png|centre|thumb|Scheme 1 Diels-Alder reaction of butadiene and ethylene to form cyclohexene]]

This reaction is a classical [4+2] cycloaddition (Diels-Alder) reaction. In this reaction, cis-butadiene reacts with ethylene to form cyclohexene with complete regioselectivity because there are no substituents attached to the reactants.

In Excercise 1, this reaction was investigated and analysed by optimising the reactants, products, and the Transition State to a minimum using PM6 Method in GaussView 5.0.9 software. In addition, their MOs and vibration frequencies, as well as the IRC (Intrinsic Reaction Coordinate) were obtained and analysed.

=== Optimisation ===

==== Optimisation of reactants and products at PM6 level====

{| class="wikitable"
|+ Table 1. Optimisation of reactants and products
| colspan="3" style="text-align: center; background: #2d918b;" | '''Optimisation of Reactants and Product'''
|-
| style="text-align: center;" |Reactant: Butadiene
| style="text-align: center;" |Reactant: Ethlyene
| style="text-align: center;" |Product: Cyclohexene
|-
| [[File:Yz20215 E1 butadiene.PNG|x400px|400px|Position:centre]]
| [[File: Yz20215 E1 ethylene.PNG|x400px|400px|Position:centre]]
| [[File:Yz20215 E1 cyclohexene.PNG|x400px|400px|Position:centre]]
|}

==== Optimisation of transition state at PM6 level====

{| class="wikitable"
|+ Table 2. Optimisation of TS
| colspan="2" style="text-align: center; background: #2d918b;"| '''Optimisation of Transition State'''
|-
| style="text-align: center;" |Transition state
|-
| [[File:Yz20215 E1 transition state.PNG|x400px|400px|Position:centre]]
|}
=== Confirmation of correst TS using frequency calculation and IRC ===

{| class="wikitable"
|+ Table 3. IRC of the transition state
| colspan="2" style="text-align: center; background: #2d918b;" | '''Frequency calculations and IRC'''
|-
| style="text-align: center;" |Vibration frequencies of the TS
| style="text-align: center;" |IRCs
|-
| [[File:Yz20215 E1 TS vibs.PNG|x400px|400px|centre|thumb|Figure 1(a) Vibration frequencies of the TS]]
| [[File:TS IRC TOTAL E.png|x400px|400px|centre|thumb|Figure 1(b) Total energy along IRC]][[File:TS IRC RMS GRA.png|x400px|400px|centre|thumb|Figure 1(c) RMS gradient along IRC]]
|}

'''Vibration frequencies:'''

'''One imaginary frequency of -948.73 cm <sup>-1</sup>''' confirming the presence of the transition state (a saddle point- the maximum point on the minimum energy path on the PES)

'''IRC:'''

RMS gradient Norm of 0 at reactants, products, as well as the transition state. The middle point with 0 gradient corresponding to the maximum energy point on IRC Total Energy curve, indicating transition state.

=== MO analysis===

[[File:Yz20215 E1 TS Mo diagram.png|centre|thumb|Figure 2. MO of transition state of this reaction]]

{| class="wikitable"
|+ Table 4. MOs of reactants, TS and products
|-
| style="text-align: center;" |
| style="text-align: center;" |Butadiene
| style="text-align: center;" |Ethylene
| style="text-align: center;" |TS(LUMO and HOMO)
| style="text-align: center;" |TS(LUMO+1 and HOMO-1)
| style="text-align: center;" |Cyclohexene
|-
| style="text-align: center;" |LUMO
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|-
| style="text-align: center;" |HOMO
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==== Symmetries and MO interactions ====

In a reaction, only orbitals with the same symmetry are able to overlap and form new MOs.

The orbital symmetry will be determined by its structure and symmetry label:

for a certain orbital to be '''symmetric''', it will have a plane of symmetry ('''σ<sub>v</sub> plane''')

for a certain orbital to be '''asymmetric''', it will have a axis of symmetry ('''C2 axis''')

The orbital overlap integral will be either zero or non-zero with different interactions between symmetric and asymmetric orbitals, zero indicating no interaction between, while non-zero integral indicates existing interaction between two orbitals.The values of orbital overlap integrals are shown as below:

'''symmetric-antisymmetric interaction: zero'''

'''symmetric-symmetric interaction: non-zero'''

'''antisymmetric-antisymmetric interaction: non-zero'''

In this reaction, as we could see from above, the asymmetric orbitals '''(diene HOMO and dienophile LUMO''') will interact with each other to give '''asymmetric HOMO-1 and LUMO+1''' orbitals of the transition state; while the symmetric orbitals ('''diene LUMO and dienophile HOMO''') will interact to give '''symmetric HOMO and LUMO''' of the transition state. The HOMO of diene interacts with the LUMO of dienophile to give a better overlap due to a smaller energy gap between these two orbitals. In addition, the antibonding MOs will be stabilised more than that the bonding MOs are stablised.

Among these two interactions, four new MOs will be formed, indicated by the dotted energy levels. However, the true MOs of the TS, indicated by the solid levels, are higher(HOMO and HOMO-1) /lower (LUMO and LUMO+1) than that predicted. This was possibly due to MO mixing, also, because of this MO is of the transition state of the reaction, which is the maximum point on the minimum energy path, therefore, the energy of the MOs will be higher.

In this [4+2] cycloaddition, two new bonds are formed on the same face of the two set of orbitals, in other words, suprafacially. This is in accordance with the Woodward-Hoffmann Rules, where the reaction is only thermally allowed when an '''odd number''' is obtained from the equation below:

{| width=30%
|<pre>
(4q + 2)s+ (4r)a

=1 + 0

=1

</pre>

s is for suprafacial, a is for antarafacial, and q and r are two constants representing the number of each component. In this reaction, the diene has 4 suprafacial pi electrons, contributing 0 to the equation; while the dienophile has 2 pi suprafacial electrons, contributing 1 to the equation.

Therefore, the reaction has a sum of 1, indicating this reaction is thermally allowed by Woodward-Hoffmann Rule.

=== Bond lengths ===
{| class="wikitable"
|+ Table 5. Bond lengths analysis
| colspan="4" style="text-align: center; background: #2d918b;"| '''Bond length values of reactants, TS, and product'''
|-
| style="text-align: center;" |Butadiene
| style="text-align: center;" |Ethylene
| style="text-align: center;" |Transition state
| style="text-align: center;" |Cyclohexene
|-
| [[File:Yz20215 E1 butadiene bond len.PNG|x300px|300px|centre]]
| [[File:Yz20215 E1 ethylene bond len.PNG|x300px|300px|centre]]
| [[File:Yz20215 E1 ts bond len.PNG|x300px|300px|centre]]
| [[File:Yz20215 E1 product bond len.PNG|x300px|300px|centre]]
|-
| style="text-align: center;" |C1=C2 1.335 Å
C2-C3 1.468 Å

C3=C4 1.335 Å
| style="text-align: center;" |C1=C2 1.331 Å
| style="text-align: center;" |C1=C2 1.380 Å
C2=C3 1.411 Å

C3=C4 1.380 Å

C4=C5 2.115 Å

C5=C6 1.382 Å

C6=C1 2.114 Å
| style="text-align: center;" |C1-C2 1.500 Å

C2=C3 1.338 Å

C3-C4 1.500 Å

C4-C5 1.540 Å

C5-C6 1.541 Å

C6-C1 1.540 Å
|}

{| class="wikitable"
|+ Table 6. Standard values of C-C bonds and VdW radius
| colspan="2" style="text-align: center; background: #2d918b;" | '''Typical C-C bond lengths and Ver der Waals' radius'''
|-
| style="text-align: center;" |C-C bond lengths / Å
| style="text-align: center;" |Van der Waals' radius of carbon / Å
|-
| style="text-align: center;" |sp<sup>3</sup>-sp<sup>3</sup> C-C: 1.54

sp<sup>3</sup>-sp<sup>2</sup> C-C: 1.50

sp<sup>2</sup>-sp<sup>2</sup> C-C: 1.47

sp<sup>2</sup>-sp<sup>2</sup> C=C: 1.34
| style="text-align: center;" |1.7
|}

The bond lengths of the reactants, product, and transition state are shown above in Table 2, while the data of typical C-C bond lengths and the Van der Waal's radius of Carbon are shown in Table 3 above.

Comparing the typical values of the carbon-carbon bonds and the experimental results obtained using Gaussview, we could see that for all three molecules of reactants and products (butadiene, ethylene and cyclohexene) have C-C bond lengths same as or very close to that of the standard values.

'''Transition State:'''

Comparison with reactants:

'''Butadiene fragment:'''

1. Lengthening of terminal C=C (C1=C2 AND C3=C4)

2. Shortening of middle C-C (C2-C3)

The terminal C=C bonds of butadiene have longer lengths of around 1.380 Å compared that of the typical value of 1.340 Å of sp<sup>2</sup>-sp<sup>2</sup> C=C bonds, and the original sp<sup>2</sup>-sp<sup>2</sup> C-C bond has a smaller observed value of 1.411 Å (standard value of 1.47 Å).

'''Ethylene fragment:'''

1. Lengthening of C=C (C1=C2, shown as C5=C6 in TS)

'''Between two reactants:'''

The distances between terminal carbon atoms of butadiene and ethylene in the TS are both at around 2.115 Å, which is much smaller than sum of two carbon atoms' Van der Waals' radius of 3.4 Å. It is also larger than the typical value of sp<sup>3</sup>-sp<sup>3</sup> C-C bond, which shows that the bonds are only partially formed.

All of these bond lengths obtained from Gaussview show that the reaction is at its transition state with two '''new sp<sup>3</sup>-sp<sup>3</sup> C-C bonds being formed''' between the terminal carbon atoms, as well as the '''dissociation of two sp<sup>2</sup>-sp<sup>2</sup> C=C bonds into sp<sup>2</sup>-sp<sup>2</sup> C-C bonds'''.

=== Vibrations ===

[[File:Yz20215 E1 TS vib.gif|x300px|300px|centre|thumb|Figure 3. Vibration of the TS]]

The vibration frequency is -948.73 x<sup>-1</sup> , as shown as the first vibration in Figure 2(a).

The formation of the two C-C bonds are synchronous, as we could see in the gif in Figure above that the terminals carbon atoms vibrate towards each other to form the new bonds.

=== Files ===

Optimised LOG File for butadiene: [[File:Yz20215 E1 BUTANDIENE MO.LOG]]

Optimised LOG File for ethylene: [[File:YZ20215 E1 ETHYLENE MO.LOG]]

Optimised LOG File for Transition State: [[File:YZ20215 E1 TS PM6 MO.LOG]]

Optimised LOG File for cyclohexene: [[File:YZ20215 E1 CYCLOHEXENE MO.LOG]]

IRC of the Transition State: [[File:Yz20215 E1 TS PM6 IRC.LOG]]

== Excercise 2- Diels-Alder reaction of cyclohexadiene and 1,3-dioxole ==

[[File:Yz20215 E2 mechanism.png|centre|thumb|Scheme 2 Diels-Alder reaction of Cyclohexadiene and 1,3-dioxole]]

In Exercise 2, the Diels-Alder/[4+2] cycloaddition of cyclohexadiene and 1,3 dioxole was investigated.

Compared with the Diels-Alder reaction in Exercise 1, in E2, both reactants in E2 are consisted of ring structures, rendering them the ability to react both in endo and exo conformations to form two products.

In this excercise, the reactants, products, and Transition states (both endo and exo) were optimised with Method 3 in tutorial using B3LYP/6-31G(d) method in Gaussview software.

In addition, the vibration frequencies, energies and MOs of the molecules were obtained and analysed through optimising the structures.

=== Optimisation of reactants and products ===

{| class="wikitable"
|+ Table 7. Optimisation of reactants and products using B3LYP/6-31G(d)
| colspan="5" style="text-align: center; background: #2d918b;" | '''Optimisation of Reactants and Products'''
|-
|
| style="text-align: center;" |Reactant: Cyclohexadiene
| style="text-align: center;" |Reactant: 1,3-dioxole
| style="text-align: center;" |Product: Endo Product
| style="text-align: center;" |Product: Exo Product
|-
| style="text-align: center;" |Structures
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|-
| style="text-align: center;" |Vibration Frequencies
| [[File:Yz20215 E2 cyclohexadiene vib.PNG|x400px|400px|Position:centre]]
| [[File:Yz20215 E2 dioxole vib freq.PNG|x400px|400px|Position:centre]]
| [[File:Yz20215 E2 Product endo vib freq.PNG|x400px|400px|Position:centre]]
| [[File: Yz20215 E2 Product exo vib freq.PNG|x400px|400px|Position:centre]]
|}

There are no imaginary frequencies for all the reactants and products, as they are at the local minimum point of energy.

=== Optimisation of Exo and Endo Transition States using B3LYP method===

{| class="wikitable"
|+ Table 8. Optimisation of TS using B3LYP/6-31G(d)
| colspan="5" style="text-align: center; background: #2d918b;" | '''Optimisation of endo and exo TS'''
|-
|
| style="text-align: center;" |Endo TS
| style="text-align: center;" |Exo TS
|-
| style="text-align: center;" |Structures
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|-
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| [[File:Yz20215 E2 TS exo vib freq.PNG|x400px|400px|Position:centre]]
|}

In either list of vibration frequency, there is one imaginary frequency, representing a saddle point of Transition State (local maximum point on the minimum energy path on PES).

=== MO analysis ===

{| class="wikitable"
|+ Table 9. MOs of reactants, TS and products
|-
| style="text-align: center;" |
| style="text-align: center;" |Endo TS(LUMO and HOMO)
| style="text-align: center;" |Endo TS(LUMO+1 and HOMO-1)
| style="text-align: center;" |Exo TS(LUMO and HOMO)
| style="text-align: center;" |Exo TS(LUMO+1 and HOMO-1)
|-
| style="text-align: center;" |LUMO
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|-
| style="text-align: center;" |HOMO
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{| class="wikitable"
|+ Table 10. MO diagrams for both TS
|-
| style="text-align: center;" |Endo TS
| style="text-align: center;" |Exo TS
|-
| [[File:Yz20215 E2 MO Endo.png|x400px|400px|centre|thumb|Figure 4(a). MO diagram for Endo TS]]
| [[File:YZ20215 E2 MO EXO.png|x400px|400px|centre|thumb|Figure 4(b). MO diagram for Exo TS]]
|}

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) There is no point in drawing the same MO diagram twice, but changing the TS schemes. You should have mentioned, showed or discussed the differences between the endo and exo conformations in terms of relative MO energies.)

==== Normal VS Inverse electron demand ====

In [4+2] cycloadditions, there are two types of electron demand:

'''Normal''' electron demand: '''Electron-rich diene''' and '''Electron-poor dienophile'''

'''Inverse''' electron demand: '''Electron-poor diene''' and '''Electron-rich dienophile'''

Therefore, in normal electron demand DA reactions, the orbitals of the electron-rich diene will be higher, therefore, favourable interaction occurs between the HOMO of diene and the LUMO of dienophile; while for inverse electron demand reactions, the LUMO of diene will interact with the HOMO of the dienophile.

Generally, carbon-based rings formed in Diels-Alder reactions usually have normal electron demand, while heterocycles formed with DA reaction tends to have an inverse electron demand, as the presence of heteroatoms contributing and changing the energies of the orbtials, leading to different interactions between HOMOs and LUMOs.

In this reaction, due to the presence of a heterocyclic reactant, 1,3-dioxole, it is postulated that the reaction could possibly have an inverse electron demand, therefore, energy calculations were done and their single point energies determined in the following section to confirm the postulation.

==== Single point energies ====

{| class="wikitable"
|+ Table 11. Single point energies of reactants
|-
| style="text-align: center; background: #2d918b;"|'''Identity'''
| style="text-align: center; background: #2d918b;"|'''Energy of HOMO/ a.u.'''
| style="text-align: center; background: #2d918b;"|'''Energy of LUMO/ a.u.'''
| style="text-align: center; background: #2d918b;"|'''Energy difference between this HOMO and the LUMO of the other reactant/ a.u.'''
|-
| style="text-align: center;" |Cyclohexadiene
| style="text-align: center;" |-0.20554
| style="text-align: center;" |-0.01711
| style="text-align: center;" |0.24349
|-
| style="text-align: center;" |Reactant: 1,3-dioxole
| style="text-align: center;" |-0.19594
| style="text-align: center;" |0.03795
| style="text-align: center;" |0.17883
|}

In this reaction, the energy difference between the HOMO of the dienophile with the LUMO of diene is smaller than that of the other pair, therefore, they will give to a larger overlap of orbitals and more favourable interaciton.

Therefore, we can conclude from the data above that the reaction has an inverse electron demand. This is due to the presence of two oxygen atoms donating their lone pairs, making the dienophile electron rich.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 11:41, 7 March 2018 (UTC) Well done for doing this. You have showed it very clearly.

=== Energy analysis ===

{| class="wikitable"
|+ Table 12. Energies of reactants, TS, and products
|-
| style="text-align: center; background: #2d918b;"|'''Identity'''
| style="text-align: center; background: #2d918b;"|'''Energy obtained using B3LYP/6-31G(d) Method/ Hartree/Particle'''
| style="text-align: center; background: #2d918b;"|'''Energy obtained using B3LYP/6-31G(d) Method/ kJ mol<sup>-1<sup>'''
|-
| style="text-align: center;" |Reactant: Cyclohexadiene
| style="text-align: center;" |-233.324375
| style="text-align: center;" |-612593.193227
|-
| style="text-align: center;" |Reactant: 1,3-dioxole
| style="text-align: center;" |-267.068642
| style="text-align: center;" |-701188.772985
|-
| style="text-align: center;" |Endo TS
| style="text-align: center;" |-500.332151
| style="text-align: center;" |-1313622.16252
|-
| style="text-align: center;" |Exo TS
| style="text-align: center;" |-500.329165
| style="text-align: center;" |-1313614.32277
|-
| style="text-align: center;" |Endo Product
| style="text-align: center;" |-500.418702
| style="text-align: center;" |-1313849.40218
|-
| style="text-align: center;" |Exo Product
| style="text-align: center;" |-500.417322
| style="text-align: center;" |-1313845.77899
|}

{| class="wikitable"
|+ Table 13. Activation energies and reaction energies
|-
| style="text-align: center; background: #2d918b;"|'''Product'''
| style="text-align: center; background: #2d918b;"|'''Activation Energy/ kJ mol<sup>-1<sup>'''
| style="text-align: center; background: #2d918b;"|'''ΔG of reaction/ kJ mol<sup>-1<sup>'''
|-
| style="text-align: center;" |Endo
| style="text-align: center;" |159.803692
| style="text-align: center;" |-67.435968
|-
| style="text-align: center;" |Exo
| style="text-align: center;" |167.643442
| style="text-align: center;" |-63.812778
|}

In a certain reaction, the kinetic product has a lower activation energy therefore will be formed faster, while the thermodynamic product is itself lower in energy, therefore, will be the major product if enough energy is provided to overcome the higher activation barrier.

In this reaction, the activation barrier is lower for the Endo product, hence it is the kinetically favoured product. Furthermore, the ΔG of reaction (which is the energy difference between reactants and the product) is also more negative, indicating a more stable product. Therefore, the Endo product is also the thermodynamically favoured product.

Therefore, the '''Endo product is both kinetically and thermodynamically favoured product.'''

==== Secondary Orbital Interaction ====

The Endo product being both the kinetic and thermodynamic product is possibly due to the stablisation from secondary orbital interactions between the p orbitals on the oxygen atoms and the π orbitals of the diene, which only takes place when the TS is in Endo conformation.

For Exo conformation, the 1,3-dioxole molecule points outwards and is unavailable to interact with the cyclohexadiene molecule.

The secondary orbital interactions of both the Endo and Exo TS are shown below in Table 9, as well as graph illustration.

{| class="wikitable"
|+ Table 14. Secondary orbital interactions
| colspan="3" style="text-align: center; background: #2d918b;" | '''MOs and graph of secondary interaction'''
|-
| style="text-align: center;" |HOMO for Endo TS
| style="text-align: center;" |HOMO for Exo TS
| style="text-align: center;" |Graph illustrating secondary orbital interactions
|-
!<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 2; mo 41; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>yz20215 E2 TS exo TS MO.log</uploadedFileContents>
</jmolApplet>
</jmol>
!<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 2; mo 41;mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>Yz20215 E2 TS endo TS MO 1.log</uploadedFileContents>
</jmolApplet>
</jmol>
| [[File:E2 Secondary orbital interaction.png|x400px|400px|centre|thumb|Figure 5. Secondary orbital interactions in both Endo and Exo TS]]
|}

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 11:44, 7 March 2018 (UTC) Well done for doing this, it was a good section. Everything was consider and nicely tabulated. Your energies are correct and you have come to the correct conclusions. You could have gone into abit more detail in some places such as for the kenetics.

=== Files ===

Optimised LOG File for cyclohexadiene: [[File:Yz20215 E2 cyclohexadiene min.LOG]]

Optimised LOG File for 1,3-dioxole: [[File:Yz20215 E2 dioxole min B3LYP.log]]

Optimised LOG File for Endo Transition State: [[File:Yz20215 E2 TS endo TS MO 1.log]]

Optimised LOG File for Exo Transition State: [[File:Yz20215 E2 TS exo TS.log]]

Optimised LOG File for Endo product: [[File:Yz20215 E2 product ENDO B3LYP MIN.LOG]]

Optimised LOG File for Exo product: [[File:Yz20215 E2 product EXO B3LYP MIN.LOG]]

== Excercise 3- Diels-Alder vs Cheletropic ==

[[File:Yz20215 E3 scheme.png|x400px|400px|thumb|centre|Scheme 3 Diels-Alder and Cheletropic reaction between Xylylene and Sulfur dioxide]]

The reaction between xylylene and sulfur dioxide was investigated.

The reaction between the reactants could occur either via a Diels-Alder reaction or cheletropic reaction as shown above in Scheme 3. For the Diels-Alder reaction between two, the product could be formed in an Endo or an Exo conformation.

In this exercise, the reactants, Transition States and products were optimised using PM6 Method using GaussView software. Also, the energies of different species were obtained to determine the activation energy and the energy change of reaction with reaction energy profile plotted. Therefore, the most thermodynamically and the most kinetically favoured product was determined.

=== Optimisation of Transitions States using PM6 Method ===

{| class="wikitable"
|+ Table 15. Optimisation of TSs using PM6
| colspan="3" style="text-align: center; background: #2d918b;" | '''Optimisation of Transition states'''
|-
| style="text-align: center;" |Diels-Alder: Endo TS
| style="text-align: center;" |Diels-Alder: Endo TS
| style="text-align: center;" |Cheletropic
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>350</size>
<script>frame 16; </script>
<uploadedFileContents>Yz20215 E3 DA endo TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>350</size>
<script>frame 16; </script>
<uploadedFileContents>Yz20215 E3 DA TS exo TS PM6.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>350</size>
<script>frame 16; </script>
<uploadedFileContents>YZ20215 E3 CHE TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

(You need to select the correct frame for JMols. These are just the 15th step of the optimisation [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:49, 6 March 2018 (UTC))

=== IRC calculations ===

{| class="wikitable"
|+ Table 16. Animations and IRC for TSs
| colspan="3" style="text-align: center; background: #2d918b;" | '''Gif animations and IRC diagrams for Transition States'''
|-
| style="text-align: center;" |Diels-Alder: Endo TS
| style="text-align: center;" |Diels-Alder: Exo TS
| style="text-align: center;" |Cheletropic TS
|-
| [[File:Yz20215 E3 endo movie.gif|x450px|450px|centre]]
| [[File:Yz20215 E3 exo movie.gif|x450px|450px|centre]]
| [[File:Yz20215 E3 Che movie.gif|x450px|450px|centre]]
|-
| [[File:Yz20215 E3 endo IRC 1.PNG|x600px|600px|centre]]
| [[File:YZ20215 E3 exo IRC.PNG|x600px|600px|centre]]
| [[File: YZ20215 E3 che IRC.PNG|x600px|600px|centre]]
|}

Looking at IRC of these three reactions, we could see that the activation barrier of the reaction is quite small, this is due to one of the reactants, xylylene, being very unstable and high in energy.

This is because, according to Huckle's Rule, it has only 8 π electrons, which is (4n) instead of (4n+2), therefore, it is antiaromatic. However, due to structure constraint, both of the dienes are cis in xylylene, which is favourable as no energy expense on converting into trans conformation.

During the reaction, the xylyene part will react with the sulfur dioxide molecule to form a bicyclic ring, containing a benzene ring, which is aromatic and stable. Therefore, making it favourable to form the products.

=== Activation and Reaction Energy calculations ===

{| class="wikitable"
|+ Table 17. Energies of reactants, TS, and products
|-
| style="text-align: center; background: #2d918b;"|'''Identity'''
| style="text-align: center; background: #2d918b;"|'''Energy obtained using PM6 Method/ Hartree/Particle'''
| style="text-align: center; background: #2d918b;"|'''Energy obtained using PM6 Method/ kJ mol<sup>-1<sup>'''
|-
| style="text-align: center;" |Reactant: Xylylene
| style="text-align: center;" |0.178047
| style="text-align: center;" |467.462434
|-
| style="text-align: center;" |Reactant: Sulfur dioxide
| style="text-align: center;" |-0.119269
| style="text-align: center;" |-313.1407834
|-
| style="text-align: center;" |DA: Endo TS
| style="text-align: center;" |0.090559
| style="text-align: center;" |237.762673
|-
| style="text-align: center;" |DA: Exo TS
| style="text-align: center;" |0.092076
| style="text-align: center;" |241.745556
|-
| style="text-align: center;" |Cheletropic: TS
| style="text-align: center;" |0.099062
| style="text-align: center;" |260.087301
|-
| style="text-align: center;" |DA: Endo Product
| style="text-align: center;" |0.021698
| style="text-align: center;" |56.9681033
|-
| style="text-align: center;" |DA: Exo Product
| style="text-align: center;" |0.021454
| style="text-align: center;" |56.3274813
|-
| style="text-align: center;" |Cheletropic Product
| style="text-align: center;" |-0.000002
| style="text-align: center;" |-0.0052510004
|}

{| class="wikitable"
|+ Table 18. Activation energies and reaction energies
|-
| style="text-align: center; background: #2d918b;"|'''Product'''
| style="text-align: center; background: #2d918b;"|'''Activation Energy/ kJ mol<sup>-1<sup>'''
| style="text-align: center; background: #2d918b;"|'''ΔG of reaction/ kJ mol<sup>-1<sup>'''
|-
| style="text-align: center;" |Endo
| style="text-align: center;" |83.4410224
| style="text-align: center;" |-97.3535473
|-
| style="text-align: center;" |Exo
| style="text-align: center;" |87.4239054
| style="text-align: center;" |-97.9941693
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |105.7656504
| style="text-align: center;" |-154.3269016
|}

(These levels of precision are far too high [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:49, 6 March 2018 (UTC))

The energies of the reactants, Transition States, and different products were obtained from the optimised LOG. files in GaussView, and the reaction profile is plotted showing their relative energies with the proceeding of the three reactions.

All TS and product energies are normalised with respect to reactant energy (reactant energy=0). The reaction profile is shown below.

=== Reaction Profiles ===

[[File:Yz20215 E3 reaction profile.png|x700px|700px|thumb|centre|Figure 6. Reaction profile of three different reaction paths]]

In this profile, we could see that the '''endo TS''' is the lowest in energy, therefore will be formed faster compared with other two, and will be the '''kinetic product''' of the reaction. The Exo TS has slightly higher but close energy, while the cheletropic TS has the largest activation barrier.

However, if we compare the energies of the products, we could see that the Endo and the Exo products are very close in energy at approximately -98 kJ mol <sup>-1</sup> , while the cheletropic product is the most stable product with energy of around -154 kJ mol <sup>-1</sup> . Therefore, if we provide enough energy for the reaction to overcome its activation barrier, the '''cheletropic product''' will be the major product as it is the '''thermodynamic product''' of the reaction.

=== Extension ===

[[File:YZ20215 E3 extension.png|x400px|400px|thumb|centre|Scheme 4 The reaction of sulfur dioxide with another diene in xylylene]]

In this reaction, the other diene of xylylene, also in cis conformation, could also reaction with sulfur dioxide to form both the endo and the exo products, and the activation energies and reaction energies of these two reactions are investigated as shown below.

{| class="wikitable"
|+ Table 19. Energies of reactants, TS, and products
|-
| style="text-align: center; background: #2d918b;"|'''Identity'''
| style="text-align: center; background: #2d918b;"|'''Energy obtained using PM6 Method/ Hartree/Particle'''
| style="text-align: center; background: #2d918b;"|'''Energy obtained using PM6 Method/ kJ mol<sup>-1<sup>'''
|-
| style="text-align: center;" |Reactant: Xylylene
| style="text-align: center;" |0.178047
| style="text-align: center;" |467.462434
|-
| style="text-align: center;" |Reactant: Sulfur dioxide
| style="text-align: center;" |-0.119269
| style="text-align: center;" |-313.1407834
|-
| style="text-align: center;" |DA: Endo TS
| style="text-align: center;" |0.102071
| style="text-align: center;" |267.987431
|-
| style="text-align: center;" |DA: Exo TS
| style="text-align: center;" |0.105054
| style="text-align: center;" |275.819298
|-
| style="text-align: center;" |DA: Endo Product
| style="text-align: center;" |0.065615
| style="text-align: center;" |172.272196
|-
| style="text-align: center;" |DA: Exo Product
| style="text-align: center;" |0.067307
| style="text-align: center;" |176.714542
|}

{| class="wikitable"
|+ Table 20. Activation energies and reaction energies
|-
| style="text-align: center; background: #2d918b;"|'''Product'''
| style="text-align: center; background: #2d918b;"|'''Activation Energy/ kJ mol<sup>-1<sup>'''
| style="text-align: center; background: #2d918b;"|'''ΔG of reaction/ kJ mol<sup>-1<sup>'''
|-
| style="text-align: center;" |Endo
| style="text-align: center;" |113.6657804
| style="text-align: center;" |17.9505454
|-
| style="text-align: center;" |Exo
| style="text-align: center;" |121.4976474
| style="text-align: center;" |22.3928914
|}

These two reaction, are endothermic reactions, with product being higher in energy than that of the reactants. The activation barrier is also around 30 kJ mol <sup>-1</sup> higher than that of the DA reaction occurring at the other diene, therefore, these two reactions are both '''thermodynamically and kinetically unfavourable'''.

=== Files ===

Optimised LOG File for Endo TS: [[File:Yz20215 E3 DA endo TS.LOG ]]

Optimised LOG File for Exo TS: [[File:Yz20215 E3 DA TS exo TS PM6.log]]

Optimised LOG File for Cheletropic TS: [[File:YZ20215 E3 CHE TS.LOG]]

'''Extension'''

Optimised LOG File for Endo TS: [[File:Yz20215 ext ENDO TS.LOG ]]

Optimised LOG File for Exo TS: [[File:Yz20215 ext exo TS.log]]

= Conclusion =

During the time span of this Transition State Computational Lab, three different pericyclic reactions were investigated.

The reactions were investigated by using PM6 and B3LYP/6-31G(d) methods in GaussView to optimise the reactants, transition states, and the products.

Other information were also extracted from the optimised molecule LOG. files: including vibration frequency calculations and IRC (Intrinsic Reaction Coordinate); bond lengths of the TS, reactants and products; the energies of different species; and the molecular orbitals.

These data were used to analyse the Transition states and the reactions, including confirmation of the TS by presence of one imaginary frequency in vibration frequencies, the activation barrier and energy change of the reaction from the energies of different species, and the MOs to determine the electron demand of a certain Diels-Alder reaction (inverse or normal electron demand).

In addition, the data confirmed that the reactions followed several existing rules and theories: Woodward-Hoffmann rules in Exercise 1, Frontier Molecular Orbital Theory, etc..

These computational methods could be applied to investigate many other pericyclic reactions, where various aspects of data could be obtained and analysed as like shown in this lab.

= References =

<references />

Rep:YR316DYN

2025-09-01T09:50:52Z

Move page script: Move page script moved page YR316DYN to Rep:YR316DYN: Move to report namespace

=Molecular Reaction Dynamics=

== H + H<sub>2</sub> system==

===Question 1: minimum and saddle===

''What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.''

In this triatomic system where the H<sub>A</sub> is approaching H<sub>B</sub>H<sub>C</sub> with an initial momentum. r<sub>1</sub> is the distance between H<sub>A</sub> and H<sub>B</sub> and r<sub>2</sub> is the distance between H<sub>B</sub> and H<sub>C</sub>. The total potential energy of the system (<math>V(r_1,r_2)</math>) is changing with respect to both r<sub>1</sub> and r<sub>2</sub>. Hence the gradient of potential energy consists of two components:<math> { \partial V(r_1,r_2)\over \partial r_1}</math> and <math> { \partial V(r_1,r_2)\over \partial r_1}</math>. The minima correspond to the minimum potential energy of the oscillating system H<sub>B</sub>H<sub>C</sub> (before collision)or H<sub>A</sub>H<sub>B</sub> (after collision). Mathematically, the minimum can be expressed as <math> { \partial V(r_1,r_2)\over \partial r_2}</math>=0 and <math> { \partial V(r_1,r_2)\over \partial r_1}</math>=0 respectively. <math> { \partial V(r_1,r_2)\over \partial r_2}</math> is partial differentiation, it differentiates V with respect to r<sub>2</sub> while treating r<sub>1</sub> as constant. The curvatures of the minima are described by the second partial derivative: V<sub>r1r1</sub> and V<sub>r1r1</sub> and smaller 0. They are the minimum point viewing from cross sections along r1 and r2 respectively. It is the same when H<sub>c</sub> is leaving after the collision.

However, as said at the beginning while H<sub>B</sub>H<sub>C</sub> are oscillating, H<sub>A</sub> is approaching and this increases the potential energy of the system. The different minima at different r<sub>1</sub> values increases to a saddle point. This is the maximum point in the trough created by joining the minima when viewing from potential surface. At this point both <math> { \partial V(r_1,r_2)\over \partial r_2}</math> and <math> { \partial V(r_1,r_2)\over \partial r_1}</math> are 0. The curvature at this point can be described by the determinant: V<sub>r1r1</sub>•V<sub>r1r1</sub>-V<sup>2</sup><sub>r1r2</sub>. This point corresponds to the transition state of the system.

[[User:Ng611|Ng611]] ([[User talk:Ng611|talk]]) 13:38, 8 June 2018 (BST) You've got the right idea for the TS, although your explanation is somewhat confusing to read. A diagram would help you significantly here.

===Question 2: estimation of transition state===

''Report your best estimate of the transition state position (rts) and explain your reasoning illustrating it with a “Inter-nuclear Distances vs Time” plot for a relevant trajectory.''

The transition state is estimated to be at r<sub>1</sub>=r<sub>2</sub>=0.90773 as shown in figure 1.

[[File:Yu_stationary_find.PNG|400px]]

figure 1. “Inter-nuclear Distances vs Time” plot at stationary point

[[File:Yu_stationary_potential.PNG|400px]]

figure2. “Inter-nuclear Distances vs Time” plot under one initial condition

Figure 1 shows under one initial condition when H<sub>A</sub> collides with H<sub>B</sub> H<sub>C</sub> the intercept of the A-B and B-C line gives a rough estimation of where the transition would be. A more accurate estimation can be done by setting the momentum to zero while changing values for distances (keep r<sub>1</sub>=r<sub>2</sub> as the potential surface of H + H<sub>2</sub> is symmetric. The trajectory would be stationary i.e. distance does not vary with time for a transition state as suggested by the definition of a stationary point(saddle as an example). while at other point it would produce a oscillation pattern over time as shown in figure 3.

[[File:Yu_stationary_try.PNG|400px]]

figure 3. “Inter-nuclear Distances vs Time” plot at stationary point at other r<sub>1</sub>=r<sub>2</sub> points other then the stationary points

===Question 3: comparing dynamics and MEP===

''Comment on how the MEP and the trajectory you just calculated differ.''

Figure 4 and 5 shows the reaction path calculation under dynamic method and MEP method respectively. Both of them have initial condition r<sub>1</sub>=0.91773, r<sub>2</sub>=0.90773, p<sub>1</sub>=p<sub>2</sub>=0. The pathway calculated with dynamic method is wavy, while the pathway calculated with MEP method is a smooth curve. MEP stands for minimum energy pathway, in each step of MEP the momentum was reset to zero, hence the inter-conversion between kinetic and the oscillation are ignored. H<sub>B</sub>H<sub>1</sub> of the molecule was ignored and was set at minimum potential energy along the pathway. The step number chosen for dynamics was 150 steps while that of the MEP calculation was chosen to be 10000. The reason for MEP has a much larger step number is also because that in each step of MEP the momentum was reset to zero and the inertia of the particles are ignored, while the momentum in the dynamic method was cumulative. Hence, if the same step number was chosen for both method, MEP result would give a much shorter of path length due to this reset of momentum.

[[User:Ng611|Ng611]] ([[User talk:Ng611|talk]]) 13:39, 8 June 2018 (BST) Good explanation!

[[File:Yu_dynamic_150steps.PNG|400px]]

Figure 4. Pathway calculation with dynamic method

[[File:Yu_MEP_10000_steps.PNG|400px]]

Figure 5. Pathway calculation with MEP method

If the initial conditions was set to r1 = rts and r2 = rts+0.01 instead, the reaction pathways would be a mirror image of r1=rts+0.01 and r2=rts condition with orthogonal plane r1=r2 as the mirror plane as shown in figure 6. This is due to the fact that the initial position of the trajectory was reversed.
If the initial positions was set to be the same as the final positions of the previous trajectory, the new resultant trajectory would represent the reverse reaction of the previous trajectory.

[[File:yu_mirror1.PNG|400px]]
[[File:yu_mirror2.PNG|400px]]

Figure 6. mirror image to each other: r1=rts+0.01 and r2=rts(left) r1 = rts and r2 = rts+0.01 (right) with steps extended to 500

=== Question 4: reactive and unreactive trajectory ===

''Complete the table by adding a column with the total energy, and another column reporting if the trajectory is reactive or unreactive. For each set of initial conditions, provide a plot of the trajectory and a small description for what happens along the trajectory.''

when setting: r<sub>1</sub>=0.74 and r<sub>2</sub>= 2.0

{| class="wikitable" border=1
! p<sub>1</sub> !! p<sub>2</sub>!!Energy!!Reactive?!!Reaction trajectory!!Comments
|-
| -1.25 || -2.5 || -99.018||reactive||[[File:yu_q3_1.PNG|400px]]||The trajectory starts at the reactant crosses the transition point and reached the product. Therefore it is reactive. More vibration observed in reactant.
|-
| -1.5 || -2.0 || -100.456||nonreactive||[[File:yu_q3_2.PNG|400px]]||The trajectory starts at the reactant, before reaching the transition point it reflects back. Therefore it is nonreactive. Vibration observed in reactant and product.
|-
| -1.5 || -2.5 || -98.956||reactive||[[File:yu_q3_3.PNG|400px]]||The trajectory starts at the reactant crosses the transition point and reached the product. Therefore it is reactive.
|-
| -2.5 || -5.0 || -84.956||nonreactive||[[File:yu_q3_4.PNG|400px]]||The trajectory starts at the reactant passes the transition point twice returned to the reactant. Therefore it is nonreactive.
|-
| -2.5 || -5.2 || -83.416||reactive||[[File:yu_q3_5.PNG|400px]]||The trajectory starts at the reactant passes the transition point three times and finally reaches the product. Therefore it is nonreactive.
|}

table 1. Energy tabel and trajectories with various initial monenta

=== Question 5: Transition state theory ===

''State what are the main assumptions of Transition State Theory. Given the results you have obtained, how will Transition State Theory predictions for reaction rate values compare with experimental values?''

My answer:

The main assumption of transition state theory <sup>[1]</sup> are give as:

1. The reactants (or products) are in equilibrium with the activated complex.

2. Quantum-tunneling effects are negligible and the Born-Oppenheimer approximation is invoked.

3. Once the system attains the transition state, with a velocity towards the product configuration, it will not reenter the initial state region again.

According to the assumptions, the transition state prediction for rate of reaction should be slower than the experimental value. It can be rationalized by the following two reasons:

1. As stated in the second assumption above. Transition State Theory predictions ignores quantum tunneling effects and treat them as noreactive, where in reality particles with energies lower than the activation energy could react by tunneling through the activation energy barrier.

2. The third assumption can be interpreted that the reaction trajectory are only allowed to cross the transition state once. However, the calculated reaction trajectory for last initial condition in question 4 shows that the reaction trajectory could cross the transition states multiple times and eventually reach the product side. This reactive condition is also ignored by the transition state prediction.

[[User:Ng611|Ng611]] ([[User talk:Ng611|talk]]) 13:41, 8 June 2018 (BST) A good explanation. It's also important to keep in mind that tunneling accelerates the real reaction compared to TS theory's predictions, while TS recrossing slows the real rate relative to TS theory. However, even for light systems, TS recrossing is a far more significant effect, and so the result is an overall overestimation in the rate.

==The study F-H-H system==

===Question 6: energetic of the reactions===

Classify the F + H2 and H + HF reactions according to their energetics (endothermic or exothermic). How does this relate to the bond strength of the chemical species involved?

My answer:

Figure 7 and figure 8 shows the potential surface of a F + H<sub>2</sub> reaction and a H + HF reaction respectively.

[[file: yu_FHH_condition.PNG|400px]]

Figure 7. Potential surface of Yu_HHF_condition.PNG reaction

In this condition the F atom is approaching the H<sub>2</sub> molecule with the momentum showing in the figure 12. The F + H<sub>2</sub> has a higher potential energy then HF+H. This shows that the reaction: F + H<sub>2</sub> ---> HF + H is exothermic. This result indicates that H-F which is formed is bond is stronger than the H-H which is broken.

[[file: Yu_HHF_condition.PNG|400px]]

Figure 8. Potential surface of H + HF reaction

In this condition the H atom is approaching the HF molecule with the momentum showing in the figure 13. The H + HF has a lower potential energy then H<sub>2</sub> + F. This shows that the reaction: H + HF ---> H<sub>2</sub> + F is endothermic. This result agrees with the previous result that H-H which is formed is bond is weaker than the H-F which is broken.

===Question 7: Transition states===

''Locate the approximate position of the transition state.''

My answer:
Hammond's postulate <sup>[2]</sup> states that the transition state of a reaction resembles either the reactants or the products, to whichever it is closer in energy.

The first reaction: F + H<sub>2</sub> = HF + H is exothermic. Hence, the transition state is closer in energy to the reactants: F + H<sub>2</sub> and it adopts the following asymmetric geometry:

F------H--H

•H-H bond is only slightly elongated

•F atom is much further away

[[file:yu_ts_FHH.PNG|400px]]

Figure 9 Distance vs time plot for the transition state of F + H<sub>2</sub> = HF + H

[[file:yu_stationary_1.PNG|400px]]

Figure 10 Surface plot for the transition state of F + H<sub>2</sub> = HF + H

This is known as an early transition state. The method used to find this transition state is similar to the one used for H-H-H transition state. The H-H (BC) distance can be interpreted to be around 0.74 Å which is a typical H-H bond length. As shown in figure the transition state is at F-H=1.8085 Å and H-H=0.7509. Figure shows that the point is a stationary point and it lies closely to the reactants.

The second reaction: H + HF= H<sub>2</sub> + F is endothermic. Hence, the transition state is closer in energy to the products: F + H<sub>2</sub> and it adopts the following asymmetric geometry:

H--H-------F

•H-F bond is largely elongated

•H-H bond is almost formed

[[file:yu_ts_HHF.PNG|400px]]

Figure 11 Distance vs time plot for the transition state of H + HF= H<sub>2</sub> + F

[[file:yu_stationary_2.PNG|400px]]

Figure 12 Surface plot for the transition state of H + HF= H<sub>2</sub> + F

This is known as a late transition state. The H-H (AB) distance can be interpreted to be around 0.74 Å. As shown in figure the transition state is at H-H=0.7495 Å and H-H=1.8080. Figure shows that the point is a stationary point and it lies closely to the products.

===Question 8: Activation energy===

''Report the activation energy for both reactions.''

My answer:

Activation energy is the energy between the reactant and the transition state. If the initial condition is set to be slightly displaced from the transition state and closer to the reactant position, the trajectory will end in the reactant condition. The energy vs time graph would start as a plateau which resembles the energy of the transition state and drops and finally reaches a plateau with resembles the energy of the reactant. The energy difference between the two plateau would give the activation energy of the reaction.

For F + H<sub>2</sub> ---> HF + H:

[[file: yu_ea_1.PNG|400px]]

[[User:Ng611|Ng611]] ([[User talk:Ng611|talk]]) 13:43, 8 June 2018 (BST) There's a slight residual gradient at the end of your simulation, which means you've underestimated the activation energy somewhat. It's probably not significant but you should comment on it.

Figure 13 Activation energy for F + H<sub>2</sub> ---> HF + H

{| class="wikitable"
|+
|-
| Energy of the transition state(kJ/mol) || Energy of the transition state(kJ/mol) || Activation energy(kJ/mol)
|-
| -103.752
| -103.987
| 0.235
|}

Table 2 Activation energy for F + H<sub>2</sub> ---> HF + H

For H + HF= H<sub>2</sub> + F:

[[file: yu_ea_2.PNG|400px]]

Figure 14 Activation energy for H + HF= H<sub>2</sub> + F

{| class="wikitable"
|+
|-
| Energy of the transition state(kJ/mol) || Energy of the transition state(kJ/mol) || Activation energy(kJ/mol)
|-
| -103.694
| -133.827
| 30.133
|}

Table 3 Activation energy for H + HF= H<sub>2</sub> + F

===Question 9: the reaction trajectory===

''In light of the fact that energy is conserved, discuss the mechanism of release of the reaction energy. How could this be confirmed experimentally?''

My answer:

The following initial condition was set:

[[file: yu_initial_3.PNG]]

[[file: yu_fhh_reactive.PNG|400px]]

Figure 15. Contour plot for visualizing the reaction trajectory

[[file: yu_traj_et.PNG|400px]]

Figure 16. Energy vs time plot for the reaction with above initial condition

[[file: yu_traj_pt.PNG|400px]]

Figure 17. Momentum vs time plot for the reaction with above initial condition

The reaction is exothermic. The reaction energy are converted to kinetic energy and specifically vibrational energy of the newly formed H-F bond in this case. As shown on the contour plot, the oscillation along AB (HF in this case) distance axis is significantly larger than the oscillation along BC (HH in this case) distance. This can be confirmed by the energy versus time plot (figure 16) as it shows a constant conversion between kinetic and potential energy. The momentum versus time plot (figure 17) also show a vibration pattern for AB after the reaction and the amplitude is much larger than that of BC before the reaction.

The course of reaction can be studied by the flow method. The reaction mixture that traveled further down in reaction tube were left reacted for a longer time. Hence, the reaction mixture furthest away from the point of mixing resembles the product and the mixture at the point of mixing resemble the reactants, between these two points are the intermediates.

Carrying out IR spectroscopy measurement the points allows the study of the vibration of chemical bonds. As said above, the reaction is exothermic and allows higher vibrational energy levels to become populated. Therefore, besides the original V<sub>n=0</sub> to V<sub>n=1</sub> transition, hot band transitions such as transition from V<sub>n=1</sub> to V<sub>n=2</sub> become possible. This gives rise to overtone signals in the spectrum. Overtones peaks are lower in energy and intensity when compared to the absorption peak representing V<sub>n=0</sub> to V<sub>n=1</sub> transition. The more vibrational energy levels populated the more overtone bands can be observed. Hence the spectrum of the product would have more overtone signals compared to the reactant.

[[User:Ng611|Ng611]] ([[User talk:Ng611|talk]]) 13:45, 8 June 2018 (BST) Good explanation and good proposal!

===Question 10: efficiency of the reaction===

''Discuss how the distribution of energy between different modes (translation and vibration) affect the efficiency of the reaction, and how this is influenced by the position of the transition state.''

My answer:
As the higher translational and vibrational energy level occupied by the molecule the higher the efficiency of the reaction in general. However, that is not always the case. More detailed discussion can be achieved by quoting the Polanyi's rule<sup>[3]</sup>:

The Polanyi's Empirical Rules states that translational energy is more efficient at promoting early transition state reactions, while vibrational energy is more efficient at promoting late transition state energy reactions.

We can look at specific examples of the reactions:

1.For a early transition state reaction: F + H<sub>2</sub> ----> HF + H

[[file: yu_t_e.PNG|400px]]

Figure 18. the contour plot for a early transition state reaction: F + H<sub>2</sub> ----> HF + H

As shown in the figure 18 above the reaction trajectory is reactive. The H<sub>2</sub> is vibrating with a small amplitude. The translation energy which can be roughly interpreted by P<sub>HF</sub> is large. The effect varying vibrational energy by varing P<sub>H2</sub> value between -3 and 3 has a small impact. This agrees with the first part rules stated above.

2.For a late transition state reaction: H + HF ----> H<sub>2</sub>+ F

[[file: yu_v_l.PNG|400px]]

Figure 19. the contour plot for a late transition state reaction: H + HF ----> H<sub>2</sub>+ F

A reactive trajectory is show as the figure above in figure 19. The H-F bond oscillates with a large amplitude this corresponds to the fact that the vibrational energy is more efficient in promoting this reaction. A much higher AB momentum is needed if BC vibration is set small, and will often cause the trajectory to bounce back resulting in a nonreactive condition.

[[User:Ng611|Ng611]] ([[User talk:Ng611|talk]]) 13:47, 8 June 2018 (BST) Some very good examples and overall a very good report. Take care with your discussion and explanations, as some of them (for example, your discussion of the TS) were a little hard to follow. Besides this, a very good piece of work. Well done.

==Reference==

1. T. Bligaard, J.K. Nørskov, Heterogeneous Catalysis in Chemical Bonding at Surfaces and Interfaces, 2008

2. 1 J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, 2001, vol. 40.

3. Z. Zhang, Y. Zhou, D. H. Zhang, G. Czakó and J. M. Bowman, J. Phys. Chem. Lett., 2012, 3, 3416–3419.

Rep:YNX19950213

2025-09-01T09:50:49Z

Move page script: Move page script moved page YNX19950213 to Rep:YNX19950213: Move to report namespace

'''Investigation of Transition States and Reactivity'''

==Introduction==

The development of computational chemistry and computational programs have provided great tools for study chemical reaction's mechanisms. One namely the study of transition states[https://en.wikipedia.org/wiki/Transition_state], which would be impossible to monitor or simulate with conventional lab experiments. This page is delicate to present some of the most well-known reactions' transition state. And discus the relationships between molecular symmetries and reactivity. Investigate how does '''Endo''' and '''Exo''' pathway in '''Diels-Alder reaction''' differ in energy profile. Which reaction pathway is more favorable in thermodynamic aspect and kinetic aspect. Is there any potential secondary orbital interactions happened during the reactions? If do, is it true for both Endo and Exo pathway? And finally how does Cheletropic reaction differ from '''Diels-Alder reaction''' in every aspect.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 18:47, 20 January 2017 (UTC) You should have defined a TS here

==Reaction of Butadiene and Ethylene==
===Reaction Scheme===
[[Image:Ash-EX1-reaction-scheme.png|700x700px|center|thumb|'''Scheme 1''':Butadiene-Ethene Diels Alder Reaction]]
This is a very typical and simple '''Diels-Alder Reaction'''[https://en.wikipedia.org/wiki/Diels%E2%80%93Alder_reaction] between Butadiene and Ethene, illustrated as the MO diagram below.

===Molecular Orbitals===
In order to have a successful reaction, symmetries of the HOMO and the LUMO orbitals must match, otherwise the reaction is forbidden. Orbitals can only be either symmetric or anti-symmetric, which can be determined by calculation of their wave functions.
As the MO diagram illustrated, the frontier orbitals of butadiene (The HOMO) and ethene (The LUMO) are both symmetric (denoted as s), thus the interaction is allowed. And the interactions between an Anti-symmetric (a) and a Symmetric (s) orbital is forbidden due to their different orbital symmetry.

[[Image:Ash-EX1-MO.jpg|700x700px|center|thumb|Butadiene-Ethene Diels Alder Reaction]]
{|align="center"
! style="background: #0D4F8B; color: white;" | MO 16
! style="background: #0D4F8B; color: white;" | MO 17
! style="background: #0D4F8B; color: white;" | MO 18
! style="background: #0D4F8B; color: white;" | MO 19
|-
| [[File:Ash-MO-16.PNG]]
| [[File:Ash-MO-17.PNG]]
| [[File:Ash-MO-18.PNG]]
| [[File:Ash-MO-19.PNG]]
|}

{|align="center"
! style="background: #0D4F8B; color: white;" | Transition state MO16
! style="background: #0D4F8B; color: white;" | Transition state MO17
! style="background: #0D4F8B; color: white;" | Transition state MO18
! style="background: #0D4F8B; color: white;" | Transition state MO19
|-
| [[File:Ash-TS-MO16.PNG]]
| [[File:Ash-TS-MO17.png]]
| [[File:Ash-TS-MO18.PNG]]
| [[File:Ash-TS-MO19.PNG]]
|}
''The corresponding MO lobes were generated by GaussView 5.0.''

For your MO did the MO of the product look like this when you checked in the MOS section?

===Reaction analysis: Symmetry and Reactivity===
It can be clearly seen that the LUMO of Butadiene interacted with the HOMO of Ethene. In order to have a successful reaction, symmetries of the HOMO and the LUMO orbitals must match, otherwise the reaction is forbidden. Orbitals can only be either symmetric or anti-symmetric, which can be determined by calculation of their wave functions.
As the MO diagram illustrated, the frontier orbitals of butadiene (The '''HOMO''') and ethene (The '''LUMO''') are both symmetric (denoted as '''s'''), thus the interaction is allowed. And the interactions between an Anti-symmetric ('''a''') and a Symmetric ('''s''') orbital is forbidden due to their different orbital symmetry.

[[File:Ash-EX1-IRC.PNG|1000x1000px|thumb|centre| '''Intrinsic Reaction Coordinate''' (IRC) of the Reaction]]

With the help of '''Intrinsic Reaction Coordinate''' calculation, we can trace the minimum energy pathway between the reactant and the product on the '''potential energy surface'''. The highest point of total energy corresponds to the Transition state.

===Visualization of the Transition State===

With the help of computationol programs, we can simulate the molecular interactions during the transition state. By looking at the ''Imaginary Vibrations''' of the '''TS''' we can have a straight forward idea of how Butadiene and Ethene were react to give the '''Cyclohexene''' .

[[File:Ash-EX1_IVibration.gif|1000px|thumb|left| The Imaginary Vibration mode]]
[[File:Ash-EX1_Reaction.gif|1000px|thumb|center| The Reaction process visualised]]

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 19:00, 20 January 2017 (UTC) You have drawn very nice MOS etc but you have missed out alot of what was asked in the script, for example you havent measured any bond lengths

==Reaction of Cyclohexadiene and 1,3-Dioxole==
===Reaction Scheme===
The reaction have two different pathway, namely '''Endo''' and '''Exo'''. Which are shown below
[[File:Ash-EX2-Scheme.png|500x500px|thumb|centre| '''Endo and Exo''' pathway of Diels-Alder Reaction]]

===Molecular Orbitals===
This is an inverse demand DA reaction because there are two new chemical bonds and a six-membered ring are formed and is between an electron-rich dienophile, namely cyclohexadiene, and an electron-poor diene, namely dioxole.
The Dioxole is electron-poor due to the two electron withdrawing Oxygen atom in adjacent to the double bond.
[[File:Ash-EX2-MO.jpg|center]]

(This MO diagram suggests that the lowest energy MO of cyclohexadiene that you've shown is higher in energy in the TS than the bonding orbitals, and becomes unoccupied. This is not the case, as the orbital is very low down in energy and it remains occupied. You can confirm this by looking at the occupied orbitals in GaussView. Also remember, the number of orbitals and electrons going into an MO diagram must remain the same [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 14:54, 4 January 2017 (UTC))

{| align="center"
! style="background: #0D4F8B; color: white;" | MO 29
! style="background: #0D4F8B; color: white;" | MO 30
! style="background: #0D4F8B; color: white;" | MO 31
! style="background: #0D4F8B; color: white;" | MO 32
|-
| [[File:Ash-EX2_MO29.PNG]]
| [[File:Ash-EX2_MO30.PNG]]
| [[File:Ash-EX2_MO31.PNG]]
| [[File:Ash-EX2_MO32.PNG]]
|}

{| align="center"
! style="background: #0D4F8B; color: white;" | Endo Transition State MO29
! style="background: #0D4F8B; color: white;" | Endo Transition State MO30
! style="background: #0D4F8B; color: white;" | Endo Transition State MO31
! style="background: #0D4F8B; color: white;" | Endo Transition State MO32
|-
| [[File:Ash-EX2-TS-Endo-MO29.PNG]]
| [[File:Ash-EX2-TS-Endo-MO30.PNG]]
| [[File:Ash-EX2-TS-Endo-MO31.PNG]]
| [[File:Ash-EX2-TS-Endo-MO32.PNG]]
|}

{| align="center"
! style="background: #0D4F8B; color: white;" | Exo Transition State MO29
! style="background: #0D4F8B; color: white;" | Exo Transition State MO30
! style="background: #0D4F8B; color: white;" | Exo Transition State MO31
! style="background: #0D4F8B; color: white;" | Exo Transition State MO32
|-
| [[File:Ash-EX2-TS-Exo-MO29.PNG]]
| [[File:Ash-EX2-TS-Exo-MO30.PNG]]
| [[File:Ash-EX2-TS-Exo-MO31.PNG]]
| [[File:Ash-EX2-TS-Exo-MO32.PNG]]
|}

(These look like PM6 orbitals, which don't show secondary orbital interactions in this case. Also quite hard to see the interactions [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 14:54, 4 January 2017 (UTC))

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 19:06, 20 January 2017 (UTC) Your energies look incorrect I assume they are PM6 energies.

===Reaction analysis: Energies===

{| class="wikitable" align="center"
| The energies of the Reaction Profile
!Endo(KJ/mol)
!Exo(KJ/mol)
|-
|Reactant
|<nowiki>-1511893.671</nowiki>
|<nowiki>-1511877.126</nowiki>
|-
|Transition State
|<nowiki>-1511963.779</nowiki>
|<nowiki>-1511970.525</nowiki>
|-
|Product
|<nowiki>-1511921.323</nowiki>
|<nowiki>-1511892.187</nowiki>
|-
|Ea
|70.108
|93.399
|-
|ΔE
|<nowiki>-27.652</nowiki>
|<nowiki>-15.061</nowiki>
|}

The Endo pathway having both a more stable product (a bigger ΔE) and a smaller activation energy (Ea), which means that the '''Endo pathway''' is '''both thermodynamically and knetically favoured.'''

[[File:Ash-EX2-EXO_IRC.PNG|1000x1000px|thumb|centre|IRC of the EXO path]]
[[File:Ash-EX2-ENDO_IRC.PNG|1000x1000px|thumb|centre|IRC of the Endo path]]

===Secondary orbital interaction===
[[File:Ash-Secondary_Orbital_Interactions.png|600x600px|thumb|centre|Secondary Orbital Interactions: Endo vs. Exo]]

(Don't forget to change the phases when flipping parts of your diagram! [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 14:54, 4 January 2017 (UTC))

The Secondary Orbital Interactions happends between the two oxygen lone pairs and the π* orbital of the diene. The '''Endo''' pathway experienced the Secondary Orbital Interactions, however,the '''Exo''' reaction pathway can't experience the Secondary Orbital Interactions however, due to the fact that the '''distance''' between the two oxygen lone pairs and the π* orbital is too far for the interactions to take place.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 19:10, 20 January 2017 (UTC) Your have clearly shown some understanding of the reaction but you have missed a few key and made a few errors. your SOO is good but you messed up your orbitals. You have also shown no understanding if it is normal or inverse demand.

==Diels-Alder vs Cheletropic==

===Reaction analysis: Reaction profile===

{| class="wikitable" align="center"
|+ Table 3: The energies of the Reaction Profile
!
!Endo(KJ/mol)
!Exo(KJ/mol)
!Cheletropic(KJ/mol)
|-
|Reactant
|176.33
|178.25
|188.56
|-
|Transition State
|241.75
|237.76
|260.07
|-
|Product
|56.3
|57.0
|0.0005
|-
|Ea
|65.42
|59.51
|71.51
|-
|ΔE
|<nowiki>-120.03</nowiki>
|<nowiki>-121.25</nowiki>
|<nowiki>-188.55</nowiki>
|}

[[File:Ash-EX3-Reaction_profile.PNG|1000x1000px|thumb|centre|Reaction Profiles]]

(Don't draw smooth lines like this - it's very misleading as you only have 3 points. If you are to use curved lines, the gradients become important and must be 0 for the stationary points. Better to draw straight lines between the points. Additionally, normalise the reactants to 0 to make comparison easier [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 14:54, 4 January 2017 (UTC))

From the Reaction Profile and the data, it can be conclude that the '''Endo''' pathway is the most '''Kinetically''' favoured as it has the lowest activation energy (Ea), thus having the fastest reaction rate. The Cheletropic pathway on the other hand, although having the highest activation energy, but has the biggest ΔE and lowest product energy. The product's free energy is almost zero, which will be extremely stable (irreversible) and thus will be the thermodynamically favourable pathway.

Below are IRC of each reaction pathway, the energy was in '''Hatree'''.(1 Hatree = 2625.5 KJ/mol)

[[File:Ash-EX3-EXO_IRC.PNG|1000x1000px|thumb|centre|IRC of the Exo path]]
[[File:Ash-EX3-ENDO_IRC.PNG|1000x1000px|thumb|centre|IRC of the Endo path]]
[[File:Ash-Cheltropic_IRC.PNG|1000x1000px|thumb|centre|IRC of the Cheltropic path]]

(Not enough text. You've missed out the question about xylylene's instability [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 14:54, 4 January 2017 (UTC))

==Conclusion==
With all the studies of the three Ring forming reactions using '''GaussView''', one can simply realized that the reaction energies are really important and useful for confirming hypothetical '''transition state'''. And being able to visualize reaction process via animation and generate actual MO lobes are also extremely helpful in the study of '''reaction mechanism''' and '''molecular symmetries'''.

With the help of computational portal, more accurate '''6-31G''' calculation can be utilized, however, in practice the '''PM6''' was much more commonly used thanks to it's fast processing. The most useful and unique feature of computational chemistry is that it costs nothing, and you have the result almost immediately, one can thus quickly get feedback and redesign his/her experiment.

Rep:Y3C Transition States and Reactivity:hjt14

2025-09-01T09:50:48Z

Move page script: Move page script moved page Y3C Transition States and Reactivity:hjt14 to Rep:Y3C Transition States and Reactivity:hjt14: Move to report namespace

==Introduction==
[[File:Intro E profile diagram thj14.png|thumb|600px|centre|Figure 1: A energy profile diagram.]]
A energy profile of a reaction (see Figure 1) describes how the energy of the system change as the reaction progress along a reaction coordinate. In Figure 1, an arbitrary reaction coordinate is chosen to describe the progress of a reaction in term of energy, from the reactant to the product via a transition state. In fact, the motion of each atom in the chemical species involved in a reaction can be describe using the 3 coordinate system (x,y,z), hence collectively a reaction involving N atoms can be describe using 3N coordinates (in other word 3N degrees of freedom).<ref name="chemwikitutorial" /> However, motions like rotation (rotating all the atoms along an axis by the same degree) and translation (moving all atoms by the same translation vector) species do not contribute to the change in energy and thus need to be factored out. As a result, we are left with 3N-6 vibrational coordinates to describe the progress of a reaction. When a reactants approach each other (This is not translation as not all the atoms are moving in the same direction and magnitude), some bonds are being broken and some are being formed. In term of nuclear motions, the nuclei involved in bond formation or bond cleavage are moving away from the equilibrium positions and this cause the energy of the system to rise, just like how the energy of a harmonic oscillator increases as it moves away from equilibrium bond distance. Then the nuclei move into new equilibrium position in the product and this cause the energy of the system to drop, just like how the energy of a harmonic oscillator decreases as it moves toward the equilibrium bond distance. In between the rise and fall of energy, there will be a point where the energy is maximum and this is where the transition state sit. It is challenging to perceive graph of more than 3 dimensions and this explains why energy profile of reaction is generally represented using 2D or 3D graph. In 2D graph, we will obtain an energy line and in 3D graph the reaction progress can be described using a surface know as potential energy surface.

Reactant and product are represented as the energy minima on the potential energy surface. In between the energy minima there is saddle point where the system have to cross to go from the reactant to the product. What a saddle point mean is that the energy is maximum along one vibrational coordinate and minimum along other vibrational coordinates. Stationary point like this on an energy surface can be located using the first derivatives of energy with respect to all the vibrational coordinates and these derivatives correspond to the forces acting on atoms along the corresponding vibrational coordinate. At energy maxima and minima, the forces acting on atoms are zero. To distinguish maxima from minima, secondary derivatives with respect to the vibrational coordinates can be used. At energy maxima, the secondary derivative will be negative; at energy minima the secondary derivative will be positive. In addition, the secondary derivative is the the force constant of the harmonic oscillator. In diatomic molecule, the force constant is related to the frequency of vibration via following equation:

[[File:Force constant thj14.PNG|thumb|center|600px|Figure 2: An equation which relates vibrational frequency of diatomic molecule to the force constant.]]

In this lab, 3 Diels-Alder reactions were studied using computational chemistry software, Gaussian. This software was used to optimise the geometry of each species (reactant, transition state, and product) by solving the Schrodinger equation and calculating the force constants along all the vibrational coordinates to locate the energy maxima (transition state) and energy minima (product and reactant).<ref name="huntcomp" />

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 16:07, 20 January 2017 (UTC) Good understanding of the normal modes of vinration (modes that cannot couple to each other). Nice intro good unstanding of TSs, Some intro about the levels of theory would have been nice.

==Calculations==
Depending on the relative position of the frontier orbitals of diene and dienophile along an energy scale, Diels-Alder reaction can be classify as neutral, normal or inverse electron demand. (See Figure 3) In each case, the TS MO symmetry order is different and this gives us an shorthand to classify the Diels-Alder reaction simply by looking at the TS MOs symmetry order. (symmetry element under study: mirror plane; S= symmetrical; AS= asymmetrical; - See Figure 6)

{| class="wikitable"
| [[File:Neutral electron demand MO hjt14.jpg|500px|]]
| [[File:Normall electron demand MO hjt14.jpg|500px|]]
| [[File:Inversel electron demand MO hjt14.jpg|500px|]]
|-
| colspan= "3"| Figure 3:MO diagrams above show how the frontier orbitals of reactants interact and also the TS MO symmetry order in each case.
|}

===Exercise 1: Diels-Alder Reaction of Butadiene with Ethylene===
In this part of the lab, a Diels-Alder reaction between butadiene and ethene was studied computationally. Figure 4 below shows the arrow pushing diagram for the Diels-Alder reaction between a butadiene and an ethene and figure 5 shows the MO diagram. Ethene, butadiene and transition state (TS) were optimised using semi-empirical method and PM6 basis set. The frontier orbitals of each species are shown in Table 1. By looking at the TS MOs symmetry order, one can deduces that this reaction is inverse electron demand.

[[File:Exercise 1 scheme hjt14.jpg|thumb|centre|400px|Figure 4: Diels-Alder reaction between a butadiene and an ethene.]]

[[File:Exercise_1_MO_diagram_thj14.jpg|thumb|centre|400px|Figure 5: MO diagram of Diels-Alder reaction between a butadiene and an ethene.]]

{| class="wikitable"
! colspan="4" |<b>Table 1: Frontier Orbitals of s-cis butadiene and ethene</b>
! style="background: #000000;" |
! colspan="5" |<b>Table 2: Frontier Orbitals of Transition State of Diels-Alder Reaction between Butadiene and Ethene</b>
|-
! <b>Species</b>
! colspan="2" |<b>Molecular Orbital</b>
! <b>Symmetry</b>
! style="background: #000000;" |
! <b>Species</b>
! colspan="2" |<b>Molecular Orbital</b>
! <b>Interating MO pair</b>
! <b>Symmetry</b>
|-
| rowspan="2" |
<jmol><jmolApplet>
<title>s-cis butadiene</title>
<color>white</color>
<script>frame 34</script>
<size>200</size>
<uploadedFileContents>S CIS BUTADIENE OPT FRE SEM PM6 BREAKSYM.LOG</uploadedFileContents>
</jmolApplet></jmol>
| HOMO
| [[File:S cis butadiene opt fre sem pm6 breaksym HOMO thj14.PNG|200px|]]
| Asymmetric
| style="background: #000000;" |
| rowspan="4"|
<jmol><jmolApplet>
<title>Transition State</title>
<color>white</color>
<script>frame 16</script>
<size>200</size>
<uploadedFileContents>42CYCLOADDITION TS OPT FRE SEM PM6 hjt14.LOG</uploadedFileContents>
</jmolApplet></jmol>
| LUMO+1
| [[File:42cycloaddition TS opt fre sem pm6 LUMO+1 hjt14.PNG|200px|]]
| Antibonding interaction between s-cis butadiene HOMO and ethene LUMO
| Asymmetric
|-
| LUMO
| [[File:S cis butadiene opt fre sem pm6 breaksym LUMO thj14.PNG|200px|]]
| Symmetric
| style="background: #000000;" |
| TS LUMO
| [[File:42cycloaddition TS opt fre sem pm6 LUMO hjt14.PNG|200px|]]
| Antibonding interaction between s-cis butadiene LUMO and ethene HOMO
| Symmetric
|-
| rowspan="2" |
<jmol><jmolApplet>
<title>Ethene</title>
<color>white</color>
<script>frame 14</script>
<size>200</size>
<uploadedFileContents>ETHENE OPT FRE SEM PM6 hjt14.LOG</uploadedFileContents>
</jmolApplet></jmol>
| HOMO
| [[File:Ethene opt fre sem pm6 HOMO thj14.PNG|200px|]]
| Symmetric
| style="background: #000000;" |
| TS HOMO
| [[File:42cycloaddition TS opt fre sem pm6 HOMO hjt14.PNG|200px|]]
| Bonding interaction between s-cis butadiene LUMO and ethene HOMO
|Symmetric
|-
| LUMO
| [[File:Ethene opt fre sem pm6 LUMO thj14.PNG|200px|]]
| Asymmetric
| style="background: #000000;" |
| TS HOMO-1
| [[File:42cycloaddition TS opt fre sem pm6 HOMO-1 hjt14.PNG|200px|]]
| Bonding interaction between s-cis butadiene HOMO and ethene LUMO
| Asymmetric
|}

<br>

[[File:Exercise 1 mirror plane hjt14.jpg|thumb|250px|Figure 6: Plane of symmetry used to to distinguish the symmetry of MO.]]
When butadiene and ethene are brought together along the reaction coordinate, the HOMO and LUMO of butadiene interact with HOMO and LUMO of ethene to form new MOs. To interact, the pair of interacting MOs need to have the same symmetry. The symmetry under question in this case is the plane of symmetry depicted in Figure 6. The HOMO of butadiene and the LUMO of ethene are both asymmetrical and they interact to form new bonding and antibonding TS MOs as shown in Table 2; the LUMO of butadiene and HOMO of ethene are symmetrical and they interact with each other to form new bonding and antibonding TS MOs as shown in Table 2.

The symmetry requirement in Diels-Alder reaction is illustrated in figure 7 and table 2. When a symmetrical MO interacts with an asymmetrical MO, antibonding and bonding configurations are equivalent to each other. Within a configuration, the bonding interaction has a positive overlap integral and the antibonding interaction has a zero overlap integral, hence the overall overlap integral is non zero. Besides, the stabilising effect of bonding interaction is cancel out by the destabilising effect of the antibonding interaction. Overall, there is no interaction between symmetrical MO and asymmetrical MO. For symmetrical pair and asymmetrical pair, the bonding configuration has non zero overlap integral and are overall stabilised, whereas the antibonding configuration has zero overlap integral and overall destabilised. In the HOMO-LUMO region, usually only the bonding orbitals are populated and result in overall stabilisation.

[[File:Exercise 1 sym requirement hjt14.jpg|thumb|centre|500px|Figure 7: Symmetry requirement for molecular orbital interactions.]]

The following graph show how the inter-nuclear distance between carbon atoms change as the reaction progress from product to reactant (from left to right).

[[File:Exercise 1 atoms label hjt14.PNG|thumb|centre|400px|Figure 8: Number labelled reactants.]][[File:Exercise 1 internuclear distance graph v2 hjt14.PNG|thumb|centre|600px|Figure 9: Graph of inter-nuclear distance against reaction coordinate.]]

{| class="wikitable"
! colspan="2"| Reactant
! colspan="2"| TS
! colspan="2"| Product
! rowspan="2" colspan= "2"| Literature Values for C-C bond length <ref name="carbon bond length" />
|-
! Bond
! Bond length (Armstrong)
! Bond
! Bond length (Armstrong)
! Bond
! Bond length (Armstrong)
|-
|C1-C4
|1.33539
|C1-C4
|1.37979
|C1-C4
|1.50085
|rowspan="2"|sp<sup>3</sup>C-sp<sup>3</sup>C
|rowspan="2"|1.54
|-
|C4-C6
|1.46833
|C4-C6
|1.41110
|C4-C6
|1.33695
|-
|C6-C8
|1.33539
|C6-C8
|1.37977
|C6-C8
|1.50079
|rowspan="2"|sp<sup>3</sup>C-sp<sup>2</sup>C
|rowspan="2"|1.50
|-
|C8-C11
|NA
|C8-C11
|2.11484
|C8-C11
|1.53716
|-
|C11-C14
|1.32737
|C11-C14
|1.38178
|C11-C14
|1.53464
|rowspan="2"|sp<sup>2</sup>C-sp<sup>2</sup>C
|rowspan="2"|1.48
|-
|C14-C1
|NA
|C14-C1
|2.11466
|C14-C1
|1.53711
|}

From reactants to product,:
*the bond order of C1-C4, C6-C8, and C11-C14 decrease from 2 to 1 and hence the bonds elongate.
*the bond order of C4-C6 changes from 1 to 2 hence the bond shortens.
*sigma bond is formed between C8 and C11, and C1 and C14, therefore the inter-nuclear distances decrease drastically.
*the sigma bonds are forming at the same time. Meaning that at any point along the reaction coordinate, the forming bonds share the same length and such reaction can be describe as synchronous.

At transition state,:
*the C-C bond length fall in between the C-C bond length in product and the C-C bond length in reactant (except C8-C11 and C14-C1).
*(for C8-C11 and C14-C1) the C-C bond length is in between 3.40 Armstrong (2*carbon van der Waal's radius and literature value for carbon van der Waal's radius<ref name="carbon van der waal radius" />: 170 Armstrong) and the sp<sup>3</sup>C-sp<sup>3</sup>C bond length.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 16:32, 20 January 2017 (UTC) Excellent first section

===Exercise 2: Diels-Alder Reaction of Cyclohexadiene and 1,3-Dioxole===
The following figure shows the reaction scheme for Diels-Alder reaction between cyclohexadiene and 1,3-dioxole and also the transition state leading to endo and exo products.

[[File:Exercise 2 reaction scheme hjt14.jpg|thumb|600px|centre|Figure 10: Diels-Alder reaction between cyclohexadiene and 1,3-dioxole]]

First, the geometry of reactants were optimised using semi-empirical method, PM6 basis set and then DFT method, B3LYP631Gd basis set. Using the optimised reactants, guess TSs were constructed and optimised using semi-empirical method, PM6 basis set and then DFT method, B3LYP631Gd basis set. After which, IRC calculation were ran for endo and exo TSs to confirm that the TSs do lead back to the correct reactants and products. The results are tabulated in table 3 (reactants) ,table 4 (endo transition state) and table 5 (exo transition state).
<br>

{| class="wikitable"
! colspan="4" |<b>Table 3: Frontier Orbitals of Cyclohexadiene and 1,3-Dioxole</b>
! style="background: #000000;" |
! colspan="4" |<b>Table 4: Frontier Orbitals of Endo Transition State of Diels-Alder Reaction between Cyclohexadiene and 1,3-Dioxole</b>
! style="background: #000000;" |
! colspan="4" |<b>Table 5: Frontier Orbitals of Exo Transition State of Diels-Alder Reaction between Cyclohexadiene and 1,3-Dioxole</b>
|-
! <b>Species</b>
! colspan="2" |<b>Molecular Orbital</b>
! <b>Symmetry</b>
! style="background: #000000;" |
! colspan="2" |<b>Molecular Orbital</b>
! <b>Interating MO pair</b>
! <b>Symmetry</b>
! style="background: #000000;" |
! colspan="2" |<b>Molecular Orbital</b>
! <b>Interating MO pair</b>
! <b>Symmetry</b>
|-
| rowspan="2" |Cyclohexadiene
| HOMO
| [[File:Cyclohexadiene opt fre dft B3LYP631Gd HOMO hjt14.PNG|200px|]]
| Asymmetric
| style="background: #000000;" |
| Endo TS LUMO+1
| [[File:Dioxole hexadiene DA endoTS opt fre dft B3LYP631Gd noeigen LUMO+1 hjt14.PNG|200px|]]
| Antibonding interaction between cyclohexadiene HOMO and 1,3-dioxole LUMO
| Asymmetric
| style="background: #000000;" |
| Exo TS LUMO+1
| [[File:Dioxole hexadiene DA exoTS opt fre dft B3LYP631Gd LUMO+1.PNG|200px|]]
| Antibonding interaction between cyclohexadiene HOMO and 1,3-dioxole LUMO
| Asymmetric
|-
| LUMO
| [[File:Cyclohexadiene opt fre dft B3LYP631Gd LUMO hjt14.PNG|200px|]]
| Symmetric
| style="background: #000000;" |
| Endo TS LUMO
| [[File:Dioxole hexadiene DA endoTS opt fre dft B3LYP631Gd noeigen LUMO.PNG|200px|]]
| Antibonding interaction between cyclohexadiene LUMO and 1,3-dioxole HOMO
| Symmetric
| style="background: #000000;" |
| Exo TS LUMO
| [[File:Dioxole hexadiene DA exoTS opt fre dft B3LYP631Gd LUMO hjt14.PNG|200px|]]
| Antibonding interaction between cyclohexadiene LUMO and 1,3-dioxole HOMO
| Symmetric
|-
| rowspan="2" |1,3-Dioxole
| HOMO
| [[File:13 dioxole opt fre dft B3LYP631Gd breaksym HOMO hjt14.PNG|200px|]]
| Symmetric
| style="background: #000000;" |
| Endo TS HOMO
| [[File:Dioxole hexadiene DA endoTS opt fre dft B3LYP631Gd noeigen HOMO hjt14.PNG|200px|]]
| Bonding interaction between cyclohexadiene LUMO and 1,3-dioxole HOMO
|Symmetric
| style="background: #000000;" |
| Exo TS HOMO
| [[File:Dioxole hexadiene DA exoTS opt fre dft B3LYP631Gd HOMO hjt14.PNG|200px|]]
| Bonding interaction between cyclohexadiene LUMO and 1,3-dioxole HOMO
|Symmetric
|-
| LUMO
| [[File:13 dioxole opt fre dft B3LYP631Gd breaksym LUMO hjt14.PNG|200px|]]
| Asymmetric
| style="background: #000000;" |
| Endo TS HOMO-1
| [[File:Dioxole hexadiene DA endoTS opt fre dft B3LYP631Gd noeigen HOMO-1 hjt14.PNG|200px|]]
| Bonding interaction between cyclohexadiene HOMO and 1,3-dioxole LUMO
| Asymmetric
| style="background: #000000;" |
| Exo TS HOMO-1
| [[File:Dioxole hexadiene DA exoTS opt fre dft B3LYP631Gd HOMO-1 hjt14.PNG|200px|]]
| Bonding interaction between cyclohexadiene HOMO and 1,3-dioxole LUMO
| Asymmetric
|}

<br>
{| class="wikitable"
! Cyclohexadiene
! 1,3-Dioxole
! Endo TS
! Endo Product
! Exo TS
! Exo Product
|-
|
<jmol><jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 18</script>
<uploadedFileContents>CYCLOHEXADIENE OPT FRE DFT B3LYP631Gd hjt14.log</uploadedFileContents>
</jmolApplet></jmol>
|
<jmol><jmolApplet>
<color>white</color>
<script>frame 18</script>
<size>250</size>
<uploadedFileContents>13 DIOXOLE OPT FRE DFT B3LYP631Gd BREAKSYM hjt14.log</uploadedFileContents>
</jmolApplet></jmol>
|
<jmol><jmolApplet>
<color>white</color>
<script>frame 30</script>
<size>250</size>
<uploadedFileContents>DIOXOLE HEXADIENE DA ENDOTS OPT FRE DFT B3LYP631Gd NOEIGEN hjt14.log</uploadedFileContents>
</jmolApplet></jmol>
|
<jmol><jmolApplet>
<color>white</color>
<script>frame 12</script>
<size>250</size>
<uploadedFileContents>DIOXOLE HEXADIENE DA DFT IRCPRODUCT ENDOTS OPT FRE DFT hjt14.log</uploadedFileContents>
</jmolApplet></jmol>
|
<jmol><jmolApplet>
<color>white</color>
<script>frame 16</script>
<size>250</size>
<uploadedFileContents>DIOXOLE HEXADIENE DA EXOTS OPT FRE DFT B3LYP631Gd.log</uploadedFileContents>
</jmolApplet></jmol>
|
<jmol><jmolApplet>
<color>white</color>
<script>frame 14</script>
<size>250</size>
<uploadedFileContents>DIOXOLE HEXADIENE DA DFT IRCPRODUCT EXOTS OPT FRE DFT hjt14.log</uploadedFileContents>
</jmolApplet></jmol>
|}

(Useful JMols with vibrations and surfaces, as some interactions are hard to see in the diagrams above [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 10:40, 4 January 2017 (UTC))

By looking at the order of the symmetry of TS MOs, one can deduce that the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole is inverse electron demand. This is expected because the lone pair on the oxygen atom in 1,3-dioxole can delocalise into the pi system, rising the energy of HOMO of dienophile. Hence, the energy gap between diene LUMO and dienophile HOMO decreases and the energy gap between diene HOMO and dienophile LUMO increases. Consequently, the orbital interactions between diene and dienophile during Diels-Alder reaction are dominated by diene LUMO and dienophile HOMO interaction.

Exo product is generally regarded as the thermodynamic product because there are less unfavourable steric interactions in exo product than in endo product. However, endo product is generally regarded as the kinetic product because secondary orbital interactions in the transition state leading to endo product lower the energy of the endo transition state and also the activation energy. Consequently, endo product is formed faster than exo product. The thermochemistry data, obtained from the gaussian output log file of reactants, transition states and products (all are optimised from IRC output) at DFT B3LYP631Gd caculation level, were summarised in following table. These data were used to calculate the activation energy and reaction energy of exo and endo pathways.

<pre>
Thermochemistry data ( obtained from DFT B3LYP631Gd optimised reactant, product, TS from IRC output ):

Temperature/ K: 298.150 Kelvin | 0 Kelvin
|
Sum of electronic and thermal | Sum of electronic and zero-
free Energies (Hartree/Particle) | point energies (Hartree/Particle)
|
Exo reactants : -500.381626 | -500.339315
Exo TS : -500.329168 | -500.294732
Exo Product : -500.417314 | -500.384595
Endo reactants : -500.381519 | -500.338832
Endo TS : -500.332154 | -500.297145
Endo product : -500.420015 | -500.387295
|
|
Exo pathway kJ/mol | kJ/mol
Activation barrier: 137.728479 | 117.0526665
Reaction energy : -93.698844 | -118.8826400
|
|
|
Endo pathway kJ/mol | kJ/mol
Activation barrier: 129.6078075 | 109.4492185
Reaction energy : -101.0712480 | -127.2396065
</pre>
<br>
[[File:Exercise 2 energy profile (endo and exo) v3 hjt14.PNG|thumb|centre|600px|Figure 11: Energy profile for exo and endo Diels-Alder reaction between cyclohexadiene and 1,3-dioxole.]]

<br>
Looking at the energy profile, the endo product is indeed the kinetic product as it has lower activation barrier. The secondary orbital interaction which lowers the energy of transition state are observed in the MO of transition state and is shown in Figure 12 (Only secondary orbital interaction in occupied MO is shown because unoccupied MO which exhibit secondary orbotal interaction do not contribute to stabalisation of TS. Interestingly, endo is also the thermodynamic product as it is lower in energy than exo product. This is probably due to the fact that the steric interaction is weak and do not contribute strongly to the energy of product.

[[File:Endo TS secondary orbital interaction hjt14.PNG|thumb|400px|centre|Figure 12: Secondary orbital interaction in endo TS.]]

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 16:41, 20 January 2017 (UTC)Good understanding of SSO. Your energies are slightly out however but you still have the correct conclusion

===Exercise 3: Diels-Alder and Cheletropic Reactions of Xylylene and Sulfur Dioxide===

In this part of the computational lab, guess exo and endo transition state of Dels-Alder reaction and guess transition state of cheletropic reaction were obtained by distorting the corresponding optimised products. The guess TSs were then optimised and IRC were carried out. After which the reactants and products from IRC output were optimised and thermochemistry data of each reaction were calculated. All the calculations were done using semi-empirical method and PM6 basis set. The gif file of each IRC output are tabulated in table 6. One feature that was common among all reaction pathway is the rearomatisation of the cyclohexadiene moeity during the course of the reaction.

{| class="wikitable"
! colspan="2"| Reactants
|-
! Xylylene
! Sulfur Dioxide
|-
|
<jmol><jmolApplet>
<color>white</color>
<script>frame 10</script>
<size>250</size>
<uploadedFileContents>XYLYLENE OPT FRE SEM BREAKSYM hjt14.LOG</uploadedFileContents>
</jmolApplet></jmol>
|
<jmol><jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 18</script>
<uploadedFileContents>SO2 OPT FRE SEM hjt14.LOG
</uploadedFileContents>
</jmolApplet></jmol>
|}

{| class="wikitable"
! colspan="2"| Cheletropic Pathway
|-
! Transition State
! Product
|-
|
<jmol><jmolApplet>
<color>white</color>
<script>frame 52</script>
<size>250</size>
<uploadedFileContents>CHELETROPIC TS OPT FRE SEM hjt14.LOG</uploadedFileContents>
</jmolApplet></jmol>
|
<jmol><jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 16</script>
<uploadedFileContents>CHELETROPIC SEM IRCPRODUCT OPT FRE SEM hjt14.LOG</uploadedFileContents>
</jmolApplet></jmol>
|}

{| class="wikitable"
! colspan="2"| Exo Pathway
|-
! Transition State
! Product
|-
|
<jmol><jmolApplet>
<color>white</color>
<script>frame 12</script>
<size>250</size>
<uploadedFileContents>42CYCLOADDITION EXOTS OPT FRE SEM hjt14.LOG</uploadedFileContents>
</jmolApplet></jmol>
|
<jmol><jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 18</script>
<uploadedFileContents>42CYCLOADDITION SEM IRCPRODUCT EXOTS OPT FRE SEM hjt14.LOG</uploadedFileContents>
</jmolApplet></jmol>
|}

{| class="wikitable"
! colspan="2"| Endo Pathway
|-
! Transition State
! Product
|-
|
<jmol><jmolApplet>
<color>white</color>
<script>frame 16</script>
<size>250</size>
<uploadedFileContents>42CYCLOADDITION TS OPT FRE SEM hjt14.LOG</uploadedFileContents>
</jmolApplet></jmol>
|
<jmol><jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 38</script>
<uploadedFileContents>42CYCLOADDITION SEM IRCPRODUCT TS OPT FRE SEM hjt14.LOG</uploadedFileContents>
</jmolApplet></jmol>
|}

{| class="wikitable"
|+ <b>Table 6: Gif file of IRC output </b>
! Endo Pathway (reactant to product)
! Exo Pathway (product to reactant)
! Cheletropic Pathway (reactant to product)
|-
| [[File:Endo TS IRC v2 hjt14.gif]]
| [[File:Exo TS hjt14.gif]]
| [[File:Cheletropic IRC hjt14.gif]]
|-
| [[File:Endo TS IRC energy profile hjt14.PNG]]
| [[File:Exo TS IRC energy profile hjt14.PNG]]
| [[File:Cheletropic TS IRC energy profile hjt14.PNG]]
|}

The thermochemistry data, obtained from the gaussian output log file of reactants, transition states and products (all species are from optimised IRC output except exo reactants) at semi-empirical PM6 caculation level, were summarised in following table. These data were used to calculate the activation energy and reaction energy of exo, endo and cheletropic pathways.
<br>
<pre>
Thermochemistry data ( obtained from semi-empirical PM6 optimised reactant, product, TS from IRC output except exo reactants ):

Temperature/ K: 298.150 Kelvin | 0 Kelvin
|
Sum of electronic and thermal | Sum of electronic and zero-
free Energies (Hartree/Particle) | point energies (Hartree/Particle)
|
Exo reactants* : 0.060496 | 0.116965
Exo TS : 0.092077 | 0.128171
Exo Product : 0.021455 | 0.056644
Endo reactants : 0.067926 | 0.114802
Endo TS : 0.090560 | 0.126589
Endo product : 0.021705 | 0.057503
Cheletropic reactants: 0.068174 | 0.114815
Cheletropic TS : 0.099062 | 0.135560
Cheletropic Product : -0.000002 | 0.034556
|
|
Exo pathway kJ/mol | kJ/mol
Activation barrier : 82.9159155 | 29.421353
Reaction energy : -102.5021455 | -158.3727855
|
|
Endo pathway kJ/mol | kJ/mol
Activation barrier : 59.425567 | 30.9467685
Reaction energy : -121.3532355 | -150.4385245
|
|
Cheletropic pathway kJ/mol | kJ/mol
Activation barrier : 81.096444 | 54.4659975
Reaction energy : -178.996088 | -210.7200045

* obtained by summing energy of individually optimised reactant molecule because it was very difficult to optimised the reactant from IRC output.

</pre>
<br>

(You must be consistent in your calculations. For your reactants, you've added a contribution of about 20 kJ/mol. This has pushed up your exo barrier and reaction energy [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 10:40, 4 January 2017 (UTC))

[[File:Exercise 3 energy profile hjt14 v2.PNG|thumb|600px|centre|Figure 13: Energy profile for endo, exo and cheletropic pathway.]]

From the energy profile (Figure 13), one can conclude that cheletropic product is the thermodynamic product and endo product is the kinetic product. If the reaction between xylylene and sulfur dioxide is irreversible, the product which forms fastest will be the major product, and in this case it is endo product.

(Not enough text. What about the instability of xylylene? [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 10:40, 4 January 2017 (UTC))

==Conclusion==
Exercise 1: The Diels-Alder reaction between butadiene and ethene is synchronous as bond is forming at the same time. During the course of the reaction, bond that is changing bond order will experience change in bond length, i.e. increase in bond order will cause the bond length to decrease and vice versa.

Exercise 2: The Diels-Alder reaction between cyclohexadiene and 1,3 dioxole is inverse electron demand. Endo product of the reaction is the kinetic and also the thermodynamic product due to secondary orbital interaction depicted in figure 12 and small steric effect.

Exercise 3: Reaction between xylylene and sulfur dioxide can occur via 3 pathways, namely endo, exo and cheletropic. Kinetic product of the reaction is endo product and thermodynamic product is cheletropic product.

==References==
<references>
<ref name="chemwikitutorial"> Transition States Tutorial,https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_tutorial</ref>
<ref name="huntcomp"> Understanding the Optimisation Process,http://www.huntresearchgroup.org.uk/teaching/teaching_comp_lab_year1/2_understand_opt_nh3.html</ref>
<ref name="carbon bond length"> E. M. Popov, G. A. Kogan and V. N. Zheltova, Theor. Exp. Chem., 1972, 6, 11–19. DOI:10.1007/BF00525890</ref>
<ref name="carbon van der waal radius"> S. S. Batsanov, Inorg. Mater. Transl. from Neorg. Mater. Orig. Russ. Text, 2001, 37, 871–885. DOI: 10.1023/A:1011625728803</ref>
</references>

Rep:Y3C MgO:hjt14

2025-09-01T09:50:44Z

Move page script: Move page script moved page Y3C MgO:hjt14 to Rep:Y3C MgO:hjt14: Move to report namespace

==Introduction==

From transition state to molecular dynamic simulation, computation chemistry allows us to explore the impossible in the experiment and provides an alternative way to understand a reaction or a material which nicely compliment the practical approach taken by the experimentalist. This computational lab aims to demonstrate the strength of computation chemistry by predicting the thermal expansion coefficient of magnesium oxide, MgO using two different theoretical approaches. The computed results were then compared and contrasted with experiment results.

The system under study is magnesium oxide, MgO. It is a white, ionic, crystalline solid and it has a rock salt crystal structure, i.e. the magnesium ions sit at the lattice point of a face-centered cubic unit cell and the oxide ions sit at all the octahedral holes in the unit cell<ref name="MgO crystal structure" />. There are several ways to represent the repeating unit of MgO crystal structure. The aforementioned face-centered cubic unit cell is one way, and other ways included primitive cell and super cell. Different unit cell representation will have different lattice parameters and the lattice parameters can be interconverted using trigonometric rules.

{| width=1002px class='wikitable'
|colspan='3'|[[File:MgO Unit cells hjt14.png|thumb|centre|1000px]]
green ion= Mg<sup>2+</sup>; red ion= O<sup>2-</sup>
|-
|width=334px |Figure 1: A primitive unit cell of MgO. The cell consists of 1 magnesium ion and 1 oxide ion.
|width=334px |Figure 2: A conventional unit cell of MgO. The cell consists of 4 magnesium ions and 4 oxide ions.
|width=334px | Figure 3: A supercell of MgO. The cell consists of 32 magnesium ions and 32 oxide ions.
|}

Thermal expansion of a material, <math>\alpha</math>, is defined as follow:

<math>
\alpha= \frac{1}{V_0} \left( \frac {\partial V} {\partial T} \right)_P
</math>

<math>\alpha</math>= thermal expansion coefficient

<math>V_0</math>= initial lattice volume

<math>V</math>= lattice volume

<math>T</math>= temperature

<math>P</math>= pressure

To obtain <math> \alpha </math>, two theoretical approaches, namely quasi-harmonic approximation, and molecular dynamics were used to construct the graph of MgO volume as a function of temperature.

'''Quasi-Harmonic Approximation''' <ref name="MgO Introduction" />

Affected by thermal energy, the ions in a lattice is not static and move about an equilibrium position. In our study, an anharmonic potential- Buckingham potential, was employed to model the interactions. However, we assumed that the displacement of an ion from equilibrium position is small. As a result, we can model the interactions between ions using a quadratic function and this is what known as the harmonic approximation. By summing up all the interaction energy between ions, we will obtain the lattice energy for the system under study. The summation of lattice energy and the zero-point energy of all the phonon states will give us the internal energy. Internal energy alone is not enough to describe the thermodynamic state of a system, what we need is the free energy. In the system where the volume is held constant, we will use the Helmholtz free energy. To calculate the free energy, a second component- entropy, is required. To do so, vibration energy level of the system, which is quantised, need to be obtained and this can be achieved by computing phonon dispersion and then derived the density of state as a function of energy. The occupancy of these phonon states will then determine the entropy of the system. Entropy together with internal energy will then allow us to compute the free energy of a system, and the following shows the expression for the Helmholtz free energy:

<math>
F=E_0+ \frac{1}{2} \sum_{k,j} \hbar \omega_{k,j}+k_BT\sum_{k,j} ln\left[1-exp(-\frac{\hbar \omega_{k,j}}{k_BT}) \right]
</math>

<math>F</math>= Helmholtz free energy

<math>E_0</math>= lattice energy of MgO

<math>k</math>= wavevector of phonon

<math>j</math>= label for branch

<math>\hbar</math>= h bar (a constant)

<math>\omega</math>= vibrational frequency of phonon (<math>k,j</math>)

<math>k_B</math>= Boltzmann constant

<math>T</math>= temperature

What is characteristic of this quadratic potential well is that the excited state and ground state have the same equilibrium position and the bonds between atoms will not break. For these reasons, this model is not particularly useful to study thermal expansion because this model does not predict thermal expansion. To account for the change in equilibrium position when the ions get excited to higher vibrational state, extra steps were taken, i.e. the lattice parameter was allowed to change at a fixed temperature and the one which gave minima in free energy was identified as the optimal lattice parameter for that particular temperature.

'''Molecular Dynamics''' <ref name="MgO Introduction" />

In this approach, the interactions between ions were modeled using Buckingham potential, however, no harmonic approximation was made and quantisation of vibrational state was not considered. After specifying the initial conditions, all the pair forces in the system will be calculated using the negative derivative of the potential surface with respect to the displacement of the ion. By using newton third law, the forces will be used to update the velocity and also the position of the ions. The system will continue to evolve within a number of timesteps. Within that timesteps, the energy and temperature will eventually converge to a constant.

==Discussion==
===Quais-Harmonic Approximation===
====Lattice Energy of MgO Crystal====

In the first part of this lab, the lattice energy <math>E_0</math> was calculated for a primitive unit cell of MgO using General Utility Lattice Program (GULP) and the results are reported as follow:

<pre>
Components of energy:

--------------------------------------------------------------------------------
Interatomic potentials = 6.72119980 eV
Monopole - monopole (real) = -5.11592628 eV
Monopole - monopole (recip)= -42.68059111 eV
Monopole - monopole (total)= -47.79651739 eV
--------------------------------------------------------------------------------
Total lattice energy = -41.07531759 eV
--------------------------------------------------------------------------------
Total lattice energy = -3963.1403 kJ/(mole unit cells)
--------------------------------------------------------------------------------
</pre>

The potential that was used to model the interaction was Buckingham potential. What the computational chemistry software did was to compute the lattice energy by summing up all the interaction energy between every pair of ions. The literature values are 3795 kJ/mol (total lattice potential energies)<ref name="MgO lattice energy literature"/> and 3791 kJ/mol (Born–Fajans–Haber cycle)<ref name="MgO lattice energy literature"/>. The discrepancy between our data with total lattice potential energies is probably because different potentials were used in modeling the interaction between ions.

====Phonon Dispersion of MgO====

Again, using a primitive unit cell of MgO, the phonon dispersion was computed using GULP along a series of special k points, namely W, L, G, W, X, K. The computed phonon dispersion and the k values are shown below:

{| class='wikitable'
!Graph
!k value
|-
|rowspan="5"| [[File:MgO Phonon Dispersion Plot hjt14.PNG|thumb|500px|Graph 1: The graph is showing the phonon dispersion of MgO calculated based on a primitive cell.]]

|W: 1/2 1/4 3/4
|-
|L: 1/2 1/2 1/2
|-
|G: 0 0 0
|-
|X: 1/2 0 1/2
|-
|K: 3/8 3/8 3/4
|}

By sampling the energy states on dispersion curve at different k points, one can construct a graph of the number of sampled energy state as a function of energy or frequency and this is equivalent to the density of state of phonon vibration (DOS). The DOS graph is crucial for zero point energy and entropy calculation. To determine the optimal number of sampling one should perform in constructing a DOS, one can do so qualitatively by computing DOS of different grid size and see how the general profile of the DOS change with grid size. The grid size determines the number of sampling done along the k points, the larger the grid size, the more k points are being sampled and the more accurate the DOS is going to be. Beyond the optimal grid size, the DOS profile will remain relatively unchanged. The followings summerise the DOSs computed for a primitive unit cell of MgO using different grid size:

{| class='wikitable'
!Grid Size/ Shrinking Factor
|1x1x1
|5x5x5
|10x10x10
|-
!Graph
|[[File:MgO Phonon DOS 1x1x1 hjt14.PNG|thumb|400px]]
|[[File:MgO Phonon DOS 5x5x5 hjt14.PNG|thumb|400px]]
|[[File:MgO Phonon DOS 10x10x10 hjt14.PNG|thumb|400px]]
|-
!Grid Size/ Shrinking Factor
|15x15x15
|20x20x20
|25x25x25
|-
!Graph
|[[File:MgO Phonon DOS 15x15x15 hjt14.PNG|thumb|400px]]
|[[File:MgO Phonon DOS 20x20x20 hjt14.PNG|thumb|400px]]
|[[File:MgO Phonon DOS 25x25x25 hjt14.PNG|thumb|400px]]
|}

For the grid size of 1x1x1, only one k point was sampled. Consequently, the DOS graph showed a group of sharp peaks and each corresponds to a phonon state at that k point. By comparing the number of peaks and the frequency of the peaks with the phonon dispersion, the k point was identified to be 1/2 1/2 1/2. By increasing the grid size, more phonon states are being sampled. As a result, the number of the peak in DOS increases and some will merge together and causes broadening of the peak. It can be seen from the table that below the grid size 15x15x15, the DOS plot changes significantly with grid size. Among the changes is the merging of the peaks at 400 cm<sup>-1</sup> to form one broad peak. For grid size beyond 15x15x15, the DOS plot remains relatively unchanged. This observation suggests that any grid size beyond 15x15x15 is suitable for free energy calculation.

The optimal grid size used in the construction of DOS depends on the material. If a material has very similar crystal structure to MgO, its DOS plot can be approximate using the MgO optimal grid size. For example, calcium oxide (CaO) has the same crystal structure as MgO and about the same lattice parameter as MgO, we can approximate the optimal grid size for CaO using the optimal grid size determined for MgO.

{| class='wikitable'
!colspan='2'|Conventional unit cell lattice parameter
|-
!MgO <ref name="MgO crystal structure" />
!CaO <ref name="CaO crystal structure" />
|-
|NaCl structure; 0.42127 nm
|NaCl structure; 0.481059(9) nm
|}

This is due to the fact that CaO and MgO have very similar phonon dispersion. Therefore using MgO optimal grid size in the construction of CaO DOS will yield a CaO DOS with the same resolution as that of MgO. For systems with different crystal structure such as Faujasite and lithium metal, a different optimal grid is needed to construct DOSs of these systems, as the phonon dispersion of these systems will be different from the that of MgO. In fact, a smaller grid size can be used in Faujasite and also lithium metal to achieve the same DOS resolution as that of MgO. Faujasite is a zeolite with a very complex structure and its formula unit is (Na<sub>x</sub>,Ca<sub>y</sub>,Mg<sub>z</sub>)<sub>3.5</sub>[Al<sub>7</sub>Si<sub>17</sub>O<sub>48</sub>].32H<sub>2</sub>O (x,y,z=1-2)<ref name="Faujasite group" />. To describe Faujasite using a unit cell, the number of atoms one must includes is at least the number of atoms appear in the formula unit of Faujasite, which is 86 atoms (excluding water molecules). Increasing the number of atoms included in a unit cell will increase the number of branches in phonon dispersion and this is equivalent to folding the phonon dispersion graph for every increase in the number of atoms. Consequently, even with grid size lower than the optimal grid size used for MgO, we can still sample enough Faujasite phonon states to achieve the same resolution as that of MgO. In the case of lithium metal, the optimal grid size will be smaller because of the narrow phonon dispersion. MgO is an ionic solid and the vibration of the ions strongly couple with their ionic neighbors. The structure of lithium metal can be described as metal cations embedded in the sea of delocalised electrons. The sea of electron screens the cation from each other and result in weak vibrational coupling and hence a narrow lithium phonon dispersion. As the phonon states of lithium spread across a relatively narrow range, it is enough to construct the DOS of lithium with grid size smaller than the optimal grid size of MgO to achieve the same resolution as that of MgO.

====Free Energy Using Harmonic Approximation====
Determining the optimal grid size to use in free energy calculation can be done qualitatively by looking at how the DOS vary with grid size, however, this approach is very subjective. To be more quantitative, one can determine the optimal grid size by looking at how the Helmholtz free energy changes with grid size as shown below:

[[File:MgO HFE vs Grid Size hjt14.PNG|thumb|500px|centre| Graph 2: This is a graph of Helmholtz free energy as a function of grid size.]]

The free energy increases significantly with grid size and then reaches the plateau at 5x5x5 and then converges to -40.926483 eV per mole of primitive unit cells at 20x20x20. Choosing an optimal grid size is a balancing act because using grid size that is too large will result in long computer calculation time and using grid size that is too small will give a large error in free energy. The free energy reaches the plateau at 5x5x5, hence grid size beyond 5x5x5 is acceptable depending on the level of accuracy one is after. All the calculations did in the following part of this lab used 20x20x20 grid size as it is the first point where the free energy converge to a constant. The following table summarises the optimal grid size which can be used for different levels of accuracy:

{| class='wikitable'
!Level of Accuracy
!Optimal Grid Size
|-
| -40.926 ± 0.001 eV
|2x2x2
|-
| -40.9265 ± 0.0005 eV
|2x2x2
|-
| -40.9265 ± 0.0001 eV
|3x3x3
|}

====Thermal Expansion of MgO====
The first three sections demonstrated how the Helmholtz free energy was calculated using GULP and in this section the relationship between Helmholtz free energy and volume will be used to plot a graph of volume as a function of temperature. As mentioned in the introduction, simple harmonic approximation does not predict thermal expansion as the equilibrium position of the system does not change with excitation. To predict thermal expansion, geometrical optimisation needs to be carried out. Helmholtz free energy is a function of temperature and volume. For the system under study, we fixed the temperature, hence to minimise the Helmholtz free energy at a particular temperature, the volume of the system must change and this is why the thermal expansion was observed even though harmonic approximation was used in computing the free energy. Because of these extra steps, this method is known as Quasi-Harmonic approximation. Using this method, the Helmholtz free energy and volume of a primitive unit cell of MgO were computed as a function of temperature and the results are shown below:

{| class='wikitable'
! Graph (Free Energy)
! Graph (Volume of a Primitive Cell)
|-
|[[File:MgO HFE vs T hjt14.PNG|thumb|500px|Graph 3: This is a graph Helmholtz free energy of MgO as a function of temperature computed using a grid size of 20x20x20.]]
|[[File:MgO LP vs T hjt14.PNG|thumb|500px|Graph 4: This is a graph lattice parameter of primitive cell of MgO as a function of temperature computed using a grid size of 20x20x20.]]
|}

As the temperature increases, the free energy of MgO decreases and the lattice parameter of primitive cell of MgO increases. In other words, the MgO crystal expands with temperature and this causes the free energy to decrease. The pressure of the system was fixed at 0 Pascal, therefore the decrease in free energy is due to the positive change in entropy and not the work done against external pressure. The change in entropy is positive because by increasing the unit cell volume we make the system more disorder as there are more ways to arrange the ions in space.

The gradient in both graphs is relatively small at low temperature. Beyond 200 K, the magnitude of both gradients increase significantly and stay constant to give an almost straight line graph. The gradient of volume graph is connected to the expansion coefficient by the following equation:

<math>
\alpha=\frac{1}{V_0} \left(\frac{\partial V}{\partial T}\right)_P
</math>

To calculate the thermal expansion coefficient, a linear fit was used to obtain the gradient of volume data points collected beyond 300K (See Graph 5). The gradient was then multiplied with <math>\frac{1}{V_0}</math> to yield <math>\alpha</math> ; <math>V_0</math> is primitive cell volume at 300K. The <math>\alpha</math> was then compared with literature value (average of a series of data point measured from 300K to 1000K). The result is shown below:

{| class='wikitable'
! colspan='2'| Thermal Expansion Coefficient, <math>\alpha</math>/K <sup>-1</sup>
|-
!Quasi-Harmonic Prediction
!Experimental Result
|-
|2.812797 x 10<sup>-5</sup>
|3.99 x 10<sup>-5</sup> <ref name="MgO alpha literature"/>
|}

Albeit the simplification in the potential surface used, the result obtained is in the right order of magnitude as literature value. The discrepancy between computed value and experimental value is due to the fact that the free energy contribution of the anharmonicity of potential surface was neglected.<ref name="Anharmonicity free energy"/>

===Molecular Dynamics Simulation===
To allow for comparison between two theoretical approaches, the volume of MgO as a function of temperature was computed using molecular dynamic and the results are shown below:

[[File:MgO V vs T hjt14.png|thumb|centre|500px| Graph 5: The results from quasi-harmonic approach and MD approach were plotted on the same graph to allow for comparison. The linear fit was performed on data from 300K to 1000K in both cases.]]

There are three interesting features in Graph 5. Within the temperature range under study, the volume of MgO computed using molecular dynamics approach is always lower than volume computed using quasi-harmonic approximation. Furthermore, molecular dynamics predicted a straight line and quasi-harmonic approximation predicted a curve line. Lastly, the difference in results obtained using different approaches decreases as the temperature increases. These observations can be attributed to the different treatment of the potential surface in different approaches. In a molecular dynamic simulation, the vibrational states are not quantised but are treated as a continuous band and there is no zero point energy in each phonon mode. In quasi-harmonic approach, quantisation of energy level and zero point energy are taken into account. In the situation where the volume of MgO at low temperature was computed using quasi-harmonic approximation, the internal energy plays the dominating role in determining the volume of the system as the entropy contribution is small. Having this zero point energy gave MgO studied using quasi-harmonic approach a 'head start in volume' over MgO computed using molecular dynamics. This explains why volume of MgO computed using quasi-harmonic approach at low temperature is significantly greater than that computed using molecular dynamics approach. To explain the curve trend obtained using quasi-harmonic approach and also the decreasing differences between two plot, again, we have to revisit the idea of quantisation of vibrational energy state. At low temperature, <math>k_BT\ll \hbar \omega</math>, the thermal energy is too low to excite the system to higher vibrational state. Consequently, the entropy contribution is close to zero at low temperature and only contribute significant when <math>k_BT\ge \hbar \omega</math>. Therefore it is important to include quantisation of vibrational energy state into our model to give a more accurate result. In fact, the curve trend obtained at low temperature in quasi harmonic approach also appears in experimental result making quasi harmonic approximation a more realistic approach at low temperature <ref name="MgO low T curve trend"/><ref name="MgO low T curve trend 2"/>. At high temperature, the quantisation of vibrational energy states become irrelevant because <math>k_BT\gg \hbar \omega</math>. Relative to the thermal energy, the energy difference of consecutive energy level is very small, making the energy states look more like a band than discrete energy level. For this reason, the volume of MgO computed using quasi-harmonic method approaches the volume of MgO computed using molecular dynamic which the energy states are continuous. When the temperature goes beyond the melting point of MgO, MgO computed using MD will undergo a sharp increase in volume which is characteristic of phase transition. Phase transition happens because the potential surface in MD is anharmonic and bonds between ions are allowed to break. However, in quasi harmonic approximation, sharp change in volume will not be observed as the potential surface is quadratic and bonds between ions will never break.

To put thing into perspective, all the computed thermal expansion coefficient and also experimental result (average of a series of data point measured from 300K to 1000K) are summarised as follow:

{| class='wikitable'
!colspan='3'| Thermal Expansion Coefficient, <math>\alpha</math>/K<sup>-1</sup>
|-
!Quasi-Harmonic Approximation
!Molecular Dynamics
!Experimental Result
|-
|2.812797 x 10<sup>-5</sup>
|3.073744 x 10<sup>-5</sup>
|3.99 x 10<sup>-5</sup> <ref name="MgO alpha literature"/>
|}

Again, the thermal expansion coefficient computed using molecular dynamics simulation is in the correct order of magnitude and is closer to the literature value than that computed using quasi harmonic approximation. The discrepancy between literature value and MD simulated data is believed to arise from the fact that the energy levels were treated as a continuous band (Not the anharmonicity free energy contribution as anharmonicity in potential surface had already been taken into account).

==Conclusion==
In conclusion, the thermal expansion coefficient, <math>\alpha</math>, obtained from quasi harmonic approach and also molecular dynamics are in close agreement with literature value for temperature ranging from 300K to 1000K. Albeit the limitation at low temperature in our case, computation chemistry is indeed a very powerful tool as it allow us to approximate properties of material. The ability to predict and approximate material properties is important in guiding the direction of material synthesis. Instead of trial and error, one can attempt to synthesis of just a few materials with desirable properties predicted by computation chemistry. The relevance of the prediction to reality rely heavily on how well the theory used fits the reality and also the approximation invoked. This lab demonstrated exactly this; at low temperature, quantisation of energy level becomes important and needs to be taken account, making molecular dynamics simulation a relatively poor model to study the thermal properties of MgO. However, at high temperature, quantisation of energy level becomes irrelevant and both methods gave very similar results. The takeaway is one should not blindly follow a particular computation method but understand the approximation and see whether the approximation is compatible the reaction condition in order to obtain meaningful results.

==References==
<references>
<ref name="MgO crystal structure">MgO Crystal Structure, http://materials.springer.com/isp/crystallographic/docs/sd_0561181 (access date: 8th March 2017)</ref>
<ref name="MgO Introduction">G. Mallia, Introduction to the Computational Laboratory, http://www.ch.ic.ac.uk/harrison/Teaching/Thermal_Expansion/Introduction.pdf </ref>
<ref name="MgO lattice energy literature">W. M. Haynes, ed., CRC Handbook of Chemistry and Physics, 97th Edition (Internet Version 2017), CRC Press/Taylor & Francis, Boca Raton, FL.</ref>
<ref name="CaO crystal structure">CaO Crystal Structure, http://materials.springer.com/isp/crystallographic/docs/sd_1400207 (access date: 8th March 2017)</ref>
<ref name="Faujasite group">Faujasite Subgroup, https://www.mindat.org/min-35126.html (access date: 8th March 2017)</ref>
<ref name="MgO alpha literature"> O. L. Anderson and K. Zou,'' J. Phys. Chem. Ref. Data'', 1990, '''19''', 69–83. </ref>
<ref name="Anharmonicity free energy">Z. Wu and R. M. Wentzcovitch, 2006, arXiv:cond-mat/0606745v1</ref>
<ref name="MgO low T curve trend"> M. A. Durand, ''Physics.'', 1936, '''7''', 297.</ref>
<ref name="MgO low T curve trend 2">D. K. Smith and H. R. Leider, ''J. Appl. Crystallogr.'', 1968, '''1''', 246–249.</ref>
</references>

Rep:Y3C Liquid Simulation:hjt14

2025-09-01T09:50:36Z

Move page script: Move page script moved page Y3C Liquid Simulation:hjt14 to Rep:Y3C Liquid Simulation:hjt14: Move to report namespace

==Section 1: Running First Simulation==
No task.

==Section 2: Introduction to Molecular Dynamics Simulation==
===Numerical Integration===
====Task====
[[Media:HO hjt14.xls| Link]] to the completed HO.xls file. The followings are the graph constructed using data from the excel worksheet. The first graph is an overlap of two graph of position of simple harmonic oscillator, <math> x(t) </math>, as a function of time, <math> t</math>; one calculated using velocity verlet algorithm, and one calculated using <math> x\left(t\right) = A\cos\left(\omega t + \phi\right)</math>. The second graph is the graph of total energy of the simple harmonic oscillator, <math> E_{total}(t) </math>, as a function of time, <math> t</math>. The third graph is the graph of absolute error (difference between in position, <math> x(t) </math>, calculated using classical harmonic oscillator solution and calculated using velocity verlet algorithm) as a function of time, <math> t</math>. The maximas were located from each parabola and were fitted with a linear function. The equation of the linear function is displayed on the graph.

{| class="wikitable" border="1"
! Graphs from excel file
|-
| [[File:Graph of Position of Simple Harmonic Oscillator Against Time (Time Step= 0.100).PNG|x500px|thumb|Graph 1]]
|-
| [[File:Graph of Total Energy of Simple Harmonic Oscillator Against Time (Time Step= 0.100).PNG|x500px|thumb|Graph 2]]
|-
| [[File:Graph of Absolute Error Against Time (Time Step= 0.100).PNG|x500px|thumb|Graph 3]]
|}

Since we are working in a close system, i.e., no energy exchange with surrounding, the total energy of the system should be a constant. What we observed from graph 2 is that the total energy of the system oscillates about an average value. The deviation from the average energy is treated as the error introduced by the approximation we made in velocity verlet algorithm. The percentage error of total energy at different timestep were computed using following equation and the results are summarized in graph 4:

Percentage Error <math> = \frac{(E_{max}-E_{min})}{2\times E_{average}} \times100 % </math>

[[File:Graph of Percentage Error of Total Energy Against Timestep.PNG|x500px|thumb|centre|Graph 4]]

It can be seen from graph 4 that for timestep beyond 0.28, the percentage error in total energy will exceed 1%.

===Atomic Forces===
====Task====
Lennard-Jones Equation: <math>\phi (r)=4\epsilon \left( \frac{\sigma ^{12}}{r^{12}}-\frac{\sigma ^6}{r^6} \right) </math>

When potential energy is set to zero, i.e.<math>\phi =0 </math>:

<math> 0=4\epsilon \left( \frac {\sigma ^{12}}{r^ {12}} - \frac {\sigma ^6}{r ^6} \right) </math>

<math>\frac {\sigma ^ {12}}{r^{12}} = \frac{\sigma ^6}{ r^6} </math>

<math>\frac {\sigma ^ 6}{r^ 6} = 1 </math>

<math>\sigma = r = r_0</math>

The value of <math>r</math> when potential energy <math>\phi</math> is set to zero,<math> r_0 </math>, is equal to <math>\sigma </math>.

The force acting on a particle a distance <math> r </math> away from another particle can be calculated using following equation:

<math> F= - \frac {d\phi(r)}{dr} </math>

<math> F= 4 \epsilon \left( \frac{12\sigma^{12}}{r^{13}} - \frac {6\sigma^6}{r^7} \right)</math>

When <math> r= r_0 =\sigma </math>:

<math>F=4\epsilon \left( \frac{12\sigma^{12}}{\sigma^{13}}-\frac{6\sigma^6}{\sigma^7} \right) </math>

<math>F=4\epsilon \left( \frac{12}{\sigma}-\frac{6}{\sigma} \right)</math>

<math>F=\frac{24\epsilon}{\sigma} </math>

At distance <math> r_0 </math>, where the potential energy, <math>\phi</math>, is zero, the force acting on a particle is equal to <math>\frac{24\epsilon}{\sigma} </math>.

When <math>F=0</math>, defines <math> r=r_{eq}</math>:

<math> F=0= 4 \epsilon \left( \frac{12\sigma^{12}}{r_{eq}^{13}} - \frac{6\sigma^6}{r_{eq}^7} \right) </math>

<math> \frac{12\sigma^{12}}{r_{eq}^{13}} = \frac{6\sigma^6}{r_{eq}^7} </math>

<math> 2\sigma^6=r_{eq}^6 </math>

<math> r_{eq}= \sqrt[6]{2} \sigma </math>

The equilibrium distance,<math> r_{eq} </math> , between 2 particles, which is defined as separation of 2 particles when the force acting on each particle is zero, is equal to <math>\sqrt[6]{2} \sigma </math>, <math> \sigma </math> is a constant and is specific to particle under study. At equalibrium distance, <math> r_{eq} </math>, the potential energy, <math> \phi(r_{eq}) </math>, is at minimum and can be caculated as follow:

<math> \phi(r_{eq}) = 4 \epsilon \left(\frac{\sigma^{12}}{(\sqrt[6]{2}\sigma)^{12}}-\frac{\sigma^6}{(\sqrt[6]{2}\sigma)^6} \right) </math>

<math> \phi(r_{eq}) = 4 \epsilon \left(\frac{1}{4}-\frac{1}{2} \right) </math>

<math> \phi(r_{eq}) = -\epsilon </math>

The following part shows the working of the integration of potential energy, <math> \phi </math>, with respect to <math> r </math>. The integral is equivalent to the area under the potential energy graph encompassed by the integration range, <math> x </math> to <math> y</math>.

<math> \int_x^y \phi(r) dr = \int_x^y 4\epsilon \left(\frac{\sigma^{12}}{r^{12}}-\frac{\sigma^6}{r^6} \right) dr </math>

<math> \int_x^y \phi(r) dr = 4\epsilon \left[\frac{-\sigma^{12}}{11r^{11}}+\frac{\sigma^6}{5r^5} \right]_x^y </math>

When <math> \epsilon =1</math> and <math> \sigma=1 </math>:

<math> \int_x^y \phi(r) dr = 4 \left[\frac{-1^{12}}{11r^{11}}+\frac{1^6}{5r^5} \right]_x^y </math>

{| class="wikitable" border="1"
|When <math> x=2\sigma=2 </math> and <math> y=\infty </math>:
|When <math> x=2.5\sigma=2.5 </math> and <math> y=\infty </math>:
|When <math> x=3\sigma=3 </math> and <math> y=\infty </math>:
|-
|<math> \int_2^\infty \phi(r) dr = 4 \left[\frac{-1^{12}}{11r^{11}}+\frac{1^6}{5r^5} \right]_2^\infty </math>

<math> \int_2^\infty \phi(r) dr = 4 \left[0-\left(\frac{-1^{12}}{11(2^{11})}+\frac{1^6}{5(2^5)} \right)\right] </math>

<math> \int_2^\infty \phi(r) dr = -\frac{699}{28160} = -0.0248 </math>
|<math> \int_{2.5}^\infty \phi(r) dr = 4 \left[\frac{-1^{12}}{11r^{11}}+\frac{1^6}{5r^5} \right]_{2.5}^\infty </math>

<math> \int_{2.5}^\infty \phi(r) dr = 4 \left[0-\left(\frac{-1^{12}}{11(2.5^{11})}+\frac{1^6}{5(2.5^5)} \right)\right] </math>

<math> \int_{2.5}^\infty \phi(r) dr = -0.00818 </math>
|<math> \int_3^\infty \phi(r) dr = 4 \left[\frac{-1^{12}}{11r^{11}}+\frac{1^6}{5r^5} \right]_3^\infty </math>

<math> \int_3^\infty \phi(r) dr = 4 \left[0-\left(\frac{-1^{12}}{11(3^{11})}+\frac{1^6}{5(3^5)} \right)\right] </math>

<math> \int_3^\infty \phi(r) dr = -0.00329 </math>
|}

===Periodic Boundary Conditions===
====Task====
The number of water molecules in 1 ml of water under standard conditions (298K and 1 atm) is calculated as follow:

Density of water under standard conditions<math> = 0.9970 g ml^{-1} </math>

Mass of 1 ml of water <math>=0.9970 g </math>

Molar mass of water <math> = 2(1.008)+15.999 = 18.015 gmol^{-1} </math>

Number of water molecule in 1 ml of water <math> = \frac{0.9970(6.022\times 10^{23})}{18.015} = 3.333\times10^{22} </math>

The volume occupied by 1000 water molecules under standard conditions is calculated as follow:

Mass of 1000 water molecules <math>=\frac{1000}{6.022\times10^{23}} (18.015) = 2.992\times 10^{-20}g </math>

Volume occupied by 1000 water molecules <math> = \frac{2.992\times 10^{-20}}{0.9970}=3.001\times 10^{-20}ml </math>

====Task====
Under periodic boundary condition, the final position of an atom, after it move from
<math>
\begin{pmatrix}
0.5\\
0.5\\
0.5
\end{pmatrix}
</math>
by
<math>
\begin{pmatrix}
0.7\\
0.6\\
0.2
\end{pmatrix}
</math>
in a cubic simulation box which runs from <math> \left(0, 0, 0\right)</math> to <math>\left(1, 1, 1\right) </math>, can be calculated as follow:

New position of atom <math> =
\begin{pmatrix}
0.5\\
0.5\\
0.5
\end{pmatrix}
+
\begin{pmatrix}
0.7\\
0.6\\
0.2
\end{pmatrix}
=
\begin{pmatrix}
1.2\\
1.1\\
0.7
\end{pmatrix}
</math>

Apply periodic boundary condition:

New position of atom
<math>=
\begin{pmatrix}
1.2-1\\
1.1-1\\
0.2
\end{pmatrix}
=
\begin{pmatrix}
0.2\\
0.1\\
0.2
\end{pmatrix}
</math>

===Reduced Units===
====Task====
{| class="wikitable" border="1"
|The real units, <math> r </math>, is calculated as follow:
|The well depth in <math> kJ mol^{-1} </math> is calculated as follow:
|The reduced temperature <math> T^* = 1.5 </math> in real units is calculated as follow:
|-
|
<math> r^*= \frac{r}{\sigma} </math>

<math> r^*= 3.2 </math> and <math> \sigma= 0.34 nm </math>

<math> r= 3.2(0.34) nm </math>

<math> r= 1.088 nm </math>
|
<math> \frac{\epsilon}{N_Ak_B} =120 K </math>

<math> \frac{\epsilon}{R} =120 K </math>

<math> \epsilon = 120(8.31) J mol^{-1} </math>

<math> \epsilon = 997 J mol^{-1}= 0.997 kJ mol^{-1} </math>
|
<math> T^*=\frac{k_BT}{\epsilon} </math>

<math> 1.5= \frac{1}{120}T </math>

<math> T= 180 K </math>
|}

==Section 3: Equilibration==
===Creating the Simulation Box===
====Task====
Within an input file for molecular simulation, the initial conditions stipulated may not be the equilibrium conditions. During a simulation, LAMMPS will try to bring the system down to the equilibrium state by iteration within the allowed timesteps specified in the input file. If random coordinate is assigned to each atom, there is a chance that two or more atoms will be assigned coordinates that are too close together. This will mean that the atoms overlap in space. According to Lennard-Jones equation, this condition will give rise to the extremely high potential energy (approach infinity) and large repulsion between atoms. As a result of this situation, calculations done using LAMMPS sumulator will run into errors and the system will not be brought to equilibrium state.

====Task====
A unit cell of a simple cubic lattice is a cube dotted with lattice point at each vertice. Each vertice is shared between 4 unit cells, hence each vertice will contribute a quarter of a lattice point to a unit cell. Consequently, each unit cell will have lattice point, 4 corners each contributes a quarter of a lattice point. The number density can be calculated by dividing the number of lattice point in a unit cell by the volume of a unit cell.

If we have a unit cell of size <math> 1.07722 \times 1.07722 \times 1.07722 </math> (in redued units):

Number density of a simple cubic lattice <math> = \frac{1}{1.07722 \times 1.07722 \times 1.07722} =0.8 </math>

If we have a face-centered cubic lattice with lattice point number density of <math>1.2</math>, we can calculate the size of unit cell as follow:

Number of lattice point in a face-centered cubic unit cell <math> =4</math>

Volume of unit cell <math>= l\times l\times l = l^3 </math>

<math> l^3= \frac{4}{1.2} =3\frac{1}{3} </math>

<math> l=\sqrt[3]{3\frac{1}{3}} =1.49 </math>

The lattice parameter of a face-centered cubic lattice with lattice point number density of <math>1.2</math> is <math> 1.49 </math> (in reduced units).

====Task====
The number of unit cell created in the defined region is <math> 1000 </math>. In each face-centered cubic unit cell, <math> 4 </math> atoms will be generated. Consequently, the total number of atom that will be generated if a face-centered cubic lattice is used instead of a simple cubic lattice is <math>4000</math>.

===Setting the Properties of the Atoms===
====Task====
{| width=50%
| <pre>
mass 1 1.0
pair_style lj/cut 3.0
pair_coeff * * 1.0 1.0
</pre>
|}

The first line of the command set the mass of the atom of type 1 to 1.0 (reduced units). The second line of the command defines the equation used to model the pairwise interaction energy between atoms. In this example, we are using Lennard-Jones Equation with a cutoff distance of 3.0 (reduced units). The third line of the command defines the coefficients of Lennard-Jones Equation, i.e <math> \epsilon =\sigma = 1.0 </math>. The asterisks (* *) mean that the Lennard-Jones model will be imposed on every pair of atoms.

====Task====
Given that we are specifying <math>x_i\left(0\right) </math> and <math>v_i\left(0\right) </math>, the integration algorithm that we will be using is velocity verlet algorithm.

===Running the Simulation===
====Task====
<pre>
### SPECIFY TIMESTEP ###
variable timestep equal 0.001
variable n_steps equal floor(100/${timestep})
variable n_steps equal floor(100/0.001)
timestep ${timestep}
timestep 0.001

### RUN SIMULATION ###
run ${n_steps}
run 100000
</pre>

By assigning a variable (timestep) to timestep (the value), it allows us to call 'timestep' at other part of the input script, meaning that we do not have to retype the exact value of timestep repeatly and allow us to call it simply by typing ${timestep}. Another advantage of assigning a variable (timestep) to timestep (the value), is that if we need to change the timestep of a simulation, we only need to change the value assigned to the variable 'timestep' and timestep in other part of the input file will be updated accordingly.

===Checking Equilibrium===
====Task====
{| class="wikitable" border="1"
|+ Graphs Obtained at 0.001 Timestep
|-
| [[File:Intro 0.001 Timestep total energy hjt14.png|thumb|500px|Graph 5]]
| [[File:Intro 0.001 Timestep Temperature hjt14.png|thumb|500px|Graph 6]]
| [[File:Intro 0.001 Timestep Pressure hjt14.png|thumb|500px|Graph 7]]
|}

Above show three graphs obtained from LAMMPS simulation ran at 0.001 timestep. It can be seen that the system did achieved equilibrium and the inset in each graph show that system achieve equilibrium in approximately 0.4 fs. LAMMPS simulations were also ran at other timestep and the results for total energy were summarised in graph 8.

[[File:Intro Total Energy Diff Timestep hjt14.png|thumb|centre|500px|Graph 8]]

From graph 8, it can be seen that the most suitable timestep is 0.0025 as it is the maximum value which still allow the system under study to achieve equilibrium (a compromise between accuracy and time). 0.015 is a particularly bad choice because the system simulated at that timestep did not converge.

==Section 4: Running Simulations under Specific Conditions==
===Thermostats and Barostats===
====Task====
In order to bring the system to target temperature, <math> \mathfrak{T} </math>, we multiply the velocity of each particle with a constant factor, <math>\gamma </math>, which will help to modulate the velocity of each atoms. The factor,<math>\gamma </math>, can be derived as follow:

<math> E_K=\frac{1}{2} \sum_i m_i v_i^2 = \frac{3}{2}NK_BT </math>

Let <math> T= \mathfrak{T} </math> and multiply velocity of each atom with a constant factor, <math>\gamma </math>

<math> E_K=\frac{1}{2} \sum_i m_i \gamma^2v_i^2 = \frac{3}{2}NK_B\mathfrak{T} </math>

<math> \gamma^2 \frac{1}{2} \sum_i m_i v_i^2 = \frac{3}{2}NK_B\mathfrak{T} </math>

<math> \gamma^2 \frac{3}{2}NK_BT = \frac{3}{2}NK_B\mathfrak{T} </math>

<math> \gamma = \sqrt{\frac{\mathfrak{T}}{T}} </math>

===Examining the Input Script===
====Task====
<pre>
fix aves all ave/time 100 1000 100000 v_dens v_temp v_press v_dens2 v_temp2 v_press2
</pre>

The numbers and its position in the series will affect how the averages are calculated. The roles that each number play are summarised as follow:

{| class="wikitable"
! Position
! Value
! Role
|-
| First
| 100
| Sample data every 100 timestep
|-
| Second
| 1000
| Perform sampling 1000 times
|-
|Third
|100000
|Average all the sampled data collected within 100000 timestep
|}

===Plotting the Equations of State===
====Task====
{| class="wikitable"
! colspan='2'|Conditions used
! rowspan='2'|Graph
|-
!Pressure
!Temperature
|-
| rowspan='5'| 2.0
| 1.8
| rowspan='10'| [[File:Section 4 NPT graph hjt14.png|thumb|500px|Graph 9]]
|-
|2.0
|-
|2.2
|-
|2.4
|-
|2.6
|-
| rowspan='5'| 3.0
| 1.8
|-
|2.0
|-
|2.2
|-
|2.4
|-
|2.6
|}

Shown in the table, are 10 phase points used in LAMMPS molecular dynamic simulation to compute density under these conditions. The theoretical value of density at these phase points were also calculated using reduced ideal gas equation. The simulated density and calculated density are summarised in graph 9. The reduced ideal gas equation used in calculation is shown below:

<math>

\rho =\frac{P}{T^*}; \rho=\frac{N}{V}

</math>

In graph 9, it can be seen that:

a) at all temperature, the simulated density are lower than the theoretical density.

Explanation: In ideal gas, we assume there are no inter-molecular interactions between particles, i.e., no attraction and no repulsion. However, in our LAMMPS simulation, we model the inter-molecular interactions using Lennord-Jones equation and this breaks the fundamental assumption made in the derivation of ideal gas equation. According to Lennard-Jones equation, below a certain bond distance, a particles pair will experience repulsive forces which will keep them at a distance away from each other. In this case, the particles are further apart from each other than they would be if they behave ideally. Consequently, the number of particle per unit volume, which is the density, is lowered.

b) the simulated density deviate considerably from ideal behavior at low temperature. At higher temperature, the magnitude of deviation decreases and the theoretical density and simulated density start to converge to a single value.

Explanation: At low temperature, the thermal energy, <math> k_BT </math>, is small relative to the interaction energy. Therefore the interaction energy can not be neglected and contribute significantly to the thermodynamic properties of our system. As temperature starts to increase, the thermal energy increases and at one point the magnitude of the thermal energy will become comparable to the interaction energy. At that point, the interactions between particles can be easily perturbed by thermal fluctuation and hence become irrelevant in determining the thermodynamic properties of our system. In other words, at higher temperature, interactions between particles become insignificant and system approaches ideal behavior. Therefore, simulated density and theoretical density converge to a single value.

==Section 5: Calculating Heat Capacities Using Statistical Physics==
====Task====
[[Media:Nvt(0.2, 2.0, 0.0025).in|Link]] to one of the input file.
{| class="wikitable"
! colspan='2'|Conditions used
! rowspan='2'|Graph
|-
!Density
!Temperature
|-
| rowspan='5'| 0.2
| 2.0
| rowspan='10'| [[File:Section 5 CvV vs T.png|thumb|500px|Graph 10]]
|-
|2.2
|-
|2.4
|-
|2.6
|-
|2.8
|-
| rowspan='5'| 0.8
| 2.0
|-
|2.2
|-
|2.4
|-
|2.6
|-
|2.8
|}

Pressure is defined as <math> \frac{N}{V} </math>, at higher pressure there will be more particle per unit volume. In section 4, we encounter following equation which correlates the instantaneous temperature of a system to total kinetic energy:

<math>

E_K = \frac{3}{2} N k_B T = \frac{1}{2}\sum_i m_i v_i^2

</math>

Occupying the same volume, a system at higher pressure will have a greater number of particles. Therefore, for the same increment in temperature, the energy we need to provide to a system at higher density is greater than that of a system at lower pressure simply because there are more particles to which we have to provide the energy. Consequently, the heat capacity per unit volume of a system at higher pressure is greater than that of system at lower pressure simply because we need to provide more energy for the same increment in temperature.

According to equipartition theorem, each degree of freedom of an atom will contribute <math> \frac{1}{2} k_B </math> to the total heat capacity of the system. As we are modeling a constant number of atoms, we expect the heat capacity of the system to be a constant. For both graphs, there is no clear reason to why the heat capacity reduces with temperature. One possible reason is that the density of state decreases with temperature. Higher density of state means that the energy levels are closer together and are easy to populate the energy states and hence correlate to lower heat capacity; Lower density of state means that the energy gap between energy levels is larger and and are difficult to populate the energy states. For same increment in temperature, greater energy need to be provided, and hence this correlate to higher heat capacity. One way to verify this is to obtain a plot of density of state versus energy.

==Section 6: Structural Properties and the Radial Distribution Function==
===Calculating <math>g(r) </math> in VMD===
====Task====
<math> g(r)</math> can be understood as the number of particle per unit volume of shell of size <math> dr </math> divided by the bulk density.

{|class="wikitable"
!Graphs
!Observed trends in graphs when going from solid to liquid and to gas
!Remark
|-
|rowspan='3'| [[File:Part 6 gr hjt14.png|thumb|600px|Graph 11]]

[[File:Part 6 int g hjt14.png|thumb|600px|Graph 12]]

|The peak intensity decreases and has greater width
|In solid, the atoms are closely packed and hence are in higher coordination environment and this translates into high <math> g(r) </math>. From solid to gas, the coordination environment drop as the atoms are more loosely packed in this case and hence <math>g(r)</math> decreases from solid to gas. The width of the peak is related to the distribution of <math> r </math>of atoms in the first coordination shell, then second coordination shell and etc.. In solid, the peaks are more confined than that of liquid and gas. This is because solid has low degree of freedom in motion, i.e., no translation, hence <math> r </math> of atoms in each coordination shell has smaller distribution and this gives rise to relatively sharp peaks. From liquid to gas, the translational degree of freedom increases, and hence the position of atoms in coordination shell becomes less and less well defined. This uncertainty in <math>r </math>gives rise to wide peaks.
|-
|The rate of decay of <math> g(r) </math> increases
|This is because solid has long range order in the spatial arrangement of atoms. In liquid and gas, the ordered structure is short range and over <math> r</math> the peaks flattened out as atoms distribution is getting more and more random.
|-
|Coordination environment decreases
|'''REFERRING TO SOLID''' The first coordination shell of solid consisted of 12 particles (cuboctahedron) and corresponds to the first peak of solid <math>g(r)</math> plot. The second coordination shell of solid consists of 6 particles (octahedron) and corresponds to the second peak of solid <math>g(r)</math> plot. The third coordination shell of solid consists of 24 particles and corresponds to the third peak of solid <math>g(r)</math> plot. The position of atoms in these shells are shown below relative to a reference point (coloured black):

[[File:FCC neighbour.jpg|thumb|400px|centre|FCC lattice made using crystal maker. Black atom is the reference atom, yellow atoms are in the first coordination shell, red atoms are in the second coordination shell and grey atoms are in the third coordination shell.]]

The lattice spacing of FCC lattice of solid is the same as the radius of the second coordination shell and is equal to 1.475 (refer graph 11)

|}

==Section 7: Dynamic Properties and the Diffusion Coefficient ==
===Mean Squared Displacement===
====Task====
{| class="wikitable"
|
[[File:Part 7 Diffusion hjt146.png|thumb|centre|500px|Graph 13]]
|
[[File:Part 7 Diffusion hjt14.png|thumb|centre|500px|Graph 14]]
|
{| class='wikitable'
!State
!Gradient (w.r.t. time)
!Diffusion Coefficient (w.r.t. time)
|-
|Gas (million)
|18.19 (linear part)
|3.032
|-
|Gas
|4.915
|0.8192
|-
|Liquid (million)
|0.5236
|0.08727
|-
|Liquid
|0.5094
|0.0849
|-
|Solid (million)
|0.00002635
|0.000004392
|-
|Solid
|0.00535
|0.000892
|}

|}

From graph 13, it can be seen that the diffusion coefficients increase from solid to gas and these are expected. In solid, the particles are closely packed, to diffuse, solid particles have to overcome large repulsion when sliding pass other particles and this contribute to the large activation energy for diffusive motion. Hence solid has negligible diffusion coefficient due to large energy barrier. From liquid to gas, the packing decreases meaning that the particles are further apart, making it easier for particle to move pass each other. The is reflected in the increase in diffusion coefficient when going from liquid to gas.

===Velocity Autocorrelation Function===
====Task====
<math>
x(t)= A \cos \left(\omega t+\phi \right) ; v(t)=\frac{dx(t)}{dt}=-A \omega \sin\left(\omega t+ \phi \right)
</math>

<math>
C\left(\tau\right) = \frac{\int_{-\infty}^{\infty} v\left(t\right)v\left(t + \tau\right)\mathrm{d}t}{\int_{-\infty}^{\infty} v^2\left(t\right)\mathrm{d}t}
</math>

<math>
C\left(\tau\right) = \frac{\int_{-\infty}^{\infty} A^2 \omega^2 \sin\left(\omega t+ \phi \right) \sin\left(\omega t+ \omega \tau +\phi \right)\mathrm{d}t}{\int_{-\infty}^{\infty} A^2 \omega^2 \sin^2\left(\omega t+ \phi \right)\mathrm{d}t}
</math>

<math>
C\left(\tau\right) = \frac{\int_{-\infty}^{\infty}A^2 \omega^2 \sin \left(\omega t+ \phi \right) \left( \sin \left( \omega t + \phi \right) \cos \omega \tau + \cos \left( \omega t + \phi \right) \sin \omega \tau \right)\mathrm{d}t }{\int_{-\infty}^{\infty} \frac {A^2 \omega^2}{2} \left(1-\cos \left( 2\omega t + 2 \phi\right)\right) \mathrm{d}t}
</math>

<math>
C\left(\tau\right) = \frac{A^2 \omega^2\int_{-\infty}^{\infty} \sin^2 \left(\omega t+ \phi \right) \cos \omega \tau + \sin \left(\omega t+ \phi \right) \cos \left( \omega t + \phi \right) \sin \omega \tau \mathrm{d}t }{\frac {A^2 \omega^2}{2}\int_{-\infty}^{\infty} \left(1-\cos \left( 2\omega t + 2 \phi\right)\right) \mathrm{d}t}
</math>

<math>
C\left(\tau\right) = \frac{\int_{-\infty}^{\infty}\left(1-\cos \left( 2\omega t + 2 \phi\right) \right) \cos \omega \tau + 2\sin \left(\omega t+ \phi \right) \cos \left( \omega t + \phi \right) \sin \omega \tau \mathrm{d}t }{\int_{-\infty}^{\infty} \left(1-\cos \left( 2\omega t + 2 \phi\right)\right) \mathrm{d}t}
</math>

<math>
C\left(\tau\right) = \frac{ \cos \omega \tau \int_{-\infty}^{\infty}\left(1-\cos \left( 2\omega t + 2 \phi\right) \right) \mathrm{d}t }{\int_{-\infty}^{\infty} \left(1-\cos \left( 2\omega t + 2 \phi\right)\right) \mathrm{d}t}
+ \frac{2\int_{-\infty}^{\infty}\sin \left(\omega t+ \phi \right) \cos \left( \omega t + \phi \right) \sin \omega \tau \mathrm{d}t }{\int_{-\infty}^{\infty} \left(1-\cos \left( 2\omega t + 2 \phi\right)\right) \mathrm{d}t}
</math>

<math>
C\left(\tau\right) = \cos \omega \tau
+ \frac{2\sin \omega \tau \int_{-\infty}^{\infty}\sin \left(\omega t+ \phi \right) \cos \left( \omega t + \phi \right) \mathrm{d}t }{\int_{-\infty}^{\infty} \left(1-\cos \left( 2\omega t + 2 \phi\right)\right) \mathrm{d}t}
</math>

<math>
C\left(\tau\right) = \cos \omega \tau
+ \frac{\sin \omega \tau \left[\sin ^2\left(\omega t+ \phi \right) \right]_{-\infty}^{\infty} }{\int_{-\infty}^{\infty} \left(1-\cos \left( 2\omega t + 2 \phi\right)\right) \mathrm{d}t}
</math>

The second term goes to zero and then we are left with:

<math>
C\left(\tau\right) = \cos \omega \tau
</math>

====Task====
{|
|[[File:Section 7 VACF graph hjt14.png|thumb|500px|Graph 15]]
|[[File:Section 7 VACF and SHO.png|thumb|500px|Graph 16]]
|}

<math>
C\left(\tau\right) = \left\langle \mathbf{v}\left(t\right) \cdot \mathbf{v}\left(t+\tau\right)\right\rangle
</math>

Equation shown above projects velocity at <math> \left(t+\tau\right) </math> onto velocity at <math> \left(t\right) </math> and what we obtain is <math> C\left(\tau\right) </math>, a number which quantifies the correlation between the two velocities. From gas to solid, the rate of decay of <math> C\left(\tau\right) </math> increases and this is related to the rate of collision. In gas and liquid, particles collide with one another, however, in liquid, the frequency of collision is higher. Rapid exchange of energy between liquid particle causes a liquid particle to lost the initial velocity component at a faster rate than that of gas. From liquid to solid, the situation changes slightly. In a solid, unlike gas and liquid, the particles only vibrate about the equilibrium position. Because the motion of solid particle is confined within a small space, it correlate to initial velocity to a longer period of time. The minima in solid and liquid VACF vs time graph (see Graph 15 and Graph 16) is related to back scattering of particle after collision, hence the <math> C(\tau) </math> is negative in sign (inverse of the initial course).

In simple harmonic oscillator the particles are vibrating about an equilibrium position and there is no collision with another particle, therefore it retain the memory of the initial velocity throughout the simulation. The simple harmonic oscillator aims to illustrate that the velocity correlation in ideal system will continue for infinite amount of time.

====Task====
{| class='wikitable'
| [[File:Section 7 Running Int hjt14.png|thumb|500px|Graph 17]]
|
{|class='wikitable'
|Phase
|Average value of plateau region of graph
|Diffusion Coefficient from VACF
|Diffusion Coefficient from MSD
|-
|Solid (million)
|0.000192943
|0.000064314
|0.000004392
|-
|Solid
|0.00345
|0.00115
|0.000892
|-
|Liquid (million)
|0.25621
|0.08540
|0.08727
|-
|Liquid
|0.2316
|0.0772
|0.0849
|-
|Gas (million)
|9.8054
|3.2685
|3.032
|-
|Gas
|2.44197
|0.81399
|0.8192
|}
|}

The diffusion coefficients calculated using two different methods are in very close agreement and this is expected.

'''Source of error'''

According to following equation:

<math>
D = \frac{1}{3}\int_0^\infty \mathrm{d}\tau \left\langle\mathbf{v}\left(0\right)\cdot\mathbf{v}\left(\tau\right)\right\rangle
</math>

The integration range is <math>0</math> to <math>\infty</math>. However in our calculation the integration was stopped at time<math> = 10 </math>, and this could contribute to the error of data. Another source of error is the integration method. By using trapezium rule, we are dividing the graph into different rectangular strips. However, the graph itself can not be be describe perfectly using rectangular strips and will eventually result in overestimation (strips bigger than graph) or underestimation (strips smaller graph).

Rep:Y3CMPCG1417

2025-09-01T09:50:33Z

Move page script: Move page script moved page Y3CMPCG1417 to Rep:Y3CMPCG1417: Move to report namespace

==Section 1 - Introduction to the Ising Model==

===TASK: Show that the lowest possible energy for the Ising model is <math>E = -DNJ</math>, where D is the number of dimensions and N is the total number of spins. What is the multiplicity of this state? Calculate its entropy.===

Consider a 1D row of lattice sites of N=3 with spin configuration [+1][+1][+1].

Mathematically the interaction energy is defined asː
<math> -\frac{1}{2} \ J \ \sum^{N}_{i} \sum^{}_{j \ \epsilon \ neighbours (i)} s_{i} s_{j} </math> where J is a constant and <math>s_{i}s_{j}</math> is the product between two spins in adjacent lattice sites.

The sum of the interaction energies <math> \sum^{N}_{i} \sum^{}_{j \ \epsilon \ neighbours (i)} s_{i} s_{j}</math> can be considered as the sum of the individual interaction energies between spinsː
<math> \sum^{N}_{i} \sum^{}_{j \ \epsilon \ neighbours (i)} s_{i} s_{j} = \epsilon_{12} + \epsilon_{23} + \epsilon_{13} + \epsilon_{21} + \epsilon_{32} + \epsilon_{31} </math>.

Although lattice sites 1 and 3 are not adjacent they are said to still interact according to the periodic boundary conditions applied.

However, <math>\epsilon_{12} = \epsilon_{21} </math> and <math>\epsilon_{23} = \epsilon_{32}</math> and <math>\epsilon_{13} = \epsilon_{31}</math> which means that all of the interactions within the system are counted twice, hence the total energy needs to be halved, resulting in the following formula being obtainedː <math> \sum^{N}_{i} \sum^{}_{j \ \epsilon \ neighbours (i)} s_{i} s_{j} = 2\epsilon_{12} + 2\epsilon_{13} + 2\epsilon_{23} </math>.

It can be determined that <math>\epsilon_{12} = (+1)(+1) = 1</math> , <math>\epsilon_{13} = (+1)(+1) = 1</math> and <math>\epsilon_{23} = (+1)(+1) = 1</math>.

Thereforeː <math>-\frac{1}{2} \ J \ \sum^{N}_{i} \sum^{}_{j \ \epsilon \ neighbours (i)} s_{i} s_{j} = -\frac{1}{2} \ J \ (2 + 2 + 2) = -\frac{1}{2} \ J \ 6 = - 3 J = -DNJ</math> for a 1D lattice with <math>D=1</math> and 3 lattice sites <math>N=3</math>

The multiplicity of the system,<math>\Omega = \frac{lattice sites!}{n. spin up!n. spin down!}</math>

Entropy, <math>S</math> is defined as <math>S = k_B ln(\Omega)</math> and so in this case <math>S = k_B ln(\frac{100!}{100!}) = 0</math>

===TASK: Imagine that the system is in the lowest energy configuration. To move to a different state, one of the spins must spontaneously change direction ("flip"). What is the change in energy if this happens <math>(D=3, N=1000)</math>? How much entropy does the system gain by doing soʔ===

In a 3D lattice system, each lattice site has three unique interactions with its neighbours to its left, top and front. In the lowest energy configuration, all spins are parallel and for a system the minimum energy is <math>E = -DNJ</math>, so for the system with <math>N=1000</math> and <math>D=3</math>, the minimum energy is <math>-3000J</math>.

If a single spin is flipped, the product of its spin with its neighbours spin reverses and becomes negative and this increases the total energy of the system. Since 3 unique spin-spin interactions are reversed in sign, the total energy increases by <math>+3J</math>, meaning the new total energy is <math>-2997J</math>

Initially the multiplicity of the system will be <math>\Omega = \frac{1000!}{1000!}</math> , and after the flip, the multiplicity becomes <math> \Omega = \frac{1000!}{999!1!}=1000</math>.

The associated change in entropy, <math>\Delta S = k_B ln(1000) - 0 = 6.91 k_B</math>, which is an expected increase in entropy as the number of possible configurations of the system increases.

===TASK: Calculate the magnetisation of the 1D and 2D lattices in Figure 1. What magnetisation would you expect to observe for an Ising lattice with <math>D = 3,\ N=1000</math> at absolute zero?===

[[File:ThirdYearCMPExpt-IsingSketch.png|thumb|left|Figure 1 - Shows 1D (N = 5), 2D (N = 5x5) and 3D (N = 5x5x5) lattices.]]

Magnetisation is defined as <math>M=\sum_{i} s_i</math>. So for the 1D lattice with <math>N = 5</math> in ''Figure 2'', <math>M = +1</math> and for the 2D lattice with <math>N = 25 , M = +1</math> too.

According to the 3rd Law of thermodynamics, entropy is 0 at absolute zero for a perfect crystalline solid, and consequently it is expected that the lattices will have follow suit and have zero entropy at 0K. To have zero entropy all spins must be parallel as such that magnetisation, <math>M = N</math>. For all the spins to be parallel, there is only one possible configuration. So, for a lattice with <math>N = 1000</math> and <math>D =3</math>, if <math>M = N</math>, then multiplicity, <math>\Omega = 1</math> and entropy, <math>S =k_B ln(1) = 0</math>.

<br clear = all >

==Section 2 - Calculating the Energy and Magnetisation==

===TASK: complete the functions energy() and magnetisation(), which should return the energy of the lattice and the total magnetisation, respectively. In the energy() function you may assume that <math>J=1.0</math> at all times (in fact, we are working in reduced units in which <math>J=k_B</math>, but there will be more information about this in later sections). Do not worry about the efficiency of the code at the moment — we will address the speed in a later part of the experiment.===

<pre>def magnetisation(self):
"Return the total magnetisation of the current lattice configuration."
lat=self.lattice #creates lattice and stores it
mag=[]
for i in range(0,len(lat)): #loops through all rows of lattice
for j in range(0,len(lat[i])): #loops through elements of each row
mag+=[lat[i][j]] #adds spin value to mag array
return sum(mag) #sums all spins from mag array
</pre>

<pre> def energy(self):
"Return the total energy of the current lattice configuration."

lat=self.lattice #creates lattice and stores it
left=[]
top=[]

for i in range(0,len(lat)):
for j in range(0,len(lat[i])):
left+=[lat[i][j]*lat[i][j-1]] #multiplies spin by spin to left
top+=[lat[i][j]*lat[i-1][j]] #multiplies spin by spin above it
int_en=left+top #sums spin products from left and top
energy=-sum(int_en) #sums all spin products for each spin to give total

return energy
</pre>

===TASK: Run the ILcheck.py script from the IPython Qt console using the command===

''Figure 2'' shows the results when ILcheck.py was ran on my IsingLattice.py file. The ILcheck.py file was ran several times to ensure the code worked for various random lattices.

[[File:Cg1417ILcheck run.png|thumb|left|500px| Figure 2 - Result from running the ILcheck.py file]]

<br clear=all>

==Section 3 - Introduction to Monte Carlo Simulation==

===TASK: How many configurations are available to a system with 100 spins? To evaluate these expressions, we have to calculate the energy and magnetisation for each of these configurations, then perform the sum. Let's be very, very, generous, and say that we can analyse <math>1\times 10^9</math> configurations per second with our computer. How long will it take to evaluate a single value of <math>\left\langle M\right\rangle_T</math>?===

For a system with 100 lattice sites and two possible spins for each site, there are <math>2^{100}</math>possible configurations for the system. <math>2^{100}= 1.27\times 10^{30} </math>, so if the computer can analyse <math>1\times 10^9</math> configurations per second, then it will take <math>\frac{1.27\times 10^{30}}{10^9} = 1.27\times 10^{21} s</math> to analyse the whole system, which is longer than the age of the universe and therefore is not a practical approach.

===TASK: Implement a single cycle of the above algorithm in the montecarlocycle(T) function. This function should return the energy of your lattice and the magnetisation at the end of the cycle. You may assume that the energy returned by your energy() function is in units of <math>k_B</math>! Complete the statistics() function. This should return the following quantities whenever it is called: <math><E>, <E^2>, <M>, <M^2></math>, and the number of Monte Carlo steps that have elapsed.===
<pre>
E = []
E2 = []
M = []
M2 = []
n_cycles = 0

def montecarlostep(self, T):
# complete this function so that it performs a single Monte Carlo step

energy = self.energy() #defines initial energy

#the following two lines will select the coordinates of the random spin for you
random_i = np.random.choice(range(0, self.n_rows))
random_j = np.random.choice(range(0, self.n_cols))
#the following line will choose a random number in the range[0,1) for you
random_number = np.random.random()

self.lattice[random_i][random_j]=(self.lattice[random_i][random_j])*(-1) #flips spin and changes lattice
energy2=self.energy() #energy of new flipped lattice
deltaE=energy2-energy #calculates change in energy

#at this point the system has the new spin config and new energy

if deltaE > 0 and random_number > e**(-deltaE/T):
self.lattice[random_i][random_j]=(self.lattice[random_i][random_j])*(-1) #reverts spin back if rejected else not changed

self.E+=[self.energy()] #records energy
self.E2+=[self.energy()**2] #records energy squared
self.M+=[self.magnetisation()] #records magnetisation
self.M2+=[self.magnetisation()**2] #records magnetisation squared
self.n_cycles=self.n_cycles+1 #adds 1 to run total

return (self.energy(),self.magnetisation())

def statistics(self):
# complete this function so that it calculates the correct values for the averages of E, E*E (E2), M, M*M (M2), and returns them

e=np.mean(self.E)
e2=np.mean(self.E2)
m=np.mean(self.M)
m2=np.mean(self.M2)

return e,e2,m,m2,self.n_cycles
</pre>

''Figure 3'' shows the results of a single run of the montecarlostep() function and the lattice the function operated upon.

[[File:cg1417MonteCarloStep_run.png|thumb|left|Figure 3 - Results from a single montecarlostep() function and the resulting lattice produced along with the correct return from the statistics() function]]

<br clear = all >

===TASK: If <math>T < T_C</math>, do you expect a spontaneous magnetisation (i.e. do you expect <math>\left\langle M\right\rangle \neq 0</math>)? When the state of the simulation appears to stop changing (when you have reached an equilibrium state), use the controls to export the output to PNG and attach this to your report. You should also include the output from your statistics() function.===

If the temperature of the system is less than the Curie Temperature, <math>T_C</math> then spontaneous magnetisation can occur and the system will tend to its lowest energy state where all of the spins are parallel - this is a property of ferromagnetic materials.

[[File:Cg1417ILanim_run.png|400px|thumb|left|Figure 4 - Results from running the ILanim.py file - shows the energy and magnetisation converging over time]]

''Figure 4'' shows that over time the the system spontaneously converges to the minimum energy state with all of the spins parallel to one another and shows, as I expected, that spontaneous magnetisation occurs and also shows that the temperature of this simulation is below the Curie Temperature, <math>T < T_C</math>.

<br clear=all>

==Section 4 - Accelerating the Code==

===TASK: Use the script ILtimetrial.py to record how long your current version of IsingLattice.py takes to perform 2000 Monte Carlo steps. This will vary, depending on what else the computer happens to be doing, so perform repeats and report the error in your average!===

''Figure 5'' show the results of running the ILtimetrial.py file on my code three timesː

{{multiple image
| align = left

| image1=cg1417ILtimetrial_run1.png
| width1=500
| image2=cg1417ILtimetrial_run2.png
| width2=500
| image3=cg1417ILtimetrial_run3.png
| width3=500
| footer = Figure 5 - Results of running the ILtimetrial.py file on my code three separate times
}}
<br clear=all>

This gave me an avergage time of <math>24.3 s \pm 0.2s</math>

===TASK: Look at the documentation for the NumPy sum function. You should be able to modify your magnetisation() function so that it uses this to evaluate M. The energy is a little trickier. Familiarise yourself with the NumPy roll and multiply functions, and use these to replace your energy double loop (you will need to call roll and multiply twice!).===

<pre> def energy(self):
"Return the total energy of the current lattice configuration."

left=multiply(roll(self.lattice,1,axis=1),self.lattice) #product of spin with spin left of it
top=multiply(roll(self.lattice,-1,axis=0),self.lattice) #product of spin with spin above it

int_en=sum(left+top) #sum of array containing sum of left and top spin products for each spin

energy = -sum(int_en) #calculates the total energy of system
return energy

def magnetisation(self):
"Return the total magnetisation of the current lattice configuration."
return sum(sum(self.lattice)) #adds up all spins in lattice
</pre>
The use of the bumpy modules reduces the amount of code required and removes the need for loops making the code significantly shorter, and is therefore expected to run ILtimetrial.py faster than the initial code developed.

===TASK: Use the script ILtimetrial.py to record how long your new version of IsingLattice.py takes to perform 2000 Monte Carlo steps. This will vary, depending on what else the computer happens to be doing, so perform repeats and report the error in your average!===

''Figure 6'' shows the result of running the ILtimetrial.py on my new accelerated code.

{{multiple image
| align = left

| image1=cg1417ILtimetrial_run1fast.png
| width1=500
| image2=ILtimetrial_run2fast.png
| width2=500
| image3=ILtimetrial_run3fast.png
| width3=500
| footer = Figure 6 - Results of running the ILtimetrial.py file on my new updated and accelerated code.
}}
<br clear=all>

The accelerated code is much faster upon using the roll, multiply and sum modules with a new average time of <math>0.790 s \pm 0.005 </math>

==Section 5 - The effect of temperature==

===TASK: The script ILfinalframe.py runs for a given number of cycles at a given temperature, then plots a depiction of the final lattice state as well as graphs of the energy and magnetisation as a function of cycle number. This is much quicker than animating every frame! Experiment with different temperature and lattice sizes. How many cycles are typically needed for the system to go from its random starting position to the equilibrium state? Modify your statistics() and montecarlostep() functions so that the first N cycles of the simulation are ignored when calculating the averages. You should state in your report what period you chose to ignore, and include graphs from ILfinalframe.py to illustrate your motivation in choosing this figure.===

''Figure 7'' below shows the results from running the ILfinalframe.py for 2x2 lattice at T=1,2,3,5.

{{multiple image
| width =350
| align = left
| image1 = cg14172x2T1.png
| image2 = cg14172x2T2.png
| image3 =cg14172x2T3.png
| image4 =cg14172x2T5.png
| footer_align = left
| footer = Figure 7 - Results of running the ILfinalframe.py file at T=1,2,3,5 for a 2x2 matrix.
}}

<br clear=all>

For a 2x2 matrix, a suitable cut-off point to exclude from the avergage energies and magnetisations is where the energy and magnetisations per spin are constant, which is 30 steps. For T=3 and T=5 the graphs do not converge because it is possible that these temperatures are higher than the Curie Temperature and as such spontaneous magnetisation will not occur and the system will not diverge to the lowest energy state. At the higher temperatures, there are larger thermal fluctations and the Boltzmann factor is more significant allowing the system to move away from the lowest energy state easier. As a result moving forwards, a suitable cut-off point will only be determined from T=1 and T=2 graphs for the larger matrices.

''Figure 8'' shows the results from running a 4x4 lattice at T=1,2 and 3.

{{multiple image
| width =350
| align = left
| image1 = cg14174x4T1.png
| image2 = cg14174x4T2.png
| image3 =cg14174x4T3.png
| footer_align = left
| footer = Figure 8 - Results of running the ILfinalframe.py file at T=1,2,3 for a 4x4 matrix.
}}

From ''Figure 8'', a suitable cut-off point for the energy and magnetisations is 200 as this is after where the energy and magnetisation has converged for T=1, and is after the initial large drop in energy for T=2, even though there are a few small fluctuations after 200 steps. The result from T=3 has been included to show the large fluctuations for the larger temperatures, and supporting my choice to determine the cut-off from T=1 and T=2 only.

''Figure 9'' shows the results for an 8x8 matrix.

{{multiple image
| width =350
| align = left
| image1 = cg14178x8T1.png
| image2 = cg14178x8T2.png
| footer_align = left
| footer = Figure 9 - Results of running the ILfinalframe.py file at T=1,2 for an 8x8 matrix
}}

From ''Figure 9'' above, a suitable cut-off point is 1000 steps as this is where the energy and magnetisation has easily converged by and is also the point after which the initial large drop in energy has been overcome for T=2 too.

''Figure 10'' shows the result of running the ILfinalframe.py for a 16x16 matrix.

{{multiple image
| width =350
| align = left
| image1 = cg141716x16T1.png
| image2 = cg141716x16T2.png
| footer_align = left
| footer = Figure 10 - Results of running the ILfinalframe.py file at T=1,2 for a 16x16 matrix
}}

From ''Figure 10'', a suitable cut-off point is 15000 steps as for T=1 the energy and magnetisations have converged significantly and will not change much, and likewise this is the same for the T=2 frame.

''Figure 11'' below shows the results from a 32x32 matrix at T=1 and T=2.

{{multiple image
| width =350
| align = left
| image1 = cg141732x32T1.png
| image2 = cg141732x32T2.png
| footer_align = left
| footer = Figure 11 - Results of running the ILfinalframe.py file at T=1,2 for a 32x32 matrix
}}

''Figure 11'' above shows the results of running the ILfinalframe.py file for a 32x32 matrix at T=1 and T=2. As a result a suitable cut-off of 50000 steps was chosen as the energy and magnetisation has significantly converged, but not as much as it could at 100000 steps. I chose a slightly lower value to ensure that the run times of my monte-carlo simulations in future tasks were not extremely time consuming.

The montecarlostep() function was changed by adding a condition that values only above the pre-determined cut-off were included when determining the average value of energy; energy squared; magnetisation and magnetisations squared from the statistics function. The statistics() function did not need to be modified.

The following code is from the 32x32 matrixː

<pre>def montecarlostep(self, T):
# complete this function so that it performs a single Monte Carlo step

energy = self.energy() #defines initial energy

#the following two lines will select the coordinates of the random spin for you
random_i = np.random.choice(range(0, self.n_rows))
random_j = np.random.choice(range(0, self.n_cols))
#the following line will choose a random number in the range[0,1) for you
random_number = np.random.random()

self.lattice[random_i][random_j]=(self.lattice[random_i][random_j])*(-1) #flips spin and changes lattice
energy2=self.energy() #energy of new flipped lattice
deltaE=energy2-energy #calculates change in energy

#at this point the system has the new spin config and new energy

if deltaE > 0 and random_number > e**(-deltaE/T):
self.lattice[random_i][random_j]=(self.lattice[random_i][random_j])*(-1) #reverts spin back

if self.n_cycles > 50000: #only adds values to array of E,E2,M and M2 above the specific cut-off
self.E+=[self.energy()]
self.E2+=[self.energy()**2]
self.M+=[self.magnetisation()]
self.M2+=[self.magnetisation()**2]
self.n_cycles=self.n_cycles+1

return (self.energy(),self.magnetisation())
</pre>

===TASK: Use ILtemperaturerange.py to plot the average energy and magnetisation for each temperature, with error bars, for an <math>8\times 8</math> lattice. Use your intuition and results from the script ILfinalframe.py to estimate how many cycles each simulation should be. The temperature range 0.25 to 5.0 is sufficient. Use as many temperature points as you feel necessary to illustrate the trend, but do not use a temperature spacing larger than 0.5. The NumPy function savetxt() stores your array of output data on disk — you will need it later. Save the file as 8x8.dat so that you know which lattice size it came from.===

Using the modified code, the file ILtemperaturerange.py was ran on an 8x8 matrix between T=0.5 and T=5 with a step of T=0.02 for 10000 Montecarlo steps and the first 1000 steps of each temperature were excluded when calculating the averages. ''Figure 12'' shows the result of the simulation and also included error bars of standard deviation.

[[File:Cg14178x8error.png|1000px|thumb|left|Figure 12 - Graph showing average energy and average magnetisation for an 8x8 lattice with error bars between T=0.5 and T=5]]

<br clear=all>

Below is the source code for the script to produce the graph from CG1417IsingModelGraphs.ipynbː

<pre>data8x8=np.loadtxt('8x8.dat') #loads data
temps8x8=data8x8[:,0] #stores temperatures
energies8x8=data8x8[:,1] #stores average energy for each T
energysq8x8=data8x8[:,2] #stores average energy squared for each T
mag8x8=data8x8[:,3] #stores magnetisation for each T
magsq8x8=data8x8[:,4] #stores magnetisation squared for each T
stde8x8=data8x8[:,5] #edited ILtemperaturerange.py to record the standard deviation of the energy for each T
stdm8x8=data8x8[:,6] #edited ILtemperaturerange.py to record the standard deviation of the magnetisation for each T

fig = pl.figure()
enerax = fig.add_subplot(2,1,1)
enerax.set_ylabel("Energy per spin")
enerax.set_xlabel("Temperature")
enerax.set_ylim([-2.5, 0.5])
enerax.set_xlim([0.5,5.1])
magax = fig.add_subplot(2,1,2)
magax.set_ylabel("Magnetisation per spin")
magax.set_xlabel("Temperature")
magax.set_ylim([-2, 2])
magax.set_xlim([0.5,5.1])
enerax.errorbar(temps8x8, np.array(energies8x8)/64,yerr=np.divide(stde8x8,64),color='black',ecolor='teal',alpha=0.8) #plots energy per spin against T
magax.errorbar(temps8x8, np.array(mag8x8)/64,yerr=np.divide(stdm8x8,64),alpha=0.8,ecolor='salmon',color='black') #plots magnetisation per spin against T on separate graph
pl.savefig('8x8error.png',bbox_inches='tight') #saves figure
pl.show()

</pre>

==Section 6 - The effect of system size==

===TASK: Repeat the final task of the previous section for the following lattice sizes: 2x2, 4x4, 8x8, 16x16, 32x32. Make sure that you name each datafile that your produce after the corresponding lattice size! Write a Python script to make a plot showing the energy per spin versus temperature for each of your lattice sizes. Hint: the NumPy loadtxt function is the reverse of the savetxt function, and can be used to read your previously saved files into the script. Repeat this for the magnetisation. As before, use the plot controls to save your a PNG image of your plot and attach this to the report. How big a lattice do you think is big enough to capture the long range fluctuations?===

The python script for this section is identical as for the 8x8 graph above in Figure 12 with the relevant files and variables changed accordingly.

Each matrix was simulated using the ILtemperaturerange.py file between T=0.5 and T=5 with a step of T=0.02.

{{multiple image
| width =350
| align = left
| image1 = cg14172x2error.png
| caption1 = 2x2 matrix - 5000 steps, cut-off = 30 steps
| image2 = 4x4error.png
| caption2 = 4x4 matrix - 1000 steps, cut-off = 200 steps
| image3 =cg141716x16error.png
| caption3 = 16x16 matrix - 50000 steps, cut-off = 15000 steps
| image4 =cg141732x32error.png
| caption4 = 32x32 matrix - 200000 steps, cut-off = 50000 steps
| footer_align = left
| footer = Figure 13 - Results of running the ILtemperaturerange.py file for 2x2, 4x4, 16x16 and 32x32 matrices.
}}

<br clear=all>

Long-range interactions are present and more significant in the smaller lattices where there are fewer stronger, short range interactions. As a result, I expect long-range interactions to be important in square lattices up to a 4x4 size.

==Section 7 - Determining the Heat Capacity==

===TASK: By definition, <math>C = \frac{\partial \left\langle E\right\rangle}{\partial T}</math>. From this, show that <math>C = \frac{\mathrm{Var}[E]}{k_B T^2}</math> (Where <math>\mathrm{Var}[E]</math> is the variance in <math>E</math>.)===

Recall from statistical thermodynamics that the average energy of a system is the sum across all microstates of the probability of that microstate multiplied by the energy of that microstate, which is defined mathematically asː <math>\langle E \rangle = \sum_i p_{i}\epsilon_{i}</math>.

The partition function <math>q</math> is defined as <math>q = \sum_{i} exp(-\beta \epsilon_{i})</math> where <math>\beta =\frac{1}{k_BT}</math>and the probability, <math>p_{i}</math> can be defined in terms of the partition function as <math>p_{i} = \frac{exp(-\beta \epsilon_{i})}{\sum_{i} exp(-\beta \epsilon_{i})} = \frac{exp(-\beta \epsilon_{i})}{q}</math>.

As a result, <math>\langle E \rangle</math> can be re-written as <math>\langle E \rangle = \sum_{i} \frac{\epsilon_{i} exp(-\beta \epsilon_{i})}{q} = -\frac{1}{q} \frac{\partial}{\partial \beta}\sum_{i}exp(-\beta \epsilon_{i}) = -\frac{1}{q} \frac{\partial q}{\partial \beta}</math>

Likewise, <math>\langle E^2 \rangle = \sum_i p_{i}\epsilon_{i}^{2} = \sum_{i} \frac{\epsilon_{i}^{2}exp(-\beta \epsilon_{i})}{q} = \frac{1}{q} \frac{\partial^{2}}{\partial \beta^{2}}\sum_{i}exp(-\beta \epsilon_{i}) = \frac{1}{q} \frac{\partial^{2} q}{\partial \beta^{2}}</math>

From definitionː <math>Var[E] = \Delta E^2 = \langle E^2 \rangle - \langle E \rangle^2 </math>

When the definition of <math>\langle E \rangle</math> and <math>\langle E^2 \rangle</math> is written in terms of partition function <math>q</math>ː <math>Var[E] = \langle E^2 \rangle - \langle E \rangle^2 = \frac{1}{q} \frac{\partial^{2} q}{\partial \beta^{2}} - \left(\frac{1}{q}\frac{\partial q}{\partial \beta}\right)^2 = \frac{1}{q} \frac{\partial^{2} q}{\partial \beta^{2}} - \frac{1}{q^2}\left(\frac{\partial q}{\partial \beta}\right)^2</math>.

According to the chain ruleː <math>\frac{\partial}{\partial \beta}\left(\frac{1}{q} \frac{\partial q}{\partial \beta}\right) = \frac{1}{q} \frac{\partial^{2} q}{\partial \beta^{2}} - \frac{1}{q^2}\left(\frac{\partial q}{\partial \beta}\right)^2 = -\frac{\partial}{\partial \beta}\langle E \rangle = Var[E]</math>

And using the chain rule againː <math>C = \frac{\partial \langle E \rangle}{\partial T}= \left(-\frac{\partial \langle E \rangle}{\partial \beta}\right) \left(-\frac{\partial \beta}{\partial T}\right) = \frac{Var[E]}{k_B T^2}</math>

===TASK: Write a Python script to make a plot showing the heat capacity versus temperature for each of your lattice sizes from the previous section. You may need to do some research to recall the connection between the variance of a variable, <math>\mathrm{Var}[X]</math>, the mean of its square <math>\left\langle X^2\right\rangle</math>, and its squared mean <math>\left\langle X\right\rangle^2</math>. You may find that the data around the peak is very noisy — this is normal, and is a result of being in the critical region. As before, use the plot controls to save your a PNG image of your plot and attach this to the report. ===

The python script for this section can be found in the Jupyter Notebook - CG1417IsingModelGraphs.ipynb

{{multiple image
| width =300
| align = left
| image1 = cg14172x2heatcap.png
| caption1 = 2x2 Matrix
| image2 = cg14174x4heatcap.png
| caption2 = 4x4 Matrix
| image3 =cg14178x8heatcap.png
| caption3 = 8x8 Matrix
| image4 =cg141716x16heatcap.png
| caption4 = 16x16 Matrix
| image5=cg141732x32heatcap.png
| caption5= 32x32 Matrix
| footer = Figure 14 - Graphs showing Heat Capacity against Temperature for each matrix size
}}

Here is the source code to produce the figuresː
<pre> def heatCap(energies,energysq,T,latsize):
#defines the heat capacity for a given temperature
energiesq=np.multiply(energies,energies) #creates array of (average energies) squared
varE=np.subtract(energysq,energiesq) #defines variance of average energy
tempsq=np.multiply(T,T) #array of temperature squared
return np.array(np.divide(varE,tempsq))/(latsize**2)

heatCap2x2=heatCap(energies2x2,energysq2x2,temps2x2,2) #creates array of heat capacity for each T

fig = pl.figure()
heatcapax = fig.add_subplot(1,1,1)
heatcapax.set_xlabel('Temperature')
heatcapax.set_ylabel('Heat Capacity')
heatcapax.plot(temps2x2,heatCap2x2,color='orange') #plots heat capacity for each T
pl.savefig('cg14172x2heatcap.png',bbox_inches='tight') #saves figure
pl.show()
</pre>

A general trend from the above graphs is that the peak of the graph shifts towards lower temperatures as the size of the matrix used increases which means the Curie Temperature decreases as matrix size increases. Also, as lattice size increases the noise around the peak becomes larger which will affect the accuracy of determining the maximum heat capacity and Curie Temperature for the larger lattices.

==Section 8 - Locating the Curie Temperature==
===TASK: A C++ program has been used to run some much longer simulations than would be possible on the college computers in Python. You can view its source code here if you are interested. Each file contains six columns: <math>T, E, E^2, M, M^2, C</math> (the final five quantities are per spin), and you can read them with the NumPy loadtxt function as before. For each lattice size, plot the C++ data against your data. For one lattice size, save a PNG of this comparison and add it to your report — add a legend to the graph to label which is which. To do this, you will need to pass the label="..." keyword to the plot function, then call the legend() function of the axis object (documentation here).===

The python code used to read and plot the C++ data is found in the Jupyter notebook CG1417IsingModelGraphs.ipynb.

''Figure 15'' below shows the C++ plotted against my own data for a 16x16 Matrix.

[[File:Cg141716x16C++.png|400px|thumb|left|Figure 15 - Graph showing my own data against the C++ data for a 16x16 matrix.]]

<br clear = all >

The curves produced using the C++ data are much smoother and have less noise than the data gained from my python code. This is likely due to the C++ code having more montecarlosteps per temperature, reducing the effect of random fluctuations on the averages and also having a smaller step gap which will make the curve smoother as the points are closer together.

Here is the source code the produce the figuresː
<pre>#reads data from C++ file
temps2x2C=data2x2C[:,0]
energies2x2C=data2x2C[:,1]
energysq2x2C=data2x2C[:,2]
mag2x2C=data2x2C[:,3]
magsq2x2C=data2x2C[:,4]
heatcap2x2C=data2x2C[:,5]

#fitting C++ data

fig = pl.figure()
enerax = fig.add_subplot(2,1,1)
enerax.set_ylabel("Energy per spin")
enerax.set_xlabel("Temperature")
enerax.set_ylim([-2.5, 0.5])
enerax.set_xlim([0.5,5.1])
magax = fig.add_subplot(2,1,2)
magax.set_ylabel("Magnetisation per spin")
magax.set_xlabel("Temperature")
magax.set_ylim([-2, 2])
magax.set_xlim([0.5,5.1])
enerax.plot(temps2x2, np.array(energies2x2)/4,color='black',alpha=0.7,label='Python Data') #python energy against T
enerax.plot(temps2x2C, energies2x2C, color='red',label='C++ Data') #C energy against T
magax.plot(temps2x2, np.array(mag2x2)/4,color='black',alpha=0.7,label='Python Data') #python magnetisation against T
magax.plot(temps2x2C, mag2x2C,color='red',label='C++ Data') #C energy against T
enerax.legend() #shows legend on energy graph
magax.legend() #shows legend on energy graph
pl.show()
</pre>

The relevant variables and dat files were changed for each matrix.

===TASK: write a script to read the data from a particular file, and plot C vs T, as well as a fitted polynomial. Try changing the degree of the polynomial to improve the fit — in general, it might be difficult to get a good fit! Attach a PNG of an example fit to your report.===

The python script to read and plot the fitted polynomial is found in CG1417PolyfitScript.ipynb

Here is the source code for ''Figure 15''

<pre>data_test = np.loadtxt("16x16C.dat")
T_test = data_test[:,0] #gets temperatures
C_test = data_test[:,5] #gets heat capacity data

#first we fit the polynomial to the data
fit_test = np.polyfit(T_test, C_test, 35) # fit a polynomial of degree 35ǃ

#now we generate interpolated values of the fitted polynomial over the range of our function
T_min_test = 0.5 #np.min(T_test)
T_max_test = 5 #np.max(T_test)

T_range_test = np.linspace(T_min_test, T_max_test, 1000) #generate 1000 evenly spaced points between T_min and T_max
fitted_C_values_test = np.polyval(fit_test, T_range_test)# use the fit object to generate the corresponding values of C

fig = pl.figure()
heatcapax = fig.add_subplot(1,1,1)
heatcapax.set_xlabel('Temperature')
heatcapax.set_ylabel('Heat Capacity')
heatcapax.plot(T_test,C_test,color='orange',label='C++ Data') #plots C data of heat capacity against temp
heatcapax.plot(T_range_test,fitted_C_values_test,label='Fitted Polynomial') #plots fitted polynomial for whole range of temp
heatcapax.legend()
pl.savefig('FIT_TEST16x16_35.png', bbox_inches='tight') #saves figure
pl.show()
</pre>

Below in ''Figure 16'' is a plot of my Heat Capacity against Temperature data for a 16x16 matrix and features a polynomial of degree 35 plotted against it. Even with a polynomial of such a high degree, it poorly fits the curve and does not fit to the peak of the curve either.

[[File:cg1417FIT_TEST16x16_35.png|thumb|left|400px|Figure 16 - Plot of Heat Capacity against Temperature along with a poorly fitted polynomial of degree 35. ]]

<br clear = all >

===TASK: Modify your script from the previous section. You should still plot the whole temperature range, but fit the polynomial only to the peak of the heat capacity! You should find it easier to get a good fit when restricted to this region===

The script was modified as such that the polynomial was fitted in a set range around the peak of the graph, this is demonstrated in ''Figure 17'' which shows a newly fitted polynomial between a much smaller range of temperatures (T = 2.15-2.55) and a much smaller degree polynomial (3).

[[File:CG1417FIT_16x16C_3.png|thumb|left|400px|Figure 17 - Graph showing Heat Capacity against Temperature for a 16x16 matrix along with a fitted polynomial between a much more restricted range of temperatures and a significantly lower degree of polynomial]]

Upon comparison with ''Figure 16'', the new fitted polynomial is a significantly better fit even for a 3rd degree polynomial and is a much more accurate representation of my data around the peak of the graph and will make it easier to determine the maximum value of Heat Capacity. However, the polyfit curve still doesn't perfectly fit the peak due to the significant amount of noise present there.

<br clear = all>

Here is the source code for ''Figure 17'' from CG1417PolyfitScript.ipynbː
<pre>data16 = np.loadtxt("16x16C.dat") #loads data to variable

T16 = data16[:,0] #gets temps
C16 = data16[:,5] # gets heat capacities

Tmin16 = 2.15 #chosen min temp
Tmax16 = 2.55 #chosen max temp

selection16 = np.logical_and(T16 > Tmin16, T16 < Tmax16) #choose only those rows where both conditions are true
peak_T_values16 = T16[selection16] #choose temp values in range chosen above
peak_C_values16 = C16[selection16] #choose heat cap values in range of t above

fit16 = np.polyfit(peak_T_values16,peak_C_values16,3) #fit 3rd order polynomial
peak_T_range16 = np.linspace(Tmin16, Tmax16, 1000) #defines 1000 temps within data range
fitted_C_values16 = np.polyval(fit16, peak_T_range16) #use the fit object to get corresponding values of heat cap

fig = pl.figure()
heatcapax = fig.add_subplot(1,1,1)
heatcapax.set_xlabel('Temperature')
heatcapax.set_ylabel('Heat Capacity')
heatcapax.plot(T16,C16,color='orange',label='C++ Data') #plots C data of heat cap against temp
heatcapax.plot(peak_T_range16,fitted_C_values16,label='Fitted Polynomial') #plots fitted polynomial for small range
heatcapax.legend()
pl.savefig('FIT_16x16C_3.png', bbox_inches='tight') #saves figure
pl.show()

</pre>

===TASK: find the temperature at which the maximum in C occurs for each datafile that you were given. Make a text file containing two colums: the lattice side length (2,4,8, etc.), and the temperature at which C is a maximum. This is your estimate of <math>T_C</math> for that side length. Make a plot that uses the scaling relation given above to determine <math>T_{C,\infty}</math>. By doing a little research online, you should be able to find the theoretical exact Curie temperature for the infinite 2D Ising lattice. How does your value compare to this? Are you surprised by how good/bad the agreement is? Attach a PNG of this final graph to your report, and discuss briefly what you think the major sources of error are in your estimate.===

''Figure 18'' below shows a graph of <math>T_{C,L}</math> against <math>\frac{1}{Lattice Size}</math> to determine the Curie Temperature of an infinite 2D Ising Model Lattice <math>T_{C,\infty}</math>. The black dots represent the raw data obtained from obtaining the temperature at which the Heat Capacity was a maximum for the lattices and the red line in a linear curve fit plotted against the data to allow the y-intercept which is the Curie Temperature for the infinite 2D lattice to be determined.

[[File:cg1417CurieTemp.png|400px|thumb|left|Figure 18 - Plot of 1/Lattice Size against Curie Temperature for that lattice size.]]

The value for <math>T_{C,\inf}</math> obtained from the data is <math>T_{C,\infty} = 2.277 \frac{J}{k_B}</math> with a literature value being <math>T_{C,\infty} = 2.269 \frac{J}{k_B}</math> <ref>L. Onsager, Phys. Rev., 1944, 65, 117--149.</ref>for an infinite square 2D lattice. This means that my result slightly over-estimates the Curie Temperature for the infinite lattice and as a result for an infinite lattice the temperature at which spontaneous magnetisation stops would actually occur at a slightly lower temperature than expected. However, the difference between my value and the literature value is only 0.008 which is incredibly small and the amount of agreement between the two values is somewhat surprising, which means that the error in my estimates of the Curie Temperature for each lattice size is relatively small. The points which have the largest residuals and deviation from the line of best fit in ''Figure 17'' corresponds to the smaller lattice sizes of 2x2 and 4x4 where longer range interactions are more significant. The longer range interactions posed by the boundary conditions are significant for the smaller sizes and causes the energy of the smaller matrices to be less accurate and have a larger associated error with the energy and the Curie Temperature for that lattice size. This affects the accuracy of the line of best fit and to increase the accuracy of this line, larger lattice sizes of 128x128, 256x256 etc should be included in the calculation for the line of best fit and the smaller matrices ignored - this should allow a more accurate value of <math>T_{C,\infty}</math> to be determined.

<br clear = all >

Below is the source code used to generate ''Figure 18'' from CG1417PolyfitScript.py
<pre>
Cmax64x64 = np.max(fitted_C_values64) #finds Cmax for 64x64 matrix - done for others already
Tmax64x64 = peak_T_range64[fitted_C_values64 == Cmax64x64] #finds Tmax corresponding to Cmax

LatSize=[2,4,8,16,32,64] #stores lattice sizes
Tmax=[Tmax2x2,Tmax4x4,Tmax8x8,Tmax16x16,Tmax32x32,Tmax64x64] #stores corresponding Tmax data
np.savetxt('CmaxVSTmax.txt', (LatSize,Tmax)) #writes data to txt file

ScalData=np.loadtxt('CmaxVSTmax.txt') #loads data
LatticeSize=ScalData[0] #gets lattice sizes
TempMax=ScalData[1] #gets max temp or curie temp for each lattice

Lmin1min = np.min(np.divide(1,LatticeSize)) #minimum of 1/LatticeSize values
Lmin1max = np.max(np.divide(1,LatticeSize)) #maximum of 1/LatticeSize values

fitTcl = np.polyfit(np.divide(1,LatticeSize),TempMax, 1) #creates fit object

Lmin1values = np.linspace(Lmin1min, Lmin1max, 1000) #finds 1000 values between min and max x-axis value of 1/LatticeSize
fitted_Tcl_values = np.polyval(fitTcl, Lmin1values) #creates corresponding Curie Temp values for each value in Lmin1values

fig = pl.figure()
scalrelax = fig.add_subplot(1,1,1)
scalrelax.set_xlabel('1/Lattice Size')
scalrelax.set_ylabel('Curie Temperature/ J/k_B')
scalrelax.plot(np.divide(1,LatticeSize),TempMax,color='black',marker='.',linestyle='') #plots Curie Temp against 1/LatticeSize
scalrelax.plot(Lmin1values,fitted_Tcl_values,color='red',marker='',linestyle='-') #plots line of best fit for data above
pl.savefig('CurieTemp.png', bbox_inches='tight') #saves figure
pl.show()

</pre>

== References ==

Rep:Y3AS6115

2025-09-01T09:50:30Z

Move page script: Move page script moved page Y3AS6115 to Rep:Y3AS6115: Move to report namespace

=Transition State Structures=

==Introduction==
In this experiment, the transition states of several pericyclic reactions where investigated using computational techniques to acquire information about the kinetics and thermodynamics of these reactions, and analyse the molecular orbitals of these chemical systems.

A potential energy surface (PES) is a plot of the potential energy of a chemical system as a function of two or more reaction coordinates. The number of dimensions of a PES is 3N-6 (where N is the number of atoms in the molecular system being considered). In the PES the first derivative of the energy, physically representing the gradient, which is related to the force acting on the atoms whilst the second derivative, a physical measure of the curvature, is related to the force constants, k. The values of k can then be used to calculate the vibrational frequencies for each of the 3N-6 modes.

A minimum point (zero first derivative, positive second derivative) along a reaction coordinate, of the PES of the molecular system of interest, corresponds to a stable species in the reaction. This could potentially be a reactant, product or intermediates of the reaction mechanism. A transition state (TS) is a maximum point (zero first derivative, negative second derivative) along the reaction coordinate, of the PES. It is also possible that the TS for a particular reaction pathway might be a saddle point on the PES which has a zero first derivative and second derivative that is positive in some directions and negative in others. The reaction path taken can be identified form the potential energy surface by keeping the system in equilibrium whilst varying the reaction coordinates, thus elucidating the minimum energy pathway linking reactants and products via a transition state; known as the intrinsic reaction coordinate (IRC).

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 10:42, 6 March 2018 (UTC) Some confusion here, A TS is always a first order saddle point. and always only has 1 dimension of negative curvature. This info is obtained by diagonalising the force constant and looking a the eigen values.

During this experiment two optimisation methods on Gaussian were used; the semi-empirical PM6 method and the density functional theory (DFT) based B3LYP. Both optimization methods are based on the Hartee-Fock model, which accounts for electron-electron interactions by assuming that any given electron in a molecule only experiences an average field from the other electrons. The relatively simpler non-ab initio PM6 method uses pre-determined empirical data in its estimation of electron density, and therefore is a relatively fast method. The B3LYP method uses 6-31(G) basis set which is essentially a mathematical representation of the atomic orbitals which make up the molecular orbitals of the molecular system being studied (i.e. LCAO theory). The B3LYP method involves the evaluation of an exchange correlation parameter term, which more accurately represents electron-electron repulsions. The purpose of these methods was to solve the time independent Schrödinger equation at each point as reaction coordinates are varied and calculate the corresponding energy, and the subsequent first and second derivatives of the energy with respect to the specific position on the PES. This allows determination of an optimised geometry. Optimised in the sense of either being a minimum and therefore a stable species involved in the reaction (by performing a minimisation calculation) or a TS (by choosing a TS(Berny) calculation). In the case of the TS optimisation calculation must have force constants calculated only once, and usually include the Gaussian keyword opt = noeigen (which in the case of multiple negative frequencies being found will prevent the calculation from terminating).

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 10:52, 6 March 2018 (UTC) Fairly good understanding, but you could have added some equations to break up the text. You have clearly read beyond the script.

Gaussian was used to locate the TSs of the reactions studied. Several different methods were employed to find to locate the TS. Firstly, a guess-TS was built in GaussView and then optimised directly to the TS, whilst this is the fastest approach it is also the least reliable and to be of any use some prior knowledge of the TS is requires. Secondly, a guess-TS could have the interatomic distances between the atoms which are broken or formed during the reaction step fixed (‘frozen coordinates’), and then preform a minimisation calculation, before optimising to the TS. Also, a third method used involved minimising the structure of the reactants, or products, and then altering the structure and freezing the coordinates to obtain a more accruate guess-TS, which can then be optimised to the TS.

==Exercise 1: Diels-Alder reaction of butadiene with ethylene==
[[File:AS6115Excercise1scheme.png |thumb|center|300px|Figure 1: Reaction scheme of for the Diels-Alder reaction between ethylene and butadiene to form cyclohexane.]]

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) Great work across the whole exercise. Well done!)

For this exercise the Diels-Alder reaction between ethylene and butadiene was studied (see figure 1). The method used was to first optimise the cyclohexene product to a minimum and then the two C-C bonds formed during the Diels-Alder were broken and frozen at a distance of 2.2 Å. This structure was then first minimised and then optimised to the TS.

===MO Diagram of transition state===
[[File:AS6115Mo-diagram-ts-1.png |thumb|center|600px|Figure 2: MO Diagram of the transition state formed from the Diels-Alder reaction between ethylene and butadiene. All MOs labelled with relative energy and described as symmetric, (s), and antisymmetric, (a).]]

The above figure illustrates the simplified molecular orbital (MO) diagram for the TS of the Diels-Alder reaction between ethylene and butadiene. An IRC calculation was run on the optimised TS for this reaction and from the log file initial frame, corresponding to the reactants an energy calculation reactants. This was necessary to obtain the relative energy levels of the MOs of the reactants and the TS which were comparable.

From the MO diagram (figure 2) it can be seen that the MOs of the TS formed from the linear combination of the frontier molecular orbitals of ethylene and butadiene are quite high in energy. Although this is to be expected since the transition state is the highest energy point along the reaction coordinate, and therefore the theoretical activated complex species which exists at this point, should be destabilised relative to the reactants.

===HOMO and LUMO of reactants and resultant MOs in transition state===

The table below displays the HOMOs and LUMOs of ethylene and butadiene as well as the four MOs of the TS which form because of the linear combination of the reactants frontier MOs, which are the same as does demonstrated in the above MO diagram. HOMO-1 (MO16) of the TS is the bonding orbital resulting from the net in-phase interaction (constructive interference) between the HOMO of butadiene and the LUMO of ethylene, and the LUMO +1 (MO19) of the TS is the corresponding antibonding orbital formed via an out-of-phase interaction (net destructive interference) between the butadiene HOMO and ethylene LUMO. The TS HOMO (MO17) is the bonding orbital formed because of the in-phase interaction between the ethylene HOMO and butadiene LUMO, with TS LUMO (MO18) being the antibonding orbital formed by the out-of-phase interaction between ethylene HOMO and butadiene LUMO.

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Name
! style="background: #0D4F8B; color: white;" | MO Jmol
! style="background: #0D4F8B; color: white;" | Comments
|-
| HOMO of ethylene
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_OPT_MIN.LOG-y3as6115</uploadedFileContents>
</jmolApplet>
</jmol>
| MO 6

Relative energy: -0.39228

Symmetric
|-
| LUMO of ethylene
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE_OPT_MIN.LOG-y3as6115</uploadedFileContents>
</jmolApplet>
</jmol>
| MO 7

Relative energy: 0.04256

Antisymmetric
|-
| HOMO of butadiene
|<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>y3as6115BUTADIENE_OPT_MIN.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| MO11

Relative energy: -0.35166

Antisymmetric
|-
| LUMO of butadiene
|<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>y3as6115BUTADIENE_OPT_MIN.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| MO 12

Relative energy: 0.01103

Symmetric
|-
| HOMO - 1 Transition state
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE-BUTADIENE_TS_OPT.LOGy3as6115</uploadedFileContents>
</jmolApplet>
</jmol>
| MO 16

Relative energy: -0.32755

Antisymmetric formed as the 'bonding' MO form the interaction of the LUMO of ethylene and the butadiene HOMO
|-
| HOMO Transition state
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE-BUTADIENE_TS_OPT.LOGy3as6115</uploadedFileContents>
</jmolApplet>
</jmol>
| MO17

Relative energy: -0.32532

Symmetric formed as the 'bonding MO' from the interaction of the ethylene HOMO and the Butadiene LUMO
|-
| LUMO Transition state
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE-BUTADIENE_TS_OPT.LOGy3as6115</uploadedFileContents>
</jmolApplet>
</jmol>
| MO 18

Relative energy: 0.01733

Symmetric formed as the 'antibonding MO' from the interaction of the ethylene HOMO and the Butadiene LUMO
|-
| LUMO + 1 Transition state
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE-BUTADIENE_TS_OPT.LOGy3as6115</uploadedFileContents>
</jmolApplet>
</jmol>
| MO 19

Relative energy: 0.0366

Antisymmetric formed as the 'antibonding MO' from the interaction of the ethylene LUMO and the Butadiene HOMO
|}

Examination of the MOs of the TS exemplifies the principle of conservation of orbital symmetry; that is that only orbitals of the same symmetry can linearly combine. This can be explained by considering the orbital overlap integral, as for an overlap between orbitals of different symmetry (i.e. symmetric-asymmetric interaction) there will be equal amounts of in-phase (constructive) overlap and out-of-phase (destructive) overlap leading to a net zero overlap, hence the overlap integral for such an interaction is therefore zero. The overlap of orbitals of the same symmetry (i.e. symmetric-symmetric and asymmetric-asymmetric) have a non-zero overlap integral, as there will be either a net in-phase or out-of-phase interaction. The LUMO of ethylene and the butadiene HOMO are both asymmetric and therefore can linearly combine to generate the two asymmetric MOs of the TS (MO16 and MO19). Conversely, the Ethylene HOMO and Butadiene LUMO are symmetric can linearly combine to give the two symmetric MOs of the TS (MO17 and MO18).

===C-C bond lengths===

====For optimized reactants:====

Throughout the course of the Diels-Alder reaction the bond lengths of all C-C bonds change due to the changes in hybridisation of some of the carbons and the changes in all the bond orders, and as two additional sigma bonds are formed by the cycloaddition. The data listed below demonstrates the changes in bond lengths that occur on going from reactants to TS to product.

Ethylene C-C bond length = 1.33 Å

Butadiene central C-C bond length = 1.46 Å

Butadiene terminal C-C bond lengths = 1.34 Å

====At the optimised transition State:====

Ethylene C-C bond length = 1.38 Å

Butadiene central C-C bond length = 1.41 Å

Butadiene terminal C-C bond lengths = 1.38 Å

Partly formed new C-C bonds = 2.11 Å

====For the optimized product====
[[File:as6115-Labelled-cyclohexene.PNG |thumb|right|200px|Figure 3: Cyclohexene product labelled]]

Ethylene C-C bond length (C4-C5) = 1.54 Å

Butadiene central C-C bond length (C1-C2)= 1.34 Å

Butadiene terminal C-C bond lengths (C2-C3 & C6-C1) = 1.50 Å

Newly formed bonds (C3-C4 & C5-C6) = 1.54 Å

Typical C-C bond lengths:

C(sp<sup>3</sup>)-C(sp<sup>3</sup>) = 1.54 Å

C(sp<sup>2</sup>)-C(sp<sup>2</sup>) = 1.50 Å

C(sp<sup>3</sup>)-C(sp<sup>2</sup>) = 1.34 Å

Carbon Van der Waals radius = 1.70 Å

The C-C bond lengths in the optimised reactants and product are as expected. The C-C bond ethylene is 1.33 Å (very close to close to the typical alkene C-C sp<sup>2</sup>-sp<sup>2</sup> double bond length of 1.34Å). During this reaction, the carbon atoms (carbon atoms 4 and 5, see figure 3) of the ethylene dienophile goes from a double bond C(sp<sup>2</sup>)-C(sp<sup>2</sup>) to a single C(sp<sup>3</sup>)-C(sp<sup>3</sup>) and therefore lengthens as the reaction progresses. The C4-C5 bond length is 1.38Å which is intermediate between the shorter C-C single bond length in the optimised TS. The C4-C5 bond length in the cyclohexene product 1.54 Å, very similar typical C(sp<sup>3</sup>)-C(sp<sup>3</sup>) bond lengths. The newly formed C-C bonds in product is also at the typical C(sp<sup>3</sup>)-C(sp<sup>3</sup>) bond length and in the TS the interatomic distances between these carbon atoms (C3-C4 and C5-C6) is within the two times the van der Waals radius of carbon which indicates that there is an interaction between these two carbon atoms in the TS of the [4+2] cycloaddition. The central C-C bond of butadiene is broadly similar to typical values for a C-C single bond between two sp<sup>2</sup>hybridised carbon atoms (1.50 Å). This bond becomes progressively shorter in the reaction as in the TS the value is intermediate between the C-C bond length in the reactant and that of the C=C of an alkene, and this illustrates how the two carbon atoms come closer together in this reaction to form the π bond in the cyclohexene product (C1-C2 bond length). Like the ethylene C-C bond, the terminal C-C bond of butadiene has the typical alkene C=C bond length, and this bond lengthens as the reaction progresses with a values in intermediate of the short the C=C alkene bond and that of the typical C(sp<sup>3</sup>)-C(sp<sup>2</sup>) single bond length, at the transition state. In the product, this C-C bond length (C2-C3 and C1-C6) is as expected for a typical C(sp<sup>3</sup>)-C(sp<sup>2</sup>) length (1.50 Å).

===Vibration at the reaction path for the transition state===

{| class="wikitable"
! [[File:AS6115Ethene-butadiene-ts-vibration1.gif |centre|x400px| IRC animation]]
|-
| Figure 4 Animation of vibration of TS at the reaction path
|}

The negative frequency at -949.20 cm<sup>-1</sup> in the TS of this reaction corresponds to the vibration at the reaction path. This negative vibration is due to the negative force constant at the TS (since the force constant is related to the second derivative and at the TS, a maximum point along the reaction path, the second derivative is negative). This vibration is animated above, figure 4, and appears to represent bond formation, and provides evidence that the bond formation in this reaction is synchronous and therefore that the cycloaddition reaction is concerted.

===LOG files===

Optimised ethylene = [[File:ETHENE_OPT_MIN.LOG-y3as6115]]

Optimised butadiene = [[File:y3as6115BUTADIENE_OPT_MIN.LOG]]

Optimised TS = [[File:ETHENE-BUTADIENE_TS_OPT.LOGy3as6115]]

Optimised cyclohexene = [[File:AS6115CYCLOHEXADIENE_MIN.LOG]]

==Excercise 2: Diels-Alder reaction of cyclohexadiene and 1,3-dioxazole==

[[File:AS6115Excercise2scheme.png |thumb|centre|400px|Figure 5: Reaction scheme for the Diels-Alder reaction between 1,3-dioxole and cyclohexa-1,3-diene]]
===MOs===

For this exercise, the endo and exo reaction pathways of the Diels-Alder reaction between 1,3-dixole and cyclohexa-1,3-diene where investigated. The method employed was to first build and optimise the structures of the products and then break the two C-C bonds which formed during the reaction and freeze them at a distance of 2.2 Å, to give the exo and endo guess TS. These guess TSs where then minimised before being optimised to give the final optimised TS. In this exercise, all minimisation and TS optimisations where first preformed using PM6 method, and then using the result optimised at B3LYP level. The IRC calculations where performed at the PM6 level using the PM6 optimised TS as the input, as the corresponding IRC calculations at the B3LYP level have a significantly long run time.

{|
|[[File:AS6115-Endo-ts-mo-diagram.PNG |thumb|centre|425px|Figure 6: MO diagram of the endo transition state]]
|[[File:AS6115Exo-ts-mo-diagram1.PNG |thumb|centre|450px|Figure 7: MO diagam of the exo transition state]]
|}
The MO diagrams for the endo (figure 6) and exo (figure 7) TSs is broadly similar to the MO diagram for exercise 1, with the TS orbitals being higher in energy relative to the reactant MO levels. In a normal electron demand process the diene HOMO is higher in energy than the dienophile HOMO, and the diene HOMO-dienophile LUMO energy gap is smaller compared to the converse dienophile HOMO-diene LUMO energy gap. However, in this reaction the HOMO of the 1,3-dioxole (dienophile) is higher in energy than the cyclohexa-1,3-diene (diene) HOMO, and this means that the reaction is an inverse electron demand Diels-Alder. This is because the dienophile is relatively electron rich, as the oxygen atom substituents can donate electron density (via lone pairs in valence p orbitals) in to the alkene π bond, thus raising the orbital energy levels of the dienophile. Again, this reaction obeys the principle of orbital symmetry.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 10:56, 6 March 2018 (UTC) What you have said is theoretically correct but you could have backed up this discussion by investigating the relative ordering of the MOS by doing a single point calc of the reactants.

Additionally it is noticeable that the energy levels of the endo and exo transition state are different. For example the TS HOMO (MO41) for the endo TS is lower in energy than that of the exo transition state. This result is to be expected since the exo TS is higher in energy than the endo, as the endo TS has stabilising secondary orbital interactions.

The secondary orbital interaction in the endo TS can be seen when viewing the TS MOs (see figure 8), as in addition to the main frontier MO interactions, the oxygen p-orbitals interact with the back of the π-orbitals of the diene component, which provides an additional stabilising factor. This lowers the reaction barrier for the reaction, relative the reaction forming the exo product.

====Reactants====

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|-
| LUMO of Cyclohexadiene
| LUMO of 1,3-Dioxole
|}

====Endo Transition-State====

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|-
| ENDO-TS HOMO
| ENDO-TS LUMO
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|-
| ENDO-TS HOMO - 1
| ENDO-TS LUMO + 1
|}

====Exo Transition State====

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| EXO-TS HOMO
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|-
| EXO-TS HOMO - 1
| EXO-TS LUMO + 1
|}

===Thermochemistry===

{| class="wikitable"
! Species
! Sum of electronic and thermal free energies (Hartee/particle)
|-
| 1,3-Dioxole
| -233.321
|-
| Cyclohexa-1,3-diene
| -267.068
|-
| Endo Transition State
| -500.332
|-
| Exo transition state
| -500.329
|-
| Endo product
| -500.419
|-
| Exo product
| -500.417
|}

{| class="wikitable"
! Reaction product
! Reaction Barrier (KJ / mol)
! Reaction Energy (KJ/mol)
|-
| Endo
| 151.04
| -76.18
|-
| Exo
| 158.86
| -72.59
|}

The thermochemistry section form the log file output of optimisation calculations contained the sum of electronic and thermal free energies which where subsequently used to determine the reaction barriers and reaction energies. The values for the reaction barriers and reaction energies for both the endo and exo reaction pathways are summarised in the above table. The results reveal that the endo reaction has a lower reaction barrier and a larger reaction energy (i.e. the endo product is lower in energy), compared to the exo reaction, thus the endo product is both the kinetic and thermodynamic product. The smaller activation energy for the formation of the endo product is due to the presence of stabilising secondary orbital interactions in the endo TS (which are absent in the exo TS), which means the product is kinetically favoured, . The endo product is the more stable product because it is less sterically encumbered than the exo. This because even though the exo product has the newly formed C-C in an equatorial position, they are in relatively close proximity to the CH2-CH2 bridge which results in steric clashes with the fused 5 membered ring (originating from the 1,3-dioxole). In the endo product, the 5-membered ring is in close proximity to a CH=CH bridge, and therefore experience less steric repulsion, making the more stable thermodynamically favoured product (see figure 9). Also, the greater steric repulsion may be an additional factor for the higher activation energy required for the exo relative to the endo Diels-Alder reaction.

[[File:AS6115Endo-ts-2nd-orb-inter.png |thumb|right|200px|Figure 8: MO41 of the endo transition state, displaying secondary orbital interactions]]

{| class="wikitable"
! Figure 9: Structue of exo and endo prodcut
|-
| [[File:AS6115-Unhindered-endo.PNG |thumb|center|200px|]]
|-
| Endo product
|-
| [[File:AS6115-Hindered-exo.PNG |thumb|center|200px|]]
|-
| Exo product
|}

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 11:05, 6 March 2018 (UTC) This was a good section, There were some part where you could have gone into more detail. Your discussion of of the SOO and sterics was particular good. You could have spoke in more detail about the kinetics. Also your energies are slightly out. I belive this is becuase you have miss calculated your reactant energies.

===LOG files===

Optimised 1,3-Dioxole = [[File:AS6115DIOXAZOLE MIN B3LYP.LOG]]

Optimised cyclohexa-1,3-diene = [[File:AS6115CYCLOHEXADIENE MIN B3LYP.LOG]]

MO energy levels of reactants = [[File:AS6115REACTANTS_ENERGY.LOG]]

Optimised Endo TS = [[File:AS6115ENDO_TS_OPT_B3LYP.LOG]]

Optimised Exo TS = [[File:AS6115EXO_TS_OPT_B3LYP.LOG]]

Optimised Endo product = [[File:AS6115ENDO_PRODUCT_MIN_B3LYP.LOG]]

Optimised Exo product = [[File:AS6115EXO_PRODUCT_MIN_B3LYP.LOG]]

==Excercise 3:o-xylylene sulfur dioxide cycloadditions: Diels-Alder v. Cheletropic==

[[File:AS6115Excercise3scheme1.png|thumb|centre|300px|Figure 10: Reaction Scheme for the different possible cycloaddition reactions between SO2 and the exocyclic diene unit of o-xylylene]]

For this experiment the TSs of different possible pericyclic (figure 10) reactions were located and optimised at the PM6 level. The method used to obtain the TS was to firs build and optimise the products of each of these reactions, individually, then break and freeze the bonds which form as a result of the cycloaddition in order to generate a relatively accurate guess TS. From the optimised TSs the IRC calculations.

From the output (log files) of the optimisation calculation the free energy values for the reactants, products and TSs were collated and used to determine the reaction barriers and energies for each reaction pathway (see tables below).

===Thermochemistry===

[[File:AS6115Reaction profile 1.png |thumb|centre|500px|Figure 11: Reaction profile for the different possible cycloaddition reactions between o-xylylene and the exocyclic diene unit of o-xylylene, with reaction barriers and energies displayed]]

The reaction profile (figure 11) illustrates the data shown in the tables below. The endo product is the kinetic product ads it has the lowest reaction barrier; this is primarily due to stabilising secondary orbital interactions in the endo TS. Also the endo product has a slightly larger reaction energy than the exo product, indicating the endo product is more thermodynamically favourable than the exo. The cheletropic reaction has the highest activation energy of the pathways but also has the highest reaction energy, which means that the cheletropic product is the less kinetically favourable but more thermodynamically favourable than either the endo or exo Diels-Alder. The cheletropic reaction has the highest activation energy because the cheletropic TS is much higher in energy than the endo or exo Diels-Alder TS. This difference can be attributed to the fact that in the cheletropiuc reaction the TS involves forming a 5-membered ring which is more strained and therefore less stable than the 6-membered chair-like TS for the Diels-Alder reactions. S=O bonds are relatively strong and therefore the fact that the cheletropic reaction has two S=O bonds means it is lower in energy compared to the endo and exo products of the Diels-Alder reaction in which one of the two S=O bonds are broken. This explains why the cheletopic reaction is the thermodynamically favoured product.

{| class="wikitable"
! Species
! Sum of thermal and electronic free energies (Hartee/Particle)
|-
| Sulfur dioxide
| -0.118614
|-
| o-Xylylene
| 0.178112
|-
| Endo Transition State
| 0.090562
|-
| Exo Transition state
| 0.092078
|-
| Cheletopic Transition State
| 0.099062
|-
| Endo Product
| 0.021697
|-
| Exo Product
| 0.023829
|-
| Cheletropic reaction
| 0.000005
|}

(Your exo product hasn't fully converged to a minimum. You need to check your geometries for imaginary frequencies [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 10:26, 6 March 2018 (UTC))

{| class="wikitable"
! Reaction Product
! Reaction Barrier (KJ/mol)
! Reaction energy (KJ/mol)
|-
| Endo
| 81.56
| -99.24
|-
| Exo
| 85.54
| -93.65
|-
| Cheletrophic
| 103.87
| -156.20
|}

As a result of the Diels-Alder and Cheltropic reaction the 6-membered ring of the xylylene reactant is converted into an aromatic benzene ring. The aromatisation of the xylylene ring is a thermodynamic major driving force for these reactions, which explains why the o-xylylene is relatively unstable, as it is readily converted to an aromatic product, as demonstrated by the IRC animations below.

===IRC Paths===

{| class="wikitable"
! Endo pathway
! Exo pathway
! Cheletrophic
|-
| [[File:AS6115Endo reactionex3.gif |left|x550px| IRC animation]]
| [[File:AS6115EXo_reactionex3.gif |left|x400px| IRC animation]]
| [[File:AS6115Cheletrophic reaction.gif |left|x450px| IRC animation]]
|-
|}

===Alternative Diels-Alder reaction===

[[File:AS6115Excercise3scheme2.png |thumb|centre|300px|Figure 12: Reaction Scheme showing the exo and endo products of the Diels Alder reaction between sulphur dioxide and the diene unit with the 6-membered ring of o-xylylene]]

In addition to the exocyclic diene unit utilised for the Cheletropic and Diels-Alder reactions describe above, there is a second s-cis diene unit with the xylylene ring which can undergo a Diels-Alder reaction (figure 12) with sulphur dioxide. The same procedure used to locate was used to locate the TS for the endo and exo pathway for this alternative Diels-Alder reaction, and the subsequent energetic analysis, summarised in the tables below.

{| class="wikitable"
! Species
! Sum of thermal and electronic free energies (Hartee/Particle)
|-
| Sulfur dioxide
| -0.118614
|-
| o-Xylylene
| 0.178112
|-
| Endo Transition State
| 0.090562
|-
| Exo Transition state
| 0.092078
|}

{| class="wikitable"
! Reaction Product
! Reaction Barrier (KJ/mol)
! Reaction energy (KJ/mol)
|-
| Alternative Endo
| 108.09
| 16.04
|-
| Alternative Exo
| 119.61
| 20.50
|}

The data shows that this alternative Diels-Alder reaction is kinetically and thermodynamically unflavoured. Both the endo and exo TSs are higher in energy than the TSs for the cheletropic and Diels-Alder reactions, with the exocyclic diene of Xylylene. The reaction barriers for the alternative Diels-Alder reaction are very high as the approach trajectory of the sulfur dioxide to the cis-diene within the xylylene ring is sterically hindered, compared to the exocyclic diene which is more sterically accessible. Also the reaction is thermodynamically unfavourable as the reaction energies are positive (both exo and endo pathways are endothermic). This is because this Diels-Alder reaction doesn’t benefit from the aromatisation of the o-xylylene ring.

===LOG files===
Optimised sulfur dioxide = [[File:AS6115SO2_MIN.LOG]]

Optimised o-xylylene = [[File:As6115O-XYLYLENE_MIN.LOG]]

Optimised endo TS = [[File:AS6115ENDO_TS_OPT.LOG]]

Optimised exo TS = [[File:AS6115EXO_TS_OPT.LOG]]

Optimised Cheletropic TS = [[File:AS6115CHELETROPHIC_REACTION_TS_OPT.LOG]]

Optimised endo Product = [[File:AS6115ENDO_PRODUCT_MIN_PM6.LOG]]

Optimised exo Product = [[File:AS6115EXO_PRODUCT_MIN.LOG]]

Optimised Cheletropic Product = [[File:AS6115CHELETROPIC_PRODUCT_MIN.LOG]]

Alternative Diels-Alder exo TS = [[File:AS6115ALT-DA_EXO_TS_OPT.LOG]]

Alternative Diels-Alder endo TS = [[File:AS6115ALT-DA_ENDO_TS_OPT.LOG]]

Alternative Diels-Alder exo product = [[File:AS6115ALT-DA_EXO_PRODUCT_MIN.LOG]]

Alternative Diels-Alder endo product = [[File:AS6115ALT-DA_ENDO_PRODUCT_MIN.LOG]]

==Extensions==
===Electrolytic ring opening/closure===
[[File:AS6115Ext1scheme1.png |thumb|centre|400px|Figure 13: Reaction scheme for the electrocylic ring closing/opening reaction between 3,4-dichlorocyclobut-1-ene and (2Z,4E)-2,5-dichlorohexa-2,4-diene]]

The 4π-electrocyclic ring opening/closing reaction which interconverts 3,4-dichlorocyclobut-1-ene and (2Z,4E)-2,5-dichlorohexa-2,4-diene was investigated. The ring open form was first built in Gaussian and minimised, and then the carbon atoms between which the bond forms during the ring closing reaction had their coordinates frozen. This guess-TS was then optimised at the PM6 level. From this an IRC calculation was run (animation shown below). The same analysis of reaction barriers and reaction energy as done in exercises 1 and 2 was performed from this reaction. (Please note that the activation energy and reaction energy quoted in the table is specifically for the ring-opening electrocyclic reaction.)

{| class="wikitable"
! [[File:AS6115-Electrocyclic-movie2.gif |centre|x400px| IRC animation]]
|-
| Figure 14 Animation the elctrocyclic ring closing reaction
|}

As expected the ring open diene form is more stable than the cyclobutene. This is two be expected as the 4-membered ring is highly strained (it suffers from significant angle strain and torsional strain). Also the high torsional strain means that the two chlorine substituents are locked in the cis conformation and therefore the molecule also suffers from steric strain since the chlorine atoms are relatively large.

(It could be worth investigating the trans- butene to see how much sterics affects the reaction energy [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 10:48, 6 March 2018 (UTC))

{| class="wikitable"
! Species
! Sum of thermal and electronic free energies (KJ/mol)
|-
| (2Z,4E)-2,5-dichlorohexa-2,4-diene
| 129.8514108
|-
| Transition State
| 326.2969913
|-
| 3,4-dichlorocyclobut-1-ene
| 177.5923287
|}

{| class="wikitable"
! Reaction Barrier (KJ / mol)
! Reaction Energy (KJ/mol)
|-
| 148.70
| -47.74
|}

The reaction energy is negative which indicates that the electrocyclic ring opening is exothermic and therefore thermodynamically favourable. This is to be expeected since the ring opening process relieves the ring strain and forms the less sterically strained diene product in which the chlorine atoms are further apart.

From analysis of the stereochemistry of the products and reactants form the IRC, it can be stated that this electrocyclic reaction is conrotation (i.e. the two chlorine substituents will rotate in the same direction during the reaction). This can be explained by considering the Woodward-Hoffman rules. The Woodward-Hoffman rules is based on the principle of conservation of orbital symmetry, and states that for a thermally allowed pericyclic reaction the number of components with the correct number of which satisfy: (4q +2)s + (4r)a must be odd. Where q and r are integers and the susbscript s represents suprafacially interacting components and a represents antrafacial components. For this electrocyclic reaction can be considered as being a single component with four π-electrons and as such this means it must interact antarafacially for this reaction to be thermally allowed (see figure 15). This antrafacial interaction can only be achieved via a conrotatory mechanism. Figure 15 shows how the it is necessary for the HOMO of the diene to undergo conrotation in order for to overlap in such a fashion to form the new C-C σ bonding orbital, in the cyclobutene product.

[[File:AS6115Ext1scheme2.png |thumb|centre|400px|Figure 15: Schematic illustrating the conrotatory ring closing mechanism]]

The frontier MOs for the reactant, TS and product for this electrocyclic reaction are displayed in the table below. The HOMO and LUMO of 2,5-dichlorohexa-2,4-diene are similar to that of butadiene, allbeit with additional contribution from the chlorine atoms. Comparing the HOMO of 2,5-dichlorohexa-2,4-diene to the HOMO of the TS, seems to support the Woodward-Hoffman analysis that this reaction proceeds with conrotation. However the frontier MOs of the TS and 3,4-dichlorocyclobut-1-ene are quite complicated which makes analysis difficult. When looking at the energies it can be seen that the HOMO of the TS is higher in energy than that of the reactant and products, which is to be expected since the TS is the highest energy point in the IRC, however the opposite trend is observed for the LUMOs.

{| class="wikitable"
! Species
! HOMO
! LUMO
|-
| (2Z,4E)-2,5-dichlorohexa-2,4-diene
| [[File:AS6115Electrocyclic-dine-HOMO.png |centre|x200px| IRC animation]]
Relative Energy: -0.3574
| [[File:AS6115Electrocyclic-dine-LUMO.png |centre|x200px| IRC animation]]
Relative Energy: -0.01145
|-
| Transition State
| [[File:AS6115Electrocyclic-TS-HOMO.png |centre|x200px| IRC animation]]
Relative Energy: -0.33726
| [[File:AS6115Electrocyclic-TS-LUMO.png |centre|x200px| IRC animation]]
Relative Energy: -0.02386
|-
| 3,4-dichlorocyclobut-1-ene
| [[File:AS6115Electrocyclic-cycle-HOMO.png |centre|x200px| IRC animation]]
Relative Energy: -0.38986
| [[File:AS6115Electrocyclic-cycle-LUMO.png |centre|x200px| IRC animation]]
Relative Energy: -0.00382
|}

(You are showing the conversion of 4 pi orbitals to 2 pi and 2 sigma orbitals, and so 4 MOs are needed for the analysis. It looks like the HOMO of the products might not be so relevant to this [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 10:48, 6 March 2018 (UTC))

===LOG files===

Optimised (2Z,4E)-2,5-dichlorohexa-2,4-diene = [[File:AS6115EXT1-RING-OPEN-MIN.LOG]]

Optimised Transition state = [[File:As6115-Ext1CORRECT-TS-OPT.LOG]]

Optimised 3,4-dichlorocyclobut-1-ene = [[File:AS6115CORRECT(2Z4E)-25-DICHLOROHEXA-24-DIENE-MIN.LOG|AS6115CYCLOBUTENE.LOG]]

===Claisen Sigmatropic rearrangement===
[[File:AS6115Extension1scheme1.png |thumb|centre|400px|Figure 16: Reaction Scheme of the Claisen rearrangement between 3-chloro-3-(vinyloxy)prop-1-ene and (E)-5-chloropent-4-enal]]

The claisen rearrangement is a [3,3]-sigmatropic rearrangement reaction which converts 3-chloro-3-(vinyloxy)prop-1-ene to (E)-5-chloropent-4-enal was also investigated in this experiment. The method used the TS was to directly carry out a TS (Berny) optimisation of a guess-TS for the sigmatropic reaction (a chair-like structure) at the PM6 level. Following this an IRC calculation was run. From the initial and final frame of the IRC reactant and product of the Claisen (i.e. the allyl vinyl ether and unsaturated aldehyde) was obtained and optimised.

{| class="wikitable"
! [[File:AS6115Clasien-film1.gif |centre|x400px| IRC animation]]
|-
| Figure 17 Claisen rearrangement IRC
|}

{| class="wikitable"
! Species
! Sum of thermal and electronic free energies (KJ/mol)
|-
| 3-chloro-3-(vinyloxy)prop-1-ene
| 82.12887106
|-
| Transition State
| 222.5397832
|-
| (E)-5-chloropent-4-enal
| 33.55569021
|}

{| class="wikitable"
! Reaction Barrier (KJ / mol)
! Reaction Energy (KJ/mol)
|-
| 140.41
| -48.57
|}

The data in the tables above reveal that the aldehyde product is more stable than the allyl vinyl ether. This is essentially due to the fact that the C=O bond is stronger than the C=C bond. (Please note the values for the reaction barrier and reaction energy are for the sigmatropic rearrangement going from the 3-chloro-3-(vinyloxy)prop-1-ene to the (E)-5-chloropent-4-enal). The reaction is exothermic, as the reaction energy is negative, and as it is thermodynamically favourable to form the aldehyde product.

The alkene in the aldehyde product has E-stereochemistry this can be explained from the analysis of the Claisen TS. The transition state has a chair like structure and the large chlorine substituent prefers to adopt a pseudo-equatorial position, and therefore proceeds to form the E-alkene product (see figure 17).

{| class="wikitable"
! Species
! HOMO
! LUMO
|-
| 3-chloro-3-(vinyloxy)prop-1-ene
| [[File:AS6115Claisen-ether-HOMO.png |centre|x200px| IRC animation]]
Relative Energy: -0.36103
| [[File:AS6115Claisen-ether-LUMO.png |centre|x200px| IRC animation]]
Relative Energy: -0.00247
|-
| Transition State
| [[File:AS6115Claisen-TS-HOMO.png |centre|x200px| IRC animation]]
Relative Energy: -0.31763
| [[File:AS6115Claisen-TS-LUMO.png |centre|x200px| IRC animation]]
Relative Energy: -0.01767
|-
| (E)-5-chloropent-4-enal
| [[File:AS6115Claisen-CHO-HOMO.png |centre|x200px| IRC animation]]
Relative Energy: -0.36658
| [[File:AS6115Claisen-CHO-LUMO.png |centre|x200px| IRC animation]]
Relative Energy: -0.00822
|}

(Again here you have 6 electrons and 6 orbitals that you'd need to track from reactants to products. There are going to be some complicated rotations going on converting sigma and pi orbitals at the reacting sites [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 10:48, 6 March 2018 (UTC))

The frontier MOs of the reactant, TS and product for this claisen rearrangement along with their corresponding relative energies are listed in the table above. The MOs for this system are quite complex and it is difficult to discern how the frontier MOs develop on going from the allyl vinyl ether through the TS to the γ,δ-unsaturated aldehyde. When considering the relative energies of the frontier MOs the trend observed, is the same as that of the electrocyclic reaction, with the TS HOMO being higher in energy than the reactant and product HOMOs, whilst the TS LUMO is lower in energy than the LUMO of the reactant and product.

===LOG files===

Optimised 3-chloro-3-(vinyloxy)prop-1-ene = [[File:AS6115VINYL-ALLYL-ETHER-MIN.LOG]]

Optimised (E)-5-chloropent-4-enal = [[File:AS6115UNSATURATED-ALDEHYDE-MIN.LOG]]

Optimised TS = [[File:AS6115CLAISEN3-GUESS_TS.LOG]]

==Conclusion==

Overall this experiment used relatively complex computational techniques on Gaussian to locate the TS structures for a variety of different pericyclic reactions, as well as reveal more information about the reactions. In exercise 1 a simple Diels-Alder reaction was investigated and the MOs of the TS and reactants were explored.

In exercise 2 the endo and exo reaction pathways of an inverse electron-demand Diels-Alder was studied and again the frontier MOs of the system were considered. Also reaction barriers and reaction energies where calculated, which revealed the endo product was both the kinetic and thermodynamic product of the reaction.

In exercise 3, a range of different possible cycloaddition reactions which can take place between o-xylene and sulfur dioxide where considered and compared, in terms of activation energies and reaction energies to probe how thermodynamically and kinetically favoured each pathway is.

In addition, further work was carried out to investigate applications of these techniques to other pericyclic reactions. This included the elucidation that the electrocyclic ring opening/closing reaction was conrotatory, in accordance with the Woodward-Hoffman rules. Furthermore, an evaluation of a [3,3]-sigmatropic rearrangement TS helped reveal the stereoselectivity of the reaction.

Ultimately, the confidence that can be placed in the findings of this experiment is limited by the accuracy of the computational methods used. The experiment could be improved by using more accurate methods (e.g repeating optimisation calculations for excercises 1 and 3 at the B3LYP level). Additionally comparing the predictions of the in sillico model with experimental data would allow the validity of this computational experiment to be assessed.

Rep:Y3.C3 og108

2025-09-01T09:50:29Z

Move page script: Move page script moved page Y3.C3 og108 to Rep:Y3.C3 og108: Move to report namespace

'''Year 3 Computational Lab Module 3: Transition State Structures'''
''Oliver Garnett'' See also [[Y3.C2_og108]]

Previous modules have explored the structures of stable organic and inorganic molecules, now analysis of transition state structures will be studied. The reactions chosen for this investigation are the Cope rearrangement and Diels Alder cycloaddition reactions.

A transition state can be defined as the geometry a molecule assumes at the saddle point on a potential energy surface. Transition state theory is focused on accounting for the kinetics of a reaction using the potential energy surface. The energy of the transition state relative to the reactants is known at the activation energy. This energy barrier dictates which product is formed fastest in a reaction and explains why some products are favoured over others despite sometimes not being the most thermodynamically stable.

=The Cope Rearrangement=

The first reaction under study for transition state analysis is the Cope rearrangement. This reaction was developed by Arthur C. Cope and is a [3,3] sigmatropic rearrangement of 1,5-dienes. The reaction can be viewed as a concerted pericyclic reaction, for the purposes of this investigation, transition state theory will be employed to understand the reaction dynamics. The rearrangement will be observed via both the boat and chair transition states.

{| class="wikitable" border="1" style="margin: 1em auto 1em auto; text-align: center"
|-
| [[image:Og108 mod3 part1 intropic2.PNG|thumb|left|400px|''Figure 2: The Cope rearrangement via chair and boat transition states'' ]]
|-
|}

==Establishing Reactants and Products==

First, a starting point for studying the transition states must be established by focusing on the reactant and product molecules (which are structurally identical). The optimised geometry and energy of different conformational isomers of 1,5-Hexadiene are studied to find which are the most stable.

===Optimisation and Confirmational Analysis===

A conformational isomer usually occurs as a result of the low kinetic barrier for rotation about a σ-bond. If the energy of the system is plotted as a function of the rotation about the σ-bond then one ore more mimia will arise along the curve. These minima have a finite lifetime and therefore have an associated zero-point energy, entropy and free energy<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref>.

[[image:og108_mod3_part1_intropic.PNG|centre|thumb|600px| '''Firgure 2: Energy as a function of rotation about a C-C bond for butane, output file provided by Henry S Rzeper'''<ref>http://hdl.handle.net/10042/to-4857</ref>]]

The variance of energy with angle of rotation about a C-C σ-bond can be attributed to three effects:
* The orbital overlap (bond orientation) effect arises from alignment of σ and σ* orbitals leading to NBOs. This favors the antiperiplanar conformation and is the most significant of the three effects.
* Pauli repulsions which disfavor any eclipsed orientations.
* Dispersion (Van der Waals) forces which are isotropic and operate at longer range than the other two effects. This effect is highly dependent on the functional groups on either side of the C-C bond.

Taking these three effects into account one can deduce that the most stable conformation about a RH<sub>2</sub>C-CH<sub>2</sub>R σ-bond should be an antiperiplanar form<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref>.

The energies of the main conformations of 1,5-Hexadiene were established first to establish a starting point structure.

{| class="wikitable" border="1" style="align: center"
|+ Table 1: 1,5-Hexadiene Conformers: Optimised at the HF/3-21G level
! scope="col" |Conformer
! scope="col" |Point Group
! scope="col" |Total Energy (Hartrees)
! scope="col" |Relative Energy*(Kcal mol<sup>-1</sup>)
|-

! Gauche 1 <jmol><jmolAppletButton>
<uploadedFileContents>Og108 15hex gauche1 opt1.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13350</ref>
| C<sub>2</sub>
| -231.68772
| +3.100
|-

! Gauche 2 <jmol><jmolAppletButton>
<uploadedFileContents>Og108 15hex gauche2 opt1.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13349</ref>
| C<sub>2</sub>
| -231.69167
| +0.621
|-
! Gauche 3 <jmol><jmolAppletButton>
<uploadedFileContents>Og108 15hex gauche3 opt1.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13345</ref>
| C<sub>1</sub>
| -231.69266
| 0.000
|-

! Gauche 4 <jmol><jmolAppletButton>
<uploadedFileContents>Og108 15hex gauche4 opt1.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13348</ref>
| C<sub>2</sub>
| -231.69153
| +0.709
|-
! Gauche 5 <jmol><jmolAppletButton>
<uploadedFileContents>Og108 15hex gauche5 opt1.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13347</ref>
| C<sub>1</sub>
| -231.68962
| +1.908
|-

! Gauche 6 <jmol><jmolAppletButton>
<uploadedFileContents>Og108 15hex gauche6 opt1.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13346</ref>
| C<sub>1</sub>
| -231.68916
| +2.196
|-

! Anti 1 <jmol><jmolAppletButton><uploadedFileContents>Og108 15hex anti1 opt1.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13311</ref>
| C<sub>2</sub>
| -231.69260
| +0.038
|-

! Anti 2 <jmol><jmolAppletButton>
<uploadedFileContents>Og108 15hex anti2 opt1.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13322</ref>
|C<sub>i</sub>
| -231.69254
| +0.075
|-

! Anti 3 <jmol><jmolAppletButton>
<uploadedFileContents>Og108 15hex anti3 opt2.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol><ref>http://hdl.handle.net/10042/to-13323</ref>
| C<sub>2h</sub>
| -231.68907
| +2.253
|-

! Anti 4 <jmol><jmolAppletButton>
<uploadedFileContents>Og108 15hex anti4 opt4.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13324</ref>
| C<sub>1</sub>
| -231.69097
| +1.060
|-
|}
*Energy relative to most stable conformer (gauche 3 in this case). 1 Hartree = 627.503 kcal mol<sup>-1</sup><ref>http://mccammon.ucsd.edu/~blu/Research-Handbook/physical-constant.html</ref>

Anti 1, anti 2, and Gauche 3 appear to be at least an order of magnitude more stable than the other conformations. However the most stable of these is Gauche 3 which is surprising since from chemical intuition one would expect the most stable conformer to be an antiperiplanar form as discussed previously.
[[image:Og108 15hex moststableconforms1.PNG|centre|pix 400| Figure 3: ''Newman projections of the most stable conformations'' ]]
This is possibly a result of unexpectedly strong Van der Waals forces stabilising the closer proximity of the two alkenes. The closest alkene hydrogen's in the gauche 3 form were measured <jmol><jmolAppletButton>
<uploadedFileContents>Og108 15hex gauche3 opt1.mol</uploadedFileContents>
<text>HERE</text>
<script>background=white; measure 10 6; measure 8 3</script>
</jmolAppletButton>
</jmol> to be 2.62A apart which is a little too far to be significantly attractive. Alternatively the calculation being performed at the HF/3-21G may be too inaccurate to distinguish accurately the energy of these conformations. Therefore, these conformers were re-optimised using the more rigorous density functional theory approach at the DFT/B3LYP/6-31G(d) level. This provided a more accurate view of the relative energies.

{| class="wikitable" border="1" style="align: center"
|+ Table 2: 1,5-Hexadiene Conformers: Further optimisation at the DFT/B3LYP/6-31G(d) level
! scope="col" |Conformer
! scope="col" |Point Group
! scope="col" |Total Energy (Hartrees)
! scope="col" |Relative Energy (Kcal mol<sup>-1</sup>)
|-

! Anti 1 <jmol><jmolAppletButton><uploadedFileContents>Og108_15hex_anti1_tightopt1.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13351</ref>
| C<sub>2</sub>
| -234.61179
| +0.000
|-

! Anti 2 <jmol><jmolAppletButton>
<uploadedFileContents>Og108_15hex_anti2_tightopt1.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13352</ref>
|C<sub>i</sub>
| -234.61171
| +0.050
|-

! Gauche 3 <jmol><jmolAppletButton>
<uploadedFileContents>Og108_15hex_gauche3_tightopt1.mol</uploadedFileContents>
<text>VIEW</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13353</ref>
| C<sub>1</sub>
| -234.61133
| +0.289
|-
|}

The Anti 1 and Anti 2 conformers now appear to be more stable than the Gauche 3 conformer by around 0.2Kcal mol<sup>-1</sup>. This is in line with previous predictions and is likely caused by the larger basis set 6-31G* which can account for the strongly stabilising stereoelectronic effects (effect 1 discussed previously). The Antiperiplanar conformations can be stabilised in this way from σ(C-C) → σ*(C-C) donation which is slightly more stable than σ(C-H) → σ*(C-C) donation seen in gauche forms because of the better orbital overlap. There are also noticable differences in some of the dihedral angles after the re-optimisation.

===Frequency Analysis===

Frequency analysis on the DFT optimised structures verified that the low energy conformations correspond to local minima.The analysis provides 42 distinct positive vibrational modes which are very similar for all 3 conformers.

[[image:Og108 15hex anti1 IR1.PNG|thumb|600 px| centre|''Figure 5: IR absorption spectrum for the anti 1 conformer'']]

Only positive vibrational modes were seen indicating the optimised structures are those of local minima. The alkene stretching modes can distinguish the conformers and have been tabulated below.

{| class="wikitable" border="1" style="align: center"
|+ Table 3: 1,5-Hexadiene Conformers: Frequency analysis at the DFT/B3LYP/6-31G(d) level
! scope="col" |Conformer
! scope="col" |Symmetrical C=C Stretching mode / cm<sup>-1</sup>
! scope="col" |Asymmetrical C=C Stretching mode / cm<sup>-1</sup>
|-

! Anti 1 <ref>http://hdl.handle.net/10042/to-13358</ref>
| 1732 [4.73][[Media:Og108 15hex anti1 freq31.gif|Show Animation]]
| 1735 [13.55][[Media:Og108 15hex anti1 freq32.gif|Show Animation]]
|-

! Anti 2 <ref>http://hdl.handle.net/10042/to-13359</ref>
| 1731 [0.00][[Media:Og108 15hex anti2 freq31.gif|Show Animation]]
| 1734 [18.13][[Media:Og108 15hex anti2 freq32.gif|Show Animation]]
|-

! Gauche 3 <ref>http://hdl.handle.net/10042/to-13360</ref>
| 1732 [6.88][[Media:Og108 15hex gauche3 freq31.gif|Show Animation]]
| 1733 [6.13][[Media:Og108 15hex gauche3 freq32.gif|Show Animation]]
|-
|}

Although these values are all fairly similar, the differences do highlight the importance of symmetry in determining the number of peaks displayed in an IR spectrum. Experimental literature values are around 1620 - 1680cm<sup>-1</sup><ref>J. Coates, Encyclopaedia of Analytical Chemistry, R. A. Meyers (Ed.), 2000, pp 10815 - 10831</ref> which are not similar to the values shown above. It should however be noted that literature values do not distinguish between different conformations as the rotations are faster than the timescale for IR spectroscopy.

===Thermochemical Analysis===
A deeper look into the log files from the frequency analysis allows thermochemical data to be uncovered.

Anti 2 Conformer at 298K
<pre>
Zero-point correction= 0.142507 (Hartree/Particle)
Thermal correction to Energy= 0.149853
Thermal correction to Enthalpy= 0.150797
Thermal correction to Gibbs Free Energy= 0.110933
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>

Modifying the input file allows the analysis to be performed at different temperatures and pressures. This was then compared to the calculation at room temperature.

Anti 2 Conformer at 0K <ref>http://hdl.handle.net/10042/to-13371</ref>
<pre>
Zero-point correction= 0.142507 (Hartree/Particle)
Thermal correction to Energy= 0.142507
Thermal correction to Enthalpy= 0.142507
Thermal correction to Gibbs Free Energy= 0.142507
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.469204
Sum of electronic and thermal Enthalpies= -234.469204
Sum of electronic and thermal Free Energies= -234.469203
</pre>

The data shows all energy sums are the same for the calculation at 0K since at this temperature no energy contributions come from thermal activity. These are in turn the same as the sum of the electronic and zero-point energies for the calculation at room temperature which is also expected since the zero point energy is the energy of the ground state of a quantum mechanical system.

From comparison of the data at 0K and room temperature it can be seen that increasing the temperature raises the thermal energy and thermal enthalpy terms, while the free energy term decreases. Clearly raising the temperature will increase the thermal contributions to the energy and enthalpy terms in lines two and three of the data. The behavior of the free energy term can be rationalised using the the Gibbs Free Energy equation: '''G(p,T)''' = '''H''' − '''TS'''. From this it can be seen that an increase in temperature results in a lower free energy.

==Transition States==

Now that the structures of the more stable reactants has been established, the transition state structures may be found.

===Optimisation===

Transition states are more difficult to optimise since a maxima rather than a minima stationary point is being located. A range of different techniques will be employed to optimise these structures and these will be compared.

Firtsly an allyl fragment was constructed and optimised which would be used to form the transition state structures and for further calculations. This calculation was performed at the HF/3-21g level and the optimised C<sub>2v</sub> structure can be seen here <jmol><jmolAppletButton>
<uploadedFileContents>Og108_allylfrag_opt1.mol</uploadedFileContents>
<text>Optimised allyl fragment</text>
<script>background=white; measure 4 6; measure 4 5; measure 1 4 6</script>
</jmolAppletButton>
</jmol> .

====Chair-like Structure====
<u>Optimisation 1</u><ref>http://hdl.handle.net/10042/to-13506</ref>: A "guess" transition structure was used as a starting point for a optimisation to a minimum TS (Berny). This was done by pasting two of the previously optimised allyl fragments and oreintating them into a chair structure on Gaussview 5.0 with the terminal alkenes being set 2.2Å apart as shown in ''figure 6''.

[[image: Og108_chair1_initialgeom1.PNG|thumb|right|px 400|''Figure 6: initial "guess structure" for the chair transition state'']]

This guess structure was then subject to an optimisation and frequency analysis using a Hessain force constant matrix. The force constant was calculated once and the parameter "Opt=NoEigen" was also added to stop the calculation crashing if more than one imaginary frequency is detected which commonly occurs if the guess structure is not close enough to the actual transition state.

<pre>
Filename = //icfs16.cc.ic.ac.uk/og108/Desktop/Labs/3rd Year Labs/
Computational/Module 3/Part 1/Chair/output/og108_chair1_optTSB3.out

chair1 optTSB1
File Name = og108_chair1_optTSB3
File Type = .log
Calculation Type = FTS
Calculation Method = RHF
Basis Set = 3-21G
Charge = 0
Spin = Singlet
E(RHF) = -231.61932246 a.u.
RMS Gradient Norm = 0.00000977 a.u.
Imaginary Freq =
Dipole Moment = 0.0005 Debye
Point Group = C1
Job cpu time: 0 days 0 hours 2 minutes 20.3 seconds.

</pre>

The above shows the molecular energy is -231.61932 Hartrees, similar to that given in [[Mod:phys3#Appendix 2|Appendix 2]]. The terminal bond lengths of the optimised structure had been reduced to 2.02Å.

Frequency analysis from this calculation shows an imaginary vibrational mode at [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Og108_chair1_freq1.gif -817.96 cm<sup>-1</sup>] representitive of the chair transition state. The negative nature of the vibrational mode arises from the second derivative of the potential energy surface implies a energy maximum has been reached.

<u>Optimisation 2</u><ref>http://hdl.handle.net/10042/to-13507</ref>: Once again the guess structure was used as a starting point in preperation for the frozen coordinate optimisation method. The terminal bonds were "frozen" at a length of 2.2Å and the rest of the molecule was optimised. A second optimisation was then performed to minimise the terminal bond lengths to a transition state (Berny) but without calculating the force constants as with the first method. The resulting structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Og108_chair1_optderivative2.mol</uploadedFileContents>
<text>Optimised Chair via the freeze coordinate method</text>
<script>background=white; measure 4 6; measure 4 5; measure 1 4 6</script>
</jmolAppletButton>
</jmol> .

<pre>
Filename = //icfs16.cc.ic.ac.uk/og108/Desktop/Labs/3rd Year Labs
/Computational/Module 3/Part 1/Chair/output/og108_chair1_optderivative2.out

chair opt derivative
File Name = og108_chair1_optderivative2
File Type = .log
Calculation Type = FTS
Calculation Method = RHF
Basis Set = 3-21G
Charge = 0
Spin = Singlet
E(RHF) = -231.61932203 a.u.
RMS Gradient Norm = 0.00004051 a.u.
Imaginary Freq =
Dipole Moment = 0.0004 Debye
Point Group = C1
Job cpu time: 0 days 0 hours 0 minutes 55.8 seconds.
</pre>

It appears that the energy for the first method was actually fractionally lower than that for the frozen coordinate method. The RMS Gradient is also closer to 0 and so the transition structure is at a flatter point on the potential energy curve. This implies that the first method was more effective for finding the optimised transition state structure. This would therefore suggest that the guess structure used was already very close to the optimised transition state. If a more complicated molecule had been used with a transition state that is more difficult to predict, then the the first method may not be the most effective or even reached a transition state at all.

====Boat-like Structure====
<u>Optimisation 3</u><ref>http://hdl.handle.net/10042/to-13512</ref><ref>http://hdl.handle.net/10042/to-13511</ref>:

A new method was used for the boat optimisation which involves interpolating between the reactant and product structures to find the transition state. This method unfortunately requires careful numbering of each atom to ensure the products correlate to the reactants. The first attempt failed to produce the correct structure: <jmol><jmolAppletButton>
<uploadedFileContents>Og108 boat1 optQST2 1 5.mol</uploadedFileContents>
<text>Failed boat optimisation via the QST2 method</text>
<script>background=white; measure 2 3; measure 4 5 6</script>
</jmolAppletButton>
</jmol>

The geometry of reactant and product structures were adjusted, changing the C2-C3-C4-C5 dihedral angle to 0° and the two inside C-C-C angles to 100°. The new geometry allowed a fully optimised boat transition structure to be obtained:<jmol><jmolAppletButton>
<uploadedFileContents>Og108_boat1_optQST2_2_7.mol</uploadedFileContents>
<text>Successful boat optimisation via the QST2 method</text>
<script>background=white; measure 2 3; measure 4 5 6</script>
</jmolAppletButton>
</jmol>

<pre>
Filename = //icfs16.cc.ic.ac.uk/og108/Desktop/Labs/3rd Year Labs/Computational
/Module 3/Part 1/Boat/output/og108_boat1_optQST2_2_7.out

boat optQST2 2 7
File Name = og108_boat1_optQST2_2_7
File Type = .log
Calculation Type = FREQ
Calculation Method = RHF
Basis Set = 3-21G
Charge = 0
Spin = Singlet
E(RHF) = -231.60280108 a.u.
RMS Gradient Norm = 0.00017457 a.u.
Imaginary Freq = 1
Dipole Moment = 0.1591 Debye
Point Group = C2V
Job cpu time: 0 days 0 hours 0 minutes 12.5 seconds.
</pre>

Frequency analysis uncovered a single imaginary vibrational mode at [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Og108_boat1_optQST2_2_7.gif -840.98 cm<sup>-1</sup>] corresponding to a boat transition state structure and is slightly different to the chair vibrational mode though different optimisation methods were used. The vibration shows how terminal σ bonds form and break during the vibration which is typical for an imaginary vibration of transition state structure.

===Intrinsic Reaction Coordinate (IRC)===

This process allows the minimum energy path to be followed from the transition state to its local minima on the PES, in this case the product. This shows which conformations the transition state is connected to and would be impossible to achieve simply through inspection of the structures. All calculations were carried out at the HF/3-21G level in the forward direction only since the reaction is reversible and symmetrical. The chair transition structure was used to find the most efficient IRC method and this method was then used for the Boat structure as well.

<u>Initial IRC</u>: The first calculation was performed with 50 points along the IRC.

This calculation failed to reach a minumum geometry so three different methods were tested to achieve this:

<u>IRC 1</u>: This calculation was simply an optimisation of the final structure from the previous IRC calculation to reach a local minima.

<u>IRC 2</u>: Another calculation was then performed but with 150 points along the IRC as opposed to only 50.

<u>IRC 3</u>: Finally the a last calculation was performed in which all the force constants were computed. This is usually by far the most effective method but is computationally expensive and generally is quite impractical for larger systems.

A summary of the 4 methods used is shown below. Once the most effective method was established then this was performed on the boat transition structure as well.

{| class="wikitable" border="1" style="align: center"
|-
|+Table 7 - Summary of IRC calculations
! IRC Method
! Transition state
! Maximum points along the IRC
! Terminated After (Steps)
! Compute Force Constant
! RMS Gradient
! Total Energy
|-
| 1
| Chair<ref>http://hdl.handle.net/10042/to-13522</ref>
| 50
| 26
| Once
| 0.00000281
| -231.69166702
|-
| 2
| Chair<ref>http://hdl.handle.net/10042/to-13526</ref>
| 150
| 26
| Once
| 0.00000975
| -231.61932246
|-
| 3
| Chair<ref>Calculated on PC</ref>
| 50
| 47
| Always
| 0.00000975
| -231.61932246
|-
| 3
| Boat<ref>http://hdl.handle.net/10042/to-13524</ref>
| 50
| 51
| Always
| 0.00001086
| -231.60280247
|-
|}

The two movies below show the structural changes that occur as the potential energy surface is followed from the transition state down to a local minima:

{| class="wikitable" border="1" style="align: center"
|-

|-
| [[file:Og108 chair1 IRC1 allforce1.gif|thumb|centre|250px|''Figure 8: Chair IRC'']]
| [[file:Og108 boat1 IRC allforce1.gif|thumb|centre|250px|''Figure 9: Boat IRC'']]
|-
| [[image:Og108 chair1 IRC allforce2 graph.PNG|thumb|centre|500px|''Figure 8: Chair IRC'']]
| [[image:Og108 boat1 IRC allforce2 graph.PNG|thumb|centre|500px|''Figure 9: Boat IRC'']]

|}

The final energy point on the chair IRC was at -231.692 and for the Boat -231.685 Hartrees. This is close enough to a local minimum for the IRC to be considered a success and from this it was determined that the boat transition state tends to the gauche 1 conformation whereas the Chair tends to gauche 2 conformer.

===Energy and the Activation Energy===

A table of energies for both transition structures was generated to determine which was the more stable. A comparison can also be made for both levels of theory by analysing the activation energies generated from each method. Therefore the reactant molecule was included for these calculations.

{| class="wikitable" border="1" style="align: center"
|+ Table 3: Transition State Energies
! rowspan="2" | Structure
! colspan="3" | HF/3-21G
! colspan="3" | B3LYP/6-31
|-
! scope="col" | Electronic Energy
! scope="col" | Sum of Electronic & Thermal Energies (298.15K)
! scope="col" | Sum of Electronic & Zero-point Energies (0K)
! scope="col" | Electronic Energy
! scope="col" | Sum of Electronic & Thermal Energies (298.15K)
! scope="col" | Sum of Electronic & Zero-point Energies (0K)
|-

! Chair
| -231.619322 <ref>http://hdl.handle.net/10042/to-13507</ref>
| -231.461339
| -231.466698 <ref>http://hdl.handle.net/10042/to-13787</ref>
| -234.556982 <ref>http://hdl.handle.net/10042/to-13781</ref>
| -234.408983
| -234.414905 <ref>http://hdl.handle.net/10042/to-13788</ref>
|-

! Boat
| -231.450929 <ref>http://hdl.handle.net/10042/to-13780</ref>
| -231.445300
| -231.450929 <ref>http://hdl.handle.net/10042/to-13783</ref>
| -234.543093 <ref> http://hdl.handle.net/10042/to-13780</ref>
| -234.396005
| -234.402338 <ref>http://hdl.handle.net/10042/to-13784</ref>
|-

! Anti 2 Conformer (reactant)
| -231.692535 <ref>http://hdl.handle.net/10042/to-13789</ref>
| -231.532566
| -231.539540
| -234.611710 <ref>http://hdl.handle.net/10042/to-13359</ref>
| -234.460913
| -234.469204 <ref>http://hdl.handle.net/10042/to-13371</ref>
|-
|}

Using the data above the activation energy can be calculated at 298.15K and at 0K by subtracting the sum of the electronic and thermal energies of the reactants from that of the transition state.

{| class="wikitable" border="1" style="align: center"
|+ Table 3: Activation Energies (in Kcal mol<sup>-1</sup>)
! rowspan="2" | Structure
! colspan="2" | HF/3-21G
! colspan="2" | B3LYP/6-31
! rowspan="2" | Experimental Literature Value at 0K <ref>3rd Year Computational Chemistry Online Lab Manual, Module 3, 2011</ref>
|-
! scope="col" | At 298.15K
! scope="col" | At 0K
! scope="col" | At 298.15K
! scope="col" | At 0K
|-

! Chair
| 44.7
| 45.7
| 32.6
| 34.0
| 33.5 ± 0.5
|-

! Boat
| 54.8
| 55.6
| 40.7
| 42.0
| 44.7 ± 2.0
|-

|}

As seen from the table, the activation energy for the chair transition structure is quite significantly lower. This is probably due to less steric clashing in this state compared with the boat transition structure. The table also shows that the DFT method gives more accurate values than the Hartree-Fock method when compared with the literature. As discussed previously this is probably also because the larger 6-31G* basis set used in the DFT method can account for more orbital interactions than the smaller 3-21G basis set.

=Diels-Alder Reaction=

A diels alder reaction is a [4+2] cycloaddition process in which an electron rich conjugated diene bonds to an electron poor dienophile, commonly a substituted alkene, to form a cyclohexene adduct. The reaction involves the cleavage of three π-bonds to form two new σ-bonds between the two reactants and a new π-bond on the diene half of the product. This reaction is said to be a thermal cycloaddition process but there are different mechanisms that are used to describe the reactivity<ref>R. B. Woodward, T. J. Katz ''Tetrahedron'' , '''1959''' , 5 , 70 - 89 </ref>. It can either proceed via a single transition state in a concerted process or a non concerted asynchronous mechanism. Real mechanisms are a mixture of the two.

The first section will focus on modeling transition state structure of the prototyical Diels Alder reaction between butadiene and ethene. Gaussview 5.0 software will be used and similar techniques to those previously used for the cope reaction study will be employed. In addition to the two levels of theory previously used for optimisations, '''HF/3-21G''' and '''DFT/B3LYP/631-G*''', a semi-empirical method will also be explored known as '''AM1'''.

The second section will analyse a more complex diels alder reaction between a cyclohexadiene and maleic anhydride. This section will pay particular attention to the sereochemistry of the cycloaddition by exploring both the ''exo-'' and ''endo-'' product formation.

==Butadiene and Ethene Cycloaddition==

The two π-systems here; butadiene and ethene, contribute 4π and 2π electrons to the system respectively. Ethene's π and π* orbitals and butadiene's HOMO and LUMO are the principle orbitals involved and these can be classified as symmetric or antisymmetric with respect to the plane bisecting the central bond in each.

[[image:Og108_da_reactionscheme.PNG |''Figure 14: A prototypical Diels-Alder reaction]]

===Optimization and analysis of reactants===

Butadiene can exist as two isomers, s-''cis'' and s-''trans'' as well as gauche forms. Only the former is reacts with the dienophile but this form is known to be the higher in energy due to steric repulsion<ref>P. G. Szalay, H. Lischka, A. Karpfen, ''J. Phys. Chem.'' '''1989''',93 (18), pp 6629–6631</ref>.''cis''-Butadiene and Ethene were both constructed in Gaussview 5.0 and initially optimised using the semi-empirical Austin Model1 (AM1) method. This method uses stored parameters from experimental data such as the ionization energies of atoms or the dipole moments of molecules and omits certain parts of the calculation in an attempt to make it less computationally demanding and more accurate than ''ab initio'' methods. Semi-empirical methods work best with systems similar to those that were used to gather experimental data and can be quite erratic with unusual systems. In general, the calculation is more efficient and accurate than using a Hartree-Fock method but less so compared wth than a density funtctional theory calculation.

To understand the molecular dynamics involved in the Diels Alder cycloaddition one can simplify the reactivity by focusing on the interactions between the HOMO on one molecule and the LUMO on the other. This approach is the underlying principle behind Frontier Molecular Orbital theory. These generally play the most important role in the bonding that takes place and therefore molecular orbital analysis was then carried out to observe these orbitals on both molecules. Orbital symmetry can be used to determine which orbitals interact as it must be maintained throughout the reaction. The density functional method was used to obtain the orbitals shown bellow and only the HOMO and LUMO were of interest here.

{|class="wikitable" align="center"
|+ Table XXX - Summary of MOs for cis-butadiene and ethene
| width="100"|.
!align="center"|''HOMO''
!align="center"|''LUMO''
! Relative MO Energies
|-
| align="center"|'''Ethene'''
|[[Image:Og108 eth MO8.PNG|200px]]
|[[Image:Og108 eth MO9.PNG|200px]]
|rowspan="2"|[[Image:Og108 eth MO relE.PNG|200px]]
|-
| align="center"|σ<sub>v</sub> symmetry
| symmetric
| asymmetric
|-
| align="center"|'''''cis''-Butadiene'''
| [[Image:Og108 cisbut MO15.PNG|200px]]
| [[Image:Og108 cisbut MO16.PNG |200px]]
| rowspan="2"| [[Image:Og108 cisbut MO relE.PNG|200px]]
|-
| align="center"|σ<sub>v</sub> symmetry
| asymmetric
| symmetric
|-
|}

The symmetry with respect to the σ<sub>v</sub> mirror planes has been noted. This determines which orbitals interact. Consequently the HOMO on one molecule interacts with the LUMO on the other and vice-versa. This is one reason why the Diels Alder reaction is so successful.

===Transition state analysis===

To obtain a transition state structure the frozen coordinate method, as used in the Cope rearrangement study, was employed.

The optimised ethene and butadiene molecules were orientated on Gaussview 5.0 at the correct angle to form the transition state. This geometry was used as a guess structure for further reactions. The angle of attack is vitally important to the success of this reaction and careful care was taken to consider how vertical it should be. The angle is similar to that of the Burgi-Dunitz angle and in this case is 102°<ref>F. Brown, K. Houk ''Tetrahedron Letters'' '''1984''' 25, 41, 4609-4612 http://dx.doi.org/10.1016/S0040-4039(01)91212-2</ref> . The bond lengths were also set to 2.2A and 'frozen' for the initial optimisation at the HF/3-21G level as with the Cope reaction. The partially optimised structure was then optimised to a transition state at the HF/3-21G level with the derivative of the new C-C bonds calculated at every step.

An imaginary vibrational mode at [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Og108_chair1_freq1.gif -817.96 cm<sup>-1</sup>] (note this is exactly the same frequency as the negative vibrational mode for the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Og108_chair1_freq1.gif chair] transition state) is the result the structure being a maxima stationary point and therefore indicates the transation optimisation calculation was a success.

IRC analysis shows the energy of the system as a function of the reaction coordinate. The calculation was performed in both directions as the reactants and products were non-symmetrical and using the HF/3-21G method with 100 points along the IRC.

[[image:Og108_da_IRCpath.PNG|thumb|600 px| centre|''Figure 8: IRC analysis'']]

The activation energy for the transition state can be seen clearly and it can also be seen that the product is the thermodynamically less stable molecule. This is expected since there is a loss of conjugation which stabilizes the diene. The gif movie shows the two molecules separating nicely and also towards the end the diene can be seen to twist into the more stable gauche form.

[[image:Og108_da_IRC2.gif|thumb|600 px| centre|''Figure 8: IRC analysis'']]

Further optimisation was also performed at both the semi-empirical AM1 level and the more rigorous DFT/B3LYP/6-31G* level. Geometry analysis of the optimised transition state structure for both the <jmol><jmolAppletButton>
<uploadedFileContents>Og108 da furtheropt3 1.mol</uploadedFileContents>
<text>DFT/B3LYP/6-31G*</text>
<script>background=white; measure 1 14; measure 1 4; measure 14 12; measure 12 10; measure 10 7 4</script>
</jmolAppletButton>
</jmol> method and the <jmol><jmolAppletButton>
<uploadedFileContents>Og108 da furtheropt2 2.mol</uploadedFileContents>
<text>AM1</text>
<script>background=white; measure 1 14; measure 1 4; measure 14 12; measure 12 10; measure 10 7 4</script>
</jmolAppletButton>
</jmol> method showed notable differences. Firstly the bond angle in the AM1 method is 99.3° which is 3° more vertical than that of the density functional method at 102.3°. The latter is closer to other computed transition stated bond angles found in literature <ref>F. Brown, K. Houk ''Tetrahedron Letters'' '''1984''' 25, 41, 4609-4612 http://dx.doi.org/10.1016/S0040-4039(01)91212-2</ref> <ref> F. Bernardi, A. Bottoni, M. Robb, ''J. Am. Chem. Soc.''' '''1988''', Vol. 110, No. 10, http://pubs.acs.org/doi/pdf/10.1021/ja00218a009</ref>(note however that these were also ''ab initio'' calculations using a STO-3G and a 4-31G basis set). Both geometries show the diene and dienophile approach each other in a non parrallel plane manner as suggested from experiment<ref>W. Roush, H. Gillis,''J- Am. Chem. Soc.'', '''1982'''. 1s 2269</ref>. The C-C bond distance for the new σ-bonds being formed is 2.27Å for the DFT method and 2.12 for the AM1 method. These two values are in between the literature value of 2.217Å<ref>F. Brown, K. Houk ''Tetrahedron Letters'' '''1984''' 25, 41, 4609-4612 http://dx.doi.org/10.1016/S0040-4039(01)91212-2</ref>. This is shorter than a typical C-C bond length at around 1.5Å but crucially shorter than twice the van der waals radiius of a carbon atom. The latter point proves there is a true bonding interaction occurring but that this is a transition state given the bond is not shorter.

Mo analysis was then carried out using each method. Below is a table of the frontier molecular orbitals as well as 1 above and below the LUMO and HOMO respectively.

{|class="wikitable" align="center"
|+ Table XXX - Summary of the transition state Molecular Orbitals given by the two different optimisation methods
| width="100"| '''Method used'''
!align="center"|''HOMO-1''
!align="center"|''HOMO''
!align="center"|''LUMO''
!align="center"|''LUMO+1''
! Relative MO Energies
|-
| align="center"|AM1
|[[Image:Og108_da_optAM_MO16.PNG|200px|thumb| ''MO 16 symmetric'']]
|[[Image:Og108_da_optAM_MO17.PNG|200px|thumb| ''MO 17 antisymmetric'']]
|[[Image:Og108_da_optAM_MO18.PNG|200px|thumb| ''MO 18 symmetric'']]
|[[Image:Og108_da_optAM_MO19.PNG|200px|thumb| ''MO 19 antisymmetric'']]
|[[Image:|[[Image:Og108_da_optAM_MOenergy.PNG|200px]]|200px]]
|-
| align="center"|DFT/B3LYP/6-31G*
|[[Image:Og108_da_MO22.PNG|200px|thumb| ''MO 22 antisymmetric'']]
|[[Image:Og108_da_MO23.PNG|200px|thumb| ''MO 23 symmetric'']]
|[[Image:Og108_da_MO24.PNG|200px|thumb| ''MO 24 symmetric'']]
|[[Image:Og108_da_MO25.PNG|200px|thumb| ''MO 25 antisymmetric'']]
|[[Image:|[[Image:Og108_da_MOenergy.PNG|200px]]|200px]]
|-
|}

The order of the HOMO and HOMO-1 is different depending on the method used. The MOs also appear to differ slightly and DFT method shows small areas of electron density where there is none in the other method, notably in the LUMO and LUMO+1 diagrams. The DFT method is likely to be the more accurate picture for the molecular orbitals since it takes more orbital interactions into account. The reaction mechanism as discussed involves the σ-overlap of two π-clouds of both unsaturated systems. This is evident from the HOMO and HOMO-1 orbitals showing the overlap of the ''cis-''Butadiene π-orbital with the Ethene π* orbital resulting in the two new σ-bonds.

The LUMO shows significant mixing creating regions of shared electron density between the atoms of the fragments.

==Maleic Anhydride and Cyclohexa-1,3-diene==

Having looked at a prototypical diels alder reaction the study will now focus on the specific example of the reaction between cyclohexa-1,3-diene (the diene) and Maleic anhydride (the dienophile). Particular attention will be paid to the stereoselectivity by building on the MO analysis carried out in the previous section and by using simple symmetry principles and FMO theory.

The total electron count for a Diels alder is 6 (4 π electrons from the diene and 2 π electrons from the dienophile) and the reaction therefore follows the 4n+2 (n=1) Huckel rule. Consequently the reaction must follow Huckel topology and thus involve only suprafacial components.

With the dienophile now being subsituted, the reactants may approach one another from two distinct orientations. The substituent on the dienophile may face away from the diene resulting in the ''exo'' stereoisomer or toward the diene resulting in the ''endo'' isomer.

[[image:Og108_cyclo2_reaction_scheme1.PNG|px 500| centre| ''Figure XXXX: '''a''' = maleic anhydride, '''b''' = Cyclohexa-1,3-diene, '''c''' = the exo product, '''d''' = endo product'']]

Hoffman proposed that the thermodynamically more stable product is the ''exo'' isomer due to the lower steric bulk compared with the ''endo''<ref>R. Woodward, R. Hoffmann, J. Am Chem. Soc., 87, 1965, pp. 4388-4389</ref>. However the kinetic product is believed to be the ''endo'' isomer and as a result a lower activation energy is expected for this form.

===Optimization and analysis of reactants===

As with the previous section the reactants Maleic Anhydride and Cyclohexa-1,3-diene were constructed on Gaussview and optimised. These fragments would then be combined to form the transition state. The calculations for this section are all carried out using a DFT/B3LYP/6-31G* level as this was established to be the most accurate method.

Final geometries for both molecules:

<jmol><jmolAppletButton>
<uploadedFileContents>Og108_mal_opt2.mol</uploadedFileContents>
<text>Maleic Anhydride</text>
<script>background=white; measure 2 3; </script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13723</ref>

<jmol><jmolAppletButton>
<uploadedFileContents>Og108_cyclohex_opt2.mol</uploadedFileContents>
<text>Cyclohexa-1,3-diene</text>
<script>background=white; measure 2 3; measure 4 5 6</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13724</ref>

<u>MO analysis of reactants</u>

The frontier molecular orbitals on both reactants were analysed to account for the two possible transition states that may form.

[[image:Og108_cyclohex2_MOenergydiagram1.PNG|500px|centre]]

The orbitals matched closely to those predicted through an LCAO MO diagram approach. The important bonding and anti bonding interactions can be clearly accounted for.

Below shows the orbitals obtained for maleic anhydride.

{|class="wikitable" align="center"
|+ Table XXX - FMOs for Maleic Anhydride
|[[Image:Og108_mal_MO26.PNG|200px|]]
| LUMO
|[[Image:Og108_mal_MO25.PNG|200px]]
| HOMO
|-
|}

Both these orbitals appear to be anti-symmetric with respect to the σ<sub>v</sub> plane of symmetry.

===Transition state analysis===

The freeze coordinate method was used to construct both transition states from the optimised reactants. The guess structures were formed in a similar manner to the previous Diels-Alder reaction.

{| class="wikitable"
|+ Table 17: Summary of Total Energies and RMS Gradients for the Transition State (AM1 and DFT optimised)
|-
! Transition State
! Point Group
! Total Energy /Ha
! RMS Gradient
|-
| <jmol><jmolAppletButton>
<uploadedFileContents>Og108_endomalcyclo_optder1.mol</uploadedFileContents>
<text>ENDO</text>
<script>background=white; measure 5 15; measure 4 5; measure 11 12 4 </script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13757</ref>
| C<sub>s</sub>
| -612.68339683
| 0.00002161
|-
| <jmol><jmolAppletButton>
<uploadedFileContents>Og108_exomalcyclo_optder1.mol</uploadedFileContents>
<text>EXO</text>
<script>background=white; measure 4 15; measure 4 5; measure 11 12 5</script>
</jmolAppletButton>
</jmol> <ref>http://hdl.handle.net/10042/to-13755</ref>

| C<sub>s</sub>
| -612.67931096
| 0.00000132
|-
|}

This table shows that the ''endo'' transition state is the more stable of the two. The explanation as to why this is the more stable structure will be explored using MO analysis later on.

Both transition state structures have one imaginary vibrational mode as expected since they are maxima on the potential energy surface. The ''endo'' vibrational mode is at [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Og108_exo_vibmode1.gif -448.51 cm<sup>-1</sup>] while the ''exo'' vibrates at [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Og108_endo_vibmode1.gif -447.83 cm<sup>-1</sup>].

{| class="wikitable"
|+ Table 19: Transition state MOs at the DFT/B3LYP/6-31G*
|-
! Transition State
! HOMO-1
! HOMO
! LUMO
! LUMO+1
! Relative Energy
|-

| '''Exo'''
| [[Image:Og108 exo MO46.PNG|thumb|centre|200px| Symmetric]]
| [[Image:Og108 exo MO47.PNG|thumb|centre|200px| Anti Symmetric]]
| [[Image:Og108 exo MO48.PNG|thumb|centre|200px| Anti Symmetric]]
| [[Image:Og108_exo_MO49.PNG|thumb|centre|200px| Symmetric]]
| [[Image:Og108 exo MOrelen.PNG|thumb|centre|200px|]]
|-
| '''Endo'''
| [[Image:Og108 endo MO46.PNG |thumb|centre|200px| Symmetric]]
| [[Image:Og108 endo MO47.PNG|thumb|centre|200px| Anti Symmetric]]
| [[Image:Og108 endo MO48.PNG|thumb|centre|200px| Anti Symmetric]]
| [[Image:Og108 endo MO49 2.PNG|thumb|centre|200px| Symmetric]]
| [[Image:Og108 endo MOrelenergy.PNG|thumb|centre|200px|]]
|-
|}

The HOMOs for each isomer are occupied bonding orbitals. As with the prototypical Diels-Alder reaction,the molecular orbitals of the reactants can be used to determine which of them make up the transition state MOs. As discussed previously only the frontier molecular orbitals were considered for this analysis. This oversimplified analysis may not be able to fully account for some of the MOs shown since contributions from lower energy reactant MOs could be present. Both transition state MOs are fairly similar. However evidence of secondary orbital interactions can also be seen from looking at the LUMO+1 orbitals. The carbonyl carbon atom play a part in the bonding through π overlap with the cyclohexene orbitals. This interaction is only present for the ''endo'' transition state and has been suggested that this is what gives rise to the regioselectivity of the reaction<ref> I. Fleming, ''Frontier Orbitals and Organic Chemical Reactions'', John Wiley & Sons Ltd, '''2002''', pp 106 - 108</ref>.

Finally, IRC analysis at the DFT/B3LYP/6-31G* level was able to give an overall picture of the reaction. The data was exported to excel and plotted onto the same graph for comparison and the key data points matched up with the corresponding geometry.

{| class="wikitable"
|+ Table 19: IRC analysis at the DFT/B3LYP/6-31G* level (endo<ref>http://hdl.handle.net/10042/to-13785</ref> exo<ref>http://hdl.handle.net/10042/to-13786</ref>)
!
!
|-
|colspan="2" | [[Image:Og108 IRC graphcomparison final.PNG |centre|1000px|]]
|-
| [[image:Og108 exomalcyclo IRC1.gif|thumb|300 px| centre|]]
| [[Image:Og108 endomalcyclo IRC1.gif|thumb| 300px| centre| ]]
|-
| align:centre| '''exo''' product formation
| align:centre| '''endo''' product formation
|-
|}

The graph shows the energy of the transtion state and the energy of the final product are both lower for the endo isomer. Therefore, according to this calculation the endo isomer is both the kinetic and thermodynamic product. The exo product was guessed to be the thermodynamic product and there may be a lack of steric interactions taken into account for this calculation which could destabilise the endo form somewhat. This calculation could be done with an even larger basis set such as the 6-311G*. However the current calculation already took over 12 hours to complete and a larger basis set would create a calculation too time consuming for this project.

The exo transition state is 0.004086 Hartrees higher than the endo form and corresponding to an activation energy that is 2.564Kcal mol<sup>-1</sup> higher. This is likely to be because of the secondary orbital interactions stabilising the endo form as discussed previously.

==Conclusion==

This study has shown the benefits of using computational calculations to study kinetics and transition state theory. Different methods clearly have their strengths and weaknesses and careful consideration should be taken into account when choosing a method for a particular calculation of a given system. Computational methods are clearly very useful for transition state studies given the difficulty in making measurements experimentally because of their short lifetime. However, this powerful tool does have its limitations and inconsistancy with experimental data was seen with the Diels Alder reaction. With ever advancing computer hardware and software, more complex calculations can be undertaken to greater accuracy and consequently this could be a very influential field for chemists in the 21st century especially when used in conjunction with experimental data.

=References=
<references/>

Rep:Xs3015 TS

2025-09-01T09:50:29Z

Move page script: Move page script moved page Xs3015 TS to Rep:Xs3015 TS: Move to report namespace

== Introduction ==

== Exercise 1: Reaction of Butadiene with Ethylene ==

== Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole ==

== Exercise 3: Diels-Alder vs Cheletropic ==

== Conclusion ==

Rep:Xnkl

2025-09-01T09:50:25Z

Move page script: Move page script moved page Xnkl to Rep:Xnkl: Move to report namespace

= Cope Rearrangement Tutorial =

== Introduction ==

A cope rearrangemnt is a [3,3] sigmatropic shift of a 1,5-diene. In this experiment 1,5-hexadiene was investigated using various computational methods in order to characterise the products/reactants and the transition state of the reaction.

== Optimising the Reactants and Products ==

The first 4 calculations were all carried out at the HF/3-21G level of theory.

The first calculation carried out was an optimisation of 1,5-hexadiene. This optimisation was carried out with the hexadiene initially in the "anti" configuration (Molecule 1).<jmol>
<jmolAppletButton>
<uploadedFileContents>DW_15hd_Anti.mol</uploadedFileContents>
<text>Molecule 1 - Pre-optimisation structure of anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>
The optimised structure had C<sub>2</sub> symmetry but wasn't significantly different from the initial structure.<jmol>
<jmolAppletButton>
<uploadedFileContents>DW_15hd_anti1.mol</uploadedFileContents>
<text>Molecule 2 - Optimised structure of anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

The second calulation was another optimisation of 1,5-hexadiene, this time with a gauche geometry about the central 4 C atoms. The optimised structure was lower in energy than the anti structure indicating that it is more stable. It had had C<sub>1</sub> symmetry. This was the lowest energy conformer of 1,5-hexadiene that was found.

A third optimisation of 1,5-hexadiene was carried out starting with the central C-C-C-C in an anti-periplanar arrangement. This led to a different gauche structure that was higher in energy, and therefore less stable, than the previous gauche structure.

The final calculation done at the HF/3-21G level of theory was to obtain another conformer with an anti configuration of the central 4 C atoms. The resulting structure had a C<sub>i</sub> point group.

The anti, C<sub>i</sub> conformer was reoptimised at the B3LYP/6-31G* level of theory. This level finds the optimum structure to a finer level of detail, using a larger basis set, at the cost of more processing time.

The structure didn't change much under the more detailed calculation and had the same point group. The sp<sub>2</sub> bond angles increased by around 2 degrees.

The fully optimised structure was subjected to a frequency analysis to determine if a potential energy minimum had been reached. All the vibrational frequencies reported were real and positive indicating that the potential minimum had been reached.

<pre>
298.15K
Sum of electronic and zero-point Energies= -234.469212
Sum of electronic and thermal Energies= -234.461856
Sum of electronic and thermal Enthalpies= -234.460912
Sum of electronic and thermal Free Energies= -234.500822
</pre>

== Optimising the "Chair" and "Boat" Transition Structures ==

Having optimised the structures of the products and reactants, the transition structures were optimised. This is a more difficult because transition structures are not potential energy minima but are saddle points, with a negative potential energy gradient along a single coordinate. Therefore different techniques are required in order to obtain the correct structures.

=== "Chair" Transition State ===
[[File:DW_chairts_negvib.gif|thumb|Chair transition state negative frequency vibration - Click to view animation]]

The first calculation carried out was an optimisation of a single allyl fragment that would make up half of the final transition structure. This calculation was carried out at the HF/3-21G level of theory. The calculation was done as an optimisation to a transition state using the Berny algorithm. This method found a transition state, it's nature as a transition state rather than a potential minimum being confirmed by the presence of a single negative frequency in the vibrational spectrum at 818cm<sup>-1</sup>. This vibrational frequency corresponded to the two allyl fragments moving into either product of the Cope rearrangement.

The second calculation was another method that can be used to obtain the transition state. In this method the atoms corresponding to the two new bonds were frozen at ~2.2Å and then a optimisation done. the bonds were then unfrozen and another optimisation to transition state performed. This method gave a structure of the transition state very similar to the first method.

<pre>
298.15K
Sum of electronic and zero-point Energies= -234.414932
Sum of electronic and thermal Energies= -234.409011
Sum of electronic and thermal Enthalpies= -234.408067
Sum of electronic and thermal Free Energies= -234.443816</pre>

The chair transition structure had an electronic energy of -234.556983 hartrees under the B3LYP/6-31G* method.

=== "Boat" Transition state ===
[[File:DW_boatts_negvib.gif|thumb|Boat transition state negative frequency vibration - Click to view animation]]

The boat transition state was found using a different method again, the QST2 method, which moves between the reactant structure and the products structure in order to find the transition state. The boat transition structure has a slightly longer distance between the allyl fragments than in the chair conformation. this is due to additional steric repulsion between the CH<sub>2</sub> at each end of the boat.

<pre>
298.15K
Sum of electronic and zero-point Energies= -234.402346
Sum of electronic and thermal Energies= -234.396011
Sum of electronic and thermal Enthalpies= -234.395067
Sum of electronic and thermal Free Energies= -234.431755</pre>

The boat transition structure had an electronic energy of -234.543093 hartrees under the B3LYP/6-31G* method.

=== Activation Energies ===

The activation energy of the reaction can be worked out by finding the energy difference between the reactants and products.

The energy of the reactants is -234.469212 hartrees at 0K and -234.461856 hartrees at 298K. By subtracting the energy of the transition state from that of the reactant, the activation energy (E<sub>a</sub>) for the process can be found.

{| class="wikitable"
|-
! Structure
! Energy of transition state (hartrees)
! E<sub>a</sub> (hartrees)
! E<sub>a</sub> (kcal/mol)
|-
| Chair 0K
| -234.414932
| 0.05428
| 34.06
|-
| Boat 0K
| -234.402346
| 0.066866
| 41.96
|-
| Chair 298.15K
| -234.409011
| 0.052845
| 33.16
|-
| Boat 298.15K
| -234.396011
| 0.065845
| 41.32
|}

These values show that the transition state is lowered in energy at higher temperatures. This would contribute to higher reaction rates at higher temperatures. The experimental values for the chair and boat transition states at 0K are 33.5 ±0.5 kcal/mol and 44.7 ±2.0 kcal/mol respectively<ref>Origin of the preference for the chair conformation in the Cope rearrangement. Effect of phenyl substituents on the chair and boat transition states
K. J. Shea, G. J. Stoddard, W. P. England, and C. D. Haffner
Journal of the American Chemical Society 1992 114 (7), 2635-2643 doi: [[http://dx.doi.org/10.1021/ja00033a042 10.1021/ja00033a042]]</ref>. These values show the same ordering as the computed values which means that the preferred transition state has been correctly identified as the chair.

== Diels Alder Cycloaddition ==

=== Optimising cis-butadiene ===
[[File:DW_ethene_HOMO.PNG|thumb|right|ethene HOMO]][[File:DW_cis-butadiene_HOMO.PNG|thumb|left|cis-butadiene HOMO]][[File:DW_ethene_LUMO.PNG|thumb|right|ethene LUMO]][[File:DW_cis-butadiene_LUMO.PNG|thumb|left|cis-butadiene LUMO]]
Cis-butadiene was optimised using the AM1 semi-empirical molecular orbital method. The resulting molecule was of the C<sub>2v</sub> point group. Ethene was also optimised using the same methods to produce the above structure

The HOMO and LUMO of the two molecules are shown in the images. Relative to the plane of relective symmetry going perpendicularly through the central C-C bond the HOMO of cis-butadiene and the LUMO of ethene are anti-symmetric but the LUMO of cis-butadiene and the HOMO of ethene are symmetric. Since for two MOs to mix they must be of the same symmetry it is clear that the HOMO of cis-butadiene and the LUMO of ethene can mix together and the LUMO of cis-butadiene and the HOMO of ethene can mix together.

=== Optimising ethylene + cis-butadiene Transition State ===
[[File:DW_DA1_HOMO.PNG|thumb|left|Ethene + butadiene Diels-Alder transition state HOMO]][[File:DW_DA1_negvib.gif|thumb|right|Ethene + butadiene Diels-Alder transition state negative vibration - Click to view animation]][[File:DW_DA1_LUMO.PNG|left|thumb|Ethene + butadiene Diels-Alder transition state LUMO]][[File:DW_DA1_lowvib.gif|thumb|right|Ethene + butadiene Diels-Alder transition state lowest positive vibration]][[File:DW_DA1_geom.PNG|right|thumb|Transition state geometry with bond lengths labelled in Å]]
The transition state of ethylene and cis-butadiene was optimised using the AM1 semi-empirical molecular orbital method to a transition state using the Berny algorithm. The nature of the negative frequency vibration shows that the transition point is correct and corresponds to the Diels-Alder reaction as wanted. The animation of the vibration shows that the reaction is concerted and the bonds form simultaneously. There is significant bond stretching involved in the vibrational mode.

The lowest positive frequency vibration (shown on right) has a very different form to the negative vibration. The positive vibration doesn't have any bond stretching as is more energetic than bond bending which is more energetic than the vibration shown. The vibration shown actually doesn't involve any bond stertches or bends but is a twist of the whole ethene molecule relative to the butadiene. The reason it is low in energy is that it doesn't require the changing of any covalent bonds.

As with the cis-butadiene, the HOMO and LUMO of the molecule are shown in the images. Relative to the plane of relective symmetry going perpendicularly through the central C-C bond the HOMO is anti-symmetric (AS) but the LUMO is symmetric (S). Since the HOMO here is AS, as was the HOMO of the butadiene, this suggests that the butadiene HOMO is involved. The ethene LUMO is AS but is part of the HOMO for the transition state. The LUMO of the transition state is made up of the HOMO of ethene and the LUMO of the butadiene. This confirms that the MOs mix as expected.

The reaction is allowed because there are 6 (which fits the 4n+2 rule) electrons involved in the reaction and there is suprafacial stereochemistry.

Given that an sp<sup>3</sup> and an sp<sup>2</sup> are 153nm and 146nm respectively<ref>Tables of bond lengths determined by X-ray and neutron diffraction. Part 1. Bond lengths in organic compounds
F. H. Allen, O. Kennard, D. G. Watson, L. Brammer, A. G. Orpen and R. Taylor, J. Chem. Soc., Perkin Trans. 2, 1987, S1
doi: [[http://dx.doi.org/10.1039/P298700000S1 10.1039/P298700000S1]]</ref>, the intramolecular distance of 212nm is much further than a single bond. This suggests the interaction in the transition state is not covalent. The van der Waals radius of carbon is 170pm<ref> van der Waals Volumes and Radii, A. Bondi, The Journal of Physical Chemistry 1964 68 (3), 441-451
doi: [[http://dx.doi.org/10.1021/j100785a001 10.1021/j100785a001]]</ref>. Therefore although the internuclear bond distance is outside that observed in covalently bonded carbon atoms, it is within twice the van der Waals radius and so there is likely to be significant orbital overlap between the two carbon atoms. The interaction being observed in this transition state is more likely to be a van der Waals type interaction.

=== Optimising cyclohexa-1,3-diene + maleic anhydride Transition State ===
[[File:DW_DA2_endo_negvib.gif|thumb|right|Endo transition state vibration - Click to view animation]][[File:DW_DA2_exo_HOMO.PNG|thumb|left|Exo transition state HOMO]][[File:DW_DA2_exo_negvib.gif|thumb|right|Exo transition state vibration - Click to view animation]][[File:DW_DA2_endo_HOMO.PNG|thumb|left|Endo transition state HOMO]]

To find the transition state of this reaction the AM1 forcefield was initially used to optimise the molecules to a transition state and then the B3LYP/6-31G* level of theory used on the pre-optimised structures. The files used in the final calculations can be found by clicking on the following two links:
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/27440 Endo D-Space]]
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/27439 Exo D-Space]]

The relative energies are -612.68339675 hartrees for the endo form and -612.67931096 hartrees for the exo form. This shows that the endo transition state is lower in energy and so the kinetically preferred product in this reaction is the endo product.

The structures of each fragment were largely similar, one difference was that the H-C-C-O dihedral angle was 160 in the endo transition state and 157 in the exo transition state. The maleic anhydride fragment was slighlty closer to planar in the endo structure (dihedral angle of 180). this is because the exo form is more strained due to the additional steric clash between the maleic part and the hydrogen atoms on sp<sup>3</sup> carbons. On the endo form, there is less repulsion because there is only one hydrogen on the sp<sup>2</sup> carbon atoms. This strain also accounts for the other significant difference between the two structures; the increased distance between the carbons that would form bonds in the reaction which are 2.269Å in the endo transition state and 2.290Å in the exo transition state.

The HOMOs of the two transition states can be seen by clicking these buttons: <jmol>
<jmolAppletButton>
<uploadedFileContents>DW_DA2_endoTSopt631G.jmol</uploadedFileContents>
<text>Endo transition state HOMO</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>DW_DA2_exoTSopt631G.jmol</uploadedFileContents>
<text>Exo transition state HOMO</text>
</jmolAppletButton>
</jmol>
The HOMO of the exo structure has a large area within the gap between the -(C=O)-O-(C=O)- fragment and the hexyl ring of low electron density, whereas the endo structure has a lot more electron density there. This suggests that there are more favourable interactions between the orbitals in the endo structure HOMO than in the exo HOMO. Thsi contributes to the increased stability of the endo state and is the secondary orbital overlap of the diene in the HOMO with the pi orbitals on the carbons α to the double bond. While this stabilises the endo transition state, it isn't present in the exo transition state.

=== Activation Energies ===
As anestimate of the reaction energies the structures of the two reactant molecules were optimised using the same B3LYP/6-31G* level of theory.

Maleic Anhydride
<pre> Sum of electronic and zero-point Energies= -379.233665
Sum of electronic and thermal Energies= -379.228481
Sum of electronic and thermal Enthalpies= -379.227537
Sum of electronic and thermal Free Energies= -379.262738</pre>

Cyclohexadiene
<pre> Sum of electronic and zero-point Energies= -233.296106
Sum of electronic and thermal Energies= -233.290933
Sum of electronic and thermal Enthalpies= -233.289989
Sum of electronic and thermal Free Energies= -233.324356</pre>

Endo TS
<pre> Sum of electronic and zero-point Energies= -612.502141
Sum of electronic and thermal Energies= -612.491787
Sum of electronic and thermal Enthalpies= -612.490843
Sum of electronic and thermal Free Energies= -612.538329</pre>

Exo TS
<pre> Sum of electronic and zero-point Energies= -612.498013
Sum of electronic and thermal Energies= -612.487662
Sum of electronic and thermal Enthalpies= -612.486717
Sum of electronic and thermal Free Energies= -612.534265</pre>

The results of the energies of the calculations are summarised in the table below:
{| class="wikitable"
|-
! Temperature
! Energy of maleic anhydride (hartrees)
! Energy of Cyclohexadiene (hartrees)
! Energy of endo transition state (hartrees)
! Energy of exo transition state (hartrees)
! E<sub>a</sub> endo (hartrees)
! E<sub>a</sub> exo (hartrees)
! E<sub>a</sub> endo (kcal/mol)
! E<sub>a</sub> exo (kcal/mol)
|-
| 0K
| -379.233665
| -233.296106
| -612.502141
| -612.498013
| 0.027630
| 0.031758
| 17.34
| 19.93
|-
| 298.15K
| -379.228481
| -233.290933
| -612.491787
| -612.487662
| 0.027627
| 0.031752
| 17.34
| 19.92
|-
|}

This shows that the activation energy for the reaction is around 2.6 kcal/mol lower via the endo transition state than via the exo transition state. These values also suggest that the activation barrier is not strongly affected by an increase in the temperature.

=== Energy of the products ===

Further optimisations using the same basis sets and methods were carried out to optimise the products of the reactions to determine the most stable thermodynamic product. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/27537 Endo D-Space]]
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/27536 Exo D-Space]]

endo product
<pre> Sum of electronic and zero-point Energies= -612.572070
Sum of electronic and thermal Energies= -612.562604
Sum of electronic and thermal Enthalpies= -612.561660
Sum of electronic and thermal Free Energies= -612.607178</pre>
exo product
<pre> Sum of electronic and zero-point Energies= -612.569381
Sum of electronic and thermal Energies= -612.559977
Sum of electronic and thermal Enthalpies= -612.559033
Sum of electronic and thermal Free Energies= -612.604283</pre>

The endo product is lower in energy and so is predicted to be both the kinetic product and the thermodynamic product of the reaction.

=== Further Study ===

These calulations of Diels-Alder reactions have all focused on symmetrical reactants. The investigation could be extended by looking at reactions where the symmetry of the reactions is broken. This would affect the interactions of the orbitals and would possibly cause a switch of the preference from endo to exo.

= References =
<references/>

Rep:Xlt15 TS

2025-09-01T09:50:24Z

Move page script: Move page script moved page Xlt15 TS to Rep:Xlt15 TS: Move to report namespace

== Transition States and Reactivity ==

=== Introduction ===
The transition state (TS) is defined as the highest energy point (maximum) on the minimum energy path linking reactants and products whereas the reactant and product are the minimum along a reaction coordinate. They are thus a stationary point with a first derivative or a gradient of zero (2). The second derivative or the curvature (4) is negative for a TS (maximum point) whereas it is positive for the reactant and product (minimum point).
[[File:Xlt15formula1.PNG|frame|center|Figure 1: Equation of potential energy and its first and second derivative, vibrational frequency and reduced mass <ref name=":0">J. J. W. McDouall, ''Computational Quantum Chemistry: Molecular Structure and Properties in Silico'', Royal Society of Chemistry, London, United Kingdom, 2013.</ref>.]]
where V(x) is the potential energy, k is the force constant, x is the displacement, ν is vibrational frequency, μ is the reduced mass of 2 atoms of mass m<sub>1</sub> and m<sub>2</sub>. The negative sign in (3) implying that the force acting in that direction lowers the potential energy <ref name=":0">J. J. W. McDouall, ''Computational Quantum Chemistry: Molecular Structure and Properties in Silico'', Royal Society of Chemistry, London, United Kingdom, 2013.</ref>.
[[File:Xlt15inroreactionprofilr.PNG|frame|center|Figure 2: Reaction profile of an exothermic reaction <ref name=":1"> J. Clayden, N. Greeves, S. Warren, P. Wothers, ''Organic Chemistry'', Oxford University Press Inc., New York, 2001.</ref>.]]
<br>
The reaction coordinate diagram provides the kinetics and thermodynamics information of a reaction. The higher the activation energy, the lower the rate of reaction. A pathway with more exothermic reaction energy is more thermodynamically favoured.
<br>
<br>
A non-linear molecule undergoing a reaction will have 3N<sub>atoms</sub>-6 reaction coordinates <ref name=":0">J. J. W. McDouall, ''Computational Quantum Chemistry: Molecular Structure and Properties in Silico'', Royal Society of Chemistry, London, United Kingdom, 2013.</ref>. The 3N<sub>atoms</sub>-6 specifies the number of internal degree of freedom or the number of normal vibrational mode that a molecule can possess <ref name=":0">J. J. W. McDouall, ''Computational Quantum Chemistry: Molecular Structure and Properties in Silico'', Royal Society of Chemistry, London, United Kingdom, 2013.</ref>. For a linear molecule, the number of internal degree of freedom is 3N<sub>atoms</sub>-5 as rotation along bond axis give the identical molecule <ref name=":0">J. J. W. McDouall, ''Computational Quantum Chemistry: Molecular Structure and Properties in Silico'', Royal Society of Chemistry, London, United Kingdom, 2013.</ref>. Hence, a reaction coordinate diagram beyond 1D is called a potential energy surface. The transition state in a PES is defined as the first order saddle point on PES with the first derivatives with respect to all coordinates are 0 <ref name=":2">A. R. Rossil, ''Reaction Paths and Transition States'', lecture notes, Department of Chemistry, The University of Connecticut, http://rossi.chemistry.uconn.edu/chem5326/files/reaction_pathways.pdf, (accessed Feb 2018).</ref>. The first order saddle point has a maximum energy for displacement between 2 minima in a direction whereas in all the remaining directions the energy is a minimum <ref name=":2">A. R. Rossil, ''Reaction Paths and Transition States'', lecture notes, Department of Chemistry, The University of Connecticut, http://rossi.chemistry.uconn.edu/chem5326/files/reaction_pathways.pdf, (accessed Feb 2018).</ref>. Hence, the transition state has only a unique normal coordinate that corresponds to reaction coordinate with second derivative < 0 (negative force constant). This negative force constant can then be related to the vibrational frequency (5) which is an imaginary frequency. The chemical reactant and products are characterized as minima and must have second derivative > 0 (positive force constant and hence positive vibrational frequency).
<br>
[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 22:27, 21 February 2018 (UTC) remeeber you must be talking about 3N-6 degrees of freedom for you minimum too. I know what you are saying. You can get your force constants from teh eigenvalues of the hessian matrix at the TS geometry. The eigen vetcors are the normal modes.
<br>
Two different electronic structure methods, the semi-empirical method PM6 and Density Functional Theory (DFT) method B3LYP were used in optimizing the reactants, transition state and product. The semi-empirical PM6 is a faster method to generate a reasonable result as it uses the experimental data whereas B3LYP is more time-consuming and accurate method <ref name=":3"> Year 3 Computational Chemistry Lab, Transition State and Reactivity, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_exercise, (accessed Feb 2018).</ref>. Hence, PM6 method was firstly used to generate an approximate structure and then reoptimize with DFT method B3LYP for Exercise 2.

=== Exercise 1: Reaction of Butadiene with Ethylene ===
[[File:Xlt15reactionscheme4ex1.PNG|frame|center|Figure 3: Reaction scheme of Diels-Alder reaction of 1,3-butadiene and ethylene.]]
Diels-Alder (D-A) reaction or [4+2]-cycloaddition occur between a conjugated diene and an alkene (dienophile). The diene component must adopt a ''s''-cis conformation. D-A reaction involved a one step, concerted formation of 2 new σ bonds. The D-A reaction proceeds with no reaction intermediates and the transition state has aromatic character with six delocalized π electron.
<br>
<br>
''' Method Used In Optimization and Analysis '''
<br>
Method 3 was employed in locating the transition state by which the product, cyclohexene was drawn and optimized to minimum at PM6 level. With the optimized cyclohexene, the C-C single bonds formed during the reaction of 1,3-butadiene and ethylene were deleted, froze at 2.20 Å and optimized to minimum at PM6 level to identify the frozen guess transition state. The distance 2.20 Å is the approximate separation between the reacting termini in guess transition state. It is a value between a C-C single bond length and their combined Van der Waals radii. The guess TS structure was then optimized at PM6 level and the PM6 optimized TS was used to run a IRC calculation. For reactants, 1,3-butadiene and ethylene, they were each obtained from the first frame of IRC calculation and optimized to minimum at PM6 level.
==== Optimized Reactants, Transition State and Product and at PM6 Level. ====
{| class="wikitable"
! colspan="4" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 1: Optimized Reactants, Transition State and Product at PM6 level
|-
! style="text-align: center;" | 1,3-Butadiene
! style="text-align: center;" | Ethylene
! style="text-align: center;" | Transition State
! style="text-align: center;" | Cyclohexene
|-
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 32</script>
<uploadedFileContents>xlt15BUTADIENE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 14</script>
<uploadedFileContents>xlt15ETH2_1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 20</script>
<uploadedFileContents>xlt15TS_1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 16</script>
<uploadedFileContents>xlt15PRODUCT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

==== Determination of The Correct TS with a Frequency and IRC Calculation ====
===== Frequency calculation =====
[[File:Xlt15tsnegative frequencyex1.PNG|center|frame|Figure 4: Screenshot of the "Display Vibrations" of transition structure.]]
<br>
[[File:Xlt15tsconvergedex1.PNG|center|frame|Figure 5: Screenshot of a section of transition structure's log file.]]
<br>
According to Figure 4 and 5, the TS structure has shown to properly converged with the presence of only one imaginary frequency at -949 cm<sup>-1</sup> and a stationary point corresponds to the transition structure is found. The imaginary frequency is then visualized to ensure a correct transition state structure is obtained (See Section 1.2.5 for the visualization of the imaginary frequency). For reactants (1,3-butadiene and ethylene) and product, their structures are checked to properly converge with no negative frequency obtained and their respective stationary points are found in log files.

===== IRC Calculation =====
{| class="wikitable"
! colspan="3" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 2. IRC Calculation of PM6 Optimized Transition Structures
|-
! style="text-align: center;" | Reaction Progress
! style="text-align: center;" | IRC Output
! style="text-align: center;" | Explanation
|-
|[[File:Xlt15irc3.gif|center]]
|[[File:Xlt15ircpathex1redo.PNG|350px|center]]
|An IRC (Intrinsic reaction coordinate) calculation is obtained for the PM6 optimized transition state structure. The plot of total energy against IRC is asymmetric and all the gradients for reactants, product and transition state (minima) are zero, indicating that it is properly converged and optimized.
|}

==== MO Analysis ====

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) Your analysis is good, but you haven't ordered the TS MOs according to their computed energies. As this is a TS MO diagram, the bonding TS MOs will be higher in energy than expected, and the antibonding ones will be lower.)

===== MO Diagram of The Formation of 1,3-Butadiene/Ethylene TS =====
[[File:Modiagram2xlt156.PNG|center|frame|Figure 6: MO diagram for the formation of the 1,3-butadiene/ethylene TS with basic symmetry labels shown; A= Antisymmetric and S= Symmetric.]]
HOMO: Highest Occupied Molecular Orbital
<br>
LUMO: Lowest Unoccupied Molecular Orbital
<br>
<br>
As seen in the MO diagram drawn in Figure 6, it is a normal electron demand Diels-Alder reaction because the HOMO of 1,3-butadiene (diene) is higher in energy than the HOMO of ethylene (dienophile). The combination of the high energy HOMO (diene) and low energy LUMO of dienophile gives a better overlap in the transition state. As their energy gap is smaller, it is more favoured than the interaction between LUMO of 1,3-butadiene and HOMO of ethylene (larger energy gap and hence worse overlap).

===== Frontier MO of Reactants and Transition State =====
{| class="wikitable"
! colspan="5" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 3: Frontier MO of 1,3-Butadiene, Ethylene and Transition State.
|-
! style="text-align: center;" | 1,3-Butadiene
! style="text-align: center;" | Ethylene
! style="text-align: center;" | Transition State
! style="text-align: center;" | Orbital Interaction and Discussion
|-
|<jmol>
<jmolApplet>
<color>white</color>
<title>LUMO, MO12, S</title>
<size>200</size>
<script>frame 32; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>xlt15BUTADIENE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO, MO6, S</title>
<size>200</size>
<script>frame 14; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>xlt15ETH2_1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO, MO17, S</title>
<size>200</size>
<script>frame 20; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>xlt15TS_1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
<jmol>
<jmolApplet>
<color>white</color>
<title>LUMO, MO18, S</title>
<size>200</size>
<script>frame 20; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>xlt15TS_1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|'''“+” stands for an in-phase combination and “-” stands for an out-of-phase interaction'''
<br>
1,3-Butadiene (LUMO, MO12, S) + Ethylene (HOMO, MO6, S) = TS (HOMO, MO17, S)
<br>
1,3-Butadiene (LUMO, MO12, S) - Ethylene (HOMO, MO6, S) = TS (LUMO, MO18, S)
<br>
<br>
MO17: Although the orbital lobes on the reacting C termini of 1,3-butadiene is not observed but there is an increased in the size of the blue lobe of ethylene MO6, implying a bonding interaction between the two.
<br>
MO18: The ethylene MO6 has diminished; Antibonding interaction.
|-
|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO, MO11, A</title>
<size>200</size>
<script>frame 32; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>xlt15BUTADIENE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>LUMO, MO7, A</title>
<size>200</size>
<script>frame 14; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>xlt15ETH2_1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO-1, MO16, A</title>
<size>200</size>
<script>frame 20; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>xlt15TS_1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
<jmol>
<jmolApplet>
<color>white</color>
<title>LUMO+1, MO19, A</title>
<size>200</size>
<script>frame 20; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>xlt15TS_1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|'''“+” stands for an in-phase combination and “-” stands for an out-of-phase interaction'''
<br>
1,3-Butadiene (HOMO, MO11, A) + Ethylene (LUMO, MO7, A) = TS (HOMO-1, MO16, A)
<br>
1,3-Butadiene (HOMO, MO11, A) - Ethylene (LUMO, MO7, A) = TS (LUMO+1, MO19, A)
<br>
<br>
MO16: A very clear bonding interaction can be observed.
<br>
MO19: Again, the orbital lobes are not observed for the reacting termini in 1,3-butadiene; Antibonding interaction.
|}

{| class="wikitable"
! colspan="2" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 4: Summary of orbital overlap integral.
|-
! style="text-align: center;" | Interaction Type
! style="text-align: center;" | Orbital Overlap Integral
|-
| style="text-align: center;" | Symmetric-symmetric
| style="text-align: center;" | Non-zero
|-
| style="text-align: center;" | Antisymmetric-antisymmetric
| style="text-align: center;" | Non-zero
|-
| style="text-align: center;" | Symmetric-antisymmetric
| style="text-align: center;" | Zero
|}
The Table 4 above summarizes the overlap integral for three different interactions. Each transition state MO is designated as symmetric (S) or antisymmetric (A) with respect to the persistent symmetry element, a mirror plane perpendicular to the central C-C bond in 1,3-butadiene. According to conservation of orbital symmetry, the orbital symmetry of reactants is smoothly transformed to an orbital of product with the same symmetry <ref name=":4"> E. V. Anslyn, D. A. Dougherty, ''Modern Physical Organic Chemistry'', University Science Books, Sausalito, United States, 2006.</ref>. Thus, this symmetry will certainly persist in transition state. In addition, the frontier molecular orbital (FMO) theory states that a reaction is only allowed if there is favourable mixing between HOMO and LUMO of the reactants <ref name=":4"> E. V. Anslyn, D. A. Dougherty, ''Modern Physical Organic Chemistry'', University Science Books, Sausalito, United States, 2006.</ref>. By referring to the MO diagram in Figure 6, the HOMO of 1,3-butadiene is of the same symmetry as the LUMO of ethylene and the vice versa. Both interactions are thus allowed because of the matching in phases. In conclusion, only orbitals of the same symmetry in reactants can combine to form the TS molecular orbitals with the same symmetry to have a significant overlap of the combining orbitals. Fragment orbitals of different symmetry combine to give zero overlap integral and thus no bond formation.

According to Woodward Hoffmann rules, the total number of (4q+2)<sub>s</sub> and (4r)<sub>a</sub> components must be odd in an allowed thermal pericyclic reaction. The suffix “s” stands for suprafacial and “a” stands for antarafacial. A suprafacial component forms new bonds at the same face at both ends. An antarafacial component forms new bonds on opposite faces at both ends <ref name=":1"> J. Clayden, N. Greeves, S. Warren, P. Wothers, ''Organic Chemistry'', Oxford University Press Inc., New York, 2001.</ref>. The thermal pericyclic reaction is forbidden if the number is even. However, a thermally forbidden pericyclic reaction is photochemically allowed. Using the Woodward Hoffmann rules, [<sub>π</sub>4<sub>s</sub> + <sub>π</sub>2<sub>s</sub>]-cycloaddition is thermally allowed as there is one (4q+2)<sub>s</sub> component (the ethylene) and no (4r)<sub>a</sub> component.

==== C-C Bond Length Measurements and Analysis ====
'''Typical C-C Bond Length''' <ref name=":5"> S. M. Mukherji, S. P. Singh, R. P.Feynman,R. Dass, ''Organic Chemistry Vol I'', New Age International, New Delhi, India, 2010.</ref> <ref name=":6"> I. L. Shabalin, ''Ultra-High Temperature Materials I Carbon (Graphene/Graphite) and Refractory Metals'', Springer, Netherlands, 2014.</ref>

{| class="wikitable"
! colspan="2" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 5: Typical C-C bond lengths.
|-
! style="text-align: center;" | Bond Type
! style="text-align: center;" | Bond Length/ Å
|-
| style="text-align: center;" | '''C–C'''
sp<sup>3</sup> C–sp<sup>3</sup> C
<br>
sp<sup>3</sup> C–sp<sup>2</sup> C
<br>
sp<sup>2</sup> C–sp<sup>2</sup> C
| style="text-align: center;" |
<br>
1.54
<br>
1.50
<br>
1.48
|-
| style="text-align: center;" | '''C=C'''
sp<sup>2</sup> C–sp<sup>2</sup> C
| style="text-align: center;" |
<br>
1.34
|}

''' C-C Bond Length Measurements'''
[[File:Xlt15reactionscheme4ex1numbering.PNG|frame|left|Figure 7: Reaction scheme with the all carbons numbered.]]
<br style="clear:right">
<br style="clear:right">
<br style="clear:right">
<br style="clear:right">
<br style="clear:right">
<br style="clear:right">
<br style="clear:right">
<br style="clear:right">
<br style="clear:right">
{| class="wikitable"
! colspan="7" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 6: C-C bond length measurements in reactants, transition state and product.
|-
! rowspan="2" style="text-align: center;" | Molecule
! colspan="6" style="text-align: center;" | Bond Length/ Å
|-
!style="text-align: center;" |C1-C2
!style="text-align: center;" |C2-C3
!style="text-align: center;" |C3-C4
!style="text-align: center;" |C4-C5
!style="text-align: center;" |C5-C6
!style="text-align: center;" |C1-C6
|-
!style="text-align: center;" |1,3-Butadiene
|style="text-align: center;" |–
|style="text-align: center;" |–
|style="text-align: center;" |1.33344
|style="text-align: center;" |1.47077
|style="text-align: center;" |1.33342
|style="text-align: center;" |–
|-
!style="text-align: center;" |Ethylene
|style="text-align: center;" |1.32731
|style="text-align: center;" |–
|style="text-align: center;" |–
|style="text-align: center;" |–
|style="text-align: center;" |–
|style="text-align: center;" |–
|-
!style="text-align: center;" |Transition State
|style="text-align: center;" |1.38173
|style="text-align: center;" |2.11470
|style="text-align: center;" |1.37976
|style="text-align: center;" |1.41112
|style="text-align: center;" |1.37973
|style="text-align: center;" |2.11478
|-
!style="text-align: center;" |Cyclohexene
|style="text-align: center;" |1.53767
|style="text-align: center;" |1.53577
|style="text-align: center;" |1.49262
|style="text-align: center;" |1.33305
|style="text-align: center;" |1.49263
|style="text-align: center;" |1.53577
|}

The C-C bond length measurement with the PM6 level optimized structures is in good accordance to the typical C-C and C=C bond length. There are 2 factors affecting the bond lengths:
<br>
(a) ''Bond order'': The higher the bond order of a given bond, the shorter the bond length and hence the stronger the bond.
<br>
(b) ''Type of Hybridization'': The size of hybrid orbitals increases in the order of sp C > sp<sup>2</sup> C > sp<sup>3</sup> C. The larger valence orbitals of sp<sup>3</sup> C has poorer overlap, forming a longer and weaker bond. The percentage of ''s''-character in hybrid orbitals decreases in the order of sp C (50%) > sp<sup>2</sup> C (33%) > sp<sup>3</sup> C (25%). The higher the ''s''-character of a given hybrid orbital, the shorter the bond. Because the electron in high ''s''-character hybrid orbital can penetrate better into the nucleus and experience a higher effective nuclear attraction, resulting in a shorter bond.
<br>
<br>
'''Atomic Van der Waals radius of C atom: 1.70 Å'''
<br>
'''Van der Waals distance between 2 C atom: 3.40 Å'''
<br>
<br>
Van der Waals distance between any 2 atoms is the minimum distance beyond which the electrostatic attractive force turn into repulsive force. The length of partially formed C–C bonds in transition state is 2.11 Å, which is smaller than the Van der Waals radii between 2 C atoms but is greater than the typical C-C bond lengths listed in Table 5. This indicates the 2p atomic orbitals of C atom are partially overlap in transition state, resulting in the partly formed C-C single bond. The bond length of a partially formed or broken C=C double bond in transition state has a value between a full C-C single and C=C double bond.

{| class="wikitable"
! colspan="2" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 7: Change in C-C bond length during Diels-Alder reaction.
|-
! style="text-align: center;" | Bond Length against Reaction Coordinate
! style="text-align: center;" | Explanation
|-
|[[File:Xlt15ex1ircexcel.PNG|center]]
|
*'''C1-C2, C3-C4, and C5-C6'''
All the plots show the bond length change when a C=C double bond breaks (in 1,3-butadiene and ethylene) into a C-C single bond (in cyclohexene). C=C bond length remains constant at 1.33 Å for a certain period of time. Then, the transition of transition state to product shows a gradual increase in bond length to 1.54 Å for C1-C2 and 1.49 Å for C3-C4 and C5-C6. The slight difference in C-C single bond length in cyclohexene is due to the difference in the hybridization type of C atom.
<br>
*'''C2-C3 and C6-C1'''
All the plots show change in bond length during the C-C single bond formation. The initial separation of terminal Cs between the 1,3-butadiene and ethylene is at 3.40 Å which is the VDW distance between 2 C atoms. As the reaction proceeds, their separation decreases almost linearly along the IRC and then it remains constant at 1.54 Å (single C-C bond formed).
<br>
*'''C4-C5'''
The plot shows the change in bond length when a C-C single bond (in 1,3-butadiene) forms a C=C double bond (in cyclohexene). C-C bond length remains constant at 1.47 Å for a certain period of time. Then, the transition of transition state to product shows a gradual decrease in bond length to 1.33 Å (C=C double bond formed).
|}

==== Vibration That Corresponds to The Reaction Path at The Transition State ====
<jmol>
<jmolApplet>
<size>300</size>
<uploadedFileContents>xlt15TS_1.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 21; rotate x -20; frank off</script>
<name>xlt15TS_1</name>
</jmolApplet>
<jmolbutton>
<script>if(vibrating==0) vibrating=1; vibration 2; else; vibrating=0; vibration off; endif</script>
<text>Vibration</text>
<target>xlt15TS_1</target>
</jmolbutton>
<jmolbutton>
<script>measure 2 3; measure 1 6</script>
<text>C-C Distances</text>
<target>xlt15TS_1</target>
</jmolbutton>
<jmolmenu>
<item>
<script>frame 7; if(vibrating==0) vibration off; else; vibration 2; endif</script>
<text>i949/cm<sup>-1</sup></text>
<target>xlt15TS_1</target>
</item>
</jmolmenu>
</jmol>
<br>
As shown in vibration mode of the imaginary vibrational frequency at -949 cm<sup>-1</sup>, the formation of the two C-C single bonds are clearly '''synchronous''' as they are formed simultaneously. Thus, the [4+2]-cycloaddition of 1,3-butadiene and ethylene is concerted and stereospecific. In addition, this is supported by the fact the C-C bond distance of the reacting termini are the same in the transition state at 2.11 Å. Also, both 1,3-butadiene and ethylene are symmetrical and so it is expected that one C-C bond will not formed faster than the other.

====Log File For IRC Calculation====
''IRC calculation of PM6 optimised TS:'' [[File:XLT15 IRCTS1.LOG]]

=== Exercise 2 ===
http://wiki.ch.ic.ac.uk/wiki/index.php?title=Xlt15_Ex2

=== Exercise 3 ===
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Xlt15_Ex3
=== Further Work ===
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Xlt15_FURTHER

=== References for Exercise 1 ===

Rep:Xlt15 FURTHER

2025-09-01T09:50:20Z

Move page script: Move page script moved page Xlt15 FURTHER to Rep:Xlt15 FURTHER: Move to report namespace

=== Further Work: Electrocyclic Reaction ===
[[File:Xlt15reactionscextraa.PNG|frame|center|Figure 1: Reaction scheme of 4π electrocyclic reaction of 1,3-butadiene derivative.]]
A 4π electrocyclic reaction in Figure 1 is investigated as a further work to the computational lab. An electrocyclic reaction is the formation or the breaking of σ bonds across the end of a conjugated π system <ref name=":0"> J. Clayden, N. Greeves, S. Warren, P. Wothers, ''Organic Chemistry'', Oxford University Press Inc., New York, 2001.</ref>. The equilibrium favoured towards left hand side because of the ring strain in the 4-membered ring.
<br>
<br>
''' Method Used In Optimization and Analysis '''
<br>
Method 3 was employed in locating the transition state by which the product, cyclobut-1-ene derivative was drawn and optimized to minimum at PM6 level. With the optimized product, the C-C single bond formed during the pericyclic reaction was deleted, froze at 2.20 Å and optimized to minimum at PM6 level to identify the frozen guess transition state. The guess TS structure was then optimized at PM6 level and the PM6 optimized TS was used to run a IRC calculation. For reactant, 1,3-butadiene derivative, it was obtained from the first frame of IRC calculation and optimized to minimum at PM6 level.
==== Optimized Reactant, Transition State, and Product at PM6 Level ====
{| class="wikitable"
! colspan="3" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 1: Optimized Reactants, Transition State and Product at PM6 Level.
|-
! style="text-align: center;" | 1,3-Butadiene Derivative
! style="text-align: center;" | Transition State
! style="text-align: center;" | 3,4-Dimethylcyclobut-1-ene
|-
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 24</script>
<uploadedFileContents>XLT15REACTANTSUE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 100</script>
<uploadedFileContents>XLT15TS SUE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 30</script>
<uploadedFileContents>XLT15PRODUCT PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
The geometry of reactants, transition structure and product are checked to properly converge with their respective stationary points found in log files. In addition, transition structure has only one imaginary frequency and that imaginary frequency is then visualized to ensure a correct transition state structure is obtained. There is no imaginary frequency obtained in reactants and products.

==== IRC Calculation ====
{| class="wikitable"
! colspan="3" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 3. IRC Calculation of PM6 Optimized Transition Structures
|-
! style="text-align: center;" | Reaction Progress
! style="text-align: center;" | IRC Output
! style="text-align: center;" | Discussion
|-
|[[File:Xlt15sueextra.gif|center]]
|[[File:Xlt15ircpathex1redo.PNG|350px|center]]
| The ring closing process of 1,3-butadiene derivative clearly proceed via conratation where both hydrogens of terminal alkene carbon rotate in the same direction, clockwise as shown in the IRC. This results in the trans-arrangement of the methyl groups in the product.
|}

==== Conrotation or Disrotation Analysis Using MO ====
{| class="wikitable"
! colspan="4" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 4. MO of Reactant, Transition State and Product.
|-
! style="text-align: center;" | Molecular Orbital
! style="text-align: center;" | 1,3 Butadiene Derivative
! style="text-align: center;" | Transition State
! style="text-align: center;" | 3,4-Dimethylcyclobut-1-ene
|-
!MO18
LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 24; mo 18; mo cutoff 0.01; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15REACTANTSUE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 100; mo 18; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15TS SUE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 30; mo 18; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15PRODUCT PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
!MO17
HOMO
|<jmol>
<jmolApplet>
<color>white</color>
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<script>frame 24; mo 17; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15REACTANTSUE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 100; mo 17; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15TS SUE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 30; mo 17; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15PRODUCT PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
{| class="wikitable"
! colspan="2" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 5. Orbital Correlation Diagram of Reactant, TS and Product.
|-
! style="text-align: center;" | Orbital Correlation Diagram
! style="text-align: center;" | Explanation
|-
|[[File:Xlt15correlationdiagra2.PNG|frame|center|Figure 2: Simplied MO of TS and orbital correlation diagram between reactant and product.]]
<br>
[[File:Xlt15cccconndissss.PNG|frame|center|Figure 3: Conrotation and disrotation under thermal and photochemical condition.]]
|The orbital correlation diagram in Figure 2 shows the simplified version of the MOs observed in reactant, transition state structure and product, with their actual HOMO and LUMO displayed in Table 4. The MOs are labelled symmetric (S) or antisymmetric (A) using the persistent symmetry element, C<sub>2</sub>-rotational axis. The type of bonding or antibonding interaction in product is also shown and coloured in orange.
<br>
The conservation of orbital symmetry introduced by Woodward and Hoffmann states that an orbital of a given symmetry in the reactant is converted smoothly to a product's orbital with the identical symmetry <ref name=":1"> E. V. Anslyn, D. A. Dougherty, ''Modern Physical Organic Chemistry'', University Science Books, Sausalito, United States, 2006.</ref> <ref name=":2"> R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res'', 1968, '''1'''(1), 17-22, {{DOI|10.1021/ar50001a003}}</ref>. Hence, the orbitals in reactant and product can correlate to one another as shown in the diagram on the left. Also, they suggested that the symmetry properties of HOMO in the open-chain conjugated π system control the stereochemical outcome of the electrocyclic reaction <ref name=":3"> B. Dinda, ''Essentials of Pericyclic and Photochemical Reactions'', Springer International Publishing, Switzerland, 2017.</ref>.
<br>
<br>
The pericyclic reaction calculated using Gussian at PM6 level assumed a thermal reaction condition. By examining the HOMO (MO17) of 1,3-butadiene derivative, under thermal condition, the conrotatory motion (anticlockwise direction) of the methyl groups allows an in-phase and constructive orbital interaction to form a σ bond between the terminal Cs. In contrast, disrotation brings the lobes of of the opposite phases for antibonding formation with a high energy transition structure and hence it is symmetry forbidden reaction path. Hence, under thermal condition, the '''4π electron''' conjugated system reacts with itself '''antarafacially''' via a '''Mobius aromatic transition state''' and is an orbital symmetry allowed process <ref name=":3"> B. Dinda, ''Essentials of Pericyclic and Photochemical Reactions'', Springer International Publishing, Switzerland, 2017.</ref>. A C<sub>2</sub> axis of symmetry is preserved during this reaction.
<br>
<br>
On the other hand, under photochemical condition, the electron in MO17 gets excited into MO18, leading to the HOMO to be considered now is MO18. The disrotatory motion of the methyl group on the terminal Cs of 1,3-butadiene derivative brings the lobes of the same phase for bonding interaction and hence it has low activation energy and is an orbital symmetry allowed process <ref name=":3"> B. Dinda, ''Essentials of Pericyclic and Photochemical Reactions'', Springer International Publishing, Switzerland, 2017.</ref>. This results in the cis-arrangement of methyl groups in the cyclobut-1ene product. In contrast, the conrotatory motion leads to an antibonding interaction and is an orbital symmetry forbidden pathway <ref name=":3"> B. Dinda, ''Essentials of Pericyclic and Photochemical Reactions'', Springer International Publishing, Switzerland, 2017.</ref>.
<br>
A summary diagram for conrotation of MO17 under thermal condition and disrotation of MO18 under photochemical condition is shown in Figure 3.
|}

(Very good work with the correlation diagram. However your labelling of the symmetries is a bit off. It is unexpected but a particular MO may switch between symmetric and antisymmetric along the reaction path as it passes the Mobius transition state [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 10:35, 26 February 2018 (UTC))

''' Woodward-Hoffmann Rule For Electrocyclic Reaction <ref name=":0"> J. Clayden, N. Greeves, S. Warren, P. Wothers, ''Organic Chemistry'', Oxford University Press Inc., New York, 2001</ref>. '''
{| class="wikitable"
! colspan="3" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 4. Orbital Symmetry Allowed Reaction by Woodward-Hoffmann Rule.
|-
! style="text-align: center;" | Number of π Electon
! style="text-align: center;" | Thermal Condition
! style="text-align: center;" | Photochemical Condition
|-
! rowspan="2" style="text-align: center;" | 4n
| style="text-align: center;" | Conrotatory
| style="text-align: center;" | Disrotatory
|-
| style="text-align: center;" | Antarafacial
| style="text-align: center;" | Suprafacial
|-
! rowspan="2" style="text-align: center;" | 4n+2
| style="text-align: center;" | Disrotatory
| style="text-align: center;" | Conrotatory
|-
| style="text-align: center;" | Suprafacial
| style="text-align: center;" | Antarafacial
|}

==== Log File For IRC Calculation ====
''IRC Calculation of PM6 Optimized TS'': [[File:XLT15SUETS IRC.LOG]]

=== Exercise 1 ===
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Xlt15_TS
=== Exercise 2 ===
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Xlt15_Ex2

=== Exercise 3 ===
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Xlt15_Ex3

=== References in Further Work ===

Rep:Xlt15 Ex3

2025-09-01T09:50:19Z

Move page script: Move page script moved page Xlt15 Ex3 to Rep:Xlt15 Ex3: Move to report namespace

=== Exercise 3: Diels-Alder vs Cheletropic ===
[[File:Xlt15reactionschemeex3complete.PNG|frame|center|Figure 1: Reaction scheme of Diels-Alder and cheletropic reaction of o-xylylene and sulfur dioxide.]]
The cycloaddition of o-xylylene and sulfur dioxide can proceed in 2 ways, [4+2]-cycloaddition (D-A exo and D-A endo pathway) and [4+1]-cheletropic cycloaddition. Cheletropic reaction is a subclass of cycloaddition involving a concerted and synchronous formation or breaking of two new σ bonds to a single atom <ref name=":0"> D. Suárez, E. Iglesias, T. L. Sordo, J. A. Sordo, ''J. Phys. Org. Chem.'', 1996, '''9''', 17–20, {{DOI|10.1002/(SICI)1099-1395(199601)9:1<17::AID-POC749>3.0.CO;2-D}}.</ref> <ref name=":1"> R. B. Woodward, R. Hoffman, ''Angew. Chem. Int. Ed. Engl.'', 1969, '''8''', 781–853, {{DOI|10.1002/anie.196907811}}.</ref>. Both D-A and cheleteropic reaction involve 6π electrons. In cheletropic reaction (giving sulfolene), the S atom contributes its lone pair of electron to the 6π pericyclic transition state whereas in the hetero-D-A reaction, the o-xylylene reacts reversibly with SO<sub>2</sub> suprafacially, producing sulfite as product <ref name=":2"> F. Monnat, P. Vogel, V. M. Rayon, J. A. Sordo, ''J. Org. Chem.'', 2002, '''67''', 1882-1889, {{DOI|10.1021/jo010998w}}.</ref>. It is worthnoting that there is an increase in the coordination number on S from 4 to 6 in cheletropic reaction whereas the S coordination number remains the same in D-A reaction.
<br>
<br>
'''Method Used In Optimization and Analysis'''
<br>
Method 3 is employed in locating the transition state by which the product is drawn and optimized to minimum at PM6 level. With the optimized product, the C-S and C-O single bonds formed during the reaction are deleted and these bonds are froze at 2.40 Å and 2.00 Å respectively. It is then optimized to minimum at PM6 level to identify the frozen guess transition state. The guess TS structure is optimized at PM6 level and the PM6 optimized TS was used to run a IRC calculation. For reactants, o-xylylene and sulfur dioxide, they are each obtained from first frame of IRC calculation and optimized to minimum at PM6 level.
<br>
<br>
For the reaction between second cis-butadiene fragemnt in o-xylylene and sulfur dioxide, method 2 was employed whereby a guess transition state structure was drawn and the C-S and C-O distances were froze at 2.40 Å and 2.00 Å respectively. This guess TS is then optimized at PM6 level and used to obtain IRC calculation. The product obtained from the last frame in the IRC calculation was optimized to minimum at PM6 level.

==== Optimized Reactants, Transition Structure and Products at B3LYP/6-31G(d) level ====
{| class="wikitable"
! colspan="2" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 1: Optimized Reactants at PM6 level
|-
! style="text-align: center;" | o-Xylylene
! style="text-align: center;" | SO<sub>2</sub>
|-
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 12</script>
<uploadedFileContents>XLT150902REDOXYLE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 16</script>
<uploadedFileContents>Xlt1522REDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

{| class="wikitable"
! colspan="4" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 2: Optimized Transition State and Product at PM6 level
|-
! style="text-align: center;" | Types of Reaction and Pathway
! style="text-align: center;" | Transition State
! style="text-align: center;" | Product
|-
! style="text-align: center;" | Diels-Alder Exo
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 76</script>
<uploadedFileContents>XLT15EXOTS_PM63.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 18</script>
<uploadedFileContents>XLT15EXOPRODUCT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
! ! style="text-align: center;" | Diels-Alder Endo
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 12</script>
<uploadedFileContents>XLT15ENDOTS PM63.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 38</script>
<uploadedFileContents>ENDOXLT15REDOPRODUCT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
! style="text-align: center;" | Cheletropic Reaction
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 12</script>
<uploadedFileContents>XLT15CHELETS PM63.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 22</script>
<uploadedFileContents>XLT15CHELEPRODUCT PM63.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
The geometry of reactants, transition state and product are checked to properly converge with their respective stationary points found in log files. In addition, the transition state has only one imaginary frequency and it is then visualized to ensure a correct transition state structure is obtained. There is no imaginary frequency obtained in reactants and products.

==== IRC Calculation ====
{| class="wikitable"
! colspan="4" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 3: IRC Calculation of PM6 Optimized Transition Structures.
|-
! rowspan="2" style="text-align: center;" | IRC Output
! colspan="3" style="text-align: center;" | Reaction Pathways
|-
! style="text-align: center;" | Diels-Alder Exo
! style="text-align: center;" | Diels-Alder Endo
! style="text-align: center;" | Cheletropic
|-
! style="text-align: center;" | Reaction Progress
|[[File:xlt15exo4.gif|center]]
|[[File:Xlt15endomovie3.gif|center]]
|[[File:Xlt15chele.gif|center]]
|-
! style="text-align: center;" | Discussion
| Asynchronous and stepwise D-A reaction with the C-O bond formed faster than the C-S bond. The forming bond lengths of C-O and C-S are 2.08 Å and 2.35 Å respectively.
| Asynchronous and stepwise D-A reaction with the C-O bond formed faster than the C-S bond. The forming bond lengths of C-O and C-S are 2.10 Å and 2.33 Å respectively.
| Synchronous and concerted C-S single bond formation. Both forming C-S bond lengths are 2.37 Å.
|-
! style="text-align: center;" | IRC Calculation
|[[File:Xlt15exo.PNG|350px|center]]
|[[File:Xlt15endoxl.PNG|350px|center]]
|[[File:Xlt15chele2.PNG|350px|center]]
|}

(This isn't a stepwise reaction unless there are intermediates along the reaction path. All you can say here is the bonds are formed at different time - you can't even really say which one is "first" as Gaussview is just using length cutoffs to determine when it should draw a line between the atoms [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 10:06, 26 February 2018 (UTC))

==== Thermochemistry ====
{| class="wikitable"
! colspan="2" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 4: Thermochemistry Data at PM6 Level.
|-
! style="text-align: center;" | Species
! style="text-align: center;" | Sum of Electronic and Thermal Free Energies/ kJmol<sup>-1</sup>
|-
! o-Xylylene
| style="text-align: center;" | +468.1083
|-
! SO<sub>2</sub>
| style="text-align: center;" | -313.1408
|-
! Sum of Reactant Energy
| style="text-align: center;" | +154.9675
|-
! Diels-Alder Exo TS
| style="text-align: center;" | +241.7456
|-
! Diels-Alder Endo TS
| style="text-align: center;" | +237.7627
|-
! Cheletropic TS
| style="text-align: center;" | +260.0847
|-
! Diels-Alder Exo Product
| style="text-align: center;" | +56.3301
|-
! Diels-Alder Endo Product
| style="text-align: center;" | +56.9839
|-
! Cheletropic Product
| style="text-align: center;" | +0.0131
|}

{| class="wikitable"
! colspan="3" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 5: Reaction Barriers and Reaction Energies.
|-
! style="text-align: center;" | Reaction Pathway
! style="text-align: center;" | Reaction Barriers/ kJmol<sup>-1</sup>
! style="text-align: center;" | Reaction Energies/ kJmol<sup>-1</sup>
|-
! Diels-Alder Exo
| style="text-align: center;" | +86.78
| style="text-align: center;" | -98.64
|-
! Diels-Alder Endo
| style="text-align: center;" | +82.80
| style="text-align: center;" | -97.98
|-
! Cheletropic Reaction
| style="text-align: center;" | +105.12
| style="text-align: center;" | -154.95
|}

'''Reaction Profile'''
{| class="wikitable"
! colspan="2" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 6: Reaction Profile For D-A and Cheletropic Reactions.
|-
! style="text-align: center;" | Reaction Profile
! style="text-align: center;" | Explanation
|-
| [[File:Xlt15reactionprofile1.PNG|center]]
| The energy level is set at 0 for reactants at infinite separation. Based on the PM6 level calculation, the D-A exo product with the non-reacting oxygen in equatorial position is the most stable D-A cycloadduct and the cheletropic product, sulfolene is more stable than both axial and equatorial sulfine. Hence, cheletropic reaction, having the most exothermic reaction energy is the most preferred route under thermodynamic control. The cheletropic reaction is favoured because it generates the more thermodynamically stable five-membered cycloadduct. D-A endo product is slightly less stable than exo product because the non-reacting oxygen in D-A endo product occupies an axial position, having some steric interaction.
<br>
The D-A endo product is the most preferred product under kinetic control as this pathway has the least activation barrier to reach transition state and hence it is more rapidly formed. As discussed in Exercise 2, it is due to the secondary orbital interaction between the reacting diene component in o-xylylene and the non-bonding p orbital in oxygen. The favourable orbital interaction through space lowers the energy of endo TS, leading to endo product. In contrast, the cheletropic pathway is the least kinetically stable product with the largest activation barrier.
<br>
<br>
All the reactions exhibit 6π electron aromaticity at the transition state, showing pericyclic character. They all are exothermic and spontaneous, probably due to the resonance stabilization owing from the aromaticity of the benzene ring in the products.
|}

==== Change in Bonding of o-Xylylene during The Reaction ====
[[File:X;t15reactionnumber.PNG|frame|left|Figure 2: Reaction scheme with the all carbons numbered.]]
{| class="wikitable"
! colspan="3" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 7: IRC Output For D-A Exo, D-A Endo and Cheletropic Reactions.
|-
! colspan="3" style="text-align: center;" | Bond Length against IRC
|-
|[[File:Xlt15ircexopathwayex34.PNG|center]]
|[[File:Xlt15endopathircex34.PNG|center]]
|[[File:Xlt15cheletropicthircex34renumber1.PNG|center]]
|-
|colspan="3" |
All C-C bond lengths against IRC are plotted using the numbering system in Figure 2.
* As the reaction progress, the C=C double bonds are lengthen and the C-C single bonds are shorten in 6-membered ring of o-xylylene.
* C=C double bond length in 6-membered ring of o-xylylene, C1-C2 and C5-C6 both are 1.347 Å. The remaining C-C single bonds have an average value of 1.473 Å. Both values are in good accordance to literature sp<sup>2</sup> C-sp<sup>2</sup> C bond lengths which are 1.34 Å (for sp<sup>2</sup> C-sp<sup>2</sup> C single bond) and 1.48 Å (for sp<sup>2</sup> C=sp<sup>2</sup> C double bond) <ref name=":3"> S. M. Mukherji, S. P. Singh, R. P.Feynman,R. Dass, ''Organic Chemistry Vol I'', New Age International, New Delhi, India, 2010.</ref>.
* Using the PM6 calculation, the C-C bond in the optimized D-A exo product, D-A endo product and cheletropic product has a maximum difference in bond length of 0.027 Å. This indicates that the C-C bond lengths are almost identical and has an average value of 1.401 Å for each individual product. Again, it is in good accordance to the benzene C-C bond length of 1.395 Å <ref name=":4"> J. Clayden, N. Greeves, S. Warren, P. Wothers, ''Organic Chemistry'', Oxford University Press Inc., New York, 2001.</ref>.
* The average bond length of 1.401 Å in product is an intermediate value between the sp<sup>2</sup> C-sp<sup>2</sup>C single bond and double bond.
* It is because the benzene ring in product has 6π electrons, exhibiting a special stability called aromaticity based on Huckel's rule. Huckel's rule states that a planar, fully conjugated, monocyclic molecule with (4n + 2) π electrons in a continuous ring of p orbitals, where n is a non-negative integer is surprisingly stable <ref name=":4"> J. Clayden, N. Greeves, S. Warren, P. Wothers, ''Organic Chemistry'', Oxford University Press Inc., New York, 2001.</ref>.
* Due to the delocalization of electron over 6 carbon atoms, the sp<sup>2</sup> C-sp<sup>2</sup> C single bond is now shorter and stronger than normal single bond. Also, the C=C double bonds are now weaker and longer than expected. The calculated C-C bond lengths in product are very similar because the 2 resonance hybrids shown in Figure 2 contributes almost equally to the benzene ring.
|}

==== Reaction of Second Cis-Butadiene in Diels-Alder Reaction ====
[[File:Xlt15secondbutareactionsche.PNG|frame|center|Figure 3: Diels-Alder reaction of second cis-butadiene and sulfur dioxide.]]
<br>
As there is a second 1,3-butadiene in ''s''-cis conformation, it can also undergo Diels-Alder (D-A) reaction with the sulfur dioxide.
===== PM6 Level Optimized Transition Structure and Product =====
{| class="wikitable"
! colspan="4" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 8: Optimized Transition State and Product at PM6 Level.
|-
! style="text-align: center;" | Types of Reaction and Pathway
! style="text-align: center;" | Transition State
! style="text-align: center;" | Product
|-
! style="text-align: center;" | Diels-Alder Exo
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 12</script>
<uploadedFileContents>XLT152BUTAEXOTS PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 16</script>
<uploadedFileContents>XLT15EXO2BUTAPRODUCT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
! style="text-align: center;" | Diels-Alder Endo
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 14</script>
<uploadedFileContents>XLT15ENDO2BUTAts PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 16</script>
<uploadedFileContents>XLT15ENDO2BUTAPRODUCT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
The geometry of reactants, transition structure and product are checked to properly converge with their respective stationary points found in log files. In addition, transition structure has only one imaginary frequency and that imaginary frequency is then visualized to ensure a correct transition state structure is obtained. There is no imaginary frequency obtained in reactants and products.

===== IRC Calculation =====
{| class="wikitable"
! colspan="3" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 9: IRC Calculation of PM6 Optimized Transition Structures.
|-
! rowspan="2" style="text-align: center;" | IRC Output
! colspan="2" style="text-align: center;" | Reaction Pathways
|-
! style="text-align: center;" | Diels-Alder Exo
! style="text-align: center;" | Diels-Alder Endo
|-
! style="text-align: center;" | Reaction Progress
|[[File:Xlt15secondbutaexoirc.gif|center]]
|[[File:Xlt15sndbutaendo.gif|center]]
|-
! style="text-align: center;" | IRC Calculation
|[[File:Xlt15seconbytaexoirc.PNG|350px|center]]
|[[File:Xlt152ndbutaendo.PNG|350px|center]]
|}

===== Thermochemistry =====
{| class="wikitable"
! colspan="2" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 10: Thermochemistry Data at PM6 Level.
|-
! style="text-align: center;" | Species
! style="text-align: center;" | Sum of Electronic and Thermal Free Energies/ kJmol<sup>-1</sup>
|-
! Sum of Reactant Energy
| style="text-align: center;" | +156.2041
|-
! Diels-Alder Exo TS
| style="text-align: center;" | +275.8245
|-
! Diels-Alder Endo TS
| style="text-align: center;" | +267.9848
|-
! Diels-Alder Exo Product
| style="text-align: center;" | +176.7067
|-
! Diels-Alder Endo Product
| style="text-align: center;" | +172.2591
|}

{| class="wikitable"
! colspan="3" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 11: Reaction Barriers and Reaction Energies
|-
! style="text-align: center;" | Reaction Pathway
! style="text-align: center;" | Reaction Barriers/ kJmol<sup>-1</sup>
! style="text-align: center;" | Reaction Energies/ kJmol<sup>-1</sup>
|-
! Diels-Alder Exo
| style="text-align: center;" | +119.62
| style="text-align: center;" | +20.50
|-
! Diels-Alder Endo
| style="text-align: center;" | +111.78
| style="text-align: center;" | +16.05
|-
|}
By comparing Table 5 and 11, the D-A reaction of second cis-butadiene and SO<sub>2</sub> has a higher activation barrier to reach TS, by about 33 kJmol<sup>-1</sup> for exo pathway and by roughly 29 kJmol<sup>-1</sup> for endo pathway. In addition, both D-A exo and endo pathway of second cis-butadiene and SO<sub>2</sub> is endothermic, meaning that these reactions require an energy input. The exo and endo products from the second-cis butadiene are rather destabilized and have higher Gibbs free energy compared to the sum of reactant energy, hence the formation of product is not spontaneous. Also, they are higher in energy than the previous D-A reaction due to the lack of aromaticity in product. Thus, the D-A reaction between the second cis-butadiene and SO<sub>2</sub> is concluded to be '''very thermodynamically and kinetically unfavourable'''.

(Very detailed section and good work. For the conclusions it's probably best to keep them in their own sections to avoid confusion [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 10:06, 26 February 2018 (UTC))

==== Log File for IRC Calculation of PM6 Optimized Transition Structures ====
''Diels-Alder Exo TS:'' [[File:XLT15EXOTS IRC3.LOG]]
<br>
''Diels-Alder Endo TS:'' [[File:XLT15ENDOTS IRC3.LOG]]
<br>
''Cheletropic reaction TS:'' [[File:XLT15CHELETS IRC3.LOG]]
<br>
''Diels-Alder Exo TS of second cis-butadiene:'' [[File:XLT152BUTAEXOTS IRC.LOG]]
<br>
''Diels-Alder Exo TS of second cis-butadiene:'' [[File:XLT15ENDO2BUTATS IRC.LOG]]

=== Exercise 1 ===
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Xlt15_TS
=== Exercise 2 ===
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Xlt15_Ex2
=== Further Work ===
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Xlt15_FURTHER

=== Conclusion ===
Both PM6 and B3LYP/6-31G(d) methods were successfully used to optimize the reactants, transition states and products. The transition state was identified by the presence of a single imaginary frequency and there is no such negative frequency in a properly optimized reactant and product. IRC calculation was then performed on the PM6 optimized transition state.
<br>
<br>
The Diels-Alder (D-A) reaction between 1,3-butadiene and ethylene is a concerted and spontaneous C-C bond formation. This is illustrated by the imaginary frequency of the transition state and the identical C-C separation of the reacting termini. The separation between the reacting termini is less than the Van der Waals distance of 2 C atoms, implying a partially formed bond. Also, only orbitals of the identical symmetry can combine to give a non-zero orbital overlap integral.
<br>
<br>
The D-A reaction between a cyclohexadiene and 1,3-dioxole is an inverse electon demand reaction, determining by a single point energy calculation to obtain the relative energy levels of the frontier molecular orbital (FMO). Based on Gibbs free energy obtained from a B3LYP/6-31G(d) calculation, the endo-adduct of D-A reaction is more kinetically and thermodynamically favoured than exo-adduct because of the favourable secondary orbital interaction and less steric hindrance in endo product.
<br>
<br>
The cycloaddition of o-xylylene and sulfur dioxide has D-A exo, D-A endo and cheletropic pathways. Based on their reaction profile, the cheletropic reaction is the most thermodynamically favoured, having the most exothermic reaction energy whereas the D-A endo product is the most kinetically favourable, having the smallest activation barrier to reach TS. All three reaction is highly exothermic due to the gain in aromaticity in product. Based on the thermochemistry data from the PM6 level calculation, the D-A exo and endo reaction of second cis-butadiene and sulfur dioxide are endothermic and require more activation barrier to reach TS, indicating that both are thermodynamically and kinetically unfavourable reaction.
<br>
<br>
For the electrocyclic reaction, MOs of reactant, TS and product was employed to determine whether it is conrotation or disrotation. It is observed that there is a C<sub>2</sub> axis of symmetry preserved during the reaction and hence it is used in symmetry labelling of the MOs. An thermal electrocyclic reaction with (4n)π reaction involve conrotation of the group on the terminal C via a Mobius aromatic TS, whereas it involved a disrotation for a photochemical eletrocyclic reaction.

=== References in Exercise 3 ===

Rep:Xlt15 Ex2

2025-09-01T09:50:19Z

Move page script: Move page script moved page Xlt15 Ex2 to Rep:Xlt15 Ex2: Move to report namespace

=== Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole ===
[[File:Xlt15reactionscheme4.PNG|center|frame|Figure 1: Reaction scheme of Diels-Alder reaction of cyclohexadiene and 1,3-dioxole.]]
<br>
The Diels-Alder (D-A) reaction of cyclohexadiene and 1,3-dioxole can proceed in 2 pathways, exo and endo pathways. In exo pathway, the oxygens in 1,3-dioxole is oriented away from the diene component of cyclohexadiene in transition state whereas in endo pathway the oxygens in 1,3-dioxole is oriented towards the diene component of cyclohexadiene in transition state.
<br>
<br>
'''Method Used In Optimization and Analysis'''
<br>
Method 3 was employed in locating the transition state by which the product was drawn and optimized to minimum at PM6 level. With the optimized product, the C-C single bonds formed during the reaction of cyclohexadiene and 1,3-dioxole were deleted, froze at 2.20 Å and optimized to minimum at PM6 level to identify the frozen guess transition state. The distance 2.20 Å is the approximate separation between the reacting termini in guess transition state. It is a value between a C-C single bond length and their combined Van der Waals radii. The guess TS structure was then optimized at PM6 level and the PM6 optimized TS was used to run a IRC calculation. The reactants, exo and endo products obtained from first and last frame of IRC calculation were optimized to minimum at PM6 level. All the PM6 optimized reactants, TS and products were then reoptimized with a more accurate calculation, B3LYP/6-31G(d).
<br>
==== Optimized Reactants, Transition Structure and Products at B3LYP/6-31G(d) Level ====
{| class="wikitable"
! colspan="2" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 1: Optimized Reactants at B3LYP/6-31G(d) Level.
|-
! style="text-align: center;" | Cyclohexadiene
! style="text-align: center;" | 1,3-Dioxole
|-
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 18</script>
<uploadedFileContents>XLT15CYCLOHEXADIENE_DPT2.LOG</uploadedFileContents>
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</jmol>
|<jmol>
<jmolApplet>
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<size>250</size>
<script>frame 24</script>
<uploadedFileContents>XLT15DIOXOLE_DPT2.LOG</uploadedFileContents>
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</jmol>
|}

{| class="wikitable"
! colspan="3" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 2: Optimized Transition State and Product of Endo and Exo Pathways at B3LYP/6-31G(d) Level.
|-
! style="text-align: center;" | Reaction Pathway
! style="text-align: center;" | Transition State
! style="text-align: center;" | Product
|-
! Exo Pathway
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 16</script>
<uploadedFileContents>XLT15EXOTS DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 14</script>
<uploadedFileContents>XLT15EXOPRODUCT DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
! Endo pathway
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 30</script>
<uploadedFileContents>XLT15ENDOTS DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>black</color>
<size>250</size>
<script>frame 18</script>
<uploadedFileContents>XLT15ENDOPRODUCT DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}
The geometry of reactants, transition state and product are checked to properly converge with their respective stationary points are found in log files. In addition, transition state has only one imaginary frequency and it is then visualized to ensure a correct transition state structure is obtained. There is no imaginary frequency obtained in reactants and products.

==== MO Analysis ====
===== MO Diagram of Exo and Endo Pathway=====

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) Your Mo diagrams are very nice. Could have expended a bit more on the differences between endo and exo conformations in term of their relative MO energies.)

{| class="wikitable"
! colspan="2" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 3: MO Diagram for The Formation of Cyclohexadiene/1,3-Dioxole TS.
|-
! style="text-align: center;" | Exo Pathway
! style="text-align: center;" | Endo Pathway
|-
|[[File:Xlt15exomo4.PNG|center|frame|Figure 1: MO diagram for the formation of the exo TS with basic symmetry labels shown; A= Antisymmetric and S= Symmetric.]]
|[[File:Xlt15endomo1.PNG|center|frame|Figure 2: MO diagram for the formation of the endo TS with basic symmetry labels shown; A= Antisymmetric and S= Symmetric.]]
|}

===== MO of Reactants, Product and Transition Structure =====
{| class="wikitable"
! colspan="5" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 4: Frontier MO of Cyclohexadiene, 1,3-Dioxole and Exo and Endo TS.
|-
! colspan="2" style="text-align: center;" | Reactants
! colspan="2" style="text-align: center;" | Transition State
! rowspan="2" style="text-align: center;" | Orbital Interaction and Discussion
|-
! style="text-align: center;" | Cyclohexadiene
! style="text-align: center;" | 1,3-Dioxole
! style="text-align: center;" | Exo
! style="text-align: center;" | Endo
|-
|rowspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<title>LUMO, MO23, S</title>
<size>200</size>
<script>frame 18; mo 23; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15CYCLOHEXADIENE_DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|rowspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO, MO19, S</title>
<size>200</size>
<script>frame 24; mo 19; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15DIOXOLE_DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO, MO41, S</title>
<size>200</size>
<script>frame 16; mo 41; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15EXOTS DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO, MO41, S</title>
<size>200</size>
<script>frame 30; mo 41; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15ENDOTS DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|rowspan="2" |'''“+” stands for an in-phase combination and “-” stands for an out-of-phase interaction'''
<br>
Cyclohexadiene (LUMO, MO23, S) + 1,3-dioxole (HOMO, MO19, S) = TS (HOMO, MO41, S)
<br>
Cyclohexadiene (LUMO, MO23, S) - 1,3-dioxole (HOMO, MO19, S) = TS (LUMO, MO42, S)
|-
|<jmol>
<jmolApplet>
<color>white</color>
<title>LUMO, MO42, S</title>
<size>200</size>
<script>frame 16; mo 42; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15EXOTS DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>LUMO, MO42, S</title>
<size>200</size>
<script>frame 30; mo 42; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15ENDOTS DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|rowspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO, MO22, A</title>
<size>200</size>
<script>frame 18; mo 22; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15CYCLOHEXADIENE_DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|rowspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<title>LUMO, MO20, A</title>
<size>200</size>
<script>frame 24; mo 20; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15DIOXOLE_DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO-1, MO40, A</title>
<size>200</size>
<script>frame 16; mo 40; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15EXOTS DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO-1, MO40, A</title>
<size>200</size>
<script>frame 30; mo 40; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15ENDOTS DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|rowspan="2" | '''“+” stands for an in-phase combination and “-” stands for an out-of-phase interaction'''
<br>
Cyclohexadiene (HOMO, MO22, A) + 1,3-dioxole (LUMO, MO20, A) = TS (HOMO-1, MO40, A)
<br>
Cyclohexadiene (HOMO, MO22, A) - 1,3-dioxole (LUMO, MO20, A) = TS (LUMO+1, MO43, A)
|-
|<jmol>
<jmolApplet>
<color>white</color>
<title>LUMO+1, MO43, A</title>
<size>200</size>
<script>frame 16; mo 43; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15EXOTS DPT2.LOG</uploadedFileContents>
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|<jmol>
<jmolApplet>
<color>white</color>
<title>LUMO+1, MO43, A</title>
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<script>frame 30; mo 43; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15ENDOTS DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

==== Normal or Inverse Demand DA Reaction ====
A single point energy calculation reveals the relative energy levels of HOMO and LUMO of the reactants, cyclohexadiene and 1,3-dioxole. An IRC calculation was first performed and the first frame which contains both reactants was used to carry out a single point calculation of both endo and exo pathway. The Table 5 below shows the HOMO and LUMO obtained from a single point energy calculation of the exo pathway. The HOMO and LUMO from a single point energy calculation for endo pathway is not shown as the shape of MOs are the same as the exo pathway but their relative energies and energy gap are tabulated in Table 6 and 7.

{| class="wikitable"
! colspan="3" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 5: MO of Reactants from Single Point Energy Calculation of Exo Pathway.
|-
! rowspan="2" style="text-align: center;" | Reactant
! colspan="2" style="text-align: center;" | Molecular Orbital
|-
! style="text-align: center;" | HOMO
! style="text-align: center;" | LUMO
|-
! Cyclohexadiene
|<jmol>
<jmolApplet>
<color>white</color>
<title>MO29, A</title>
<size>200</size>
<script>frame 2; mo 29; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SINGLEPOINTREDOXLT15exo.LOG</uploadedFileContents>
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</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>MO31, S</title>
<size>200</size>
<script>frame 2; mo 31; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SINGLEPOINTREDOXLT15exo.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
! 1,3-Dioxole
|<jmol>
<jmolApplet>
<color>white</color>
<title>MO30, S</title>
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<script>frame 2; mo 30; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SINGLEPOINTREDOXLT15exo.LOG</uploadedFileContents>
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</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>MO32, A</title>
<size>200</size>
<script>frame 2; mo 32; mo cutoff 0.02; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SINGLEPOINTREDOXLT15exo.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

{| class="wikitable"
! colspan="5" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 6: Relative Energy Levels of Reactants obtained from Single Point Energy Calculation For Exo and Endo Pathways.
|-
! rowspan="3" style="text-align: center;" | Reactant
! colspan="4" style="text-align: center;" | Orbital Energy/ a.u.
|-
! colspan="2" style="text-align: center;" |Exo Pathway
! colspan="2" style="text-align: center;" |Endo Pathway
|-
! HOMO
! LUMO
! HOMO
! LUMO
|-
! style="text-align: center;" |Cyclohexadiene
| style="text-align: center;" |-0.32217
| style="text-align: center;" |+0.02111
| style="text-align: center;" |-0.32135
| style="text-align: center;" |+0.02288
|-
! style="text-align: center;" |1,3-Dioxole
| style="text-align: center;" |-0.32207
| style="text-align: center;" |+0.02979
| style="text-align: center;" |-0.31696
| style="text-align: center;" |+0.03219
|}

{| class="wikitable"
! colspan="4" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 7: Energy Gap of Normal and Inverse Electron Demand D-A Reaction.
|-
! rowspan="2" style="text-align: center;" | Reaction Type
! rowspan="2" style="text-align: center;" | Orbital Combination
! colspan="2" style="text-align: center;"| Energy Gap/ a.u.
|-
! Exo Pathway
! Endo pathway
|-
! style="text-align: center;" | Normal Electron Demand
| style="text-align: center;" | HOMO (Cyclohexadiene) + LUMO (1,3-Dioxole)
| style="text-align: center;" | 0.352
| style="text-align: center;" | 0.354
|-
! style="text-align: center;" | Inverse Electron Demand
| style="text-align: center;" | HOMO (1,3-Dioxole) + LUMO (1,3-Cyclohexadiene)
| style="text-align: center;" | 0.343
| style="text-align: center;" | 0.340
|}
The reactivity or the outcome of pericyclic reaction is controlled by the relative energies of the Frontier Molecular Orbitals (FMOs) which are the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital). Frontier molecular orbital theory states that a reaction is only allowed if there is favourable mixing (constructive and in-phase combination) between HOMO and LUMO of the reactants <ref name=":0">E. V. Anslyn, D. A. Dougherty, ''Modern Physical Organic Chemistry'', University Science Books, Sausalito, United States, 2006.</ref>. This results in 2 types of Diels-Alder (D-A) reaction. The normal electron demand Diels-Alder reaction occurs between the electron deficient dienophile (low energy LUMO) and the electron rich diene (high energy HOMO) <ref name=":1">K.N. Houk, ''‎Acc. Chem. Res.'', 1975, '''8'''(11), 361-369, {{DOI|10.1021/ar50095a001}}.</ref>. In contrast, the species involved in an inverse electron demand D-A reaction is the electron rich dienophile (high energy HOMO) and the electron deficient diene (low energy LUMO) <ref name=":1">K.N. Houk, ''‎Acc. Chem. Res.'', 1975, '''8'''(11), 361-369, {{DOI|10.1021/ar50095a001}}</ref>. Both combinations are symmetry allowed and result in a small energy difference between the FMOs, hence enhancing their orbitals interaction.
<br>
<br>
By referring to the MO diagram in Table 3, the HOMO of cyclohexadiene is of the same symmetry as the LUMO of 1,3-dioxole and the vice versa. Both interactions are thus allowed because of the matching in phases. However, based on the single point energy calculation in Table 6, the HOMO and LUMO of 1,3-dioxole is higher in energy than that of cyclohexadiene. This is due to the electron donating ability of oxygen lone pair of electron into the π system in 1,3-dioxole, raising its HOMO and LUMO energy. Hence, the calculated energy gap in Table 7 between the 1,3-dioxole HOMO and the cyclohexadiene LUMO (inverse electron demand D-A) is smaller than that between cyclohexadiene HOMO and 1,3-dixole LUMO and (normal electron demand D-A). Since a smaller energy gap giving rise to a stronger HOMO-LUMO interaction, hence the D-A reaction between the 1,3-dioxole and cyclohexadiene is an '''inverse electron demand''' D-A reaction.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 22:38, 21 February 2018 (UTC) This is an excellent section well done. Excellent single point energy analysis of the reactant FMOS

==== Thermochemistry ====
{| class="wikitable"
! colspan="2" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 8: Thermochemistry Data at B3LYP/6-31G(d) Level.
|-
! style="text-align: center;" | Species
! style="text-align: center;" | Sum of Electronic and Thermal Free Energies/ 10<sup>5</sup> kJmol<sup>-1</sup>
|-
! Cyclohexadiene
| style="text-align: center;" | -6.1259319
|-
! 1,3-Dioxole
| style="text-align: center;" | -7.0118878
|-
! Sum of Reactant Energy
| style="text-align: center;" | -13.1378197
|-
! Exo TS
| style="text-align: center;" | -13.1361433
|-
! Endo TS
| style="text-align: center;" | -13.1362216
|-
! Exo Product
| style="text-align: center;" | -13.1384578
|-
! Endo Product
| style="text-align: center;" | -13.1384938
|}

{| class="wikitable"
! colspan="3" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 9: Reaction Barriers and Reaction Energies
|-
! style="text-align: center;" | Reaction Pathway
! style="text-align: center;" | Reaction Barriers/ 10<sup>5</sup> kJmol<sup>-1</sup>
! style="text-align: center;" | Reaction Energies/ 10<sup>5</sup> kJmol<sup>-1</sup>
|-
! Exo Pathway
| style="text-align: center;" | +167.64
| style="text-align: center;" | -63.81
|-
! Endo Pathway
| style="text-align: center;" | +159.81
| style="text-align: center;" | -67.41
|}
Activation energy is the minimum kinetic energy that reactant(s) must acquire to overcome the energy barrier to have a productive chemical reaction. It is the Gibbs free energy difference between the total of reactant’s energies and transition structure. The lower the activation energy of a reaction, the higher the rate of reaction and hence it is more kinetically favourable and vice versa. The reaction energy determined exclusively by the Gibbs free energy difference between reactants and product and is completely independent of the reaction pathway as Gibbs free energy is a state function.
ΔG = ΔH − TΔS, where ΔG is the Gibbs free energy, ΔH is the enthalpy change, T is the Kelvin temperature and ΔS is the entropy change.
An exothermic reaction, with an enthalpy change < 0, the Gibbs free energy change will also be < 0 (a spontaneous process) unless entropy change is large and negative. An endothermic reaction, with an enthalpy change > 0, the Gibbs free energy change will also be > 0 (a non-spontaneous process) unless entropy change is large and positive. Hence, the product of a more exothermic reaction is thermodynamically more stable (lower in energy) and always favoured.
<br>
<br>
In the D-A reaction between a cyclohexadiene and 1,3-dioxole, the endo pathway has a lower reaction barrier and is more exothermic than the exo pathway. The endo TS and product both have lower (more negative) Gibbs free energy and thus are more stable than the respective exo TS and product. Hence, it is concluded that '''the kinetically and thermodynamically favoured product is endo product''' based on B3LPY/6-31G(d) calculation. This is quite different from the most famous D-A reaction of cyclopentadiene and maleic anhydride in which the exo product is thermodynamic product and the endo product is kinetic product.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 22:39, 21 February 2018 (UTC) Your energies are correct and your analysis has included an exclennt discussion on thermodynamics.

==== Secondary Orbital Interaction or Sterics <ref name=":2"> J. Clayden, N. Greeves, S. Warren, P. Wothers, ''Organic Chemistry'', Oxford University Press Inc., New York, 2001.</ref>====
{| class="wikitable"
! colspan="5" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 10: Orbital Interaction in Transition Structures and Products and Sterics in Products of Exo and Endo Pathway.
|-
! rowspan="2" style="text-align: center;" | Reaction Pathway
! colspan="2" style="text-align: center;" | Orbital Interaction
! rowspan="2" style="text-align: center;" | Sterics Interaction in Product
! rowspan="2" style="text-align: center;" | Discussion
|-
! style="text-align: center;" | Transition Structure
! style="text-align: center;" | Product
|-
!Exo
|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO, MO41, S</title>
<size>200</size>
<script>frame 16; mo 41; mo cutoff 0.01; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15EXOTS DPT2.LOG</uploadedFileContents>
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</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO, MO41, S</title>
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<uploadedFileContents>XLT15EXOPRODUCT DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Xlt15exost2.PNG|280px|center]]
|rowspan="2"| The HOMOs in TS and product for both pathways is the combination of 1,3-dioxole HOMO and cyclohexadiene LUMO. In exo pathway, there is no secondary orbital interaction observed in HOMO of TS and product as the oxygen is orientated away from the diene component. Contrastingly, a significant secondary orbital interaction is observed between the lone pair in non-bonding p orbital of oxygen and diene component of cyclohexadiene. This in-phase interaction across the space between the orbitals although no real bonds are formed is favourable and stabilizes the endo TS to a lower energy, leading to decrease in the activation barrier and hence it is favoured under kinetic control D-A reaction. This stabilizing secondary orbital interaction is also observed in the endo product and lowering its energy. Hence, the endo pathway is more exothermic and is favoured under thermodynamic control D-A reaction.
<br>
As shown in diagram, there is small amount of steric clash between H6, H8 and H23 of tetrahedral sp<sup>3</sup> carbon atoms in exo product and destabilizes the exo product slightly. The corresponding steric clash is less in endo product because H5 and H6 bonded to a trigonal planar sp<sup>2</sup> (Note: The endo product is oriented in this way to compare with the exo product). Hence, the exo product is less stable and higher in energy than endo product, which is against the outcome of the most famous D-A reaction between cyclopentadiene and maleic anhydride.
|-
!Endo
|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO, MO41, S</title>
<size>200</size>
<script>frame 30; mo 41; mo cutoff 0.01; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15ENDOTS DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<title>HOMO, MO41, S</title>
<size>200</size>
<script>frame 18; mo 41; mo cutoff 0.01; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XLT15ENDOPRODUCT DPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
||[[File:Xlt15endost2.PNG|280px|center]]
|}

'''Simplified Diagram of Primary and Secondary Orbital Interaction'''
{| class="wikitable"
! colspan="3" style="text-align: center;" style="background: #0D4F8B; color: white;" | Table 11: Primary and Secondary Orbital Interaction in Exo and Endo Pathway.
|-
! Reaction Pathway
! Exo
! Endo
|-
! Orbital Interaction
|[[File:Exoooxlt15simplified2orb.PNG|250px|center]]
|[[File:Enodxlt15simplified2orb.PNG|345px|center]]
|}

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 22:41, 21 February 2018 (UTC) These diagrams are excllent and your discussion is again really good. There is not much room for improvement in the section well done.

==== Log File for PM6 Level and IRC Calculation ====
''Cyclohexadiene:'' [[File:XLT15CYCLOHEXADIENE pm62.LOG]]
<br>
''1,3-Dioxole:'' [[File:XLT15DIOXOLE PM62.LOG]]
<br>
''Exo transition structure:'' [[File:XLT15EXOTS PM62.LOG]]
<br>
''Endo transition structure:'' [[File:XLT15ENDOTS PM62.LOG]]
<br>
''Exo product:'' [[File:XLT15EXOPRODUCT PM62.LOG]]
<br>
''Endo product:'' [[File:XLT15ENDOPRODUCT PM62.LOG]]
<br>
''IRC calculation for Exo TS:'' [[File:XLT15EXOTS IRS2.LOG]]
<br>
''IRC calculation for Endo TS:'' [[File:XLT15ENDOTS IRC2.LOG]]
<br>
''Single Point Energy Calculation using Endo Pathway:'' [[File:SINGLEPIENDOREDOXLT15.LOG]]

=== Exercise 1 ===
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Xlt15_TS
=== Exercise 3 ===
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Xlt15_Ex3
=== Further Work ===
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Xlt15_FURTHER

=== References In Exercise 2 ===

Rep:Xlt15 Ex

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Rep:Xizi

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==Introduction==
===Transition State===
Transition state on a 1D potential energy surface is the highest point. But a 2D PES is in 3D-6 dimensions (degrees of freedom). It has lots of local minimas and maximas. Reactants and products corrsponding to local minimas, stationary point, with dV/dq=0 for all q and d2V/dq2>0 for all q, q is the geometric parameter. Intrinsic reaction coordinate, the lowest energy pathway linking two minima, is the path that would be followed by a molecule in going from one minimum to another when it just get enough energy to overcome the activation barrier, pass through the transition state, and reach the other minimum. A transition it is located on the maxima of the reaction path, which corresponding to reaction corrdinate, but minima of all other demesions, defined by dV/dq=0 and d2V/dq2>0 when q= reaction coordinate but d2V/dq2<0 for all other q.
The geometric parameter corresponding to the reaction coordinate is usually a composite of several parameters like bond lengths, angles and dihedrals. This will result in that sometimes trasntion state can be maxima on more than one reaction coordinate, which could be called a hilltop. Because two reaction path will lead to the same transition state, like the first exercise in the extension work. (1)

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 22:58, 8 February 2018 (UTC) q at the transition state is in the basis of the normal modes which is a linear combination of all bond rotations, streches and bends. and hnce looks like a vibration.

===Born–Oppenheimer Approximation===
When talking about the geometric coordiante, it is defined by nuclear coordinates the parameter. Which is to say, a potential energy surface is a plot of the energy of a collection of nuclei and electrons against the geometric coordinates of the nuclei. The reason is in the BO approximation which say nuclei could be assumed stationary with respect to electrons. So Schrcodinger equation for a molecule could be separated into an electronic and a nuclear equation. Two solved and added together to get the total internal energy.
===Computational method===
Computational method finds potential energy by computing Haltonian <Ψ|H|Ψ>=E, direction vectors, r=ax+by+cz=(1,3)Σci|i> in 3D space. <br>
LCAO gives |Ψ>=(1,N)Σci|Φ> in N dimension. Then E=(c1,...)(<Φ1|H|Φ1>...)(c1,...) and |Φ1>=aexp(-ar2) <br>
There are two methods used in this computational lab, semi-empirical paramazation method 6 (PM6) and density functional method Becke-3-LYP (B3lYP) with basis set 631G. <br>
PM6 is under Hartree–Fock formalism, but make many approximations and obtain some parameters from empirical data which allows some electron correlations. <br>
631g is a basis set of basis functions represent the electronic wave function in the Hartree–Fock method or DFT in order to turn the partial differential equations of the model into algebraic equations for computer. Molecular orbitals could then be expressed as linear combinations of the basis functions. <br>
Becke-3-LYP: a hybrid functional uses a mixing scheme of 3 parameters <br>
EXC = 0.2*EX(HF) + 0.8*EX(LSDA) + 0.72*DEX(B88) + 0.81*EC(LYP) + 0.19*EC(VWN) <br>
===Geometry Optimization===
Geometry optimization is the process of starting with an input structure and finding a stationary point on the PES. The stationary point found will normally be the one closest to the input structure. If calculating TS, analytical first and section derivative should as algorithms to be used.
===Frequency calculation and zero point energy===
Calculating vibrational spectrum (normal-mode vibrations) is often needed to check if the geometry is the desired one. The algorithm for this works by calculating an accurate
Hessian (force constant matrix) and diagonalizing it to give a matrix with the direction vectors,ř, of the normal modes, and a diagonal matrix with the force
constants,k, of these modes. Mass weighted force constants give the normal-mode vibration frequencies, v=h/2πc* (k/m)1/2. A transition state has one imaginary vibration, corresponding to motion along the reaction coordinate. K values calculated for transition state is all positive except one on reaction coordinate. <br>
The equations used for computing thermochemical data in Gaussian are equivalent to those in thermodynamics. One of the most important approximations to be aware of throughout this analysis is that all the equations assume non-interacting particles and therefore apply only to an ideal gas. This added up contribution of translation, electronic, vibrational and rotational motion. <br>
Sum of electronic and thermal free energies = ɛ0 + Gcorr

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 23:03, 8 February 2018 (UTC) Nice section, you could have made the equations ab it more clear by using the wiki formatting.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Constructions and Data Presenting ===
====MO and FMO anlysis====
[[File:MO of EXP1.png|thumb|center|500px|Molecular Orbital Diagram for the formation of the transition state.]]

{| class="wikitable" style="border: none; background: none;"
|+|table 1: table of correlations of the visualised MOs and MO diagrams in MO analysis (fig 1)
! colspan="2"| S-cis Butadiene !! colspan="2"| Ethene !! colspan="4"| Transition state
|-
| HOMO || LUMO || HOMO || LUMO || HOMO-1 || HOMO || LUMO || LUMO+1
|-
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|-
! colspan="8"| Correlation with MO diagram (Fig1)
|-
|align=center|[[File:YXZ_DIENE_HOMO.png|100px|]]
|align=center|[[File:YXZ_DIENE_LUMO.png|100px|]]
|align=center|[[File:YXZ_ETHENE_HOMO.png|100px|]]
|align=center|[[File:YXZ_ETHENE_LUMO.png|100px|]]
|align=center|[[File:YXZ_TS16.png|100px|]]
|align=center|[[File:YXZ_TS17.png|100px|]]
|align=center|[[File:YXZ_TS18.png|100px|]]
|align=center|[[File:YXZ_TS19.png|100px|]]
|}

====Bond Changes in Diels Alder Reaction====
{| class="wikitable" style="border: none; background: none;"
|+|table 2: Bond Length Display
|-
! colspan="4"| carbon atom annoted
|-
|align=center|[[File:YXZ_DIENE_CARBON.png|400px|]]
|align=center|[[File:YXZ_ETHENE_CARBON.png|400px|]]
|align=center|[[File:YXZ_TS_CARBON.png|400px|]]
|align=center|[[File:YXZ_PRODUCT_CARBON.png|400px|]]
|-
! colspan="2"| BOND LENGTH AND BOND CHANGE
|-
|align=center|[[File:YXZ_BONDLENGTH.png|400px|]]
|align=CENTER|[[File:YXZ_BONDCHANGE.png|500px|]]
|}

The vibration corresponds to the reaction path at transition state is shown below:

[[File:YXZ_VIBRATIONMOVIE.gif]]

<br>

===Explanation of Graph and Table and Answers to Questions===
====Requirements for Diels Alder Reaction====
Quantitative analysis of MO with energy will be discussed in Exercise 2.
In this [4+2] cycloaddition reaction, interactions were between the conjugated pi system of butadiene and pi bond of alkene to form an aromatic transition state with 6 pi electrons. Two new C-C bonds were generated between two species at ends of each. Molecular diagram for the formation of transition state was tried to be constructed. HOMO and LUMO of alkene and butadiene were used for formation of the transition state for their right symmetry of the bonding orbital.
By inspection of the frontier molecular orbital of the transition state in the Jmol, it could be found that all molecular orbitals constructed in the transition state are formed supra-facially. That is to say two reactants approach each other with their pi orbitals parallel head to head via sigma interactions. P orbitals of both ends of both reactants dis-rotate to form two new sigma bonds symmetrically. This is supported by Woodward–Hoffmann rules which stated that a [4+2] cycloadditon reaction is thermally allowed if ps+qs= 4n+2 or pa+qa=4n+2. 4n+2 is the number of pi electrons used, s and a notations mean that a fragment uses it molecular orbital supra-facially or antasupra-facially. For this reaction, must be 4s+2s which means this reaction only happens if two new bonds formed symmetrically with C2v plane.
Orbital overlap integrals provide a quantitative understanding about how much two orbitals/ wave functions overlap. As shown in the MO diagram, overlap integral between symmetric-symmetric (u-u) and asymmetrical-asymmetric (g-g)are non-zero. Overlap of symmetric-asymmetric interaction (u-g) is 0 because the interaction is asymmetric.
====Bond Change====
All numbers in this section saved the unit Angstrom for convenience.
The table outlines all carbon-carbon bond lengths in reactants, transition states and products and how they change. Explanation about the color in the table: red the double bond, orange the single bond, yellow the partial double bond, pale yellow the partial single bond. It is found all double bond length increases and bond order decreases. The only single bond lengthens and become a double bond. Two newly formed single bonds have decreased the length from infinity to partial single bond in the tradition state. All bond lengths changed dramatically near transition state. Typical bond length: sp3-sp3 1.54, sp2-sp2 1.34, sp2-sp3 (in product)1.50, Van der Waal radius of carbon 1.7. Two carbon with Vdw radius 1.7 have 3.4 interaction length, any length shorter than this means two atoms are in interaction like 2.11 in partial single bond in TS, of course, longer than typical C-C single bond.
====Synchronous Reaction====
At transition state, the vibration is as shown with a negative vibration frequency along the reaction coordinate. This frequency in transition state is an imaginary number but represented as a negative value. It is imaginary number from square root of -1 when solving equations for force constant, k, a negative value.

The vibration with the lowest positive frequency is along their plane. Comparing with the transition state vibration, these vibrations are orthogonal of each other, which can be described as the energy exchange only happens within one normal mode and never exchanges to another. The one with negative frequency describes the formation of two bonds synchronously (hence Diels-Alder reactions are concerted reactions).
===Calculation output by PM6===
optimized reactant: [[Media:YXZ_EXP1_DIENE_MINIPM6.LOG]] [[Media:YXZ EXP1 DIENE MINIPM6.LOG]]

optimized product: [[Media:YXZ_EXP_PRODUCT.LOG]]

otimized/frequency calculated transition state:[[Media:YXZ_EXP1_REAL_TS_WITH_MO.LOG]]

IRC to confirm the transition state: [[Media:YXZ_EXP2_IRC2.LOG]]

== Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole ==
===MO construction and data presentation===
====MO and FMO====
([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) The MO diagrams are well constructed, but there should have been some discussion on the difference between the endo and exo transition state and on the electron demand of this reaction. Additionally, you have not labelled the TS MOs consistently in the diagrams and in the table.)
MO diagrams of transition states were approximated quantitatively by IRC and single point energy calculation which were demonstrated as well.
[[File:MO of endo TS.png|thumb|left|500px|Molecular Orbital Diagram for endo transition state.]]
[[File:MO of exo TS.png|thumb|right|500px| Molecular Orbital Diagram for exo transition state.]]

{| class="wikitable" style="border: none; background: none;"
|+|Visualisation of Frontier Molecular Orbital of Transition State
! colspan="4"| Endo-transition state !! colspan="4"| Exo-transition state
|-
| HOMO-1 || HOMO || LUMO || LUMO+1 || HOMO-1 || HOMO || LUMO || LUMO+1
|-
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|-
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====Single point energy and IRC calculation====
{| class="wikitable" style="border: none; background: none;"
|+|Energy Calculation of Reactants: Display of MO and Energy
! colspan="4"| reactants to form exo-product !! colspan="4"| reactants to form endo-product
|-
| MO 40:HOMO of cyclohexadiene || MO 41:HOMO of dioxole || MO 42:LUMO of cyclohexadiene || MO 43:LUMO of dioxole || MO 40:HOMO of cyclohexadiene || MO 41:HOMO of dioxole || MO 42:LUMO of cyclohexadiene || MO 43:LUMO of dioxole
|-
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|-
! colspan="4"|[[File:YXZ_EXP2_EXO_ENERGYLEVEL.png |400px|thumb|left|ENERGY OF MO 40-43]]!!colspan="4"|[[File:YXZ_EXP2_ENDO_ENERGYLEVEL.png|400px|thumb|left|ENERGY OF MO 40-43]]
|-
|}

IRC Calculation
[[File:YXZ EXP2 ENDO IRCPIC.png|800px|thumb|left|IRC of endo TS]]
[[File:YXZ EXP2 EXO IRCPIC.png|800px|thumb|left|IRC of exo TS]]

====Reaction Thermodynamics====
{| class="wikitable"
|+|thermochemistry data from Gaussian calculation
! !!1,3-cyclohexadiene!! 1,3-dioxole!! endo-transition state!! endo-product!! exo-transition state!! exo-product
|-
!style="text-align: left;"| Sum of electronic and thermal energies at 298k (Hartree/Particle)
| -233.303306 || -267.044172 || -500.306014 || -500.395004 || -500.303864 || -500.394008
|-
|}

====Secondary Orbital Interactions====
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[[File:MO of secon orbital.png|200px|left|orbital interactions in exo and endo approach]]
<br>
<br>
<br>
<br>
<br>
<br>
<br>
<br>
===Explanation of data and answers to questions===
====MO analysis====
By using single point energy calculation, energies of MOs could be compared under same potential energy surfaces. In the Energy Calculation of reactants, diene and dienophile called for simplicty, (displacement of C-C set larger than 2xVdw radius: >3.5 A) it was found that MOs of their frontier orbitals for bonding are arranged low to high: HOMO of diene, HOMO of dienophile, LUMO of diene, LUMO of dienophile with energy gap 3:4:4 approximately. This is beacuse diene has conjugated pi bond thus smaller HOMO-LUMO energy gap(consistent with calcualtion: 3+4<4+4). Electron donating groups (heteroatoms) in dienophile raised energies of MOs thus its HOMO and LUMO all higher than diene and normal dienophile. <br>
For the FMO in transition state, by doing the Electron Calculation and setting C-C displace ment just at 2xVdw radius, reactants MOs began to overlap. Reactants MOs under calculation were not pure one reactant, sometimes orbitals small in size of another reactant and its enrgy is between two pure MOs of reactants. Finally when setting C-C displacement to distance in transition state in exo transition state, the energy gap between HOMO-1, HOMO, LUMO, LUMO+1 are 4:10:3. <br>
It could be concluded that this is an inverse demand DA reaction for EDG in dienophile. When orbitals began to interact, filled orbitals(bonding orbitals in reactants) energy raise and unfilled orbitals(anti bonding orbitals in reactant) energy lowered because electrons began to appear in the overlapped region. When forming the transition state, new bonding orbitals energies reached a maximum and began to lower to a minimum, new anti-bonding orbital energy lowered to a minimum and began to raise to the highest. In endo-transition sate, because of its lower Gibbs energy, energy gaps between frontier orbitals should be larger, in other words, bonding orbitals lower in energy than exo-TS and anti-bonding higher because interaction of this extent is enough for it to form new bonds and damp energy to form a product. Other energies of stabilization are from secondary orbitals in endo-TS

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 23:19, 8 February 2018 (UTC) IT is very good that you have done this analysis well done. You could have presented it much clear rather than just having a single picture. But well done for doing this.

==== Reaction Thermodynamics====

Kinetic product and thermodynamic product are ones have smallest activation energy and largest reaction ernergy. In this case, both are endo product, numerical calculation will be shown soon.
However, Diesl Alder reaction could be reversed by taking in and releasing energy. Also interesting are the fastest formed product and the most stable product. <br>
The fastest formed product. <br>
The Arrhenius Equation states that reaction rate constant is related to Ea by k=Ae−Ea/RT. A is frequency factor constant or also known as pre-exponential factor or Arrhenius factor. It indicates the rate of collision and the fraction of collisions with the proper orientation for the reaction to occur.
There are two ways to achieve the fastest reaction path. One is the lowest free energy of TS and one is higher rate of collision. <br> So thermodynamic control involves raising temperature.
The most stable product, having less chance of going reversible reaction to products or other dissociation reaction. One has lowest free energy of product and lowest rate of collision to go back to product. <br>
<br>
Free energy information could be extracted from Thermochemistry data in output file from Gaussview. and gradient of free energy could be estimate by the vibration frequency in the TS. Gradient from IRC provides gradient in the intrinsic reaction coordinate so not always working right. <br>
Sum energy of reactants: -233.303306+(-267.044172)=-500.347478 Hartree <br>
1 Hartree = 2625.50 KJ/mol <br>
The reaction barrier of a reaction is the energy difference between the transition state and the reactants. <br>
For the endo reaction: -500.347478-(-500.306014)=0.041464 Hartree = 108.8636841 KJ/mol <br>
For the exo reaction: -500.347478-(-500.303864)=0.043614 Hartree = 114.5085066 KJ/mol <br>
The reaction energy of a reaction is the energy difference between the product and the reactants. <br>
For the endo reaction: -500.347478-(-500.395004)=-0.047526 Hartree = -124.7794581 KJ/mol <br>
For the exo reaction: -500.347478-(-500.394008)=-0.046530 Hartree = -122.1644612 KJ/mol <br>
Endo product has both lower activation energy and reaction energy because of secondary orbital interaction. <br>
<br>
Collision rate of both directions to get to TS could be understood by the inverse of frequency of vibration the forming bond in TS. Frequency equals square root of k/m. K is the gradient of potential energy. In TS, k from reactants is negative so frequency calculated by Gauss is imaginary. Thus gradient of free energy around TS could also provides an idea about the collision rate but gradient from IRC provides gradient in the intrinsic reaction corrdinate so not always working right. <br>
In this case exo-TS has negative frequency of -530 which is larger than -523 of endo-TS which means the collision rate or dissociation rate to achieve exo-TS is higher. This is because it has less steric to prevent bond forming. However, the difference is small which may be outweighed by free energy. Negative frequency is from negative k because bonds do not exist in TS, it is an imaginary vibration of bond. <br>
It is concluded that endo product is both kinetic and thermodynamic product, not like reaction of simple dienophile. It is also the fastest and most stable product. <br>

====Secondary orbital interation====

Secondary orbital interactions arises in TS when oxygen in dienophile can donate one pair of electrons in its nonbonding orbital of the correct orrientation into diene's LUMO. This is consistent with bonding interaction of HOMOs of dienophile with diene`s LUMO. This further lowers energy of LUMO of diene and HOMO of TS and helps bonding breaking of HOMO of diene. This is possilble when TS adopt a more steric arrangement in the endo way. However, this stabilization of TS energy outweighs the steric hindrance destabilization in endo TS. <br>

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 23:17, 8 February 2018 (UTC) This is not formatted very well and it makes it extremely difficult to read. Sadly your energies are incorrect. I dont know what you have done because it looks like you have calculated them at B3LYP. Other than that you have come to the correct conclusions and had a balanced argument.

===Calculation output by B3LYP 621g.d.p===
Optimized endo transition state (frequency calculation is done in the same file): [[Media:YXZ_EXP2_ENDO_TS_631_WITH_MO3.log]]

Optimized exo transition state (frequency calculation is done in the same file): [[Media:YXZ_EXP2_EXO_TS631_MOLATEST.log]]

Energy calulation exo transition state (frequency calculation is done in the same file): [[Media:YXZ_EXP2_EXO_ENERGY4WITHMO.log]]

Energy calculation endo transition state (frequency calculation is done in the same file): [[Media:YXZ_EXP2_ENDO_ENERGY2.log]]

IRC endo transition state (frequency calculation is done in the same file): [[Media:YXZ_EXP2_ENDO_IRC4.log.log]]

IRC exo transition state (frequency calculation is done in the same file): [[Media:YXZ_EXP2_EXO_IRC4.log]]

B3LYP optimized dioxole (frequency calculation is done in the same file): [[Media:YXZ_EXP2_DIXOLE631.LOG]]

B3LYP optimized cyclohexadiene (frequency calculation is done in the same file): [[Media:YXZ_EXP2_DIENE631.LOG]]

B3LYP optimized endo product (frequency calculation is done in the same file): [[Media:EXP2_ENDO_MINI631_FREQUENCY_CHECK.LOG]]

B3LYP optimized exo product (frequency calculation is done in the same file): [[Media:EXP2_EXO_621_MINI.LOG]]

==Exercise 3. Diels-Alder vs Cheletropic==
All this exercise use method PM6 <br>
===Data reprensatation===
====Optimisation of TS====
{| class="wikitable"
|+ Optimisation
| style="text-align: centre;" | Optimisation of TS of EXO
| style="text-align: centre;" | Optimisation of TS of Endo
| style="text-align: centre;" | Optimisation of TS of cheletropic
| style="text-align: centre;" | Optimisation of TS of Second Exo
| style="text-align: centre;" | Optimisation of TS of Second Endo
|-
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 10; </script>
<uploadedFileContents>YXZ_EXP3_EXOPM6MINI.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 30; </script>
<uploadedFileContents>YXZ_EXP3_DAENDO_TS1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 16; </script>
<uploadedFileContents>YXZ_EXP3_CHELE_TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 16; </script>
<uploadedFileContents>YXZ EXTENSION TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 16; </script>
<uploadedFileContents>YXZ_EXTENSION_ENDO_TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

(You have used random frames for your Jmols which suggests you've used someone else's code without understanding what it does [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 13:36, 4 February 2018 (UTC))

====IRC analysis====
{| class="wikitable"
|+ Table9. Approach trajectory of 3 reactions and their IRC
| style="text-align: centre;" | Approach trajectory of endo product formation
| style="text-align: centre;" | Endo product formation IRC
|-
|[[File:YXZ_EXP3_ENDO_IRCMOVIE.gif]]
|[[File:YXZ EXP3 ENDO IRCGRAPH.png]]
|-
| style="text-align: centre;" | Approach trajectory of exo product formation
| style="text-align: centre;" | Exo product formation IRC
|-
|[[File:YXZ EXP3 EXO IRCMOVIE.gif]]
|[[File:YXZ EXP3 EXO IRCGRAPH.png]]
|-
| style="text-align: centre;" | Approach trajectory of cheletropic product formation
| style="text-align: centre;" | Cheletropic product IRC
|-
|[[File:YXZ EXP3 CHELE IRCMOVIE.gif]]
|[[File:YXZ EXP3 CHELE IRCGRAPH.png]]
|-
| style="text-align: centre;" | Approach trajectory of exo product using second diene
| style="text-align: centre;" | Second diene exo product IRC
|-

|[[File:YXZ EXTENSION EXO IRCMOVIE1.gif]]
|[[File:YXZ EXTENSION EXO IRCPIC.png]]
|-
| style="text-align: centre;" | Approach trajectory of endo product using second diene
| style="text-align: centre;" | Second diene endo product IRC
|-
|[[File:YXZ EXTENSION ENDO IRCMOVIE.gif]]
|[[File:YXZ EXTENSION ENDO IRCPIC.png]]
|-
|}

====Thermochemistry data====
{| class="wikitable"
|+ Sum of electronic and thermal energies of reactants, TS, and products by Calculation PM6 at 298K
! style="background: #0D4F8B; color: yellow;" | Components
! style="background: #0D4F8B; color: yellow;" | Energy/Hatress
! style="background: #0D4F8B; color: yellow;" | Energy/kJmol<sup>-1</sup>
|-
|SO2
| style="text-align: center;" |-0.091033
| style="text-align: center;" |-239.0071415
|-
|Xylylene
| style="text-align: center;" |0.21851
| style="text-align: center;" |573.698005
|-
|Reactants energy
| style="text-align: center;" |0.127008
| style="text-align: center;" |333.459504
|-
|ExoTS
| style="text-align: center;" |0.138518

| style="text-align: center;" |363.679009

|-
|EndoTS
| style="text-align: center;" |0.136783

| style="text-align: center;" |359.1237665

|-
|Cheletropic TS
| style="text-align: center;" |0.146

| style="text-align: center;" |383.323
|-
|Exo TS from second butadiene
| style="text-align: center;" |0.151054
| style="text-align: center;" |396.592277

|-
|Endo TS from second butadiene
| style="text-align: center;" |0.148031
| style="text-align: center;" |388.6553905
|-
|Exo product
| style="text-align: center;" |0.066032
| style="text-align: center;" |173.367016
|-
|Endo Product
| style="text-align: center;" |0.067003
| style="text-align: center;" |175.9163765
|-
|Cheletropic product
| style="text-align: center;" |0.043585
| style="text-align: center;" |114.4324175
|-
|Exo product from second butadiene
| style="text-align: center;" |0.112672
| style="text-align: center;" |295.820336
|-
|Endo product from second butadiene
| style="text-align: center;" |0.10999
| style="text-align: center;" |288.778745
|-

(You were asked to use the sum of electronic and thermal free energies. Your results are a bit off as a result [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 13:36, 4 February 2018 (UTC))

|}
{| class="wikitable"
|+ Activation energy and reaction energy(KJ/mol) of five reaction paths at 298K
! style="background: #0D4F8B; color: yellow;" |
! style="background: #0D4F8B; color: yellow;" | Exo
! style="background: #0D4F8B; color: yellow;" | Endo
! style="background: #0D4F8B; color: yellow;" | Cheletropic
! style="background: #0D4F8B; color: yellow;" | Exo from second butadiene
! style="background: #0D4F8B; color: yellow;" | Endo from second butadiene
|-
|Activation energy
| style="text-align: center;" |30.219505
| style="text-align: center;" |25.6642625
| style="text-align: center;" |49.863496
| style="text-align: center;" |63.132773
| style="text-align: center;" |55.1958865
|-
|Reaction energy
| style="text-align: center;" |-160.092488
| style="text-align: center;" |-157.5431275
| style="text-align: center;" |-219.0270865
| style="text-align: center;" |-37.639168
| style="text-align: center;" |-44.680759
|}

====Reaction profile====
[[File:Graph2.png|500px|thumb|left|Reaction Energy Profile]]

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===Explanation of data and answers to questions===
====Competing reactions analyse with help of IRC====
This exercise is focused on comparing five possible paths: Diesl alder reaction by exo, DA by endo, cheletropic, DA by endo by second diene, DA by exo by second diene and how they they copetes.
By inspection of energy graph, the kinetic product is DA endo with least activation energy and second lowest product energy. The thermodynamic product is the cheletropic product which has lowest product energy and third lowest activation energy. The preferred reaction path should only be specified under kintic or thermodynamic control. Details in Exercise 2. <br>
IRC can give generalised idea about how stable the product is and how hard TS can be reached because it measures many steps TS needed to go to reactant and product. <br>
====Rationalization of free energy surface by FMO theory====
These five reaction paths can be firstly divided into two classes accroding to different parts of conjugated 4 pi bond. Here for simplicity only considering two dienes inside and outside the ring separately. And also the MO of two dienes treated separately. <br>
1. diene outisde ring <br>
The first three reactions used diene outside the ring or at end. These three reactions are more competing than the other two because diene outside the ring is more accessible for less steric and reorganization of structure and benzene formation from xylylene giving a larger stabilization in energy. <br>
Two DA reactions in this class uses HOMO of SO2 and cheletropic reaction uses LUMO. Then diene uses HOMO and LUMO according to the SO2. For DA endo reaction, the lowest TS of three benefited from secondary orbital interaction of one pi cloud localized on oxygen, the one not in bonding interaction. Cheletropic reaction uses pi cloud delocalzed on soft sufurhas higher TS for larger gap of its LUMO and diene`s HOMO. However it has most stable product because two strong S=O bonds remain there. All reactions form new two bonds but cheletropic reaction destory least bonds and has least steric hindrance. <br>
2. diene in ring <br>
These two DA reactions uses dienes in ring and are far less competitive for the unstable product and high activation energy. <br>
====Dimerization and isomerization====
O-xylylene is highly reactive because it can dimerize to a dimer and isomerize via electrocylic reaction to form a benzocyclobutenes. This two reactions can also benefited from formation of benzene in product like the first three reactions. And, product of these five reactions are all more stable than the two second DA mechanism reactions for the aromatic product instead of the aromatic TS only. So the second two DA reactions can hardly be seen.

(You must back up claims about reactivity with calculations or literature [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 13:36, 4 February 2018 (UTC))

===Calculation output by PM6===
PM6 optimized endo transition state: [[Media:YXZ_EXP3_EXOPM6MINI.LOG]]

PM6 optimized exo transition state: [[Media: YXZ_EXP3_DAENDO_TS1.LOG]]

PM6 optimized cheletropic transition state:[[Media:YXZ_EXP3_CHELE_TS.LOG]]

PM6 optimized Second exo transition state: [[Media:YXZ EXTENSION TS.LOG]]

PM6 optimized Second endo transition state:[[Media:YXZ_EXTENSION_ENDO_TS.LOG]]

PM6 optimized endo transition state IRC: [[Media:YXZ_EXP3_ENDO_IRC.log]]

PM6 optimized exo transition state IRC:[[Media:YXZ_EXP3_EXO_IRC.log]]

PM6 optimized cheletropic transition state IRC: [[Media:YXZ_EXP3_CHELE_IRC.log]]

PM6 optimized second endo transition state IRC: [[Media:YXZ_EXTENSIONENDO_IRC.log]]

PM6 optimized second exo transition state IRC:[[Media:YXZ EXTENSIONEXO IRC.log]]

PM6 optimized endo product: [[Media:YXZ_EXP3_ENDO_PRODUCTLATEST.LOG]]

PM6 optimized exo product:[[Media: YXZ_EXP3_EXOPM6MINI.LOG]]

PM6 optimized cheletropic product:[[Media:YXZ_EXP3_CHELE_MINI2.LOG]]

PM6 optimized second endo product: [[Media:YXZ_EXTENSION_ENDO_PRODUCT.LOG]]

PM6 optimized second exo product:[[Media: YXZ_EXTENSION_PRODUCT1.LOG]]

PM6 optimized SO<sub>2</sub>:[[Media: YXZ_EXP3_SO2MINI2.LOG]]

PM6 optimized xylylene:[[Media: YXZ_EXP3_DIENEMINI2.LOG]]

==Further work==
1. Ring opening of a cyclobutene <br>
Reaction proceeds with conrotation. This reaction involves forming of Möbius aromatic transition state involving one antarafacial four-electron component. Pi bond of double bond overlap with sp3 C-C anti bond followed by the rotation of sigma bond of double and rotation of sp3 bonding of C-C. To form a monocyclic array of molecular orbitals which has odd number of out-of-phase overlaps, Mobius aromatic TS, forming HOMO of butadiene, rotation is restricted to conrotation. Diene formed then have different phases of p orbitals on terminal. <br>
2. Photochemical reaction <br>
Reaction proceeds with disrotation. When such a diene absorbs light of right frequency, an electron will be exiced from HOMO to LUMO. Now this diene has new HOMO and LUMO of ψ3 and ψ4 of ground state. In this case, To form Möbius aromatic transition state, sigma bond will choose disrotation to form the new bond p orbitals of different phase at ends. And now the transition state is better understood by a conical intersection in 2D PES accompanied by a change of symmetry. <br>
3.Opening of the cyclopropyl cation <br>
Reaction proceeds with dis-rotation. It forms a Huckel aromatic TS of 2 electrons and form a product of allylic cation which has same phases on terminal p orbital and stabilized by conical form. HOMO of OTs- helps stabilizing form of TS and leaves just after that. <br>
4. Opening of the cyclopropyl cation with hetero-atoms <br>
Conrotation needed to form Möbius aromatic transition state and gives a product with different phases at ends and has HOMO of C2v and pi cloud localized. <br>

==Conclusion==
In this work, different calculations are studied. Optimization to find the right structure. Frequency to check if it is correct. IRC to visualize free energy surface on intrinsic coordinate. Energy to compare different molecules on one potential surface. <br>
Important information can be extracted by using Gaussview like visualization of MO to study the FMO theory and symmetry of reaction. Thermochemistry data like Gibbs free enrgy, zero point energy and enthalpy an so on. Visualization of vibration modes can give the reaction coordinate. <br>
For pericyclic reactions, symmetry rules, Hammond's postulate, bond forming and rotation have been studied and there is still something to be explored like other thermal pericyclic reactions and light-induced pericyclic reactions.

==Reference==
1. file://icnas4.cc.ic.ac.uk/xy3513/9789048138609-c2.pdf

Rep:Xiaojie102820

2025-09-01T09:50:18Z

Move page script: Move page script moved page Xiaojie102820 to Rep:Xiaojie102820: Move to report namespace

==Module 3==
In this page, I will characterise transition structures on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition reactions.

==The Cope Rearrangement Tutorial==
In this section,I will use the Cope rearrangement of 1,5-hexadiene as an example.And my objectives are to locate the low-energy minima and transition structures on the C6H10 potential energy surface, to determine the preferred reaction mechanism.

[[File:Reaction_core_rearrangement.PNG|thumb|[3,3]-sigmatropic shift rearrangement]]

===Molecule1,5-hexadiene===
==== HF/3-21G Optimised (Anti 1) 1,5-hexadiene====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>HEX_ANTI_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Aniti opt xiaojie.PNG]]

'''Point group Analysis'''

[[File:Anti pointgroup xiaojie.PNG]]

'''General Information'''

{| class="wikitable" border="1"
| Energy || -231.69260235a.u.
|-
| Point Group || C2
|}
By comparing to the Appendix 1, this structure should be Anti-1.

'''Output File'''

<pre> Item Value Threshold Converged?
Maximum Force 0.000027 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.000825 0.001800 YES
RMS Displacement 0.000314 0.001200 YES
Predicted change in Energy=-2.682036D-08
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/7/75/HEX_ANTI_OPT.LOG The log. file of HF/3-21G optimised (anti-1) 1,5-hexadiene]

==== HF/3-21G Optimised (Gauche1) 1,5-hexadiene====
Now I will optimised 1,5-hexadiene molecule with a "gauche" linkage for the central four C atoms. With the previous knowledage from the conformational analysis course(2nd year), the gauche form should be lower in the energy. Unlike small straight molecules which prefer app(anti-periplanar) due to large orbital overlap, the large molecules shift balance to the gauche conformation(given highly proportion of gauche form in large molecules). This incursion of gauche forms is due to the longer distances favouring folding of the chain back upon itself, and hence setting up van der Waals attractions.

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>HEX_GAU1_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Hex_gauche1_xiaojie.PNG]]

'''Point Group Analysis'''

[[File:Gauche1_pointgroup_xiaojie.PNG]]

'''General Information'''

{| class="wikitable" border="1"
| Energy || -231.68771617a.u.
|-
| Point Group ||C2 with a twofold symmetry axis
|}
By checking the Appendix 1, this structure is Gauche-1.

Now we can compare the final energy of this two structures. The 'gauche' linkage is 0.00488618a.u.(about3.066kcal/mol) higher in energy than the 'anti' linkage. This result is different from my prediction. I think there is overestimation of the enetgy for gauche form when we using the 3-21G basis set.

'''Output File'''

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000466 0.001800 YES
RMS Displacement 0.000111 0.001200 YES
Predicted change in Energy=-1.790991D-09
Optimization completed.
-- Stationary point found.</pre>
'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/8/87/HEX_GAU1_OPT.LOG Log.file of HF/3-21G optimised (Gauche-1) structure]

==== Lowest energy conformation-(gauche3)====
Prediction: Based on the results I got from above, I think the anti/gauche/anti conformer might be the lowest energy conformation of 1,5-hexadiene. In order to check the result, I need to optimised the guess structure.

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>HEX_GAUCHE3.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Gauche3_summary_Xiaojie.PNG]]

'''Point Group Analysis'''

[[File:Pointgropu_gauch3_Xiaojie.PNG]]

{| class="wikitable" border="1"
| Energy || -231.69266120a.u.
|-
| Point Group || C1
|}

'''The energy I get is -231.69266120a.u. which is lower than the above two structures(check!).'''

'''Comparing the structure in the Appendix table, the conformer which I predict is Gauche-3.'''

'''Output File'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000009 0.000300 YES
Maximum Displacement 0.001317 0.001800 YES
RMS Displacement 0.000491 0.001200 YES
Predicted change in Energy=-3.610321D-08
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/d/d6/HEX_GAUCHE.LOG LOG. file of HF/3-21G optimised (gauche-3)structure]

==== HF/3-21G Optimised (Anti2) 1,5-hexadiene====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>HEX_ANTI2_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Tble'''

[[File:Hex_anti2_summary_xiaojie.PNG]]

'''Point Group Analysis'''

[[File:Anti2_pointgroup_xiaojie.PNG]]

'''The energy I got is -231.69253528a.u. which is very colse to the value in the table(-231.69254a.u.).'''

'''The point group is Ci(identical to the one in table).'''

'''Output File'''

<pre> Item Value Threshold Converged?
Maximum Force 0.000060 0.000450 YES
RMS Force 0.000010 0.000300 YES
Maximum Displacement 0.000476 0.001800 YES
RMS Displacement 0.000171 0.001200 YES
Predicted change in Energy=-2.037252D-08
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/f/f3/HEX_ANTI2_OPT.LOG LOG. file of the HF/3-21G optimised (Anti-2) conformer]

==== B3LYP/6-31G* Optimised (Anti2) 1,5-hexadiene====
'''This time, I will optimise the anti 2 structure at a higher level of theory.'''

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>HEX_ANTI2_REOPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Hex_anti_reopt_xiaojie.PNG]]

'''Point Group'''

[[File:Anti2 repot pg xiaojie.PNG]]

'''The energy decreased to -234.61170280a.u. And the point group is Ci.'''

'''Output file'''

<pre> Item Value Threshold Converged?
Maximum Force 0.000015 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000219 0.001800 YES
RMS Displacement 0.000079 0.001200 YES
Predicted change in Energy=-1.588886D-08
Optimization completed.
-- Stationary point found.</pre>

'''geometry Comparison'''

{| class="wikitable" border="1"
|+
! No. !! HF/3-21G !! B3LYP/6-31*
|-
| c=c Bond length || 1.31613Å || 1.33352Å
|-
| C13-C10-C7 || 126.806<sup>0</sup>C || 125.287<sup>0</sup>C
|}

The bond length of the carbon-carbon double bond is lengthened, while, the bond length of the carbon-carbon single bond is shortened on the B3LYP/6-31G* calculation.

The B3LYP/6-31G* method has a better basis set than the HF/3-21G method,gives a better aggrement with the experiment value.

'''File link'''

[https://wiki.ch.ic.ac.uk/wiki/images/9/9a/HEX_ANTI2_REOPT.LOG LOG. file of B3LYP/6-31G* Optimised (Anti2)conformer]

==== Frequency Analysis of B3LYP/6-31G* Optimised (Anti2) 1,5-hexadiene====

'''The final energies given represent the energy of th molecule on the bare potential energy surface. In order to compare these energies with experimental value, the frequency calculation is required.'''

'''vibrational frequencies'''

[[File:Anti_631.PNG]]

'''It is clear that there is no imaginary frequencies, only real ones which confirms it is a minimum(the frequency analysis is essentially the second derivative of the potential energy surface, if the frequencies are all positive then we have a minimum).'''

'''IR spectrum'''

[[File:IR_anti2__xiaojie.PNG|400px]]

'''File'''

<pre>Sum of electronic and zero-point Energies= -234.469212
Sum of electronic and thermal Energies= -234.461856
Sum of electronic and thermal Enthalpies= -234.460912
Sum of electronic and thermal Free Energies= -234.500821</pre>

'''File link'''

[https://wiki.ch.ic.ac.uk/wiki/images/6/6e/-HEX_ANTI2_631G_FRE-_view_file.txt Frequency file of anti2 conformer]

=="Chair" and "Boat" Transition Structures==
In this section I will set up a transition structure optimization.

=== allyl fragment ===

''' HF/3-21G ''' optimised allyl fragement

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>ALLYLCHAIR_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Allyl_summary.PNG]]

'''Output file'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000141 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277267D-08
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/b/b9/ALLYLCHAIR_OPT.LOG Log. file of optimised allyl]

===Chair transition state-guess structure===
Now we are going to optimise the transition state. Actually,the transition states correspond to saddle points with one negative second derivative on the potential energy surface. So in order to locate the transition state we can find a point with one negative second derivative(which is one imaginary frequency in this experiment). The reason why transition state optimisation is more difficult than that of minimum is that a successful search should start off in a region where the reaction coordinate already has a negative curvature.(which I did at second year lab: search for a transition state should start near the transition state!)

====Optimisation of chair-ts in 3-21G====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>GUESS_TS_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Guess_ts_opt.PNG]]

'''Output file'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000024 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001401 0.001800 YES
RMS Displacement 0.000201 0.001200 YES
Predicted change in Energy=-1.246508D-07
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/7/75/GUESS_TS_OPT.LOG Log. file of 3-21G optimised chair ts]

'''Frequency Analysis'''

'''Vibrational Frequency'''

[[File:Vf_321g_chair.PNG]]

The frequency calculation gives an(only one) imaginary frequency of magnitude 817.98 cm-1. It means that the guess transition state is reasonable(reason explained above).

'''Animation'''

{| class="wikitable" border="1"
|+
! No. !! Form of the vibration !! Discription !! frequency !! intensity !! symmetry D3h point group
|-
| 1 || [[File:Animation_321g_chair.PNG |150px]] || The two pair terminal carbons are moving in concerned direction, but in opposite direction( bond making and bond breaking occur synchronously). || -817.98 || 5.8659 || C<sub></sub>2h
|}

'''File'''

<pre>Sum of electronic and zero-point Energies= -231.466702
Sum of electronic and thermal Energies= -231.461342
Sum of electronic and thermal Enthalpies= -231.460398
Sum of electronic and thermal Free Energies= -231.495208</pre>

====Optimisation of chair-ts by using the frozen coordinate method====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>GUESS_TS_FROZEN_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''CHK file'''

[[File:CHK.PNG]]

The optimized structure looks a lot like the transition I got by using 3-21G. However, bond forming/breaking distances are fixed to 2.2 Å.

'''Summary table'''

[[File:Fro_chair_xiaojie.PNG]]

The final energy here is -231.61499866a.u.which is greater than the one in the HF/3-21G.

'''Output file'''

<pre> Item Value Threshold Converged?
Maximum Force 0.000011 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000512 0.001800 YES
RMS Displacement 0.000088 0.001200 YES
Predicted change in Energy=-3.015680D-08
Optimization completed.
-- Stationary point found.</pre>
'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/8/86/GUESS_TS_FROZEN_OPT.LOG File of frozen coordinate method optimised chair ts]

====Reoptimisation after Redundant Coord Editor====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>GUESS_TS_FROZEN_reOPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:2ndfro_chair_xiaojie.PNG]]

'''Output File'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000051 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001240 0.001800 YES
RMS Displacement 0.000244 0.001200 YES
Predicted change in Energy=-6.076100D-07
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/9/9b/GUESS_TS_FROZEN_OPT2ND.LOG File of reoptimised chair ts after RCE]

'''chk'''

[[File:Bong_length_chair_frozen_2nd.PNG]]

'''bond forming/bond breaking bond length is 2.01923 Å'''

====Comparison Table====

{| class="wikitable" border="1"
|+
! hf/3-21g !! After RCE
|-
| 2.02049Å || 2.01923Å
|}

===Boat transition state===
We are going to optimise the boat transition state using QST2 method. It requires that the reactants and products are numbered in the same way.

====Reactant/product-QST2 calculation====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Reactant_boat_optfrequ.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Reactant_freopt.PNG]]

'''Output file'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000094 0.000450 YES
RMS Force 0.000016 0.000300 YES
Maximum Displacement 0.001626 0.001800 YES
RMS Displacement 0.000525 0.001200 YES
Predicted change in Energy=-4.528634D-08
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/c/cd/REACTANT_BOAT_OPTFRE.LOG File of QST2 optimised reactant]

'''Vibrational Analysis'''

[[File:Boat__vf.PNG]]

There is only one imaginary frequency of magnitude of 839.84cm-1(confirms it is a transition state).

'''Animation'''

'''IR Spectrum'''

[[File:Reactant.PNG]]

'''Output File'''
<pre>Sum of electronic and zero-point Energies= -231.450926
Sum of electronic and thermal Energies= -231.445297
Sum of electronic and thermal Enthalpies= -231.444352
Sum of electronic and thermal Free Energies= -231.479772</pre>

===IRC Analysis of Chair Transition state===
Since I have known the optimised chair and boat structure, here is a problem raised: Which conformers of 1,5-hexadiene do they connect? And I find it is hard to predict which conformer the reaction paths from the transitions structures will lead to. So in this section, I will run the IRC method which allows me to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface.

[[File:Chair_irc.gif]]

[[File:Irc_1_summary.PNG]]

'''irc pathway'''

[[File:IRC_pathway.PNG|800px]]

There are 27 intermediate geometries. According to the IRC pathway diagram ,we start from the transition state(red circled point on the graph) and the energy goes down towards the minimum with the deepest gradient.However, it hasn't reached the minimum geometry and it stopped! So In order to get the minimum, I can take the last point on the IRC and run a normal minimization(optimisation).

'''Run minimization'''

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>Optafterirc.mol</uploadedFileContents> </jmolApplet></jmol>
'''Summary Table'''

[[File:IRC_MINIUM_SUMMARY.PNG|300px]]

'''Point Group'''

[[File:Ater_ircpoint_group.PNG]]

After the optimisation, the final gradient decreased to 0.00000256. In addition,the final energy is -231.69167a.u. and the point group is C2. Now we can look up the Appendix table, the structure we get is Gauche-2!(the conformer that chair and boat connect).

===Optimisation of Chair and Boat by using B3LYP/6-31G*===

Finally I will calculate the activation energies for our reaction via both transition structures.

====B3LYP/6-31G* optimised Chair transition state====

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>CHAIR TS GUESS XIAOJIE 631g.mol</uploadedFileContents> </jmolApplet></jmol>

'''Summary Table'''

[[File:CHAIR TS GUESS XIAOJIE 631g summary.PNG]]

'''Output File'''

<pre> Item Value Threshold Converged?
Maximum Force 0.000024 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000829 0.001800 YES
RMS Displacement 0.000115 0.001200 YES
Predicted change in Energy=-7.523575D-08
Optimization completed.
-- Stationary point found.
</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/CHAIR_TS_GUESS_XIAOJIE.LOG B3LYP/6-31G* optimised Chair ts log. file B3LYP/6-31G optimised chair ts]

====B3LYP/6-31G* optimised Boat transition state====

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>PRODUCT_OPTFRE_631G.mol</uploadedFileContents> </jmolApplet></jmol>

'''Summary Table'''

[[File:Boat_summary.PNG‎]]

'''Output File'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000304 0.000450 YES
RMS Force 0.000074 0.000300 YES
Maximum Displacement 0.001756 0.001800 YES
RMS Displacement 0.000780 0.001200 YES
Predicted change in Energy=-1.694834D-06
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/8/8f/PRODUCT_OPTFRE_631G.LOG Log. file of B3LYP/6-31G optimised Boat ts]

===Frequency analysis of B3LYP/6-31G*optimised transition state===
====Chair TS====

'''Output File'''

<pre>Sum of electronic and zero-point Energies= -234.414931
Sum of electronic and thermal Energies= -234.409010
Sum of electronic and thermal Enthalpies= -234.408066
Sum of electronic and thermal Free Energies= -234.443817</pre>

'''IR spectrum'''

[[File:IR_631gfre.PNG]]

'''Vibration frequency'''

[[File:Chair__631g_xiaojie.PNG]]

It is clear that there is only one imaginary frequency(-565.72) which again confirms it is a transition state.

'''Vibrational animation'''

{| class="wikitable" border="1"
|+
! No. !! Form of the vibration !! Discription !! frequency !! intensity !! symmetry D3h point group
|-
| 1 || [[File:Chair_ts_631g.PNG |150px]] || The two pair terminal carbons are moving in concerned direction, but in opposite direction( bond making and bond breaking occur synchronously). || -565.72 || 0.0803 || C<sub></sub>2h
|}

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/f/f0/CHAIR_TS_GUESS_XIAOJIE_631G_FRE.LOG Frequency output file of 6-31G* optimised chair ts]

====Boat TS====

'''Output File'''

<pre>Sum of electronic and zero-point Energies= -234.402318
Sum of electronic and thermal Energies= -234.395985
Sum of electronic and thermal Enthalpies= -234.395041
Sum of electronic and thermal Free Energies= -234.431729</pre>

'''Vibrational Frequency'''

[[File:Boat_631g_vf.PNG]]

'''IR Spectrum'''

[[File:Boat_IR631g.PNG|500px]]

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/7/79/PRODUCT_OPTFRE_631G_FRE.LOG Frequency output file of optimised boat ts]

====Result Table====
<br />

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932243
| align="center" | -231.466702
| align="center" | -231.461342
| align="center" | -234.55698283
| align="center" | -234.414931
| align="center" | -234.409010
|-
| '''Boat TS'''
| align="center" | -231.60280245
| align="center" | -231.450926
| align="center" | -231.445297
| align="center" | -234.54309090
| align="center" | -234.402318
| align="center" | -234.395985
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

<br /> *1 hartree = 627.509 kcal/mol <br /><br />

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.70
| align="center" | 44.69
| align="center" | 34.06
| align="center" | 33.16
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.98
| align="center" | 41.33
| align="center" | 44.7 ± 2.0
|}

<br />
Ater comparing the activation energy of two basis set, it is clear that although the geometries are reasonably similar, energy differences are markedly different! Moreover, activation energies calculated at BYLYP/6-31G* are in good aggrement with the experiment value. Finally, the activation energy of the chair ts is about10kJ/mol lower than that of the boat transition state(small barrier!). So the reaction will more likely to proceed via a chair transition state.

==The Diels Alder Cycloaddition==
In this exercise, I will characterise transition structures using the method I have learned in the tutorial part. The Diels–Alder reaction is an organic chemical reaction (specifically, a cycloaddition) between a conjugated diene and a substituted alkene,to form a substituted cyclohexene system. The HOMO-LUMO overlap can be used to perdict whether the reaction occur or not.

=== cis butadiene===

====AM1 optimised ciabutadiene====

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>AM1_OPT_CISBUTDIENE.mol</uploadedFileContents> </jmolApplet></jmol>

'''Summary Table'''

[[File:AM1_OPT_cisbutadiene_summary.PNG]]

[[File:Pg.PNG]]
{| class="wikitable" border="1"
| Energy || 0.04879719a.u.
|-
| Point group || C2v
|}

'''Output File'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000360 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691168D-09
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/6/69/AM1_OPT_CISBUTDIENE.LOG File Link of AM1 optimised cisbutadiene]

====Molecule orbital====

{| class="wikitable" border="1"
| HOMO || LUMO
|-
| [[File:HOMO cis butadiene xiaojiet.PNG|550px]] || [[File:CIS_BUTA._LUMO.PNG]]
|-
| '''It is clear that the HOMO orbital of cic-butadiene is asymmetric(a) with repect to the plane.''' || '''While the LUMO orbital is symmetric(s) with respect to the plane.'''
|}

===Transition State geometry for the prototype reaction===

[[File:REACTION_222.PNG|300px]]

====Optimisation of the ts====

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>Ts opt_xiaojie.mol</uploadedFileContents> </jmolApplet></jmol>

'''Summary Table'''

[[File:Ts_opt_summary.PNG|300px]]

'''Point group'''

[[File:Cs.PNG]]

'''General information'''

{| class="wikitable" border="1"
| Energy || 0.11165468a.u.
|-
| Point Group || Cs
|}

'''Output file'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000065 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.001005 0.001800 YES
RMS Displacement 0.000240 0.001200 YES
Predicted change in Energy=-2.489584D-08
Optimization completed.
-- Stationary point found.</pre>

====Geometry Information====
[[File:Ts_alder.PNG|250px|thumb|A Gaussview image(chk.file) of an optimised transition state]]
{| class="wikitable" border="1"
|+ Bond length Comparison
! partly formed sigma C-C bond !! SP3 C-C bond !! SP2 C-C bond !! Van Der Waal radius of C atom
|-
| 2.1195 Å || 1.54000 Å<ref name="1.54000" /> || 1.35520 Å<ref name="1.35520" /> || 1.70 Å<ref name="1.70" />
|}

The new C-C sigma bond length is 2.1195 Å. It is shorter than the twice of Van Der Waal redius(1.70*2=3.40 Å) which indicates there do has bonds forming between the terminal carbons. However, It is longer than the SP3 and SP2 C-C bond which means that the new sigma bond in the transition state is just partly formed.

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/c/c9/TS_opt.LOG File of optimised transition state]

'''Vibrational frequency'''

[[File:TS_VF.PNG]]

There is only one imaginary frequency of magnitude 955.70cm<sup>-1</sup>, again indicates it is a transition state.

{| class="wikitable" border="1"
|+
! Animation for imaginary frequency !! Animation for lowest positive frequency
|-
| [[File:IMAGINARY_FREQUENCY.gif]] || [[File:LOWEST FREQUENCY.gif]]
|-
| Two pairs of terminal c-c bonds are moving in and out at the same time. In addition,the two forming bonds had the same bond length of 2.1195. So both bonds are formed to exactly the same extent in the transition statewhich means it is a synchronous concerned reaction. || For the lowest vibrational frequency, the two molecules are just vibrating ,not forming bonds(not interacting) actually.'''
|}

====Molecule Orbital====

{| class="wikitable" border="1"
| HOMO || LUMO
|-
| [[File:HOMO_TS_XIAOJIE.PNG|500px]] || [[File:LUMO TS xiaojie.PNG|500px]]

|-
| The HOMO is asymmetric(a) with respect to the plane. || While the LUMO is symmetric(s).
|}

'''Conclusion'''

The symmetry of the HOMO at transition state can be found from the MO diagram above. It is clear that the HOMO is asymmetric.The LUMO of ethylene and the HOMO of the butadiene are both a. Thurs it is the HOMO-LUMO pairs of orbital that interact. The reaction is allowed because the HOMO of butadiene can interact with the LUMO of the ethylene;in other words, the HOMO(butadiene)and LUMO(ethylene) have the same symmetry.

{| class="wikitable" border="1"
|+
! ethylene !! butadiene
|-
| [[File:LUMO_ETHYLENE.PNG|200px|thumb|LUMO of ethylene]] || [[File:HOMO cis butadiene xiaojiet.PNG|200px|thumb|HOMO of cis butadiene]]
|-
| asymmetric || asymmetric
|}

'''HOMO-LUMO interact-chemdraw(HOMO Of cis butadiene and LUMO of ethylene)'''

[[File:MO_xiaojie.bmp]]

'''Approach with the ethylene under diene.'''

=== The regioselectivity of the Diels Alder Reaction===
====Exo transition state====

=====Optimisation of exo-ts=====

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>EXO 3.mol</uploadedFileContents> </jmolApplet></jmol>

'''Summary Table'''

[[File:Exo__3opt_summary.PNG]]

'''Point group'''

[[File:Cs2.PNG]]

'''General Information'''

{| class="wikitable" border="1"
| Energy || -0..5041985a.u.
|-
| point group || Cs
|}
'''Output File'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000002 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000075 0.001800 YES
RMS Displacement 0.000015 0.001200 YES
Predicted change in Energy=-1.459715D-10
Optimization completed.
-- Stationary point found.</pre>

'''Vibrational frequency'''

[[File:Exo_vf.PNG]]

There is only one imaginary frequency of magnitude of 812.19cm<sup>-1</sup>, again confirms it is a transition state.

'''File link'''

[https://wiki.ch.ic.ac.uk/wiki/images/3/32/EXO_3.LOG File of optimised exo-transition state]

=====Molecule orbital Analysis=====

{| class="wikitable" border="1"
| HOMO || LUMO
|-
| [[File:HOMO_EXO.PNG|450px]] || [[File:LUMO_EXO.PNG|450px]]

|-
| The HOMO is asymmetric w.r.t the plane || The LUMO is asymmetric w.r.t the plane as well
|}

====ENDO transition state====

=====Optimisation of endo ts=====

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>ENDO_TS_OPT.mol</uploadedFileContents> </jmolApplet></jmol>

'''Summary Table'''

[[File:Summary_endo.PNG]]

'''Point group'''

[[File:Cs3.PNG]]

'''General Information'''

{| class="wikitable" border="1"
| point group || Cs
|-
| Energy || -0.05150480 a.u.
|}

'''Output File'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000010 0.000450 YES
RMS Force 0.000002 0.000300 YES
Maximum Displacement 0.000194 0.001800 YES
RMS Displacement 0.000054 0.001200 YES
Predicted change in Energy=-2.030039D-09
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/c/c9/ENDO_TS_OPT.LOG File link of optimised endo transition state]

=====Molecule Orbital Analysis=====

{| class="wikitable" border="1"
| HOMO || LUMO
|-
| [[File:HOMO_ENDO.PNG|300px]] || [[File:LUMO_ENDO.PNG|300px]]

|-
| The HOMO is asymmetric with respect to the plane || The LUMO is asymmetric with respect to the plane as well
|}

====Discussion====

{| class="wikitable" border="1"
| ts || C-C space || partly formed C-C bond || other C-C bond
|-
| exo || 2.94510Å ||2.17040Å ||1.48819Å

|-
| endo || 2.89215Å ||2.16232Å ||1.48923Å

|}

'''Structure comparison'''

For the exo transition state, the large maleic anhydride group is located near to the -CH2-CH2- fragment which leads to high steric hindrance.
so the exo-ts is higher in energy(more strained). However, this steric hindrance does not exists for the endo form as the maleic anhydride group is located far away from the beidging carbon fragments. Consequently, the endo adduct will be the major product under kinetic control due to the lower activation barrier.

'''HOMO-LUMO overlap'''

[[File:Momomo.bmp]]

The reaction is allowed due to perfect interaction of the HOMO-LUMO.

'''secondary orbital overlap effect-MO overlap'''

The secondary orbital overlap effect is defined as the positive overlap of the nonactive frontier molecule orbital of a pericyclic reaction.

[[File:Secondary.PNG|450px]]

[[File:Sc.PNG|450px]]

According to the diagram above, it is clear that the P orbitals on the carbonyl carbon atom can overlap with the p orbitals of carbon on diene which leads to the stabilisation of the endo-transition state. While in exo-form, there is no such interaction.

==Further discussion==
Ans: The conditions of the solvent used is neglect when we run the calculation.

== References ==

<references> <ref name="1.70">^ a b c Bondi, A. (1964). "Van der Waals Volumes and Radii". J. Phys. Chem. 68 (3): 441–51.[http://pubs.acs.org/doi/abs/10.1021/j100785a001 ] </ref>
<ref name="1.35520">Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. .</ref>
<ref name="1.54000">Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. </ref>
</references>

Rep:Xiaojie102818

2025-09-01T09:50:12Z

Move page script: Move page script moved page Xiaojie102818 to Rep:Xiaojie102818: Move to report namespace

==Module 3==
In this page, I will characterise transition structures on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition reactions.

==The Cope Rearrangement Tutorial==

[[File:Reaction_core_rearrangement.PNG|thumb|[3,3]-sigmatropic shift rearrangement]]

===Molecule1,5-hexadiene===
==== HF/3-21G Optimised (Anti 1) 1,5-hexadiene====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>HEX_ANTI_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Aniti opt xiaojie.PNG]]

'''Point group Analysis'''

[[File:Anti pointgroup xiaojie.PNG]]

'''General Information'''

{| class="wikitable" border="1"
| Energy || -231.69260235a.u.
|-
| Point Group || C2
|}
By comparing to the Appendix 1, this structure should be Anti-1.

'''Output File'''

<pre> Item Value Threshold Converged?
Maximum Force 0.000027 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.000825 0.001800 YES
RMS Displacement 0.000314 0.001200 YES
Predicted change in Energy=-2.682036D-08
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/7/75/HEX_ANTI_OPT.LOG The log. file of HF/3-21G optimised (anti-1) 1,5-hexadiene]

==== HF/3-21G Optimised (Gauche1) 1,5-hexadiene====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>HEX_GAU1_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Hex_gauche1_xiaojie.PNG]]

'''Point Group Analysis'''

[[File:Gauche1_pointgroup_xiaojie.PNG]]

'''General Information'''

{| class="wikitable" border="1"
| Energy || -231.68771617a.u.
|-
| Point Group ||C2 with a twofold symmetry axis
|}
By checking the Appendix 1, this structure is Gauche-1.

Now we can compare the final energy of this two structures. The 'gauche' linkage is 0.00488618a.u.(about3.066kcal/mol) higher in energy than the 'anti' linkage. This result is different from my prediction. I think there is overestimation of the enetgy for gauche form when we using the 3-21G basis set.

'''Output File'''

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000466 0.001800 YES
RMS Displacement 0.000111 0.001200 YES
Predicted change in Energy=-1.790991D-09
Optimization completed.
-- Stationary point found.</pre>
'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/8/87/HEX_GAU1_OPT.LOG Log.file of HF/3-21G optimised (Gauche-1) structure]

==== Lowest energy conformation-(gauche3)====
Prediction: Based on the results I got from above, I think the anti/gauche/anti conformer might be the lowest energy conformation of 1,5-hexadiene. In order to check the result, I need to optimised the guess structure.

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>HEX_GAUCHE3.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Gauche3_summary_Xiaojie.PNG]]

'''Point Group Analysis'''

[[File:Pointgropu_gauch3_Xiaojie.PNG]]

{| class="wikitable" border="1"
| Energy || -231.69266120a.u.
|-
| Point Group || C1
|}

'''The energy I get is -231.69266120a.u. which is lower than the above two structures(check!).'''

'''Comparing the structure in the Appendix table, the conformer which I predict is Gauche-3.'''

'''Output File'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000009 0.000300 YES
Maximum Displacement 0.001317 0.001800 YES
RMS Displacement 0.000491 0.001200 YES
Predicted change in Energy=-3.610321D-08
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/d/d6/HEX_GAUCHE.LOG LOG. file of HF/3-21G optimised (gauche-3)structure]

==== HF/3-21G Optimised (Anti2) 1,5-hexadiene====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>HEX_ANTI2_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Tble'''

[[File:Hex_anti2_summary_xiaojie.PNG]]

'''Point Group Analysis'''

[[File:Anti2_pointgroup_xiaojie.PNG]]

'''The energy I got is -231.69253528a.u. which is very colse to the value in the table(-231.69254a.u.).'''

'''The point group is Ci(identical to the one in table).'''

'''Output File'''

<pre> Item Value Threshold Converged?
Maximum Force 0.000060 0.000450 YES
RMS Force 0.000010 0.000300 YES
Maximum Displacement 0.000476 0.001800 YES
RMS Displacement 0.000171 0.001200 YES
Predicted change in Energy=-2.037252D-08
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/f/f3/HEX_ANTI2_OPT.LOG LOG. file of the HF/3-21G optimised (Anti-2) conformer]

==== B3LYP/6-31G* Optimised (Anti2) 1,5-hexadiene====
'''This time, I will optimise the anti 2 structure at a higher level of theory.'''

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>HEX_ANTI2_REOPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Hex_anti_reopt_xiaojie.PNG]]

'''Point Group'''

[[File:Anti2 repot pg xiaojie.PNG]]

'''The energy decreased to -234.61170280a.u. And the point group is Ci.'''

'''Output file'''

<pre> Item Value Threshold Converged?
Maximum Force 0.000015 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000219 0.001800 YES
RMS Displacement 0.000079 0.001200 YES
Predicted change in Energy=-1.588886D-08
Optimization completed.
-- Stationary point found.</pre>

'''geometry Comparison'''

{| class="wikitable" border="1"
|+
! No. !! HF/3-21G !! B3LYP/6-31*
|-
| c=c Bond length || 1.31613Å || 1.33352Å
|-
| C13-C10-C7 || 126.806<sup>0</sup>C || 125.287<sup>0</sup>C
|}

The bond length of the carbon-carbon double bond is lengthened, while, the bond length of the carbon-carbon single bond is shortened on the B3LYP/6-31G* calculation.

The B3LYP/6-31G* method has a better basis set than the HF/3-21G method,gives a better aggrement with the experiment value.

'''File link'''

[https://wiki.ch.ic.ac.uk/wiki/images/9/9a/HEX_ANTI2_REOPT.LOG LOG. file of B3LYP/6-31G* Optimised (Anti2)conformer]

==== Frequency Analysis of B3LYP/6-31G* Optimised (Anti2) 1,5-hexadiene====

'''The final energies given represent the energy of th molecule on the bare potential energy surface. In order to compare these energies with experimental value, the frequency calculation is required.'''

'''vibrational frequencies'''

[[File:Anti_631.PNG]]

'''It is clear that there is no imaginary frequencies, only real ones which confirms it is a minimum(the frequency analysis is essentially the second derivative of the potential energy surface, if the frequencies are all positive then we have a minimum).'''

'''IR spectrum'''

[[File:IR_anti2__xiaojie.PNG|400px]]

'''File'''

<pre>Sum of electronic and zero-point Energies= -234.469212
Sum of electronic and thermal Energies= -234.461856
Sum of electronic and thermal Enthalpies= -234.460912
Sum of electronic and thermal Free Energies= -234.500821</pre>

'''File link'''

[https://wiki.ch.ic.ac.uk/wiki/images/6/6e/-HEX_ANTI2_631G_FRE-_view_file.txt Frequency file of anti2 conformer]

=="Chair" and "Boat" Transition Structures==
In this section I will set up a transition structure optimization.

=== allyl fragment ===

''' HF/3-21G ''' optimised allyl fragement

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>ALLYLCHAIR_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Allyl_summary.PNG]]

'''Output file'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000141 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277267D-08
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/b/b9/ALLYLCHAIR_OPT.LOG Log. file of optimised allyl]

===Chair transition state-guess structure===

====Optimisation of chair-ts in 3-21G====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>GUESS_TS_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Guess_ts_opt.PNG]]

'''Output file'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000024 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001401 0.001800 YES
RMS Displacement 0.000201 0.001200 YES
Predicted change in Energy=-1.246508D-07
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/7/75/GUESS_TS_OPT.LOG Log. file of 3-21G optimised chair ts]

'''Frequency Analysis'''

'''Vibrational Frequency'''

[[File:Vf_321g_chair.PNG]]

The frequency calculation gives an(only one) imaginary frequency of magnitude 817.98 cm-1. It means that the guess transition state is reasonable(reason explained above).

'''Animation'''

{| class="wikitable" border="1"
|+
! No. !! Form of the vibration !! Discription !! frequency !! intensity !! symmetry D3h point group
|-
| 1 || [[File:Animation_321g_chair.PNG |150px]] || The two pair terminal carbons are moving in concerned direction, but in opposite direction( bond making and bond breaking occur synchronously). || -817.98 || 5.8659 || C<sub></sub>2h
|}

'''File'''

<pre>Sum of electronic and zero-point Energies= -231.466702
Sum of electronic and thermal Energies= -231.461342
Sum of electronic and thermal Enthalpies= -231.460398
Sum of electronic and thermal Free Energies= -231.495208</pre>

====Optimisation of chair-ts by using the frozen coordinate method====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>GUESS_TS_FROZEN_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''CHK file'''

[[File:CHK.PNG]]

The optimized structure looks a lot like the transition I got by using 3-21G. However, bond forming/breaking distances are fixed to 2.2 Å.

'''Summary table'''

[[File:Fro_chair_xiaojie.PNG]]

The final energy here is -231.61499866a.u.which is greater than the one in the HF/3-21G.

'''Output file'''

<pre> Item Value Threshold Converged?
Maximum Force 0.000011 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000512 0.001800 YES
RMS Displacement 0.000088 0.001200 YES
Predicted change in Energy=-3.015680D-08
Optimization completed.
-- Stationary point found.</pre>
'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/8/86/GUESS_TS_FROZEN_OPT.LOG File of frozen coordinate method optimised chair ts]

====Reoptimisation after Redundant Coord Editor====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>GUESS_TS_FROZEN_reOPT.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:2ndfro_chair_xiaojie.PNG]]

'''Output File'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000051 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001240 0.001800 YES
RMS Displacement 0.000244 0.001200 YES
Predicted change in Energy=-6.076100D-07
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/9/9b/GUESS_TS_FROZEN_OPT2ND.LOG File of reoptimised chair ts after RCE]

'''chk'''

[[File:Bong_length_chair_frozen_2nd.PNG]]

'''bond forming/bond breaking bond length is 2.01923 Å'''

====Comparison Table====

{| class="wikitable" border="1"
|+
! hf/3-21g !! After RCE
|-
| 2.02049 || 2.01923
|}

===Boat transition state===
We are going to optimise the boat transition state using QST2 method. It requires that the reactants and products are numbered in the same way.

====Reactant/product-QST2 calculation====

<jmol><jmolApplet>
<title>test molecule</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Reactant_boat_optfrequ.mol</uploadedFileContents>
</jmolApplet></jmol>

'''Summary Table'''

[[File:Reactant_freopt.PNG]]

'''Output file'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000094 0.000450 YES
RMS Force 0.000016 0.000300 YES
Maximum Displacement 0.001626 0.001800 YES
RMS Displacement 0.000525 0.001200 YES
Predicted change in Energy=-4.528634D-08
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/c/cd/REACTANT_BOAT_OPTFRE.LOG File of QST2 optimised reactant]

'''Vibrational Analysis'''

[[File:Boat__vf.PNG]]

There is only one imaginary frequency of magnitude of 839.84cm-1(confirms it is a transition state).

'''Animation'''

'''IR Spectrum'''

[[File:Reactant.PNG]]

'''Output File'''
<pre>Sum of electronic and zero-point Energies= -231.450926
Sum of electronic and thermal Energies= -231.445297
Sum of electronic and thermal Enthalpies= -231.444352
Sum of electronic and thermal Free Energies= -231.479772</pre>

===IRC Analysis of Chair Transition state===

[[File:Chair_irc.gif]]

[[File:Irc_1_summary.PNG]]

'''irc pathway'''

[[File:IRC_pathway.PNG|800px]]

There are 27 intermediate geometries. According to the IRC pathway diagram ,we start from the transition state(red circled point on the graph) and the energy goes down towards the minimum with the deepest gradient.However, it hasn't reached the minimum geometry and it stopped! So In order to get the minimum, I can take the last point on the IRC and run a normal minimization(optimisation).

'''Run minimization'''

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>Optafterirc.mol</uploadedFileContents> </jmolApplet></jmol>
'''Summary Table'''

[[File:IRC_MINIUM_SUMMARY.PNG|300px]]

'''Point Group'''

[[File:Ater_ircpoint_group.PNG]]

===Optimisation of Chair and Boat by using B3LYP/6-31G*===

Finally I will calculate the activation energies for our reaction via both transition structures.

====B3LYP/6-31G* optimised Chair transition state====

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>CHAIR TS GUESS XIAOJIE 631g.mol</uploadedFileContents> </jmolApplet></jmol>

'''Summary Table'''

[[File:CHAIR TS GUESS XIAOJIE 631g summary.PNG]]

'''Output File'''

<pre> Item Value Threshold Converged?
Maximum Force 0.000024 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000829 0.001800 YES
RMS Displacement 0.000115 0.001200 YES
Predicted change in Energy=-7.523575D-08
Optimization completed.
-- Stationary point found.
</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/CHAIR_TS_GUESS_XIAOJIE.LOG B3LYP/6-31G* optimised Chair ts log. file B3LYP/6-31G optimised chair ts]

====B3LYP/6-31G* optimised Boat transition state====

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>PRODUCT_OPTFRE_631G.mol</uploadedFileContents> </jmolApplet></jmol>

'''Summary Table'''

[[File:Boat_summary.PNG‎]]

'''Output File'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000304 0.000450 YES
RMS Force 0.000074 0.000300 YES
Maximum Displacement 0.001756 0.001800 YES
RMS Displacement 0.000780 0.001200 YES
Predicted change in Energy=-1.694834D-06
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/8/8f/PRODUCT_OPTFRE_631G.LOG Log. file of B3LYP/6-31G optimised Boat ts]

===Frequency analysis of B3LYP/6-31G*optimised transition state===
====Chair TS====

'''Output File'''

<pre>Sum of electronic and zero-point Energies= -234.414931
Sum of electronic and thermal Energies= -234.409010
Sum of electronic and thermal Enthalpies= -234.408066
Sum of electronic and thermal Free Energies= -234.443817</pre>

'''IR spectrum'''

[[File:IR_631gfre.PNG]]

'''Vibration frequency'''

[[File:Chair__631g_xiaojie.PNG]]

It is clear that there is only one imaginary frequency(-565.72) which again confirms it is a transition state.

'''Vibrational animation'''

{| class="wikitable" border="1"
|+
! No. !! Form of the vibration !! Discription !! frequency !! intensity !! symmetry D3h point group
|-
| 1 || [[File:Chair_ts_631g.PNG |150px]] || The two pair terminal carbons are moving in concerned direction, but in opposite direction( bond making and bond breaking occur synchronously). || -565.72 || 0.0803 || C<sub></sub>2h
|}

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/f/f0/CHAIR_TS_GUESS_XIAOJIE_631G_FRE.LOG Frequency output file of 6-31G* optimised chair ts]

====Boat TS====

'''Output File'''

<pre>Sum of electronic and zero-point Energies= -234.402318
Sum of electronic and thermal Energies= -234.395985
Sum of electronic and thermal Enthalpies= -234.395041
Sum of electronic and thermal Free Energies= -234.431729</pre>

'''Vibrational Frequency'''

[[File:Boat_631g_vf.PNG]]

'''IR Spectrum'''

[[File:Boat_IR631g.PNG|500px]]

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/7/79/PRODUCT_OPTFRE_631G_FRE.LOG Frequency output file of optimised boat ts]

====Result Table====
<br />

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932243
| align="center" | -231.466702
| align="center" | -231.461342
| align="center" | -234.55698283
| align="center" | -234.414931
| align="center" | -234.409010
|-
| '''Boat TS'''
| align="center" | -231.60280245
| align="center" | -231.450926
| align="center" | -231.445297
| align="center" | -234.54309090
| align="center" | -234.402318
| align="center" | -234.395985
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

<br /> *1 hartree = 627.509 kcal/mol <br /><br />

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.70
| align="center" | 44.69
| align="center" | 34.06
| align="center" | 33.16
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.98
| align="center" | 41.33
| align="center" | 44.7 ± 2.0
|}

<br />
Ater comparing the activation energy of two basis set, it is clear that although the geometries are reasonably similar, energy differences are markedly different! Moreover, activation energies calculated at BYLYP/6-31G* are in good aggrement with the experiment value. Finally, the activation energy of the chair ts is about10kJ/mol lower than that of the boat transition state(small barrier!). So the reaction will more likely to proceed via a chair transition state.

==The Diels Alder Cycloaddition==

=== cis butadiene===

====AM1 optimised ciabutadiene====

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>AM1_OPT_CISBUTDIENE.mol</uploadedFileContents> </jmolApplet></jmol>

'''Summary Table'''

[[File:AM1_OPT_cisbutadiene_summary.PNG]]

[[File:Pg.PNG]]
{| class="wikitable" border="1"
| Energy || 0.04879719a.u.
|-
| Point group || C2v
|}

'''Output File'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000360 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691168D-09
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/6/69/AM1_OPT_CISBUTDIENE.LOG File Link of AM1 optimised cisbutadiene]

====Molecule orbital====

{| class="wikitable" border="1"
| HOMO || LUMO
|-
| [[File:HOMO cis butadiene xiaojiet.PNG|550px]] || [[File:CIS_BUTA._LUMO.PNG]]
|-
| '''It is clear that the HOMO orbital of cic-butadiene is asymmetric(a) with repect to the plane.''' || '''While the LUMO orbital is symmetric(s) with respect to the plane.'''
|}

===Transition State geometry for the prototype reaction===

[[File:REACTION_222.PNG|300px]]

====Optimisation of the ts====

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>Ts opt_xiaojie.mol</uploadedFileContents> </jmolApplet></jmol>

'''Summary Table'''

[[File:Ts_opt_summary.PNG|300px]]

'''Point group'''

[[File:Cs.PNG]]

'''General information'''

{| class="wikitable" border="1"
| Energy || 0.11165468a.u.
|-
| Point Group || Cs
|}

'''Output file'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000065 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.001005 0.001800 YES
RMS Displacement 0.000240 0.001200 YES
Predicted change in Energy=-2.489584D-08
Optimization completed.
-- Stationary point found.</pre>

====Geometry Information====
[[File:Ts_alder.PNG|250px|thumb|A Gaussview image(chk.file) of an optimised transition state]]
{| class="wikitable" border="1"
|+ Bond length Comparison
! partly formed sigma C-C bond !! SP3 C-C bond !! SP2 C-C bond !! Van Der Waal radius of C atom
|-
| 2.1195 Å || 1.54000 Å<ref name="1.54000" /> || 1.35520 Å<ref name="1.35520" /> || 1.70 Å<ref name="1.70" />
|}

The new C-C sigma bond length is 2.1195 Å. It is shorter than the twice of Van Der Waal redius(1.70*2=3.40 Å) which indicates there do has bonds forming between the terminal carbons. However, It is longer than the SP3 and SP2 C-C bond which means that the new sigma bond in the transition state is just partly formed.

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/c/c9/TS_opt.LOG File of optimised transition state]

'''Vibrational frequency'''

[[File:TS_VF.PNG]]

There is only one imaginary frequency of magnitude 955.70cm<sup>-1</sup>, again indicates it is a transition state.

{| class="wikitable" border="1"
|+
! Animation for imaginary frequency !! Animation for lowest positive frequency
|-
| [[File:IMAGINARY_FREQUENCY.gif]] || [[File:LOWEST FREQUENCY.gif]]
|-
| Two pairs of terminal c-c bonds are moving in and out at the same time. In addition,the two forming bonds had the same bond length of 2.1195. So both bonds are formed to exactly the same extent in the transition statewhich means it is a synchronous concerned reaction. || For the lowest vibrational frequency, the two molecules are just vibrating ,not forming bonds(not interacting) actually.'''
|}

====Molecule Orbital====

{| class="wikitable" border="1"
| HOMO || LUMO
|-
| [[File:HOMO_TS_XIAOJIE.PNG|500px]] || [[File:LUMO TS xiaojie.PNG|500px]]

|-
| The HOMO is asymmetric(a) with respect to the plane. || While the LUMO is symmetric(s).
|}

'''Conclusion'''

The symmetry of the HOMO at transition state can be found from the MO diagram above. It is clear that the HOMO is asymmetric.The LUMO of ethylene and the HOMO of the butadiene are both a. Thurs it is the HOMO-LUMO pairs of orbital that interact. The reaction is allowed because the HOMO of butadiene can interact with the LUMO of the ethylene;in other words, the HOMO(butadiene)and LUMO(ethylene) have the same symmetry.

{| class="wikitable" border="1"
|+
! ethylene !! butadiene
|-
| [[File:LUMO_ETHYLENE.PNG|200px|thumb|LUMO of ethylene]] || [[File:HOMO cis butadiene xiaojiet.PNG|200px|thumb|HOMO of cis butadiene]]
|-
| asymmetric || asymmetric
|}

=== The regioselectivity of the Diels Alder Reaction===
====Exo transition state====

=====Optimisation of exo-ts=====

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>EXO 3.mol</uploadedFileContents> </jmolApplet></jmol>

'''Summary Table'''

[[File:Exo__3opt_summary.PNG]]

'''General Information'''

'''Output File'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000002 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000075 0.001800 YES
RMS Displacement 0.000015 0.001200 YES
Predicted change in Energy=-1.459715D-10
Optimization completed.
-- Stationary point found.</pre>

'''Vibrational frequency'''

[[File:Exo_vf.PNG]]

There is only one imaginary frequency of magnitude of 812.19cm<sup>-1</sup>, again confirms it is a transition state.

'''File link'''

[https://wiki.ch.ic.ac.uk/wiki/images/3/32/EXO_3.LOG File of optimised exo-transition state]

=====Molecule orbital Analysis=====

{| class="wikitable" border="1"
| HOMO || LUMO
|-
| [[File:HOMO_EXO.PNG|450px]] || [[File:LUMO_EXO.PNG|450px]]

|-
| The HOMO is asymmetric w.r.t the plane || The LUMO is asymmetric w.r.t the plane as well
|}

====ENDO transition state====

=====Optimisation of endo ts=====

<jmol><jmolApplet> <title>test molecule</title> <color>black</color> <size>200</size> <uploadedFileContents>ENDO_TS_OPT.mol</uploadedFileContents> </jmolApplet></jmol>

'''Summary Table'''

[[File:Summary_endo.PNG]]

'''Output File'''

<pre>Item Value Threshold Converged?
Maximum Force 0.000010 0.000450 YES
RMS Force 0.000002 0.000300 YES
Maximum Displacement 0.000194 0.001800 YES
RMS Displacement 0.000054 0.001200 YES
Predicted change in Energy=-2.030039D-09
Optimization completed.
-- Stationary point found.</pre>

'''File Link'''

[https://wiki.ch.ic.ac.uk/wiki/images/c/c9/ENDO_TS_OPT.LOG File link of optimised endo transition state]

=====Molecule Orbital Analysis=====

''''HOMO'''

[[File:HOMO_ENDO.PNG|300px]]

'''The HOMO is asymmetric with respect to the plane.'''

'''LUMO'''

[[File:LUMO_ENDO.PNG|300px]]

'''The LUMO is asymmetric with respect to the plane as well.'''

====Conclusion====

== References ==

<references> <ref name="1.70">^ a b c Bondi, A. (1964). "Van der Waals Volumes and Radii". J. Phys. Chem. 68 (3): 441–51.[http://pubs.acs.org/doi/abs/10.1021/j100785a001 ] </ref>
<ref name="1.35520">Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. .</ref>
<ref name="1.54000">Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. </ref>
</references>

Rep:XP715TS3

2025-09-01T09:50:12Z

Move page script: Move page script moved page XP715TS3 to Rep:XP715TS3: Move to report namespace

='''Transition states and reactivity'''=

== Introduction ==
Along a reaction pathway, a transition state is the maximum of energy, while the minima are reactants and products.

== Exercise 1: Reaction of Butadiene with Ethylene ==

Rep:XP715TS

2025-09-01T09:50:09Z

Move page script: Move page script moved page XP715TS to Rep:XP715TS: Move to report namespace

='''Transition states and reactivity'''=

==Introduction==

For a potential energy curve with only one variable, the curve is considered to be the energy profile. If more than one geometric coordinates associate with potential energy, the three-dimensional surface is called '''potential energy surface'''. For a nonlinear molecule, there is 3N-6 independent geometric variables.
The transition state is the maximum point on the surface, connecting two minima, reactant and product. These stationary points (transition state, reactant and product) are defined as zero first derivative (<math>\frac{\partial E}{\partial R}</math>= 0), showing that the gradient is zero.

{|class=wikitable
|-
|<math>\frac{\partial E}{\partial R}=-F</math>
|-
|Equation.1, first derivative
|}

The gradient is related to force acts on the atoms, and the negative sign indicates the force is in the direction of lowering potential energy.
In order to distinguish them, curvatures (frequencies) at these points are determined by second derivative.

{|class=wikitable
|-
|<math>\dfrac{\partial ^2 E}{\partial R^2} =k</math>
|-
|Equation.2, second derivative
|}
[[File:XP715 Eqn ferq.PNG|thumb|center|300px|Equation.3, frequency calculation <ref>J. McDouall, Computational Quantum Chemistry: Molecular Structure and Properties in Silico, Royal Society of Chemistry, Cambridge, 2013, ch.1, pp.1-62</ref>]]

Second derivatives are hold in the Hessian matrix, and by diagonalizing the Hessian matrix, force constant k can be determined as well as the frequency by Eqn.3.
The saddle point (transition state) has negative curvature (<math>\dfrac{\partial ^2 E}{\partial R^2} < 0</math>), while the minima have positive curvature (<math>\dfrac{\partial ^2 E}{\partial R^2} > 0</math>).

Energy, electronic structure and properties of molecules can be determined by solving Schrödinger’s equation. The computational method is used to solve the equation by deciding to use different level of theory (Hamiltonian operator) and basis set (mathematical description of wavefunction). In this page, two optimisation methods were adopted, PM6 and B3LYP/6-31G(d). For PM6, it is a semi-empirical method, which is based on Hartree-Fock theory. <ref>C.A. Coulson, B.O’Leary, R.B. Mallion, Hückel theory for organic chemists, Academic Press, London, New York, 1978</ref> It solves the many-electron equation by expanding the coefficient of linear combination of atomic orbitals (LCAO) and simplifies with Born–Oppenheimer approximation. The full HF calculation is too expensive, therefore PM6 is simplified by neglecting two-electron part of Hamiltonian, and further simplification can be applied for π-electron system by Hückel method. It is overall a quick but not reliable method. B3LYP/6-31G(d) is based on density functional theory (DFT), which associates with HF theory and an additional term, exchange-correlation energy.<ref>K. Kim and K. D. Jordan, J. Phys. Chem., 1994, 98, 10089–10094.</ref> B3LYP is the choice of exchange-correlation energy and 6-31G is the basis set. DFT is sufficient accurate but it is an expensive method.

In this page, Gaussian, the computational method, is used to interpret the mechanisms of four pericyclic reactions. This technique is able to identify whether the bond formation is synchronous or asynchronous, formation of kinetic or thermodynamic product and whether the proposed reaction pathway is favourable.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 23:43, 22 March 2018 (UTC) You have clearly read beyond the script here well done. Some equations would have been good. When you diagonalise the hessian your are changing your coordinate basis into the noraml modes. which are then linear combinations of the degrees of freedom.

==Exercise 1: Reaction of Butadiene with Ethylene==

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) Overall you have done a good job. However you have used your B3LYP optimisation of ethene instead of the PM6 one, which led you to the wrong MO energies.)

[[File:XP715_Scheme_EX1.PNG|thumb|center|500px|Scheme.1, reaction scheme of butadiene and ethylene with annotated bond length]]

The first reaction is the classical [4+2] cycloadditon (Scheme.1), which is also called Diels-Alder reaction. This reaction was investigated by guessing the transition state first and finding the optimised product. Both reactants and TS were optimised at PM6 level, and a frequency calculation and Intrinsic Reaction Coordinate (IRC) were analysed to ensure that a correct TS was obtained. Finally, the product was optimised at PM6 level.

===Optimisation and Calculation===

{| class="wikitable"
| colspan="4" style="text-align: center; background: #3b5998; color: white" | '''Optimised structures of reactants, TS and product at PM6 level'''
|-
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38 ; </script>
<uploadedFileContents>XP715_DIENE_MINPM6_NEW.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14; </script>
<uploadedFileContents>XP715_ETHENE_MINPM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6; </script>
<uploadedFileContents>XP715 2MOL TSPM6 JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 42; </script>
<uploadedFileContents>XP715_PROD_MINPM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Butadiene
| Ethene
| TS
| Product
|}

{| class="wikitable"
|[[File:XP715 TS PM6 freq.PNG|thumb|left|700px|Figure.1, Frequency calculation of TS]]
|[[File:XP715 EX1 IRC.png|thumb|center|500px|Figure.2, IRC (total energy and RMS gradient) of TS]]
|}
Fig. 1 shows that only one frequency is negative, indicating the transition state. IRC is the minimum energy pathway on the potential energy surface, starting from the first-derivative stationary point, TS, and calculating in both direction until reaching two minima, reactants and products. Fig. 2 illustrates the total energy and RMS gradient along IRC, and the gradients of reactants, products and TS are all zero, confirming a successful and asymmetric IRC was performed.

=== MO Analysis===
{| class="wikitable"
| colspan="4" style="text-align: center; background: #3b5998; color: white" | '''HOMO and LUMO of reactants and HOMO/-1, LUMO/+1 of transition states'''
|-
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715_DIENE_MINPM6_JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715_DIENE_MINPM6_JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 10; mo 8; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715_ETHENE_MINPM6_JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 10; mo 9; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715_ETHENE_MINPM6_JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Butadiene (HOMO)
| Butadiene (LUMO)
| Ethene (HOMO)
| Ethene (LUMO)
|-
! <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>XP715_2MOL_TSPM6_JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <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>XP715_2MOL_TSPM6_JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <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>XP715_2MOL_TSPM6_JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <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>XP715_2MOL_TSPM6_JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| TS (HOMO-1)
| TS (HOMO)
| TS (LUMO)
| TS (LUMO+1)
|}

By visualising MO of reactants and TS, part of the MO was constructed in Fig.3 (MO). The calculated orbital energies at PM6 level are labelled in grey, however, due to the low optimisation level, these energies are only a rough guide to the MO diagram. Although the energies are not accurate, it still provides an evidence of mixing. The dotted orbitals are the MOs without mixing, while the solid-line orbitals are the ones with predicted mixing.
[[File:XP715_MO.PNG|thumb|center|800px|Figure.3, MO diagram]]
[[File:XP715 MO sym.PNG|thumb|center|500px|Figure.4, symmetry label for HOMO and LUMO of diene]]
[[File:XP715 Eqn orbital.PNG|thumb|center|500px|Equation.4, orbital overlap integral]]

The symmetry of each orbital can be identified with its symmetry axis or plane. The HOMO of butadiene is antisymmetric as it has C2 symmetry, while the LUMO is symmetric as it contains σ(v) symmetry plane. (Fig.4) Only the orbitals with same symmetry could combine to form new MOs. The orbital overlap integral is represented by Eqn.4<ref>P. W. Atkins and J. De Paula, Physical Chemistry, 2009.</ref>, and it is zero when the overall interaction is antisymmetric. The symmetric-antisymmetric interaction is '''antisymmetric''', integrating to '''zero''' (forbidden reaction). The symmetric-symmetric and antisymmetric-antisymmetric interaction are '''symmetric''', resulting to '''non-zero''' integral (allowed reaction).
The Woodward-Hoffmann rules states that in a thermally allowed reaction, the total number of (4q+2)<sub>s</sub> and (4r)<sub>a</sub> components must be odd, where the suffix s stands for suprafacial (forming bond on same face), and a for antarafacial (forming bond on opposite face).
By applying Woodward-Hoffmann rules, this reaction is proved to be '''thermally allowed'''.
<pre>(4q+2)s+(4r)a
=1+0
=1
=thermally allowed reaction
</pre>

===Bond Length Analysis===
{| class="wikitable"
|+ Table 1. Bond length of reactants, transition states and product
''(refer to Scheme.1)''
|-
| style="background: #c8d9ec; color: black;" | '''Structure'''
| style="background: #c8d9ec; color: black;" | '''sp<sup>3</sup>-sp<sup>3</sup> C-C /Å'''
| style="background: #c8d9ec; color: black;" | '''sp<sup>3</sup>-sp<sup>2</sup> C-C /Å'''
| style="background: #c8d9ec; color: black;" | '''sp<sup>2</sup>-sp<sup>2</sup> C=C /Å'''
| style="background: #c8d9ec; color: black;" | '''sp<sup>2</sup>-sp<sup>2</sup> C-C /Å'''
|-
|Butadiene
|n/a
|n/a
|1.34, 1.34
|1.47
|-
| Ethene
| n/a
| n/a
| 1.33
| n/a
|-
| TS
| 2.11, 2.11 (forming single bond)
| n/a
| 1.38, 1.38 (partially double bond);
1.38 (partially double bond)
|1.41 (partially double bond)
|-
|Product
|1.54, 1.54
|1.50, 1.50
|1.34
|n/a
|-
|Typical value
| 1.54
|1.50
|1.34
|1.47
|}

{| class="wikitable"
|+ Table 2. Van der Waals radius of Carbon
| colspan="4" style="text-align: center; background: #3b5998; color: white" |''' Van der Waals radius of Carbon'''
|-
|One carbon atom /Å
|1.70
|Two carbon atoms /Å
|3.40
|}
Comparing the bond length of reactants and TS, the reactants show typical bond length of sp<sup>2</sup>-sp<sup>2</sup> C=C, sp<sup>3</sup>-sp<sup>2</sup> C-C and sp<sup>2</sup>-sp<sup>2</sup> C=C. In the TS, C10-C12 is shortened due the change from sp<sup>3</sup>-sp<sup>3</sup> single bond to sp<sup>2</sup>-sp<sup>2</sup> double bond, while C7-C10 and C12-14 show an elongation because sp<sup>2</sup>-sp<sup>2</sup> double bonds are changed into sp<sup>2</sup>-sp<sup>3</sup> single bonds. C1-C4 becomes longer as it converts from sp<sup>2</sup>-sp<sup>2</sup> double bond to sp<sup>3</sup>-sp<sup>3</sup> single bond. The distance between C4 and C7/ C1 and C14 is both 2.11 Å, which is shorter than sum of Van der Waals radius of two carbon atoms (Table.2), indicating that two molecules are approaching to each other and forming a partial bond. The product shows typical sp<sup>3</sup>-sp<sup>3</sup> C-C, sp<sup>2</sup>-sp<sup>3</sup> C-C and sp<sup>2</sup>-sp<sup>2</sup> C=C bond length.

{| class="wikitable"
| [[File:XP715 EX1 Bond length.png|thumb|center|1000px|a]]
| [[File:XP715 EX1 Prod label.jpg|thumb|center|300px|b]]
|-
| colspan="2" | Figure.5, a) The change of bond length with respect to reaction coordinate. b) Numbering of atoms of the product
|}
Fig.5 illustrated the change of bond length along the reaction coordinate by analysing IRCs of each bond. C1-C4 (purple) and C4-C7 (black) starts from 3.40 Å, where no bond is formed, and then reaching TS at 2.11 Å. The product is formed when the bond length is at 1.54 Å. The rest of the bonds corresponds to the explanation in the previous section.

===Vibration===
{| class="wikitable"
!<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 7; vibration 2</script>
<uploadedFileContents>XP715_2MOL_TSPM6_JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
![[File:XP715 EX1 TS formbond.gif|thumb|center|500px|Figure.6, Video of forming and breaking bonds]]
|}
By visualising the vibration of TS and motion picture of Fig.6, the formation of two bonds are '''synchronous'''.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

[[File:XP715 Scheme ex2.JPG|thumb|center|500px|Scheme.2, reaction schemes of Cyclohexadiene and 1,3-Dioxole to form endo and exo products.]]

This Diels-Alder reaction is stereospecific, leading to endo and exo adducts. The more favourable reaction pathway is examined by the calculating activation energy and free energy. Reactants, TS and products were optimised first with PM6 following by using B3LYP/6-31G(d).

===Optimisation and Calculation===
{| class="wikitable"
| colspan="3" style="text-align: center; background: #3b5998; color: white" | '''Optimisation of reactants, TS and products'''
|-
|Cyclohexadiene
|1,3-Dioxole
|Endo TS
|-
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6</script>
<uploadedFileContents>XP715 DIENE 631G JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6</script>
<uploadedFileContents>XP715 DIOXOLE 631G JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 8</script>
<uploadedFileContents>XP715 ENDO TSPM6 631G 3 JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|[[File:XP715 Diene freq.PNG|thumb|center|500px]]
|[[File:XP715 Dioxole freq.PNG|thumb|center|500px]]
|[[File:XP715 ENDO TS 631G freq.PNG|thumb|center|500px]]
|-
|Exo TS
|Endo Product
|Exo Product
|-
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 102</script>
<uploadedFileContents>XP715_EXO_TS_jmol.log</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14</script>
<uploadedFileContents>XP715 Endo prod 631G.log</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 8</script>
<uploadedFileContents>XP715 EXO 631G jmol.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|[[File:XP715 EXO TS 631G freq.PNG|thumb|center|500px]]
|[[File:XP715_ENDO_prod_freq.PNG|thumb|center|500px]]
|[[File:XP715_EXO_prod_freq.PNG|thumb|center|500px]]
|}
There is no imaginary frequency for all the reactants and products, and there is only one negative frequency for each TS, confirming that all of them were well optimised.

===MO Analysis===
{| class="wikitable"
| colspan="4" style="text-align: center; background: #3b5998; color: white" | '''HOMO and LUMO of reactants and HOMO/-1, LUMO/+1 of ENDO/EXO transition states'''
|-
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6; mo 22; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715 DIENE 631G JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6; mo 23; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715 DIENE 631G JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <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>XP715 DIOXOLE 631G JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6; mo 20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715 DIOXOLE 631G JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Cyclohexadiene (HOMO)
| Cyclohexadiene (LUMO)
| 1,3-Dioxole (HOMO)
| 1,3-Dioxole (LUMO)
|-
! <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>XP715 ENDO TSPM6 631G 3 JMOL.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>XP715 ENDO TSPM6 631G 3 JMOL.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>XP715 ENDO TSPM6 631G 3 JMOL.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>XP715 ENDO TSPM6 631G 3 JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| ENDO TS (HOMO-1)
| ENDO TS (HOMO)
| ENDO TS (LUMO)
| ENDO TS (LUMO+1)
|-
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 102; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715_EXO_TS_jmol.log</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 102; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715_EXO_TS_jmol.log</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 102; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715_EXO_TS_jmol.log</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 102; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715_EXO_TS_jmol.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| EXO TS (HOMO-1)
| EXO TS (HOMO)
| EXO TS (LUMO)
| EXO TS (LUMO+1)
|}

{| class="wikitable"
| colspan="2" style="text-align: center; background: #3b5998; color: white" | '''MOs of reactants and transition states'''
|-
|[[File:XP715_Endo_MO.png|thumb|center|700px|Figure.7, MO diagram of Endo TS]]
|[[File:XP715Exo MO.png|thumb|center|700px|Figure.8, MO diagram of Exo TS]]
|}
By visualising MOs of reactants and TSs, the MO diagrams of endo and exo TSs were constructed. The calculated orbital energies (in grey) gave a rough guide to the energy difference between orbitals. The orbital energies of HOMO/HOMO-1 and LUMO/LUMO+1 of two TSs are similar, but the actual shapes shown in jmol are different (in later section). The HOMO of endo TS is slightly more stabilised than that of exo TS.

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) Good MO diagrams. Could have discussed more the differences between exo and endo conformations in terms of relative MO energies.)

====Inverse Demand DA Reaction====
For a standard DA reaction, the electron rich component is diene and the electron poor component is dienophile. The HOMO of diene and the LUMO of dienophile are similar in energy and interact strongly. However, for a DA reaction with inverse electron demand, the electron rich component is dienophile and the electron poor component is diene.<ref>A. T. Dang, D. O. Miller, L. N. Dawe and G. J. Bodwell, Org. Lett., 2008, 10, 233–236</ref> Then the more strongly interacting frontier orbitals are the HOMO of dienophile and the LUMO of diene. In this reaction, the dienophile is 1,3-dioxole, and the electron donation from lone pair of oxygen atoms results in more electron rich dienophile. The single point energy calculation confirms this suggestion.

{| class="wikitable"
|+ Table.3, Single point energy of HOMO/LUMO of reactants
|-
| style="background: #c8d9ec; color: black;" | '''Molecule'''
| style="background: #c8d9ec; color: black;" | '''Energy of HOMO/a.u.'''
| style="background: #c8d9ec; color: black;" | '''Energy of LUMO/a.u.'''
| style="background: #c8d9ec; color: black;" | '''Difference of LUMO<sub>diene</sub> and HOMO<sub>dienophile</sub>'''
| style="background: #c8d9ec; color: black;" | '''Difference of HOMO<sub>diene</sub> and LUMO<sub>dienophile</sub>'''
|-
|Cyclohexadiene
|style="text-align: center;"| -0.20601
|style="text-align: center;"|-0.01800
|rowspan="2" style="text-align: center;"| 0.17815
|rowspan="2" style="text-align: center;"| 0.24265
|-
|1,3-Dioxole
|style="text-align: center;"|-0.19615
|style="text-align: center;"| 0.03664
|}

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 23:49, 22 March 2018 (UTC) Nice this is well done and clear.

===Energy Analysis===

{| class="wikitable"
|+ Table.4, Gibbs free energies of reactants, TSs and products using B3LYP/6-31G(d)
|-
| style="background: #c8d9ec; color: black;" | '''Molecule'''
| style="background: #c8d9ec; color: black;" | '''Gibbs free energy/Hartrees'''
| style="background: #c8d9ec; color: black;" | '''Gibbs free energy/kJmol<sup>-1</sup>'''
|-
|Cyclohexadiene
|style="text-align: center;" |-233.324375
|style="text-align: center;" |-612593.193227
|-
|1,3-Dioxole
|style="text-align: center;" |-267.068644
|style="text-align: center;" |-701188.778236
|-
|Reactants (total)
|style="text-align: center;" |-500.393019
|style="text-align: center;" |-1313781.971463
|-
|Endo TS
|style="text-align: center;" |-500.332149
|style="text-align: center;" |-1313622.15727
|-
|Exo TS
|style="text-align: center;" |-500.329163
|style="text-align: center;" |-1313614.31752
|-
|Endo Product
|style="text-align: center;" |-500.418694
|style="text-align: center;" |-1313849.381181
|-
|Exo Product
|style="text-align: center;" |-500.417319
|style="text-align: center;" |-1313845.77112
|}

{| class="wikitable"
|+ Table.5, Activation energies and ΔG of two reactions using B3LYP/6-31G(d)
|-
| style="background: #c8d9ec; color: black;" | '''State'''
| style="background: #c8d9ec; color: black;" | '''Activation energy /kJmol<sup>-1</sup>'''
| style="background: #c8d9ec; color: black;" | '''ΔG /kJmol<sup>-1</sup>'''
|-
|Endo
|style="text-align: center;" |159.8
|style="text-align: center;" |-67.4
|-
|Exo
|style="text-align: center;" |167.7
|style="text-align: center;" |-63.8
|}

The kinetic product is the one with lower activation energy, leading to faster reaction, and the thermodynamic product is the one with more negative ΔG, which forms more stable product. The calculation of energies in Table.5 illustrates that the endo product has lower activation energy and more negative ΔG, indicating that '''endo product''' is the '''kinetic''' product as well as '''thermodynamic''' product

{| class="wikitable"
| colspan="3" style="text-align: center; background: #3b5998; color: white" | '''HOMOs of endo and exo TSs'''
|-
!<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 8; mo 41; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715 ENDO TSPM6 631G 3 JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
!<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 102; mo 41; mo cutoff 0.01 mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>XP715_EXO_TS_jmol.log</uploadedFileContents>
</jmolApplet>
</jmol>
![[File:XP715 Secondary interaction.PNG]]
|-
|Endo TS
|Exo TS
|Figure.9, graphic illustration of primary/secondary interactions of HOMOs.
|}

There is only primary interaction in exo TS, while the secondary interaction is also observed in endo TS. The secondary interaction stabilises the endo TS (Fig.9), resulting in faster formation of endo TS and confirming that the endo product is more kinetically favourable.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 23:53, 22 March 2018 (UTC) Good section, you could have gone into more detail about the thermo and kenetic theory. But otherwise a very good section.

==Exercise 3: Diels-Alder vs Cheletropic==

[[File:XP715 Scheme ex3.JPG|thumb|center|700px|Scheme.3, reaction schemes between Xylylene and SO<sub>2</sub> through Diels-Alder reaction and Cheletropic reaction]]

For this reaction, three products were examined, including endo and exo products of DA reactions and cheletropic product. Energy calculations were carried out to identify the most favourable reaction pathway. All the reaction species were optimised at PM6 level. The extension investigated the possibility of DA reaction of a second cis-butadiene in o-xylylene. The activation energies and Gibbs free energies were calculated to suggest the viability of the reactions.

===Optimisation and Calculation===
{| class="wikitable"
| colspan="3" style="text-align: center; background: #3b5998; color: white" | '''Optimisation of three TSs'''
|-
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 12</script>
<uploadedFileContents>XP715 M3 MOL1 SPLIT TSPM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 12</script>
<uploadedFileContents>XP715 DA ENDO SPLIT TSPM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 12</script>
<uploadedFileContents>XP715_CHE_SPLIT_TSPM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|DA-Exo
|DA-Endo
|Cheletropic
|-
|[[File:XP715 DA ENDO freq.PNG|thumb|center|500px]]
|[[File:XP715_DA_EXO_freq.PNG|thumb|center|500px]]
|[[File:XP715_CHE_freq.PNG|thumb|center|500px]]
|}

{| class="wikitable"
| colspan="3" style="text-align: center; background: #3b5998; color: white" | '''IRC of three TSs'''
|-
|DA-Endo TS
|[[File:XP715_DA_ENDO_TS_IRC.png|thumb|center|900px]]
|[[File:XP715 DA ENDO TS.gif|thumb|center|500px]]
|-
|DA-Exo TS
|[[File:XP715_DA_EXO_TS_IRC.png|thumb|center|900px]]
|[[File:XP715_DA_EXO_TS.gif|thumb|center|500px]]
|-
|Cheletropic TS
|[[File:XP715_Che_TS_IRC.png|thumb|center|900px]]
|[[File:XP715_CHE_TS.gif|thumb|center|500px]]
|-
|colspan="3"|Figure.10, IRCs of three TSs
|}

All IRCs were successful asymmetric graphs. The endo and exo DA TS starts from product to reactant and the cheletropic TS starts from reactant to product. The approach trajectories are shown as motion pictures on the right.

===Energy Analysis===

{| class="wikitable"
|+ Table.6, Gibbs free energies of reactants, TSs and products at PM6
|-
| style="background: #c8d9ec; color: black;" | '''Molecule'''
| style="background: #c8d9ec; color: black;" | '''Gibbs free energy/Hartrees'''
| style="background: #c8d9ec; color: black;" | '''Gibbs free energy/kJmol<sup>-1</sup>'''
|-
|o-Xylylene
|style="text-align: center;" |0.178816
|style="text-align: center;" |469.481444
|-
|SO<sub>2</sub>
|style="text-align: center;" |-0.119268
|style="text-align: center;" |-313.1381579
|-
|Reactants (total)
|style="text-align: center;" |0.059548
|style="text-align: center;" |156.343286
|-
|Endo TS
|style="text-align: center;" |0.090559
|style="text-align: center;" |237.762673
|-
|Exo TS
|style="text-align: center;" |0.092077
|style="text-align: center;" |241.748182
|-
|Cheletropic TS
|style="text-align: center;" |0.099059
|style="text-align: center;" |260.079424
|-
|Endo Product
|style="text-align: center;" |0.021697
|style="text-align: center;" |56.9654778
|-
|Exo Product
|style="text-align: center;" |0.021452
|style="text-align: center;" |56.3222303
|-
|Cheletropic product
|style="text-align: center;" |0.000007
|style="text-align: center;" |0.0183785014
|}

{| class="wikitable"
|+ Table.7, Activation energies and ΔG of two reactions at PM6
|-
| style="background: #c8d9ec; color: black;" | '''State'''
| style="background: #c8d9ec; color: black;" | '''Activation energy /kJmol<sup>-1</sup>'''
| style="background: #c8d9ec; color: black;" | '''ΔG /kJmol<sup>-1</sup>'''
|-
|Endo
|style="text-align: center;" |81.4
|style="text-align: center;" |-99.4
|-
|Exo
|style="text-align: center;" |85.4
|style="text-align: center;" |-100.0
|-
|Cheletropic
|style="text-align: center;" |103.7
|style="text-align: center;" |-156.3
|}

[[File:XP715_Energy_profile.png|thumb|center|700px|Figure.11, reaction profile of three reactions]]

By plotting the energy profile (Fig.11), the '''endo''' product is the '''kinetic product''' as the activation barrier is the lowest. The ΔG of exo product is similar to endo product, indicating that endo and exo products have same thermodynamic stability. The '''thermodynamic product''' is the '''cheletropic''' product as the ΔG is the most negative one. The energy of o-xylylene is very high, indicating that it is highly unstable. Therefore, by examining IRCs, the 6-membered ring is converted from 8π electrons (4n, '''antiaromatic''') to 6π electrons (4n+2, '''aromatic'''), resulting in more stable structures. The required cis-butadiene structure is already present in the o-xylylene, so it accelerates the DA reactions.

===Extension===

[[File:XP715_Scheme_ext.PNG|thumb|center|500px|Scheme.4, reaction scheme of o-Xylylene with a second cis-butadiene fragment and SO<sub>2</sub>]]

====Optimisation====
{| class="wikitable"
| colspan="4" style="text-align: center; background: #3b5998; color: white" | '''Optimisation of Endo and Exo TSs and products'''
|-
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 12</script>
<uploadedFileContents>XP715_EXT_ENDO_SPLIT_TSPM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 12</script>
<uploadedFileContents>XP715_EXT_EXO_SPLIT_TSPM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 12</script>
<uploadedFileContents>XP715 EXT ENDO MINPM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
! <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 12</script>
<uploadedFileContents>XP715 EXT EXO MINPM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|Endo TS
|Exo TS
|Endo Product
|Exo Product
|}

====Energy Analysis====

{| class="wikitable"
|+ Table.8, Gibbs free energies of reactants, TSs and products at PM6
|-
| style="background: #c8d9ec; color: black;" | '''Molecule'''
| style="background: #c8d9ec; color: black;" | '''Gibbs free energy/Hartrees'''
| style="background: #c8d9ec; color: black;" | '''Gibbs free energy/kJmol<sup>-1</sup>'''
|-
|Endo TS
|style="text-align: center;" |0.102071
|style="text-align: center;" |267.987431
|-
|Exo TS
|style="text-align: center;" |0.105053
|style="text-align: center;" |275.816673
|-
|Endo Product
|style="text-align: center;" |0.065611
|style="text-align: center;" |172.261694
|-
|Exo Product
|style="text-align: center;" |0.067306
|style="text-align: center;" |176.711916
|}

{| class="wikitable"
|+ Table.9, Activation energies and ΔG of two reactions at PM6
|-
| style="background: #c8d9ec; color: black;" | '''State'''
| style="background: #c8d9ec; color: black;" | '''Activation energy /kJmol<sup>-1</sup>'''
| style="background: #c8d9ec; color: black;" | '''ΔG /kJmol<sup>-1</sup>'''
|-
|Endo
|style="text-align: center;" |111.6
|style="text-align: center;" |15.9
|-
|Exo
|style="text-align: center;" |119.5
|style="text-align: center;" |20.4
|}

Both of the reactions has positive ΔG, which requires energy from the environment to proceed the reaction, and the activation energies are much higher than previous DA reactions, suggesting that the DA reaction of cis-butadiene within the ring is '''kinetically and thermodynamically unfavourable'''.

==Conclusion==

Gaussian is powerful in applying the computational method to carry out pericyclic reactions. Different optimisation methods (PM6 and B3LYP) can be adopted to optimise reactants, TSs and products to the required level of precision. The structures can be checked by frequency calculation, as one imaginary frequency appears in TS. The IRC shows the energy profile, and the activation energy and Gibbs free energy can be calculated to predict the most favourable reaction pathway. Information including shape of MOs and bond length is also available, so a MO diagram is constructed easily.

In exercise 1, Woodward-Hoffmann rules and Frontier molecular orbital theory are confirmed experimentally. The reactions in exercise 2 concludes that the endo product is the kinetic and thermodynamic product, and the DA reaction is with inverse electron demand. The reactions in exercise 3 infer that the endo product is the kinetic product and the cheletropic product is the thermodynamic product. Due to the high activation energy barrier of cheletropic TS, the endo product is more likely to form. The cis-butadiene fragment within the ring is too steric to perform DA reactions.

Gaussian is also viable for other pericyclic reactions such as electrocyclic reactions.

==Reference==

<references/>

==Appendix==

===Exercise 1===
Butadiene: [[File:XP715_DIENE_MINPM6_NEW.LOG]]

Ethene: [[File:XP715_ETHENE_MINPM6.LOG]]

TS: [[File:XP715 2MOL TSPM6 JMOL.LOG]]

Product:[[File:XP715_PROD_MINPM6.LOG]]

IRC: [[File:XP715 2mol IRC.log]]

===Exercise 2===
Cyclohexadiene:[[File:XP715 DIENE 631G JMOL.LOG]]

1,3-Dioxole:[[File:XP715 DIOXOLE 631G JMOL.LOG]]

Endo TS:[[File:XP715 ENDO TSPM6 631G 3 JMOL.LOG]]

Exo TS:[[File:XP715_EXO_TS_jmol.log]]

Endo Product:[[File:XP715 Endo prod 631G.log]]

Exo product: [[File:XP715 EXO 631G jmol.log]]

IRC (Endo):[[File:ENDO TSPM6 IRC.log]]

IRC (Exo):[[File:XP715 EXO SPLIT TSPM6 IRC.log]]

===Exercise 3===
Exo TS:[[File:XP715 M3 MOL1 SPLIT TSPM6.LOG]]

Endo TS: [[File:XP715 DA ENDO SPLIT TSPM6.LOG]]

Cheletropic TS: [[File:XP715_CHE_SPLIT_TSPM6.LOG]]

IRC (Exo):[[File:XP715 M3 mol1 IRC.log]]

IRC (Endo):[[File:XP715 DA ENDO SPLIT TSPM6 IRC.LOG]]

IRC (Cheletropic):[[File:XP715 CHE SPLIT TSPM6 IRC protal.log]]

====Extension====
Endo TS:[[File:XP715_EXT_ENDO_SPLIT_TSPM6.LOG]]

Exo TS:[[File:XP715_EXT_EXO_SPLIT_TSPM6.LOG]]

Endo Product: [[File:XP715 EXT ENDO MINPM6.LOG]]

Exo Product: [[File:XP715 EXT EXO MINPM6.LOG]]

IRC (Endo): [[File:XP715 EXT ENDO SPLIT TSPM6 IRC.LOG]]

IRC (Exo): [[File:XP715 EXT EXO SPLIT TSPM6 IRC.LOG]]

Rep:Wtc14

2025-09-01T09:50:08Z

Move page script: Move page script moved page Wtc14 to Rep:Wtc14: Move to report namespace

== Introduction<ref><nowiki>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_exercise</nowiki></ref> ==
Gaussian can been used to locate and characterize transition structures for a variety of pericyclic reactions. It is able to optimize the molecular geometry and calculate the vibrational frequency of a compound based on Born Oppenheimer approximation where the electronic distribution of a molecule move instantaneously to the movement of nuclei. Potential energy surface can be plotted using quantum chemistry by calculating the energy as a function of positions of the nuclei.

Geometry optimization will lead to a local minimum which may be a global minimum or a transition state of the product. We can tell which of the minimum we got using to methods.
First, we can tell from the potential energy surface(PES). For a global minimum, energy will rise in all direction from that point of minimum on the PES. Whereas for a transition state, the energy will decrease in one direction which is the reaction path on the PES.
Secondly, by performing a frequency calculation on Gaussian. For a global minimum, we will have all positive frequency whereas for a transition state we will have one imaginary frequency which is due to the negative value of secondary derivative of energy(a maximum on the PES). <ref><nowiki>Lecture notes: Quantum Mechanics 3/3rd Year Computational Chemistry Laboratory, Michael Bearpark, Imperial College.</nowiki></ref>

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 22:49, 14 December 2016 (UTC) You need to talk about derivatives here

== Exercise 1 ==
=== MO diagram for the formation of butadiene/ethene TS ===
[[File:MO wtc.png|700px]]
Diagram 1: MO diagram for the formation of the butadiene/ethene TS.
{| class="wikitable" border="1"
|+ MOs:
! Name of Compound !! Orbital !! Symmetry !! MO Images !! JMol Images
|-
| Ethene || HOMO || u || [[File:Ethene HOMO wtc.png|150px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Ethene || LUMO || g || [[File:Ethene LUMO wtc.png|150px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>ETHENE OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Butadiene || HOMO || g || [[File:Butadiene HOMO wtc.png|150px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 22; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>DIENOPHILE OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Butadiene || LUMO || u || [[File:Butadiene LUMO wtc.png|150px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 22; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>DIENOPHILE OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| TS || second HOMO || symmetric || [[File:TS 2nd HOMO wtc.png|150px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>CYCLOHEXENE TS PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| TS || HOMO || unsymmetric || [[File:TS HOMO wtc.png|150px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>CYCLOHEXENE TS PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| TS || LUMO || unsymmetric || [[File:TS LUMO wtc.png|150px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>CYCLOHEXENE TS PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| TS || Second LUMO || symmetric || [[File:TS 2nd LUMO wtc.png|150px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>CYCLOHEXENE TS PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|colspan=5 |Table 1: MOs of the reactants and TS.
|}
Symmetry of the reacting orbitals of both reactants must be the same for the reaction to be allowed. The HOMO of butadiene(g) interact with the LUMO of ethene(g) to produce the second HOMO and second LUMO of TS. LUMO of butadiene(u) interact with the HOMO of ethene(u) to produce the HOMO and LUMO of the TS. The overlap integral equals to 1 for same symmetry overlap of orbitals(ie ungerand-ungerand orbitals and gerand-gerand orbitals) and equals to 0 for different symmetry overlap of orbitals(ie gerand-ungerand orbitals and vice versa).

=== Bond Length Measurement ===

{| class="wikitable" border="1"
|+ Bond Length
! Compound !! Images !! Bond Length
|-
| Ethene || [[File:Ethene bondlength wtc.png|300px]] || C1C2=1.32732Å
|-
| Butadiene || [[File:Butadiene bondlength 1234.png|300px]] || C1C2=C3C4=1.33526Å
|-
| Butadiene || [[File:Butadiene bondlength 1.png|300px]] || C1C2=1.46834Å
|-
| TS || [[File:TS bondlength.png|300px]] || C1C2=C3C4=1.37977Å, C2C3=1.41110Å
|-
| TS || [[File:TS bondlength 1.png|300px]] || C1C2=1.38177Å
|-
| TS || [[File:TS bondlength 2.png|300px]] || C1C2=C3C4=2.11478Å
|-
| Product || [[File:Product bondlength 1.png|300px]] || C2C3=1.33702Å, C1C2=1.50077Å, C3C4=1.50095Å
|-
| Product || [[File:Butadiene bondlength 2.png|300px]] || C1C2=1.53711Å, C2C3=1.53450Å, C3C4=1.53720Å
|-
|colspan=8 | Table 2: Bond length measurements.
|}

All double bonds elongate to form a single bond and all single bonds shorten to form a double bond
Typical sp3 and sp2 C-C bond lengths are 1.544 Å and 1.337 Å <ref><nowiki>Bartell, L. S. (1959). Journal of the American Chemical Society, 81(14), 3497–3498. doi:10.1021/ja01523a002</nowiki></ref>.
The Van der Waals radii of carbon is 1.7 Å <ref><nowiki>Batsanov, S. S. (2001). Van der Waals Radii of Elements. Inorganic Materials, 37(9), 871–885. doi:10.1023/A:1011625728803</nowiki></ref>.
The interatomic distance between the reacting carbons (2.11 Å) are shorter than the Van der Waals radii (3.4 Å). This means that the reacting carbons are approaching each other to react.

=== Vibration ===

{| class="wikitable" border="1"
|+ Vibration
! Type of Frequency !! Vibration !! Description
|-
| Imaginary Frequency
|<jmol>
<jmolApplet>
<uploadedFileContents>CYCLOHEXENE TS PM6.LOG</uploadedFileContents>
<color>black</color>
<script>frame 17; vibration 2;</script>
</jmolApplet>
</jmol>
| The imaginary frequency is taken from the negative frequency. It shows the vibration mode that lead to the transition state of the reaction. Stretching of C=C and shortening of C-C bonds in both reactants can be seen. This compliments to the trend of the change in bond lengths from reactants to the final product shown above. There is also an out-of-plane bend involving the reaction centres, bring the terminal carbons of butadiene and ethene close together to form bonds.
|-
| Lowest positive frequency
|<jmol>
<jmolApplet>
<uploadedFileContents>CYCLOHEXENE TS PM6.LOG</uploadedFileContents>
<color>black</color>
<script>frame 18; vibration 2;</script>
</jmolApplet>
</jmol>
| From the vibration,the reacting centers do not approach each other in the vibration and hence does not lead to the reaction taking place.
|-
|colspan=3 |Table 3: Vibration of Trasition State.
|}

=== Reaction Path ===
{| class="wikitable" border="1"
|+ Reaction Path
! Formation of Cyclohaxene(IRC)!! Energy Curve
|-
| [[File:Reaction path movie.gif|350px]] || [[File:Reaction path of cyclohexene.png|350px]]
|-
|colspan=2 | Table 4: Reaction path of cyclohexene
|}
The formation of cyclohexene is synchronous because the bond between the reacting centers break at the same time. This can be proved by the presence of a single maximum in the potential energy curve.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 22:57, 14 December 2016 (UTC) Good section however you could have stated if it was normal or invest demand.

== Exercise 2 ==
=== Transition State MO ===
{| class="wikitable" border="1"
|+ Molecular Orbitals
! Type of Product !! MO !! Transition State MO
|-
| Exo Product || HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 1.8; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; mo cutoff 0.02</script>
<uploadedFileContents>Cyclo dioxo exo b3lyp jmol.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Exo Product || Second HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 1.8; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; </script>
<uploadedFileContents>Cyclo dioxo exo b3lyp jmol.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Exo Product || LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 1.8; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; </script>
<uploadedFileContents>Cyclo dioxo exo b3lyp jmol.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Exo Product || Second LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 1.8; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; </script>
<uploadedFileContents>Cyclo dioxo exo b3lyp jmol.log</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| ENDO Product || HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 1.6; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; mo cutoff 0.02</script>
<uploadedFileContents>CYCLO DIOXOLE TS B3LYP JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| ENDO Product || Second HOMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 1.6; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; </script>
<uploadedFileContents>CYCLO DIOXOLE TS B3LYP JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| ENDO Product || LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 1.6; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; </script>
<uploadedFileContents>CYCLO DIOXOLE TS B3LYP JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| ENDO Product || Second LUMO
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 1.6; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; </script>
<uploadedFileContents>CYCLO DIOXOLE TS B3LYP JMOL.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|colspan=3 |Table 5: Transition state MO from cyclohexene and 1,3-dioxole.
|}

The HOMO and LUMO TS of both exo and endo products are symmetric because they are formed by HOMO of dienophile and LUMO of diene that are also symmetric. Whereas both the LUMO of dienophile(antisymmetric)and HOMO of the diene(antisymmetric) form the second HOMO and second LUMO of the TS which are both antisymmetric.
The HOMO of transition state is formed between the HOMO of 1,3-dioxole(dienophile) and the LUMO of hexacyclodiene(diene). From the HOMO TS shown above the orbitals of 1,3-dioxole dominate over the orbitals from hexacyclodiene. This is because 1,3-dioxole has electron donating groups that pulls up the potential energy of the HOMO of 1,3-dioxole. Therefore, this reaction is an inverse electron demand Diels-Alder reaction.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 23:00, 14 December 2016 (UTC) Excellent use of the MOs to come to this conclusion

=== Thermochemistry ===
{| class="wikitable" border="1"
|+ Sum of electronic and thermal Free Energies
! !! Cyclopentadiene(Hartree)
| '''1,3-dioxole (Hartree)'''
| '''Reactants (Hartree)'''
| '''Product (Hatree)'''
| '''Transition State (Hartree)'''
|'''Energy Barrier (KJ/mol)'''
| '''Reaction Energy(KJ/mol)'''
|-
| '''exo''' || -233.324377 || -267.068644 || -500.393021 || -500.417324 || -500.329164 || 167.65
| 63.84
|-
| '''endo''' || -233.324377 || -267.068644 || -500.393021 || -500.418691 || -500.332149 || 159.82
| 67.40
|-
| colspan=8 |(Table 6:Values of the sum of electronic and thermal Free Energies, calculation ran at DFT (B3LYP))

|}

The endo product has a more negative energy than the exo product. Therefore it is more thermodinamically stable compare to the exo product.
The energy barrier of the endo product is lower than the exo product. Hence endo product is both kinetically and thermodinamically favourable product.

From table 5, we can see that in the endo HOMO transition state, there are secondary (non-bonding) orbital interactions between the p orbital of oxygens in 1,3-dioxole and the p orbitals in diene. This will lower its energy barrier and make it more kinetically favorable.
The exo product only has a primary (bonding) orbital interactions.Besides that, the sp3 carbon centres of both cyclohexene and 1,3-dioxole might have cause some steric hindrance which cause it to be both unfavourable thermodynamically and kinetically. For the endo product, the sp3 carbon centres of the cyclohexene and 1,3-dioxole are pointing at different direction, hence no steric factors that will cause in increased of energy barrier.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 23:05, 14 December 2016 (UTC) Correct energies, very well done section.,

== Exercise 3 ==
=== Intrinsic Reaction Coordinate ===
{| class="wikitable" border="1"
|+ Transition State of o-Xylylene-SO2 Cycloaddition
! Reaction !! Type of Product !! Transition state !! IRC !! Total Energy along IRC
|-
| Diels-Alder || Exo || [[File:Exo product TS wtc.png|500px]]
| [[File:Product exo ric.gif|500px]]
| [[File:Diels alder irc reaction pathway exo wtc.png|400px]]
|-
| colspan=5 | Backward Diels-Alder Path of Exo product.
|-
| || Endo || [[File:Endo product TS wtc.png|500px]]
| [[File:Product endo ric.gif|500px]]
| [[File:Diels alder irc reaction pathway endo.png|400px]]
|-
| colspan=5 | Foward Diels-Alder Path of endo product.
|-
| Cheletropic || - || [[File:Cheletropic TS wtc.png|500px]]
| [[File:Cheletropic ric wtc.gif|500px]]
| [[File:Cheletropic ric rxn pathway.png|400px]]
|-
| colspan=5 | Forward cheletropic reaction
|-
|colspan=5 | Table 7: Transition state and IRC of Diels-Alder and cheletropic pathway.
|}

=== Energy Profile ===
{| class="wikitable" border="1"
|+ Sum of electronic and thermal Free Energies
! !! Xylylene(Hartree)
| '''SO<sub>2</sub> (Hartree)'''
| '''Reactants (Hartree)'''
| '''Product (Hartree)'''
| '''Transition State (Hartree)'''
|'''Energy Barrier (KJ/mol)'''
| '''Reaction Energy(KJ/mol)'''
|-
| '''exo Diels-Alder''' || 0.178764 || -0.119269 || 0.059495 || 0.021455 || 0.092074 || 101.44 || -99.87
|-
| '''endo Diels-Alder''' || 0.178764 || -0.119269 || 0.059495 || 0.021695 || 0.090560 || 81.56 || -99.24
|-
| '''Cheletropic''' || 0.178764 || -0.119269 || 0.059495 || 0.000005 || 0.099055 || 103.86 || -156.19
|-
| colspan=8 |(Table 8:Values of the sum of electronic and thermal Free Energies, calculation ran at semi-empirical(PM6))
|}

(Calculation for exo barrier is incorrect [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 12:21, 6 December 2016 (UTC))

The energy of reactant is calculated by adding both the energy of SO<sub>2</sub> and xylylene. (In vacuum, no interaction between xylylene and SO<sub>2</sub>.)
Compare to the exo product in exercise 2, there is no steric problem in the exo product. For the diels-Alder pathway, the exo product has a lower energy than the endo product, hence it is the thermodynamically favoured product.
The energy barrier(activation energy) of the exo product is higher that the endo product. Therefore, endo product is the kinetically favoured product.
Due to small difference between the endo and exo product, and with the secondary interaction between the p orbital of the oxygen in the SO<sub>2</sub> and the p orbital of the diene, the endo product will probably be the major product.
Within Diels-Alder product and cheletropic product, the energy barrier of cheletropic product is around 79 kJ/mol higher than the Diels-Alder product but the cheletropic product has a much lower energy than the Diels-Alder product. Hence, the Diels-Alder pathway is more preferred at low temperature and the cheletropic pathway is more preferred at high temperature so that it has enough energy to overcome the high activation energy.

Xylylene is unstable and this can be proven by its high Gibbs free energy. According to the IRC pathway, after the bonding of the 6-membered ring the energy of the product drops. This is due to the formation of delocalised pi system that forms resonance which lowers the energy of the product.

=== Reaction Profile ===
[[File:Reaction profile wtc.png|600px]]

Diagram 2: Energy profile between Dielse-Alder pathway and cheletropic pathway.

=== Diels-Alder Reaction for the second cis-butadiene fragment in o-xylylene ===
{| class="wikitable" border="1"
|+ Sum of electronic and thermal Free Energies second cis-butadiene fragment in o-xylylene
! !! Xylylene(Hartree)
| '''SO<sub>2</sub> (Hartree)'''
| '''Reactants (Hartree)'''
| '''Product (Hartree)'''
| '''Transition State (Hartree)'''
|'''Energy Barrier (KJ/mol)'''
| '''Reaction Energy(KJ/mol)'''
|-
| '''exo Diels-Alder''' || 0.178764 || -0.119269 || 0.059495 || 0.065610 || 0.102066 || 111.77 || 16.05
|-
| '''endo Diels-Alder''' || 0.178764 || -0.119269 || 0.059495 ||0.067304 || 0.105053 || 119.61 || 20.50
|-
| colspan=8 |(Table 9:Values of the sum of electronic and thermal Free Energies for second cis-butadiene fragment in o-xylylene, calculation ran at semi-empirical(PM6))
|}

(Not too important here, but it's a good idea to explicitly say the reaction energies are positive to prevent ambiguity [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 12:17, 6 December 2016 (UTC))

This reaction is endothermic for both endo and exo product. The endo(0.067304 kJ/mol) and exo(0.065610 kJ/mol) products of Diels-Alder Reaction between second cis-butadiene fragment in o-xylylene and SO<sub>2</sub> are unfavourable because both has higher energy than the reactants (0.059495 kJ/mol).

== Conclusion ==

The partly formed C-C bonds in the TS of the reaction between butadiene and ethene is shorter than the Van der Waals radii(3.4 Å). This shows that the reacting carbon from butadiene and ethene are approaching together to form a bond. Along the reaction, all double bond shortens to form single bond and all single bond elongate to form double bond. New bonds are formed between the reacting carbons at the same time for the Diels-Alder reaction between butadiene and ethene, therefore it is a synchronous reaction.

The endo product of the reaction between cyclopentadiene and 1,3-dioxole is kinetically and thermodynamically favoured product because the secondary interaction between the p orbital of oxygen in 1,3-dioxole and p orbital of diene provides extra stability of product in the formation of transition state. The exo product has higher energy than expected due to steric problem caused by the sp3 carbon centres of both cyclohexene and 1,3-dioxole. From the HOMO of TS from table 5, it is clearly dominated by the HOMO of 1,3-dioxole(dienophile). The HOMO and LUMO TS of both exo and endo products are symmetric because they are formed by HOMO of dienophile and LUMO of diene that are also symmetric.

Diels Alder reaction between xylylene and SO<sub>2</sub> will only occur at the outer cis butadiene fragment of the xylylene. The product obtained through the reaction with the second cis-butadiene fragment in o-xylylene is unfavourable because the it has higher energy than the reactant which might due to increase steric hindrance of the product compare to it's reactants.

== Reference ==

Rep:Wm1415TransitionStatesWorking

2025-09-01T09:50:08Z

Move page script: Move page script moved page Wm1415TransitionStatesWorking to Rep:Wm1415TransitionStatesWorking: Move to report namespace

==First Bit==
<jmol> //Group all the HTML within "jmol"
<jmolApplet>
<color>white</color>
<size>400</size> //Initialise the applet
<uploadedFileContents>wm1415_DIELS_ALDER_TS_PM6_GFPRINT.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<name>DielsAlderTS</name> //The name of the applet must be set. This is the name that the controls refer to (the target)
</jmolApplet>
<jmolbutton> //Adding a jmol button, which executes code on the target
<script>frame 14; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script> //Turn on MO
<text>MO ON</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>frame 14; mo off</script>
<text>MO OFF</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==14) frame=15; vibrating=0; endif</script>
<script>frame 15; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 4 6 8 14 11]; label display; color label blue; font label 16</script>
<text>Carbon Labels ON</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 16</script>
<text>MO 16</text>
<target>DielsAlderTS</target>
</item>
<item>
<script>mo 17</script>
<text>MO 17</text>
<target>DielsAlderTS</target>
</item>
<item>
<script>mo 18</script>
<text>MO 18</text>
<target>DielsAlderTS</target>
</item>
<item>
<script>mo 19</script>
<text>MO 19</text>
<target>DielsAlderTS</target>
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<jmolmenu>
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<script>mo cutoff 0.02</script>
<text>Cutoff 0.02</text>
<target>DielsAlderTS</target>
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<item>
<script>mo cutoff 0.01</script>
<text>Cutoff 0.01</text>
<target>DielsAlderTS</target>
</item>
</jmolmenu>
</jmol>

==Second Bit==
{| class="wikitable"
!Some title
!<jmolFile text="Show PM6-Optimised Last IRC Structure Diels-Alder Reaction">wm1415_DA_LAST_IRC_STRUCTURE_PM6_OPT.LOG</jmolFile>
|-
|<jmolFile text="Show PM6-Optimised Last IRC Structure Diels-Alder Reaction">wm1415_DA_LAST_IRC_STRUCTURE_PM6_OPT.LOG</jmolFile>
|<jmolFile text="Show PM6-Optimised Last IRC Structure Diels-Alder Reaction">wm1415_DA_LAST_IRC_STRUCTURE_PM6_OPT.LOG</jmolFile>
|}

<jmolFile>wm1415_DA_LAST_IRC_STRUCTURE_PM6_OPT.LOG</jmolFile>

{| class="wikitable"
!Ethene
!Energy / a.u.
!Butadiene
!Energy / a.u.
!Transition State
!Energy / a.u.
|-
|
|
|MO 13
|0.0636
|MO 19
|0.0307
|-
|LUMO
|0.0426
|LUMO
|0.0194
|MO 18
|0.0173
|-
|HOMO
|-0.3923
|HOMO
|-0.3590
|MO 17
|-0.3253
|-
|
|
|MO 10
|-0.4133
|MO 16
|-0.3276
|}

<jmol>
<jmolApplet>
<UploadedFileContents>wm1415_EXT_TS_PM6.LOG</UploadedFileContents>
</jmolApplet>
</jmol>

==Some other bit==

{| class = "wikitable" align = "center"
!Electrocyclic TS
!Gif
|-
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_EXT_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 92; mo 24; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<name>extension</name>
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<jmolbutton>
<script>vibrating=0; spinning=0; frame 92; mo 24; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>extension</target>
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<jmolbutton>
<script>frame 92; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script>
<text>MO ON</text>
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<script>frame 92; mo off</script>
<text>MO OFF</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==92) frame=93; vibrating=0; endif</script>
<script>frame 93; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>extension</target>
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<jmolbutton>
<script>select atomno=[1 3 5 7]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>extension</target>
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<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>extension</target>
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<script>vibrating=0; spinning=0; frame 92; mo 24; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
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<target>extension</target>
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<item>
<script>mo 24</script>
<text>MO 24</text>
<target>extension</target>
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<item>
<script>mo 25</script>
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<text>MO 26</text>
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<script>mo 27</script>
<text>MO 27</text>
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<text>Cutoff 0.01</text>
<target>extension</target>
</item>
</jmolmenu>
</jmol></center>
|some gif
|}

==Fourth Bit==

Rep:Wm1415TransitionStatesBackup

2025-09-01T09:50:08Z

Move page script: Move page script moved page Wm1415TransitionStatesBackup to Rep:Wm1415TransitionStatesBackup: Move to report namespace

'''This is William Micou's report on the Transition States computational lab, starting Monday 11th December 2017.'''
==Introduction==

Transition state analysis is a powerful technique to predict major products and observed stereochemistry; to rationalise reaction rates with different substituents; to examine reaction mechanisms and to understand catalytic cycles. Transition states are very difficult to probe experimentally: here, transition states for a variety of pericyclic reactions will be calculated computationally.

The potential energy surface of a molecule consisting of N atoms exists in (3N-6) dimensions. The potential energy can be expressed as a function of (3N-6) independent nuclear coordinates, <math>q_{i}</math>:

<center><math>E = f(q_{1}, q_{2}, ..., q_{3N-6})</math></center>

A stable structure sits in a local minimum on the potential energy surface, whereas a transition state is a saddle point on the PES. These are both stationary points, where:

<center><math>\frac{\partial E(q_{i})}{\partial q_{i}}=0</math> along every coordinate <math>q_{i}</math>.</center>

Minima and transition states can be distinguished by their second derivatives. At a minimum, the energy must increase in all directions from that point on the PES: therefore,

<center><math>\frac{\partial^{2}E}{\partial q_{i}^{2}}>0</math> for all <math>q_{i}</math>.</center>

At a transition state, the energy increases in all directions except one: the energy decreases along the reaction path. A transition state is hence defined as a stationary point on the PES with one negative Hessian eigenvalue.<sup>[1]</sup>

The second derivatives of the PES correspond to the force constants of the vibrations along the chosen nuclear coordinates:

<center><math>k=\frac{\partial^{2}E}{\partial q^{2}}</math>, with the frequency of the vibration <math>\omega=\frac{1}{2\pi}\sqrt{\frac{k}{\mu}}</math></center>

Therefore, a transition state will have an optimised structure with a single imaginary frequency - this corresponds to the collective motions of the atoms forming the products and lowering the energy. In contrast, optimised reactants or products which sit at a local minimum have only positive frequency vibrations.

==Methodology==

Calculations were executed with Gaussian 09, Revision D.01, with input files prepared in GaussView 5.0.9. Transition states were located and reaction paths calculated using the following sequence:
*Products optimised at the desired level of approximation.
*Bonds formed in the reaction are broken and these bond lengths are frozen to around 2.2 Å for C-C bonds, 2.0 Å for C-O and 2.4 Å for C-S bonds. The structure is reoptimised.
*Bonds are unfrozen and the transition state is located using the Berny algorithm, calculating the force constants only once.
*The transition state structure is checked with an IRC (intrinsic reaction coordinate) calculation at the PM6 level (as more accurate calculations are computationally expensive) and a frequency calculation.

The B3LYP level of approximation makes use of Density Functional Theory (DFT)<sup>[2]</sup>. The electron energy is a functional<sup>[3]</sup> of the electronic density, <math>E[\rho]</math>. The B3LYP force field is a hybrid density functional method<sup>[3]</sup>, incorporating the exchange correlation predicted by gradient-corrected functionals<sup>[2]</sup> (GGA methods - Generalised Gradient Approximation) along with exact, Hartree-Fock exchange<sup>[2]</sup>. These calculations are considerably heavier and time allowed for B3LYP calculations in Exercise 2 only.

The PM6 approximation is a semi-empirical model: reference data is used<sup>[4]</sup> in combination with calculations based on a Hartree-Fock model (with considerable simplifications). These calculations are fast and successful for reasonably small molecules, although the PM6 method has been modified to model proteins<sup>[5]</sup>.

==Exercise 1: Butadiene + Ethene Diels-Alder Reaction==

===Optimised structures of the reactants and transition state (PM6 level)===

The table below contains Jmol applets of the reactants and transition state structures optimised at the PM6 level.

{| class = "wikitable" align = "center"
!Ethene
!Butadiene (s-cis)
!Transition state
|-
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>350</size>
<uploadedFileContents>wm1415_ETHENE_PM6_OPT_GFPRINT.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<name>ethene</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>ethene</target>
</jmolbutton>
<jmolbutton>
<script>frame 6; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script>
<text>MO ON</text>
<target>ethene</target>
</jmolbutton>
<jmolbutton>
<script>frame 6; mo off</script>
<text>MO OFF</text>
<target>ethene</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ethene</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==6) frame=7; vibrating=0; endif</script>
<script>frame 7; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Lowest Frequency</text>
<target>ethene</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 4]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>ethene</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ethene</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>ethene</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 6</script>
<text>HOMO</text>
<target>ethene</target>
</item>
<item>
<script>mo 7</script>
<text>LUMO</text>
<target>ethene</target>
</item>
</jmolmenu>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>350</size>
<uploadedFileContents>wm1415_BUTADIENE_SCIS_PM6_OPT_GFPRINT.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 6; mo 10; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<name>butadiene</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 6; mo 10; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>frame 6; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script>
<text>MO ON</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>frame 6; mo off</script>
<text>MO OFF</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==6) frame=7; vibrating=0; endif</script>
<script>frame 7; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Lowest Frequency</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 4 6 8]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 6; mo 10; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>butadiene</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 10</script>
<text>MO 10</text>
<target>butadiene</target>
</item>
<item>
<script>mo 11</script>
<text>HOMO</text>
<target>butadiene</target>
</item>
<item>
<script>mo 12</script>
<text>LUMO</text>
<target>butadiene</target>
</item>
<item>
<script>mo 13</script>
<text>MO 13</text>
<target>butadiene</target>
</item>
</jmolmenu>
</jmol></center>
|<center><jmol> //Group all the HTML within "jmol"
<jmolApplet>
<color>#F9F9F9</color>
<size>350</size> //Initialise the applet
<uploadedFileContents>wm1415_DIELS_ALDER_TS_PM6_GFPRINT.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<name>DielsAlderTS</name> //The name of the applet must be set. This is the name that the controls refer to (the target)
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton> //Adding a jmol button, which executes code on the target
<script>frame 14; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script> //Turn on MO
<text>MO ON</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>frame 14; mo off</script>
<text>MO OFF</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==14) frame=15; vibrating=0; endif</script>
<script>frame 15; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 4 6 8 14 11]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 16</script>
<text>MO 16</text>
<target>DielsAlderTS</target>
</item>
<item>
<script>mo 17</script>
<text>MO 17</text>
<target>DielsAlderTS</target>
</item>
<item>
<script>mo 18</script>
<text>MO 18</text>
<target>DielsAlderTS</target>
</item>
<item>
<script>mo 19</script>
<text>MO 19</text>
<target>DielsAlderTS</target>
</item>
</jmolmenu>
</jmol></center>
|}

===Confirmation of Transition State===

The frequency calculation on the TS structure yielded an imaginary vibration at -948 cm<sup>-1</sup>: this corresponds to the motion of atoms that leads to the formation of products. Further evidence for a correct TS was obtained through an IRC calculation: this structure is indeed the maximum along the minimum energy path. The table below provides a plot of the C-C bond lengths during the reaction; an animation of the reaction calculated reaction path; and a plot of the energy, along with its derivative, against the reaction coordinate. It is worth noting that in this case, the IRC path was arbitrarily calculated from the products to the reactants: hence the reversed animation. The same atom labels were used as in the Jmol structures above.

{| class = "wikitable"
!Internuclear distances in IRC path
!Diels-Alder Reaction Animation
!IRC Path Energy Plot
|-
|<center>[[File:wm1415_Diels_alder_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Diels_alder_resized_gif.gif]]</center>
|<center>[[File:wm1415_Irc_energy_diels_alder.PNG|200 px]]</center>
|}

The animation shows that this Diels-Alder reaction is synchronous: the new σ bonds (C8-C14 and C11-C1) are formed at the same time. These bond lengths are identical in the internuclear distances plot, which also show shortening of the C4-C6 bond as it becomes a double bond (around 1.35 Å). Also shown is the lengthening of the 3 double bonds involved in the cycloaddition, which all become single bonds (C14-C11, C1-C4 and C6-C8) with lengths of between 1.50-1.54 Å (C1-C4 and C6-C8, which are adjacent to the double bond in the product, are slightly shorter). Carbon has a van der Waals radius of 1.70 Å.

In the transition state, the C8-C14/C11-C1 bonds measure around 2.12 Å - the new σ bonds are not yet fully formed, but well within twice the van der Waals radius. The original double bonds are all elongated - around 1.38 Å, and the originally singly-bonded C4-C6 is shorted to 1.41 Å.

===Optimisation and Analysis of Product===
To reduce the number of Jmol applets on this page, these structures have not been embedded - they can be opened in a new window by clicking the following links. A PM6-level optimisation of the products of the IRC yields <jmolFile text="boat cyclohexene">wm1415_DA_LAST_IRC_STRUCTURE_PM6_OPT.LOG</jmolFile>: this was compared to a PM6-level optimisation of <jmolFile text="chair cyclohexene.">wm1415_CYCLOHEXENE_PM6_OPT.LOG</jmolFile> The boat structure features an imaginary frequency: this corresponds to the collective motion of atoms that would yield the chair cyclohexene structure, lowering the energy.

===Orbital Description of the Diels-Alder Reaction===

{| align="left"
|[[File:wm1415_Final_diels_alder_MO_diagram.PNG|550 px|thumb|left|MO Diagram showing the orbitals involved in the Diels-Alder reaction. While the ordering of the energies of the reactant frontier orbitals has been respected, the diagram is not drawn to scale and the transition state energy levels are skewed for clarity. Approximate energy levels calculated at the PM6-level approximation are listed in the table opposite.]]
|
{| class="wikitable" align="center"
!Ethene
!Energy / a.u.
!Butadiene
!Energy / a.u.
!Transition State
!Energy / a.u.
|-
|
|
|MO 13
|0.0636
|MO 19
|0.0307
|-
|LUMO
|0.0426
|LUMO
|0.0194
|MO 18
|0.0173
|-
|HOMO
| -0.3923
|HOMO
| -0.3590
|MO 17
| -0.3253
|-
|
|
|MO 10
| -0.4133
|MO 16
| -0.3276
|}
LUMO<sub>dienophile</sub> - HOMO<sub>diene</sub> = 0.4016 a.u.

LUMO<sub>diene</sub> - HOMO<sub>dienophile</sub> = 0.4117 a.u.

The smallest energy gap is that between the LUMO of ethene and the HOMO of butadiene (the antisymmetric orbitals). The interaction between these orbitals is stronger as there is greater orbital overlap: the reaction proceeds by electrons from the HOMO of the diene populating the LUMO of the dienophile (MO 16) - normal electron demand. The other bonding orbital in the TS, MO 17, is formed from the interaction between the HOMO of ethene and the LUMO of butadiene.
|}

The Woodward-Hoffmann rules can be used to predict whether a pericyclic reaction is allowed by symmetry. According to the Woodward-Hoffman rules, a cycloaddition is thermally allowed if the total number of (4q+2)<sub>s</sub> and (4r)<sub>a</sub> components is odd<sup>[6]</sup>. Here, the 's' label refers to a suprafacial component, which forms new bonds on the same side at both ends ('a' label - antarafacial, opposite sides). The Diels-Alder reaction is a [<sub>π</sub>4<sub>s</sub>+<sub>π</sub>2<sub>s</sub>] cycloaddition: (4q+2)<sub>s</sub> + (4r)<sub>a</sub> = 1, so it is allowed.

==Exercise 2==

===Optimised Reactants: B3LYP/6-31G(d)===

These Jmol files have been excluded from this page to speed up loading time. The applets can be accessed here: <jmolFile text="cyclohexadiene">wm1415_CYCLOHEXADIENE_B3LYP_OPT.LOG</jmolFile> and <jmolFile text="1,3-dioxole">wm1415_LAST_DIOXOLE_TRY_B3LYP.LOG</jmolFile>. There are no negative frequencies as these are stable structures.

===Endo/Exo Transition States and Products: B3LYP/6-31G(d)===

{| class = "wikitable" align = "center"
!Endo TS
!Endo Product
!Exo TS
!Exo Product
|-
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_ENDO_TS_B3LYP.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 36; mo 40; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<name>ex2_endoTS</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 36; mo 40; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>frame 36; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script>
<text>MO ON</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>frame 36; mo off</script>
<text>MO OFF</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==36) frame=37; vibrating=0; endif</script>
<script>frame 37; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 2 3 4 5 6]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>ex2_endoTS</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 36; mo 40; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 40</script>
<text>MO 40</text>
<target>ex2_endoTS</target>
</item>
<item>
<script>mo 41</script>
<text>MO 41</text>
<target>ex2_endoTS</target>
</item>
<item>
<script>mo 42</script>
<text>MO 42</text>
<target>ex2_endoTS</target>
</item>
<item>
<script>mo 43</script>
<text>MO 43</text>
<target>ex2_endoTS</target>
</item>
</jmolmenu>
<jmolmenu>
<item>
<script>mo cutoff 0.02</script>
<text>Cutoff 0.02</text>
<target>ex2_endoTS</target>
</item>
<item>
<script>mo cutoff 0.01</script>
<text>Cutoff 0.01</text>
<target>ex2_endoTS</target>
</item>
</jmolmenu>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_ENDO_PRODUCTS_B3LYP_OPT.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 18; frank off</script>
<name>ex2_endoproduct</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 18; frank off</script>
<text>Reload</text>
<target>ex2_endoproduct</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex2_endoproduct</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==18) frame=19; vibrating=0; endif</script>
<script>frame 19; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Lowest Frequency</text>
<target>ex2_endoproduct</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 2 3 4 5 6]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>ex2_endoproduct</target>
</jmolbutton>
<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex2_endoproduct</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 18; frank off</script>
<text>Reload</text>
<target>ex2_endoproduct</target>
</jmolbutton>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_EXO_TS_B3LYP.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 18; mo 40; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<name>ex2_exoTS</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 18; mo 40; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>frame 18; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script>
<text>MO ON</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>frame 18; mo off</script>
<text>MO OFF</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==18) frame=19; vibrating=0; endif</script>
<script>frame 19; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 2 3 4 5 6]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>ex2_exoTS</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 18; mo 40; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 40</script>
<text>MO 40</text>
<target>ex2_exoTS</target>
</item>
<item>
<script>mo 41</script>
<text>MO 41</text>
<target>ex2_exoTS</target>
</item>
<item>
<script>mo 42</script>
<text>MO 42</text>
<target>ex2_exoTS</target>
</item>
<item>
<script>mo 43</script>
<text>MO 43</text>
<target>ex2_exoTS</target>
</item>
</jmolmenu>
<jmolmenu>
<item>
<script>mo cutoff 0.02</script>
<text>Cutoff 0.02</text>
<target>ex2_exoTS</target>
</item>
<item>
<script>mo cutoff 0.01</script>
<text>Cutoff 0.01</text>
<target>ex2_exoTS</target>
</item>
</jmolmenu>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_EXO_PRODUCTS_B3LYP_OPT.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 14; frank off</script>
<name>ex2_exoproduct</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 14; frank off</script>
<text>Reload</text>
<target>ex2_exoproduct</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex2_exoproduct</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==14) frame=15; vibrating=0; endif</script>
<script>frame 15; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Lowest Frequency</text>
<target>ex2_exoproduct</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 2 3 4 5 6]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>ex2_exoproduct</target>
</jmolbutton>
<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex2_exoproduct</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 14; frank off</script>
<text>Reload</text>
<target>ex2_exoproduct</target>
</jmolbutton>
</jmol></center>
|}

===Endo/Exo IRC Paths (PM6 Level)===
{| class = "wikitable"
!Endo Internuclear distances
!Endo Reaction Animation
!Endo IRC Energy Plot
|-
|<center>[[File:wm1415_Ex2_endo_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Ex2_endo_resized.gif]]</center>
|<center>[[File:wm1415_ex2_Irc_energy_endo_cyclo.PNG|200 px]]</center>
|-
!Exo Internuclear distances
!Exo Reaction Animation
!Exo IRC Energy Plot
|-
|<center>[[File:wm1415_Ex2_exo_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Ex2_exo_resized.gif]]</center>
|<center>[[File:wm1415_ex2_Irc_energy_exo_cyclo.PNG|200 px]]</center>
|}

===MO Diagram: Diels-Alder Inverse Electron Demand===
{| align="left"
|[[File:wm1415_Ex2_mo_diagram.PNG|550 px|thumb|left|MO Diagram showing the FMOs. While the ordering of the energies of the reactant FMOs has been respected, the diagram is not drawn to scale and the transition state energy levels are skewed for clarity. Approximate energy levels calculated at the B3LYP 6-31G(d)-level approximation are listed in the table opposite.]]
|
{| class="wikitable" align="center"
!1,3-dioxole
!Energy / a.u.
!Cyclohexadiene
!Energy / a.u.
!Endo TS
!Energy / a.u.
|-
|
|
|MO 24
| 0.0874
|MO 43
| -0.0154
|-
|LUMO
|0.0380
|LUMO
| -0.0171
|MO 42
| -0.0046
|-
|HOMO
| -0.1959
|HOMO
| -0.2055
|MO 41
| -0.1905
|-
|
|
|MO 21
| -0.2994
|MO 40
| -0.1965
|}
LUMO<sub>dienophile</sub> - HOMO<sub>diene</sub> = 0.2435 a.u.

LUMO<sub>diene</sub> - HOMO<sub>dienophile</sub> = 0.1788 a.u.

In this case, it is the energy gap between the LUMO of the diene and the HOMO of the dienophile that is smallest and leads to the greatest orbital overlap. The dienophile, 1,3-dioxole, is electron-rich: electrons flow from its HOMO to the diene LUMO - this is inverse electron demand. This interaction can be visualised as MO 41 in both the endo and exo transition states. Interactions between the dienophile LUMO and diene HOMO form MO 40 in the transition state.
|}

Because the energies of the interacting orbitals are a better match than in the butadiene and ethene reaction, there is greater orbital overlap and hence greater stabilisation of the transition state: the reaction with the electron-rich 1,3-dioxole would be much faster.

===Energy Profile: Secondary Orbital Interactions===

{| align="left"
|[[File:wm1415_Ex2_energy_profile.png|300px]]
|The endo product is both the kinetic and thermodynamic product of the reaction: it is formed from a lower energy TS (difference of 8 kJ/mol) and releases more energy (difference of 3.6 kJ/mol) in the reaction. The exo product is comparatively destabilised because of the unfavourable steric clash between the heteroatomic 5-membered ring and the bridging ethyl group across the newly-formed 6-membered ring from the Diels-Alder reaction.

The endo transition state is stabilised by secondary orbital interactions (SOI)<sup>[7]</sup>. These interactions come about between atomic orbitals that are not involved in the formation or cleavage of σ bonds<sup>[7]</sup>. In the endo TS, the O atoms are aligned with the back of the diene. While this configuration is sterically unfavourable, it allows for the SOI between the O lone pairs and the orbitals at the back end of the diene LUMO. This is a favourable interaction which is only accessible to the endo TS, and more than makes up for the increased steric clash - resulting in a lower energy transition state. These interactions can be clearly seen in the Jmol files above: select MO 41 in the transition state structures (and adjusting the MO cutoff value if necessary).
|[[File:wm1415_SOI_interaction.PNG|200 px]]
{| class="wikitable"
|+ '''Free energies / kJ/mol'''
! <math>\Delta G^{o}_{endo}</math>
| -67.4
! <math>\Delta G^{o}_{exo}</math>
| -63.8
|-
! <math>\Delta G^{*}_{endo}</math>
| 160
! <math>\Delta G^{*}_{exo}</math>
| 168
|}
|}

[[Wm1415TransitionStates#Introduction|Back to Introduction.]]

==Exercise 3==
===The Reaction between SO<sub>2</sub> and Xylylene===
Sulfur dioxide can undergo Diels-Alder and Cheletropic pericyclic reactions with dienes. Xylylene contains two diene fragments: what will be deemed the main fragment undergoes aromatisation in the considered reactions. The following sections contain calculations for the formation of the each of the possible products for both the main and second fragments of xylylene: from these calculations, a reaction profile will be constructed.

Jmol applets for PM6-optimised structures of <jmolFile text="xylylene">wm1415_XYLYLENE_OPT2_PM6.LOG</jmolFile> and <jmolFile text="sulfur dioxide">wm1415_SO2_OPT_PM6.LOG</jmolFile> have been excluded from the report, but can be opened in a new window from these links.

===Main Fragment: Optimised TS (PM6-level) and IRC Paths===

[[File:wm1415_Main_Fragment_Scheme.PNG|400 px]]

{| class = "wikitable" align = "center"
!Endo TS
!Exo TS
!Cheletropic TS
|-
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_ENDO_PRODUCT_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<name>ex3_endoTS</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<text>Reload</text>
<target>ex3_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex3_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==22) frame=23; vibrating=0; endif</script>
<script>frame 23; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex3_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[11 4 3 10 16 15]; label display; color label lime; font label 16</script>
<text>Labels ON</text>
<target>ex3_endoTS</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex3_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<text>Reload</text>
<target>ex3_endoTS</target>
</jmolbutton>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_ENDO_PRODUCT_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<name>ex3_exoTS</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<text>Reload</text>
<target>ex3_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex3_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==22) frame=23; vibrating=0; endif</script>
<script>frame 23; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex3_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[11 4 3 10 16 15]; label display; color label lime; font label 16</script>
<text>Labels ON</text>
<target>ex3_exoTS</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex3_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<text>Reload</text>
<target>ex3_exoTS</target>
</jmolbutton>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_CHELO_PM6_TS.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 12; frank off</script>
<name>ex3_cheloTS</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 12; frank off</script>
<text>Reload</text>
<target>ex3_cheloTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex3_cheloTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==12) frame=13; vibrating=0; endif</script>
<script>frame 13; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex3_cheloTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[11 4 3 10 16 15 17]; label display; color label lime; font label 16</script>
<text>Labels ON</text>
<target>ex3_cheloTS</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex3_cheloTS</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 12; frank off</script>
<text>Reload</text>
<target>ex3_cheloTS</target>
</jmolbutton>
</jmol></center>
|-
!Endo IRC Path
!Exo IRC Path
!Cheletropic IRC Path
|-
|<center>[[File:wm1415_Ex3_endo1_regif.gif|300 px]]</center>
|<center>[[File:wm1415_Ex3_exo1_regif.gif|300 px]]</center>
|<center>[[File:wm1415_Ex3_chelo1_regif.gif|300 px]]</center>
|-
!Endo Internuclear Distances
!Exo Internuclear Distances
!Cheletropic Internuclear Distances
|-
|<center>[[File:wm1415_Endo31_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Exo31_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Che1_internucdist.png|400 px]]</center>
|-
!<jmolFile text="Endo Product Jmol">wm1415_ENDO_PRODUCT_OPT_PM6.LOG</jmolFile>
!<jmolFile text="Exo Product Jmol">wm1415_INITIAL_PM6_OPT.LOG</jmolFile>
!<jmolFile text="Cheletropic Product Jmol">wm1415_CHELO_PRODUCTS_INITIAL_PM6_OPT.LOG</jmolFile>
|}

===Second Fragment: Optimised TS (PM6-level) and IRC Paths===

[[File:wm1415_Second_fragment_scheme.PNG|400 px]]

{| class = "wikitable" align = "center"
!Endo TS
!Exo TS
!Cheletropic TS
|-
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_SECOND_ENDO_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 12; frank off</script>
<name>ex3_endoTS2</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 12; frank off</script>
<text>Reload</text>
<target>ex3_endoTS2</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex3_endoTS2</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==12) frame=13; vibrating=0; endif</script>
<script>frame 13; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex3_endoTS2</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 2 3 6 16 17 18]; label display; color label lime; font label 16</script>
<text>Labels ON</text>
<target>ex3_endoTS2</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex3_endoTS2</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 12; frank off</script>
<text>Reload</text>
<target>ex3_endoTS2</target>
</jmolbutton>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_SECOND_EXO_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 24; frank off</script>
<name>ex3_exoTS2</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 24; frank off</script>
<text>Reload</text>
<target>ex3_exoTS2</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex3_exoTS2</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==24) frame=25; vibrating=0; endif</script>
<script>frame 25; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex3_exoTS2</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 2 3 6 16 17 18]; label display; color label lime; font label 16</script>
<text>Labels ON</text>
<target>ex3_exoTS2</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex3_exoTS2</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 24; frank off</script>
<text>Reload</text>
<target>ex3_exoTS2</target>
</jmolbutton>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_SECOND_CHELO_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<name>ex3_cheloTS2</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<text>Reload</text>
<target>ex3_cheloTS2</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex3_cheloTS2</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==22) frame=23; vibrating=0; endif</script>
<script>frame 23; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex3_cheloTS2</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 4 5 6 11 12 13]; label display; color label lime; font label 16</script>
<text>Labels ON</text>
<target>ex3_cheloTS2</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex3_cheloTS2</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<text>Reload</text>
<target>ex3_cheloTS2</target>
</jmolbutton>
</jmol></center>
|-
!Endo IRC Path
!Exo IRC Path
!Cheletropic IRC Path
|-
|<center>[[File:wm1415_Ex3_endo2_regif.gif|300 px]]</center>
|<center>[[File:wm1415_Ex3_exo2_regif.gif|300 px]]</center>
|<center>[[File:wm1415_Ex3_chelo2_regif.gif|300 px]]</center>
|-
!Endo Internuclear Distances
!Exo Internuclear Distances
!Cheletropic Internuclear Distances
|-
|<center>[[File:wm1415_Endo32_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Exo32_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Che2_internucdist.png|400 px]]</center>
|-
!<jmolFile text="Endo Product Jmol">wm1415_SECOND_ENDO_PRODUCTS_PM6_OPT.LOG</jmolFile>
!<jmolFile text="Exo Product Jmol">wm1415_SECOND_EXO_PRODUCTS_PM6_OPT.LOG</jmolFile>
!<jmolFile text="Cheletropic Product Jmol">wm1415_SECOND_CHELO_PRODUCTS_OPT_PM6.LOG</jmolFile>
|}

===Reaction Profile===

[[File:wm1415_Energy_profile_ex3.png|400 px|left]]
{| align="left"
|
{| class="wikitable"
|+ '''Main Fragment: Free energies / kJ/mol'''
! <math>\Delta G^{o}_{endo}</math>
| -99.2
! <math>\Delta G^{o}_{exo}</math>
| -84.1
! <math>\Delta G^{o}_{chel}</math>
| -156
|-
! <math>\Delta G^{*}_{endo}</math>
| 81.6
! <math>\Delta G^{*}_{exo}</math>
| 85.6
! <math>\Delta G^{*}_{chel}</math>
| 104
|}

{| class="wikitable"
|+ '''Second Fragment: Free energies / kJ/mol'''
! <math>\Delta G^{o}_{endo}</math>
| 16.1
! <math>\Delta G^{o}_{exo}</math>
| 20.6
! <math>\Delta G^{o}_{chel}</math>
| 47.1
|-
! <math>\Delta G^{*}_{endo}</math>
| 112
! <math>\Delta G^{*}_{exo}</math>
| 120
! <math>\Delta G^{*}_{chel}</math>
| 141
|}
|
|}

Reaction with the second fragment are clearly very thermodynamically and kinetically unfavourable: the overall reactions are endothermic, and all activation energies are larger than in reactions with the main fragment. The products from these theoretical reactions would not be observed experimentally.

Reaction with the main fragment is much more favourable on account of the formation of the benzene ring. The aromatisation confers stability to both the products and the transition states. The cheletropic reaction yields the thermodynamic product, which contains the more stable 5-membered ring and an additional strong S=O bond. However, the chelotropic rearrangement proceeds via a considerably higher reaction barrier, and at very low temperatures one might observe the kinetic product: the endo Diels-Alder product. The endo configuration has a stabilised TS due to secondary orbital interactions, as discussed previously. Endo-selectivity for this Diels-Alder reaction is reported by the literature<sup>[8]</sup>.

The reaction between xylylene and SO<sub>2</sub> has been performed in the literature<sup>[8]</sup>. At 20 <sup>o</sup>C, SO<sub>2</sub> adds reversibly and a 1:9 mixture of the Diels-Alder and cheletropic products was obtained. Low temperatures are required for the reaction to proceed, because of the reduction in entropy as SO<sub>2</sub> gas is consumed. Prior simulations of this reaction have included a second molecule of SO<sub>2</sub> that may help stabilise the Diels-Alder transition state<sup>[9]</sup>.

==Extension: 4π Electrocyclic Ring Closure with Conrotation (PM6-Level)==
According to the Woodward-Hoffmann rules, an electrocyclic reaction involving 4n π electrons should proceed via a conrotary mechanism<sup>[6]</sup>. The subject of this extension is to demonstrate this stereospecifity, using the same methods as previously in this experiment:

[[File:wm1415_Extension_reaction_scheme.PNG|400 px]]

{| class = "wikitable" align = "center"
!Electrocyclic TS
!IRC Path Animation
!IRC Energy Plot
|-
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>350</size>
<uploadedFileContents>wm1415_EXT_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 92; mo 24; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<name>extension</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 92; mo 24; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>frame 92; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script>
<text>MO ON</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>frame 92; mo off</script>
<text>MO OFF</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==92) frame=93; vibrating=0; endif</script>
<script>frame 93; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 3 5 7]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>extension</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 92; mo 24; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>extension</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 24</script>
<text>MO 24</text>
<target>extension</target>
</item>
<item>
<script>mo 25</script>
<text>MO 25</text>
<target>extension</target>
</item>
<item>
<script>mo 26</script>
<text>MO 26</text>
<target>extension</target>
</item>
<item>
<script>mo 27</script>
<text>MO 27</text>
<target>extension</target>
</item>
</jmolmenu>
<jmolmenu>
<item>
<script>mo cutoff 0.02</script>
<text>Cutoff 0.02</text>
<target>extension</target>
</item>
<item>
<script>mo cutoff 0.01</script>
<text>Cutoff 0.01</text>
<target>extension</target>
</item>
</jmolmenu>
</jmol></center>
|[[File:wm1415_Ext_gif.gif]]
|[[File:wm1415_ext_Irc_path_energy.PNG|250 px]]
|}

The calculated IRC path clearly shows the conrotation about the migrating carbon atoms, confirming the predicted stereochemistry of the products.

==Conclusion==
In this experiment, the transition states were successfully calculated for a variety of pericyclic reactions, using both the semi-empirical PM6 and the DFT B3LYP 6-31G(d) levels of approximation. The results agreed with predictions laid out by the Woodward-Hoffmann rules. The Diels-Alder reaction between butadiene and ethene was found to proceed via normal electron demand; using an electron-rich dienophile, 1,3-dioxole, resulted in inverse electron demand. In the reaction between 1,3-dioxole and cyclohexadiene, the endo product was found to be both the kinetic and thermodynamic product: the endo transition state is stabilised by secondary orbital interactions, and the exo product is destabilised by unfavourable steric clashes.

The rearomatisation was confirmed to be the driving force for the pericyclic reactions of SO<sub>2</sub> with xylylene, as reactions with the second diene fragment yielded kinetically and thermodynamically very unfavourable products. The thermodynamic product arises from the cheletropic mechanism, but the kinetic product arises from the faster, reversible Diels-Alder cycloaddition - this finding was backed up with experimental observations from literature<sup>[8]</sup>.

The Woodward-Hoffmann rules were further investigated: the predicted stereochemistry of the 4π electrocyclic reaction was obtained from these theoretical calculations.

==References==

1 D. Wales, in Energy Landscapes: Applications to Clusters, Biomolecules and Glasses, Cambridge University Press, 2004, pp. 192–240.

2 P. J. Stephens, F. J. Devlin, C. F. Chabalowski and M. J. Frisch, J. Phys. Chem., 1994, '''98''', 11623–11627.

3 S. F. Sousa, P. A. Fernandes and M. J. Ramos, J. Phys. Chem. A, 2007, '''111''', 10439–10452.

4 J. J. P. Stewart, J. Mol. Model., 2007, '''13''', 1173–1213.

5 J. J. P. Stewart, J. Mol. Model., 2009, '''15''', 765–805.

6 R. B. Woodward and R. Hoffmann, Angew. Chemie Int. Ed. English, 1969, '''8''', 781–853.

7 A. Arrieta, F. P. Cossío and B. Lecea, J. Org. Chem., 2001, '''66''', 6178–6180.

8 B. Deguin and P. Vogel, J. Am. Chem. Soc., 1992, '''114''', 9210–9211.

9 T. Fernandez, J. A. Sordo, F. Monnat, B. Deguin and P. Vogel, J. Am. Chem. Soc., 1998, '''120''', 13276–13277.

Rep:Wm1415TransitionStates

2025-09-01T09:50:07Z

Move page script: Move page script moved page Wm1415TransitionStates to Rep:Wm1415TransitionStates: Move to report namespace

'''This is William Micou's report on the Transition States computational lab, starting Monday 11th December 2017.'''
==Introduction==

Transition state analysis is a powerful technique to predict major products and observed stereochemistry; to rationalise reaction rates with different substituents; to examine reaction mechanisms and to understand catalytic cycles. Transition states are very difficult to probe experimentally: here, transition states for a variety of pericyclic reactions will be calculated computationally.

The potential energy surface of a molecule consisting of N atoms exists in (3N-6) dimensions. The potential energy can be expressed as a function of (3N-6) independent nuclear coordinates, <math>q_{i}</math>:

<center><math>E = f(q_{1}, q_{2}, ..., q_{3N-6})</math></center>

A stable structure sits in a local minimum on the potential energy surface, whereas a transition state is a saddle point on the PES. These are both stationary points, where:

<center><math>\frac{\partial E(q_{i})}{\partial q_{i}}=0</math> along every coordinate <math>q_{i}</math>.</center>

Minima and transition states can be distinguished by their second derivatives. At a minimum, the energy must increase in all directions from that point on the PES: therefore,

<center><math>\frac{\partial^{2}E}{\partial q_{i}^{2}}>0</math> for all <math>q_{i}</math>.</center>

At a transition state, the energy increases in all directions except one: the energy decreases along the reaction path. A transition state is hence defined as a stationary point on the PES with one negative Hessian eigenvalue.<sup>[1]</sup>

The second derivatives of the PES correspond to the force constants of the vibrations along the chosen nuclear coordinates:

<center><math>k=\frac{\partial^{2}E}{\partial q^{2}}</math>, with the frequency of the vibration <math>\omega=\frac{1}{2\pi}\sqrt{\frac{k}{\mu}}</math></center>

Therefore, a transition state will have an optimised structure with a single imaginary frequency - this corresponds to the collective motions of the atoms forming the products and lowering the energy. In contrast, optimised reactants or products which sit at a local minimum have only positive frequency vibrations.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 12:07, 16 January 2018 (UTC) This is excellently written and you have clearly understood how the hessian matrix is used. Well done. Nice use of the equations to make it clear.

==Methodology==

Calculations were executed with Gaussian 09, Revision D.01, with input files prepared in GaussView 5.0.9. Transition states were located and reaction paths calculated using the following sequence:
*Products optimised at the desired level of approximation.
*Bonds formed in the reaction are broken and these bond lengths are frozen to around 2.2 Å for C-C bonds, 2.0 Å for C-O and 2.4 Å for C-S bonds. The structure is reoptimised.
*Bonds are unfrozen and the transition state is located using the Berny algorithm, calculating the force constants only once.
*The transition state structure is checked with an IRC (intrinsic reaction coordinate) calculation at the PM6 level (as more accurate calculations are computationally expensive) and a frequency calculation.

The B3LYP level of approximation makes use of Density Functional Theory (DFT)<sup>[2]</sup>. The electron energy is a functional<sup>[3]</sup> of the electronic density, <math>E[\rho]</math>. The B3LYP force field is a hybrid density functional method<sup>[3]</sup>, incorporating the exchange correlation predicted by gradient-corrected functionals<sup>[2]</sup> (GGA methods - Generalised Gradient Approximation) along with exact, Hartree-Fock exchange<sup>[2]</sup>. These calculations are considerably heavier and time allowed for B3LYP calculations in Exercise 2 only.

The PM6 approximation is a semi-empirical model: reference data is used<sup>[4]</sup> in combination with calculations based on a Hartree-Fock model (with considerable simplifications). These calculations are fast and successful for reasonably small molecules, although the PM6 method has been modified to model proteins<sup>[5]</sup>.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 12:13, 16 January 2018 (UTC) I would be careful calling B3LYP a force field method as this would imply a molecular mechanics method. But good understanding on the exchange correlation correction. PM6 is correct, the integrals are replaces by some experimentally fitted parameters,

==Exercise 1: Butadiene + Ethene Diels-Alder Reaction==

===Optimised structures of the reactants and transition state (PM6 level)===

The table below contains Jmol applets of the reactants and transition state structures optimised at the PM6 level.

{| class = "wikitable" align = "center"
!Ethene
!Butadiene (s-cis)
!Transition state
|-
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>350</size>
<uploadedFileContents>wm1415_ETHENE_PM6_OPT_GFPRINT.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<name>ethene</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>ethene</target>
</jmolbutton>
<jmolbutton>
<script>frame 6; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script>
<text>MO ON</text>
<target>ethene</target>
</jmolbutton>
<jmolbutton>
<script>frame 6; mo off</script>
<text>MO OFF</text>
<target>ethene</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ethene</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==6) frame=7; vibrating=0; endif</script>
<script>frame 7; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Lowest Frequency</text>
<target>ethene</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 4]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>ethene</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ethene</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>ethene</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 6</script>
<text>HOMO</text>
<target>ethene</target>
</item>
<item>
<script>mo 7</script>
<text>LUMO</text>
<target>ethene</target>
</item>
</jmolmenu>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>350</size>
<uploadedFileContents>wm1415_BUTADIENE_SCIS_PM6_OPT_GFPRINT.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 6; mo 10; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<name>butadiene</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 6; mo 10; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>frame 6; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script>
<text>MO ON</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>frame 6; mo off</script>
<text>MO OFF</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==6) frame=7; vibrating=0; endif</script>
<script>frame 7; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Lowest Frequency</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 4 6 8]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>butadiene</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 6; mo 10; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>butadiene</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 10</script>
<text>MO 10</text>
<target>butadiene</target>
</item>
<item>
<script>mo 11</script>
<text>HOMO</text>
<target>butadiene</target>
</item>
<item>
<script>mo 12</script>
<text>LUMO</text>
<target>butadiene</target>
</item>
<item>
<script>mo 13</script>
<text>MO 13</text>
<target>butadiene</target>
</item>
</jmolmenu>
</jmol></center>
|<center><jmol> //Group all the HTML within "jmol"
<jmolApplet>
<color>#F9F9F9</color>
<size>350</size> //Initialise the applet
<uploadedFileContents>wm1415_DIELS_ALDER_TS_PM6_GFPRINT.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<name>DielsAlderTS</name> //The name of the applet must be set. This is the name that the controls refer to (the target)
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton> //Adding a jmol button, which executes code on the target
<script>frame 14; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script> //Turn on MO
<text>MO ON</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>frame 14; mo off</script>
<text>MO OFF</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==14) frame=15; vibrating=0; endif</script>
<script>frame 15; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 4 6 8 14 11]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>DielsAlderTS</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 16</script>
<text>MO 16</text>
<target>DielsAlderTS</target>
</item>
<item>
<script>mo 17</script>
<text>MO 17</text>
<target>DielsAlderTS</target>
</item>
<item>
<script>mo 18</script>
<text>MO 18</text>
<target>DielsAlderTS</target>
</item>
<item>
<script>mo 19</script>
<text>MO 19</text>
<target>DielsAlderTS</target>
</item>
</jmolmenu>
</jmol></center>
|}

===Confirmation of Transition State===

The frequency calculation on the TS structure yielded an imaginary vibration at -948 cm<sup>-1</sup>: this corresponds to the motion of atoms that leads to the formation of products. Further evidence for a correct TS was obtained through an IRC calculation: this structure is indeed the maximum along the minimum energy path. The table below provides a plot of the C-C bond lengths during the reaction; an animation of the reaction calculated reaction path; and a plot of the energy, along with its derivative, against the reaction coordinate. It is worth noting that in this case, the IRC path was arbitrarily calculated from the products to the reactants: hence the reversed animation. The same atom labels were used as in the Jmol structures above.

{| class = "wikitable"
!Internuclear distances in IRC path
!Diels-Alder Reaction Animation
!IRC Path Energy Plot
|-
|<center>[[File:wm1415_Diels_alder_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Diels_alder_resized_gif.gif]]</center>
|<center>[[File:wm1415_Irc_energy_diels_alder.PNG|200 px]]</center>
|}

The animation shows that this Diels-Alder reaction is synchronous: the new σ bonds (C8-C14 and C11-C1) are formed at the same time. These bond lengths are identical in the internuclear distances plot, which also show shortening of the C4-C6 bond as it becomes a double bond (around 1.35 Å). Also shown is the lengthening of the 3 double bonds involved in the cycloaddition, which all become single bonds (C14-C11, C1-C4 and C6-C8) with lengths of between 1.50-1.54 Å (C1-C4 and C6-C8, which are adjacent to the double bond in the product, are slightly shorter). Carbon has a van der Waals radius of 1.70 Å.

In the transition state, the C8-C14/C11-C1 bonds measure around 2.12 Å - the new σ bonds are not yet fully formed, but well within twice the van der Waals radius. The original double bonds are all elongated - around 1.38 Å, and the originally singly-bonded C4-C6 is shorted to 1.41 Å.

===Optimisation and Analysis of Product===
To reduce the number of Jmol applets on this page, these structures have not been embedded - they can be opened in a new window by clicking the following links. A PM6-level optimisation of the products of the IRC yields <jmolFile text="boat cyclohexene">wm1415_DA_LAST_IRC_STRUCTURE_PM6_OPT.LOG</jmolFile>: this was compared to a PM6-level optimisation of <jmolFile text="chair cyclohexene.">wm1415_CYCLOHEXENE_PM6_OPT.LOG</jmolFile> The boat structure features an imaginary frequency: this corresponds to the collective motion of atoms that would yield the chair cyclohexene structure, lowering the energy.

===Orbital Description of the Diels-Alder Reaction===

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) Very confused on why you didn't draw the TS LUMO and LUMO+1 according to the calculated relative energies: you know the order is (by decreasing energy) MO13 - ethene LUMO - TS LUMO+1 - butadiene LUMO - TS LUMO, so why not draw them this way?)

{| align="left"
|[[File:wm1415_Final_diels_alder_MO_diagram.PNG|550 px|thumb|left|MO Diagram showing the orbitals involved in the Diels-Alder reaction. While the ordering of the energies of the reactant frontier orbitals has been respected, the diagram is not drawn to scale and the transition state energy levels are skewed for clarity. Approximate energy levels calculated at the PM6-level approximation are listed in the table opposite.]]
|
{| class="wikitable" align="center"
!Ethene
!Energy / a.u.
!Butadiene
!Energy / a.u.
!Transition State
!Energy / a.u.
|-
|
|
|MO 13
|0.0636
|MO 19
|0.0307
|-
|LUMO
|0.0426
|LUMO
|0.0194
|MO 18
|0.0173
|-
|HOMO
| -0.3923
|HOMO
| -0.3590
|MO 17
| -0.3253
|-
|
|
|MO 10
| -0.4133
|MO 16
| -0.3276
|}
LUMO<sub>dienophile</sub> - HOMO<sub>diene</sub> = 0.4016 a.u.

LUMO<sub>diene</sub> - HOMO<sub>dienophile</sub> = 0.4117 a.u.

The smallest energy gap is that between the LUMO of ethene and the HOMO of butadiene (the antisymmetric orbitals). The interaction between these orbitals is stronger as there is greater orbital overlap: the reaction proceeds by electrons from the HOMO of the diene populating the LUMO of the dienophile (MO 16) - normal electron demand. The other bonding orbital in the TS, MO 17, is formed from the interaction between the HOMO of ethene and the LUMO of butadiene.
|}

The orbital overlap integral is non-zero for symmetric-symmetric and antisymmetric-antisymmetric orbital interactions; but the orbital overlap integral is zero for symmetric-antisymmetric interactions.
([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) You are missing the final part of this reasoning: this is why only S-S and AS-AS orbital interactions are allowed.)

The Woodward-Hoffmann rules can be used to predict whether a pericyclic reaction is allowed by symmetry. According to the Woodward-Hoffman rules, a cycloaddition is thermally allowed if the total number of (4q+2)<sub>s</sub> and (4r)<sub>a</sub> components is odd<sup>[6]</sup>. Here, the 's' label refers to a suprafacial component, which forms new bonds on the same side at both ends ('a' label - antarafacial, opposite sides). The Diels-Alder reaction is a [<sub>π</sub>4<sub>s</sub>+<sub>π</sub>2<sub>s</sub>] cycloaddition: (4q+2)<sub>s</sub> + (4r)<sub>a</sub> = 1, so it is allowed.

==Exercise 2==

===Optimised Reactants: B3LYP/6-31G(d)===

These Jmol files have been excluded from this page to speed up loading time. The applets can be accessed here: <jmolFile text="cyclohexadiene">wm1415_CYCLOHEXADIENE_B3LYP_OPT.LOG</jmolFile> and <jmolFile text="1,3-dioxole">wm1415_LAST_DIOXOLE_TRY_B3LYP.LOG</jmolFile>. There are no negative frequencies as these are stable structures.

===Endo/Exo Transition States and Products: B3LYP/6-31G(d)===

{| class = "wikitable" align = "center"
!Endo TS
!Endo Product
!Exo TS
!Exo Product
|-
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_ENDO_TS_B3LYP.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 36; mo 40; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<name>ex2_endoTS</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 36; mo 40; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>frame 36; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script>
<text>MO ON</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>frame 36; mo off</script>
<text>MO OFF</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==36) frame=37; vibrating=0; endif</script>
<script>frame 37; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 2 3 4 5 6]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>ex2_endoTS</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 36; mo 40; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>ex2_endoTS</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 40</script>
<text>MO 40</text>
<target>ex2_endoTS</target>
</item>
<item>
<script>mo 41</script>
<text>MO 41</text>
<target>ex2_endoTS</target>
</item>
<item>
<script>mo 42</script>
<text>MO 42</text>
<target>ex2_endoTS</target>
</item>
<item>
<script>mo 43</script>
<text>MO 43</text>
<target>ex2_endoTS</target>
</item>
</jmolmenu>
<jmolmenu>
<item>
<script>mo cutoff 0.02</script>
<text>Cutoff 0.02</text>
<target>ex2_endoTS</target>
</item>
<item>
<script>mo cutoff 0.01</script>
<text>Cutoff 0.01</text>
<target>ex2_endoTS</target>
</item>
</jmolmenu>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_ENDO_PRODUCTS_B3LYP_OPT.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 18; frank off</script>
<name>ex2_endoproduct</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 18; frank off</script>
<text>Reload</text>
<target>ex2_endoproduct</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex2_endoproduct</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==18) frame=19; vibrating=0; endif</script>
<script>frame 19; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Lowest Frequency</text>
<target>ex2_endoproduct</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 2 3 4 5 6]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>ex2_endoproduct</target>
</jmolbutton>
<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex2_endoproduct</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 18; frank off</script>
<text>Reload</text>
<target>ex2_endoproduct</target>
</jmolbutton>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_EXO_TS_B3LYP.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 18; mo 40; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<name>ex2_exoTS</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 18; mo 40; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>frame 18; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script>
<text>MO ON</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>frame 18; mo off</script>
<text>MO OFF</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==18) frame=19; vibrating=0; endif</script>
<script>frame 19; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 2 3 4 5 6]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>ex2_exoTS</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 18; mo 40; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>ex2_exoTS</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 40</script>
<text>MO 40</text>
<target>ex2_exoTS</target>
</item>
<item>
<script>mo 41</script>
<text>MO 41</text>
<target>ex2_exoTS</target>
</item>
<item>
<script>mo 42</script>
<text>MO 42</text>
<target>ex2_exoTS</target>
</item>
<item>
<script>mo 43</script>
<text>MO 43</text>
<target>ex2_exoTS</target>
</item>
</jmolmenu>
<jmolmenu>
<item>
<script>mo cutoff 0.02</script>
<text>Cutoff 0.02</text>
<target>ex2_exoTS</target>
</item>
<item>
<script>mo cutoff 0.01</script>
<text>Cutoff 0.01</text>
<target>ex2_exoTS</target>
</item>
</jmolmenu>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_EXO_PRODUCTS_B3LYP_OPT.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 14; frank off</script>
<name>ex2_exoproduct</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 14; frank off</script>
<text>Reload</text>
<target>ex2_exoproduct</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex2_exoproduct</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==14) frame=15; vibrating=0; endif</script>
<script>frame 15; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Lowest Frequency</text>
<target>ex2_exoproduct</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 2 3 4 5 6]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>ex2_exoproduct</target>
</jmolbutton>
<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex2_exoproduct</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 14; frank off</script>
<text>Reload</text>
<target>ex2_exoproduct</target>
</jmolbutton>
</jmol></center>
|}

===Endo/Exo IRC Paths (PM6 Level)===
{| class = "wikitable"
!Endo Internuclear distances
!Endo Reaction Animation
!Endo IRC Energy Plot
|-
|<center>[[File:wm1415_Ex2_endo_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Ex2_endo_resized.gif]]</center>
|<center>[[File:wm1415_ex2_Irc_energy_endo_cyclo.PNG|200 px]]</center>
|-
!Exo Internuclear distances
!Exo Reaction Animation
!Exo IRC Energy Plot
|-
|<center>[[File:wm1415_Ex2_exo_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Ex2_exo_resized.gif]]</center>
|<center>[[File:wm1415_ex2_Irc_energy_exo_cyclo.PNG|200 px]]</center>
|}

===MO Diagram: Diels-Alder Inverse Electron Demand===
([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) As before, even though you state that you are respecting the order of your calculated energies, you are not. The highest energy interacting orbital is the dioxole LUMO, not the TS LUMO +1 (which by the way has an energy of +0.0154, not -0.0154). Additionally, you make no comment on the difference in terms of relative energies between the exo and endo cases.)
{| align="left"
|[[File:wm1415_Ex2_mo_diagram.PNG|550 px|thumb|left|MO Diagram showing the FMOs. While the ordering of the energies of the reactant FMOs has been respected, the diagram is not drawn to scale and the transition state energy levels are skewed for clarity. Approximate energy levels calculated at the B3LYP 6-31G(d)-level approximation are listed in the table opposite.]]
|
{| class="wikitable" align="center"
!1,3-dioxole
!Energy / a.u.
!Cyclohexadiene
!Energy / a.u.
!Endo TS
!Energy / a.u.
|-
|
|
|MO 24
| 0.0874
|MO 43
| -0.0154
|-
|LUMO
|0.0380
|LUMO
| -0.0171
|MO 42
| -0.0046
|-
|HOMO
| -0.1959
|HOMO
| -0.2055
|MO 41
| -0.1905
|-
|
|
|MO 21
| -0.2994
|MO 40
| -0.1965
|}
LUMO<sub>dienophile</sub> - HOMO<sub>diene</sub> = 0.2435 a.u.

LUMO<sub>diene</sub> - HOMO<sub>dienophile</sub> = 0.1788 a.u.

In this case, it is the energy gap between the LUMO of the diene and the HOMO of the dienophile that is smallest and leads to the greatest orbital overlap. The dienophile, 1,3-dioxole, is electron-rich: electrons flow from its HOMO to the diene LUMO - this is inverse electron demand. This interaction can be visualised as MO 41 in both the endo and exo transition states. Interactions between the dienophile LUMO and diene HOMO form MO 40 in the transition state.
|}

Because the energies of the interacting orbitals are a better match than in the butadiene and ethene reaction, there is greater orbital overlap and hence greater stabilisation of the transition state: the reaction with the electron-rich 1,3-dioxole would be much faster.

===Energy Profile: Secondary Orbital Interactions===

{| align="left"
|[[File:wm1415_Ex2_energy_profile.png|300px]]
|The endo product is both the kinetic and thermodynamic product of the reaction: it is formed from a lower energy TS (difference of 8 kJ/mol) and releases more energy (difference of 3.6 kJ/mol) in the reaction. The exo product is comparatively destabilised because of the unfavourable steric clash between the heteroatomic 5-membered ring and the bridging ethyl group across the newly-formed 6-membered ring from the Diels-Alder reaction.

The endo transition state is stabilised by secondary orbital interactions (SOI)<sup>[7]</sup>. These interactions come about between atomic orbitals that are not involved in the formation or cleavage of σ bonds<sup>[7]</sup>. In the endo TS, the O atoms are aligned with the back of the diene. While this configuration is sterically unfavourable, it allows for the SOI between the O lone pairs and the orbitals at the back end of the diene LUMO. This is a favourable interaction which is only accessible to the endo TS, and more than makes up for the increased steric clash - resulting in a lower energy transition state. These interactions can be clearly seen in the Jmol files above: select MO 41 in the transition state structures (and adjusting the MO cutoff value if necessary).
|[[File:wm1415_SOI_interaction.PNG|200 px]]
{| class="wikitable"
|+ '''Free energies / kJ/mol'''
! <math>\Delta G^{o}_{endo}</math>
| -67.4
! <math>\Delta G^{o}_{exo}</math>
| -63.8
|-
! <math>\Delta G^{*}_{endo}</math>
| 160
! <math>\Delta G^{*}_{exo}</math>
| 168
|}
|}

[[Wm1415TransitionStates#Introduction|Back to Introduction.]]

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 12:21, 16 January 2018 (UTC) Your energies are correct well done. And your JMols ae very very nicely done. It is clear and concise too. When you are comparing the energies of the MOs of the different reactants you need to be careful here. This is because they are on different potential energy surfaces and hence the relative energies are different as the Hamiltonian has different amounts of terms in. To do this properly you would do an energy calculation at the last point of the reactants on the IRC and then look at the relative ordering of the MOs there with both reactants on the potential energy surface.

==Exercise 3==
===The Reaction between SO<sub>2</sub> and Xylylene===
Sulfur dioxide can undergo Diels-Alder and Cheletropic pericyclic reactions with dienes. Xylylene contains two diene fragments: what will be deemed the main fragment undergoes aromatisation in the considered reactions. The following sections contain calculations for the formation of the each of the possible products for both the main and second fragments of xylylene: from these calculations, a reaction profile will be constructed.

Jmol applets for PM6-optimised structures of <jmolFile text="xylylene">wm1415_XYLYLENE_OPT2_PM6.LOG</jmolFile> and <jmolFile text="sulfur dioxide">wm1415_SO2_OPT_PM6.LOG</jmolFile> have been excluded from the report, but can be opened in a new window from these links.

===Main Fragment: Optimised TS (PM6-level) and IRC Paths===

[[File:wm1415_Main_Fragment_Scheme.PNG|400 px]]

{| class = "wikitable" align = "center"
!Endo TS
!Exo TS
!Cheletropic TS
|-
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_ENDO_PRODUCT_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<name>ex3_endoTS</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<text>Reload</text>
<target>ex3_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex3_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==22) frame=23; vibrating=0; endif</script>
<script>frame 23; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex3_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[11 4 3 10 16 15]; label display; color label lime; font label 16</script>
<text>Labels ON</text>
<target>ex3_endoTS</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex3_endoTS</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<text>Reload</text>
<target>ex3_endoTS</target>
</jmolbutton>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_ENDO_PRODUCT_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<name>ex3_exoTS</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<text>Reload</text>
<target>ex3_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex3_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==22) frame=23; vibrating=0; endif</script>
<script>frame 23; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex3_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[11 4 3 10 16 15]; label display; color label lime; font label 16</script>
<text>Labels ON</text>
<target>ex3_exoTS</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex3_exoTS</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<text>Reload</text>
<target>ex3_exoTS</target>
</jmolbutton>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_CHELO_PM6_TS.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 12; frank off</script>
<name>ex3_cheloTS</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 12; frank off</script>
<text>Reload</text>
<target>ex3_cheloTS</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex3_cheloTS</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==12) frame=13; vibrating=0; endif</script>
<script>frame 13; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex3_cheloTS</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[11 4 3 10 16 15 17]; label display; color label lime; font label 16</script>
<text>Labels ON</text>
<target>ex3_cheloTS</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex3_cheloTS</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 12; frank off</script>
<text>Reload</text>
<target>ex3_cheloTS</target>
</jmolbutton>
</jmol></center>
|-
!Endo IRC Path
!Exo IRC Path
!Cheletropic IRC Path
|-
|<center>[[File:wm1415_Ex3_endo1_regif.gif|300 px]]</center>
|<center>[[File:wm1415_Ex3_exo1_regif.gif|300 px]]</center>
|<center>[[File:wm1415_Ex3_chelo1_regif.gif|300 px]]</center>
|-
!Endo Internuclear Distances
!Exo Internuclear Distances
!Cheletropic Internuclear Distances
|-
|<center>[[File:wm1415_Endo31_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Exo31_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Che1_internucdist.png|400 px]]</center>
|-
!<jmolFile text="Endo Product Jmol">wm1415_ENDO_PRODUCT_OPT_PM6.LOG</jmolFile>
!<jmolFile text="Exo Product Jmol">wm1415_INITIAL_PM6_OPT.LOG</jmolFile>
!<jmolFile text="Cheletropic Product Jmol">wm1415_CHELO_PRODUCTS_INITIAL_PM6_OPT.LOG</jmolFile>
|}

(Very nice JMol scripting and Python. Try to be consistent with respect to the direction along the IRC [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 16:00, 9 January 2018 (UTC))

===Second Fragment: Optimised TS (PM6-level) and IRC Paths===

[[File:wm1415_Second_fragment_scheme.PNG|400 px]]

{| class = "wikitable" align = "center"
!Endo TS
!Exo TS
!Cheletropic TS
|-
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_SECOND_ENDO_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 12; frank off</script>
<name>ex3_endoTS2</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 12; frank off</script>
<text>Reload</text>
<target>ex3_endoTS2</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex3_endoTS2</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==12) frame=13; vibrating=0; endif</script>
<script>frame 13; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex3_endoTS2</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 2 3 6 16 17 18]; label display; color label lime; font label 16</script>
<text>Labels ON</text>
<target>ex3_endoTS2</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex3_endoTS2</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 12; frank off</script>
<text>Reload</text>
<target>ex3_endoTS2</target>
</jmolbutton>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_SECOND_EXO_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 24; frank off</script>
<name>ex3_exoTS2</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 24; frank off</script>
<text>Reload</text>
<target>ex3_exoTS2</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex3_exoTS2</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==24) frame=25; vibrating=0; endif</script>
<script>frame 25; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex3_exoTS2</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 2 3 6 16 17 18]; label display; color label lime; font label 16</script>
<text>Labels ON</text>
<target>ex3_exoTS2</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex3_exoTS2</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 24; frank off</script>
<text>Reload</text>
<target>ex3_exoTS2</target>
</jmolbutton>
</jmol></center>
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>300</size>
<uploadedFileContents>wm1415_SECOND_CHELO_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<name>ex3_cheloTS2</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<text>Reload</text>
<target>ex3_cheloTS2</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>ex3_cheloTS2</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==22) frame=23; vibrating=0; endif</script>
<script>frame 23; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>ex3_cheloTS2</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 4 5 6 11 12 13]; label display; color label lime; font label 16</script>
<text>Labels ON</text>
<target>ex3_cheloTS2</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>ex3_cheloTS2</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 22; frank off</script>
<text>Reload</text>
<target>ex3_cheloTS2</target>
</jmolbutton>
</jmol></center>
|-
!Endo IRC Path
!Exo IRC Path
!Cheletropic IRC Path
|-
|<center>[[File:wm1415_Ex3_endo2_regif.gif|300 px]]</center>
|<center>[[File:wm1415_Ex3_exo2_regif.gif|300 px]]</center>
|<center>[[File:wm1415_Ex3_chelo2_regif.gif|300 px]]</center>
|-
!Endo Internuclear Distances
!Exo Internuclear Distances
!Cheletropic Internuclear Distances
|-
|<center>[[File:wm1415_Endo32_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Exo32_internucdist.png|400 px]]</center>
|<center>[[File:wm1415_Che2_internucdist.png|400 px]]</center>
|-
!<jmolFile text="Endo Product Jmol">wm1415_SECOND_ENDO_PRODUCTS_PM6_OPT.LOG</jmolFile>
!<jmolFile text="Exo Product Jmol">wm1415_SECOND_EXO_PRODUCTS_PM6_OPT.LOG</jmolFile>
!<jmolFile text="Cheletropic Product Jmol">wm1415_SECOND_CHELO_PRODUCTS_OPT_PM6.LOG</jmolFile>
|}

===Reaction Profile===

[[File:wm1415_Energy_profile_ex3.png|400 px|left]]
{| align="left"
|
{| class="wikitable"
|+ '''Main Fragment: Free energies / kJ/mol'''
! <math>\Delta G^{o}_{endo}</math>
| -99.2
! <math>\Delta G^{o}_{exo}</math>
| -84.1
! <math>\Delta G^{o}_{chel}</math>
| -156
|-
! <math>\Delta G^{*}_{endo}</math>
| 81.6
! <math>\Delta G^{*}_{exo}</math>
| 85.6
! <math>\Delta G^{*}_{chel}</math>
| 104
|}

{| class="wikitable"
|+ '''Second Fragment: Free energies / kJ/mol'''
! <math>\Delta G^{o}_{endo}</math>
| 16.1
! <math>\Delta G^{o}_{exo}</math>
| 20.6
! <math>\Delta G^{o}_{chel}</math>
| 47.1
|-
! <math>\Delta G^{*}_{endo}</math>
| 112
! <math>\Delta G^{*}_{exo}</math>
| 120
! <math>\Delta G^{*}_{chel}</math>
| 141
|}
|
|}

(Use straight lines for reaction profiles [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 16:00, 9 January 2018 (UTC))

Reaction with the second fragment are clearly very thermodynamically and kinetically unfavourable: the overall reactions are endothermic, and all activation energies are larger than in reactions with the main fragment. The products from these theoretical reactions would not be observed experimentally.

Reaction with the main fragment is much more favourable on account of the formation of the benzene ring. The aromatisation confers stability to both the products and the transition states. The cheletropic reaction yields the thermodynamic product, which contains the more stable 5-membered ring and an additional strong S=O bond. However, the chelotropic rearrangement proceeds via a considerably higher reaction barrier, and at very low temperatures one might observe the kinetic product: the endo Diels-Alder product. The endo configuration has a stabilised TS due to secondary orbital interactions, as discussed previously. Endo-selectivity for this Diels-Alder reaction is reported by the literature<sup>[8]</sup>.

The reaction between xylylene and SO<sub>2</sub> has been performed in the literature<sup>[8]</sup>. At 20 <sup>o</sup>C, SO<sub>2</sub> adds reversibly and a 1:9 mixture of the Diels-Alder and cheletropic products was obtained. Low temperatures are required for the reaction to proceed, because of the reduction in entropy as SO<sub>2</sub> gas is consumed. Prior simulations of this reaction have included a second molecule of SO<sub>2</sub> that may help stabilise the Diels-Alder transition state<sup>[9]</sup>.

==Extension: 4π Electrocyclic Ring Closure with Conrotation (PM6-Level)==
According to the Woodward-Hoffmann rules, an electrocyclic reaction involving 4n π electrons should proceed via a conrotary mechanism<sup>[6]</sup>. The subject of this extension is to demonstrate this stereospecifity, using the same methods as previously in this experiment:

[[File:wm1415_Extension_reaction_scheme.PNG|400 px]]

{| class = "wikitable" align = "center"
!Electrocyclic TS
!IRC Path Animation
!IRC Energy Plot
|-
|<center><jmol>
<jmolApplet>
<color>#F9F9F9</color>
<size>350</size>
<uploadedFileContents>wm1415_EXT_TS_PM6.LOG</uploadedFileContents>
<script>vibrating=0; spinning=0; frame 92; mo 24; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<name>extension</name>
</jmolApplet>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 92; mo 24; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>frame 92; mo on; mo nodots nomesh fill translucent; mo titleformat ""</script>
<text>MO ON</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>frame 92; mo off</script>
<text>MO OFF</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>if(spinning==0) spinning=1; spin; else; spinning=0; spin off; endif</script>
<text>Spin</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>if(frame==92) frame=93; vibrating=0; endif</script>
<script>frame 93; if(vibrating==0) vibrating=1; vibration 1.5; else; vibrating=0; vibration off; endif</script>
<text>Toggle Imaginary Frequency</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>select atomno=[1 3 5 7]; label display; color label lime; font label 16</script>
<text>Carbon Labels ON</text>
<target>extension</target>
</jmolbutton>

<jmolbutton>
<script>label hide</script>
<text>Labels OFF</text>
<target>extension</target>
</jmolbutton>
<jmolbutton>
<script>vibrating=0; spinning=0; frame 92; mo 24; mo nodots nomesh fill translucent; mo cutoff 0.02; mo titleformat ""; frank off</script>
<text>Reload</text>
<target>extension</target>
</jmolbutton>
<jmolmenu>
//The dropdown menu. Each item has to be declared individually and can execute script
<item>
<script>mo 24</script>
<text>MO 24</text>
<target>extension</target>
</item>
<item>
<script>mo 25</script>
<text>MO 25</text>
<target>extension</target>
</item>
<item>
<script>mo 26</script>
<text>MO 26</text>
<target>extension</target>
</item>
<item>
<script>mo 27</script>
<text>MO 27</text>
<target>extension</target>
</item>
</jmolmenu>
<jmolmenu>
<item>
<script>mo cutoff 0.02</script>
<text>Cutoff 0.02</text>
<target>extension</target>
</item>
<item>
<script>mo cutoff 0.01</script>
<text>Cutoff 0.01</text>
<target>extension</target>
</item>
</jmolmenu>
</jmol></center>
|[[File:wm1415_Ext_gif.gif]]
|[[File:wm1415_ext_Irc_path_energy.PNG|250 px]]
|}

The calculated IRC path clearly shows the conrotation about the migrating carbon atoms, confirming the predicted stereochemistry of the products.

(Good additional section. What would happen on the excited state? [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 16:04, 9 January 2018 (UTC))

(+6%)

==Conclusion==
In this experiment, the transition states were successfully calculated for a variety of pericyclic reactions, using both the semi-empirical PM6 and the DFT B3LYP 6-31G(d) levels of approximation. The results agreed with predictions laid out by the Woodward-Hoffmann rules. The Diels-Alder reaction between butadiene and ethene was found to proceed via normal electron demand; using an electron-rich dienophile, 1,3-dioxole, resulted in inverse electron demand. In the reaction between 1,3-dioxole and cyclohexadiene, the endo product was found to be both the kinetic and thermodynamic product: the endo transition state is stabilised by secondary orbital interactions, and the exo product is destabilised by unfavourable steric clashes.

The rearomatisation was confirmed to be the driving force for the pericyclic reactions of SO<sub>2</sub> with xylylene, as reactions with the second diene fragment yielded kinetically and thermodynamically very unfavourable products. The thermodynamic product arises from the cheletropic mechanism, but the kinetic product arises from the faster, reversible Diels-Alder cycloaddition - this finding was backed up with experimental observations from literature<sup>[8]</sup>.

The Woodward-Hoffmann rules were further investigated: the predicted stereochemistry of the 4π electrocyclic reaction was obtained from these theoretical calculations.

==References==

1 D. Wales, in Energy Landscapes: Applications to Clusters, Biomolecules and Glasses, Cambridge University Press, 2004, pp. 192–240.

2 P. J. Stephens, F. J. Devlin, C. F. Chabalowski and M. J. Frisch, J. Phys. Chem., 1994, '''98''', 11623–11627.

3 S. F. Sousa, P. A. Fernandes and M. J. Ramos, J. Phys. Chem. A, 2007, '''111''', 10439–10452.

4 J. J. P. Stewart, J. Mol. Model., 2007, '''13''', 1173–1213.

5 J. J. P. Stewart, J. Mol. Model., 2009, '''15''', 765–805.

6 R. B. Woodward and R. Hoffmann, Angew. Chemie Int. Ed. English, 1969, '''8''', 781–853.

7 A. Arrieta, F. P. Cossío and B. Lecea, J. Org. Chem., 2001, '''66''', 6178–6180.

8 B. Deguin and P. Vogel, J. Am. Chem. Soc., 1992, '''114''', 9210–9211.

9 T. Fernandez, J. A. Sordo, F. Monnat, B. Deguin and P. Vogel, J. Am. Chem. Soc., 1998, '''120''', 13276–13277.

Rep:Wm1415CMPbackup

2025-09-01T09:50:01Z

Move page script: Move page script moved page Wm1415CMPbackup to Rep:Wm1415CMPbackup: Move to report namespace

'''<big>This is William Micou's report on the compulsory CMP experiment: Ising Model.</big>'''

==Introduction to the Ising Model==

'''<big>TASK</big>: Show that the lowest possible energy for the Ising model is <math>E\ =\ -DNJ</math>, where <math>D</math> is the number of dimensions and <math>N</math> is the total number of spins. What is the multiplicity of this state? Calculate its entropy.'''

In the absence of an external magnetic field, the energy of the lattice depends on the interaction energy <math>E</math>between adjacent sites<sup>[1]</sup>:

<center><math>E = - \frac{1}{2} J \sum_i^N \sum_{j\ \in\ \mathrm{neighbours}\left(i\right)} s_i s_j</math></center>

Thus, the energy is minimised when neighbouring spins are aligned. <math>J</math> is the exchange energy and comes about from the Pauli Exclusion Principle<sup>[2]</sup>: electrons cannot occupy the same quantum state. Electrons on neighbouring atoms with spins aligned cannot occupy the same space - electron repulsion is reduced and the total energy is lowered<sup>[2]</sup>.

The state with the lowest possible energy is the state with all spins aligned. To calculate its energy, the total number of interactions must be counted.

For an infinite lattice, there are 2 nearest neighbours per dimension (degree of freedom): number of pairs of interactions <math> =\ 2ND</math>.

Hence, total number of interactions <math>=\frac{1}{2}\times2ND = ND</math>.

Since all spins are aligned, every interaction will contribute <math>-J</math> to the energy:

<math>E=-J\times ND=-DNJ</math> as required.

This lowest energy state is comprised of 2 microstates: all spins up, and all spins down. Thus, the entropy <math>S</math> is:

<math>S=k_{B}\ln\Omega=k_{B}\ln2</math>

'''<big>TASK</big>: Imagine that the system is in the lowest energy configuration. To move to a different state, one of the spins must spontaneously change direction ("flip"). What is the change in energy if this happens (<math>D=3,\ N=1000</math>)? How much entropy does the system gain by doing so?'''

Lowest energy state is -3000J

The spin being flipped has 6 nearest neighbours. Its 6 interactions with the lattice contribute -6J to the energy. After flipping, the spin now contributes +6J to the energy.

Energy of one spin flipped state: -2988J

Difference of +12J

There are 2000 microstates for this system: 1000 all but one spin up states, or 1000 all but one spin down states.

<math>S=k_{B}\ln\Omega=k_{B}\ln2000-k_{B}\ln2=k_{B}\ln1000</math>

'''<big>TASK</big>: Calculate the magnetisation of the 1D and 2D lattices in figure 1. What magnetisation would you expect to observe for an Ising lattice with <math>D = 3,\ N=1000</math> at absolute zero?'''

1D lattice: magnetisation = +1

2D lattice: magnetisation = +1

At absolute zero, entropic effects vanish and the system will remain in the lowest energy state, with all spins aligned. Thus the magnetisation would be either +N or -N (+1000 or -1000).

==Calculating the energy and magnetisation==

'''<big>TASK</big>: complete the functions energy() and magnetisation(), which should return the energy of the lattice and the total magnetisation, respectively. In the energy() function you may assume that <math>J=1.0</math> at all times (in fact, we are working in ''reduced units'' in which <math>J=k_B</math>, but there will be more information about this in later sections). Do not worry about the efficiency of the code at the moment — we will address the speed in a later part of the experiment.'''

Below is the initial, inefficient code for energy() and magnetisation() using indented for loops.

<pre>
def energy(self):
"Return the total energy of the current lattice configuration."
energy = 0.0
for i in range(0,self.n_rows):
for j in range(0,self.n_cols):
start = int(self.lattice[i][j])
top = int(self.lattice[i-1][j])
bot = int(self.lattice[(i+1)%self.n_rows][j])
left = int(self.lattice[i][j-1])
right = int(self.lattice[i][(j+1)%self.n_cols])
total_interaction = (start*top)+(start*bot)+(start*left)+(start*right)
energy = energy + (-0.5*total_interaction)
return energy

def magnetisation(self):
"Return the total magnetisation of the current lattice configuration."
magnetisation = 0
for i in range(0,self.n_rows):
for j in range(0,self.n_cols):
magnetisation = magnetisation + int(self.lattice[i][j])
return magnetisation
</pre>

The energy() method loops over every spin in the lattice and adds up every interaction. The modulus allows for boundary conditions for spins at the edges of our lattice. The magnetisation() method simply adds up every spin value in the lattice.

'''<big>TASK</big>: Run the ILcheck.py script from the IPython Qt console using the command <pre>%run ILcheck.py</pre> The displayed window has a series of control buttons in the bottom left, one of which will allow you to export the figure as a PNG image. Save an image of the ILcheck.py output, and include it in your report.'''

[[File:wm1415cmp_first_check.png|500px|thumb|center|ILcheck.py generates 3 lattices and compares the expected energy and magnetisation with the values calculated from the above methods. The initial energy() and magnetisation() methods are in agreement with expected values: the code works.]]

==Introduction to the Monte Carlo simulation==

'''<big>TASK</big>: How many configurations are available to a system with 100 spins? To evaluate these expressions, we have to calculate the energy and magnetisation for each of these configurations, then perform the sum. Let's be very, very, generous, and say that we can analyse <math>1\times 10^9</math> configurations per second with our computer. How long will it take to evaluate a single value of <math>\left\langle M\right\rangle_T</math>?'''

For a system of 100 spins, there are <math>2^{100}</math> configurations. Evaluating <math>10^9</math> calculations per second, it would take around <math>1.3\times10^{21}</math> seconds to finish the calculation - around 4 orders of magnitude larger than the age of the Universe. Clearly, this is impractical: importance sampling is a method of circumventing this.

'''<big>TASK</big>: Implement a single cycle of the above algorithm in the montecarlocycle(T) function. This function should return the energy of your lattice and the magnetisation at the end of the cycle. You may assume that the energy returned by your energy() function is in units of <math>k_B</math>! Complete the statistics() function. This should return the following quantities whenever it is called: <math><E>, <E^2>, <M>, <M^2></math>, and the number of Monte Carlo steps that have elapsed.'''

<pre>
def montecarlostep(self, T):
"Performs a single cycle of the Monte Carlo algorithm presented in the lab script."
# first: increase the number of steps elapsed
self.n_cycles = self.n_cycles + 1
# introduce a variable that will indicate whether the new configuration should be rejected or kept
accept_configuration = True
# energy of the original lattice (E_0)
energy_0 = self.energy()
# select coordinates for a random spin in the lattice
random_i = np.random.choice(range(0, self.n_rows))
random_j = np.random.choice(range(0, self.n_cols))
# flip the spin in the lattice
self.lattice[random_i, random_j] = -1 * self.lattice[random_i, random_j]
# calculate new lattice energy (E_1)
energy_1 = self.energy()
# calculate the difference in energy
delta = energy_1 - energy_0
# if the new configuration is lower in energy, keep the new configuration
if delta < 0:
accept_configuration = True

# choose a random number in the range [0,1)
random_number = np.random.random()
boltzmann_factor = np.exp(-delta / T)
# if the new configuration is higher in energy, compare random number to Boltzmann factor
if delta >= 0:
if random_number <= boltzmann_factor:
accept_configuration = True
else:
accept_configuration = False

if accept_configuration == False:
# return to original lattice
self.lattice[random_i, random_j] = -1 * self.lattice[random_i, random_j]

# keep running totals of data for n_cycles > 250000 steps
discount = 250000
if self.n_cycles > discount:
self.E = self.E + self.energy()
self.E2 = self.E2 + (self.energy()) ** 2
self.M = self.M + self.magnetisation()
self.M2 = self.M2 + (self.magnetisation()) ** 2
# return the post-MMC step energy and magnetisation as a list
return [self.energy(), self.magnetisation()]

def statistics(self):
"Calculates the average values of E, E*E (E2), M, M*M(M2) and returns them."
# Discount the first 250000 steps.
discount = 250000
avg_E = self.E / (self.n_cycles-discount)
avg_E2 = self.E2 / (self.n_cycles-discount)
avg_M = self.M / (self.n_cycles-discount)
avg_M2 = self.M2 / (self.n_cycles-discount)
return [avg_E, avg_E2, avg_M, avg_M2, self.n_cycles]
</pre>

This code comes from later in the assignment - when the methods are modified to discard the first 250000 steps from the running averages of energy and magnetisation.

'''<big>TASK</big>: If <math>T < T_C</math>, do you expect a spontaneous magnetisation (i.e. do you expect <math>\left\langle M\right\rangle \neq 0</math>)? When the state of the simulation appears to stop changing (when you have reached an equilibrium state), use the controls to export the output to PNG and attach this to your report. You should also include the output from your statistics() function.'''

The original ILanim.py expected only 4 entries from the statistics() method:

<pre>
E, E2, M, M2 = il.statistics()
print("Averaged quantities:")
print("E = ", E/spins)
print("E*E = ", E2/spins/spins)
print("M = ", M/spins)
print("M*M = ", M2/spins/spins)
</pre>

I modified the code to expect 5 entries from the statistics() method, which also outputs the number of cycles elapsed:

<pre>
E, E2, M, M2, n = il.statistics()
print("Averaged quantities:")
print("E = ", E/spins)
print("E*E = ", E2/spins/spins)
print("M = ", M/spins)
print("M*M = ", M2/spins/spins)
print("No. cycles = ", n)
</pre>

The figure below shows the outcome of running ILanim.py for an 8x8 lattice, at temperature = 1. This is below the critical temperature: the figure shows the system converged to the lowest energy state, with all spins aligned. The atoms do not have sufficient thermal energy to flip their spins: the exchange effect dominates over the entropic effect at low temperatures - spontaneous magnetisation is expected.

<center>[[File:wm1415_Second_animpy.png|600px]]</center>

The output of statistics():

<pre>
Averaged quantities:
E = -1.44239045383
E*E = 2.44085607394
M = 0.732834507042
M*M = 0.641262287265
No. cycles = 639
</pre>

These are averaged quantities over the course of the rearrangement of the lattice: no cycles were discarded in the calculation of the averages (discount variable set to zero).

==Accelerating the code==

'''<big>TASK</big>: Use the script ILtimetrial.py to record how long your ''current'' version of IsingLattice.py takes to perform 2000 Monte Carlo steps. This will vary, depending on what else the computer happens to be doing, so perform repeats and report the error in your average!'''

9.74 <math>\pm</math> 0.26 seconds before optimisation (10 trials)

'''<big>TASK</big>: Look at the documentation for the [http://docs.scipy.org/doc/numpy/reference/generated/numpy.sum.html NumPy sum function]. You should be able to modify your magnetisation() function so that it uses this to evaluate M. The energy is a little trickier. Familiarise yourself with the NumPy [http://docs.scipy.org/doc/numpy/reference/generated/numpy.roll.html roll] and [http://docs.scipy.org/doc/numpy/reference/generated/numpy.multiply.html multiply] functions, and use these to replace your energy double loop (you will need to call roll and multiply twice!).'''

<pre>
def energy(self):
"Return the total energy of the current lattice configuration."
# Roll the lattice vertically and multiply with original to find total vertical interactions
vertical_interactions = np.multiply(self.lattice, np.roll(self.lattice, 1, axis=0))
# Roll the lattice horizontally and multiply with original to find total horizontal interactions
horizontal_interactions = np.multiply(self.lattice, np.roll(self.lattice, 1, axis=1))
return (-1)*(np.sum(vertical_interactions + horizontal_interactions))

def magnetisation(self):
"Return the total magnetisation of the current lattice configuration."
# Sum every spin in the lattice to obtain the total magnetisation.
return np.sum(self.lattice)
</pre>

'''<big>TASK</big>: Use the script ILtimetrial.py to record how long your ''new'' version of IsingLattice.py takes to perform 2000 Monte Carlo steps. This will vary, depending on what else the computer happens to be doing, so perform repeats and report the error in your average!'''

0.21 <math>\pm</math> 0.01 seconds after second optimisation (10 trials) (new code)

==The effect of temperature==

'''<big>TASK</big>: The script ILfinalframe.py runs for a given number of cycles at a given temperature, then plots a depiction of the ''final'' lattice state as well as graphs of the energy and magnetisation as a function of cycle number. This is much quicker than animating every frame! Experiment with different temperature and lattice sizes. How many cycles are typically needed for the system to go from its random starting position to the equilibrium state? Modify your statistics() and montecarlostep() functions so that the first N cycles of the simulation are ignored when calculating the averages. You should state in your report what period you chose to ignore, and include graphs from ILfinalframe.py to illustrate your motivation in choosing this figure.'''
{| class="wikitable"
!Lattice
!Temperature
!ILfinalframe.py
!Comments
|-
|6x6
|1.0
|[[File:wm1415_Final_frame5_1.png|200px|thumb|left]][[File:wm1415_Final_frame5_2.png|200px|thumb|right]]
|System stabilises after ~170 steps
|-
|8x8
|1.0
|[[File:wm1415_Final_frame1_1.png|200px|thumb|left]][[File:wm1415_Final_frame1_2.png|200px|thumb|right]]
|System stabilises after 600-700 steps
|-
|8x8
|2.0
|[[File:wm1415_Final_frame_9_1.png|200px|thumb|left]][[File:wm1415_Final_frame_9_2.png|200px|thumb|right]]
|System stabilises after ~2600 steps
|-
|8x8
|3.0
|[[File:wm1415_Final_frame8_1.png|200px|thumb|center]]
|Above the critical temperature: spins flip rapidly and the system starts fluctuating from the first step.
|-
|10x10
|0.5
|[[File:wm1415_Final_frame4_1.png|200px|thumb|left]][[File:wm1415_Final_frame4_2.png|200px|thumb|right]]
|System stabilises after ~1800 steps
|-
|10x10
|1.5
|[[File:wm1415_Final_frame3_1.png|200px|thumb|left]][[File:wm1415_Final_frame3_2.png|200px|thumb|right]]
|System stabilises after ~2200 steps
|-
|10x10
|3.0
|[[File:wm1415_Final_frame2_1.png|200px|thumb|center]]
|Above the critical temperature: spins flip rapidly and the system starts fluctuating from the first step.
|-
|15x15
|1.0
|[[File:wm1415_Final_frame7_1.png|200px|thumb|center]]
|Magnetisation appears to converge after 1-2000 steps, but then much later on the final state is found.
|-
|32x32
|0.5
|[[File:wm1415_32x32_t05_clump.png|200px|thumb|left|'Clumping']][[File:wm1415_32x32_t05_noclump2.png|200px|thumb|right|No 'clumping']]
|The larger 32x32 lattice can either find the lowest possible energy state,

or get stuck in a metastable clumped state - even with the same initial conditions.
|-
|32x32
|1.0
|[[File:wm1415_32x32_t1_clump2.png|200px|thumb|left|'Clumping']][[File:Wm1415_32x32_t1_noclump.png|200px|thumb|right|No 'clumping']]
|The larger 32x32 lattice can either find the lowest possible energy state,

or get stuck in a metastable clumped state - even with the same initial conditions.
|-
|32x32
|3.0
|[[File:wm1415_32x32_T3.png|200px|thumb|center]]
|Above the critical temperature, the larger 32x32 lattice reaches a

'final' state after around 5000 steps: many more steps required cf smaller lattices.
|}

The number of steps required to converge to the 'final' state increases with lattice size. Consider the case below the critical temperature - we expect to observe spontaneous magnetisation and for all spins to align. For larger lattices, it may take many steps to flip the remaining sites since they are chosen entirely at random - and the probability of choosing the 'correct' site to flip decreases with lattice size. For the smaller lattice sizes (<16x16), the system behaves as expected within at most a few thousand steps.

Larger lattices are prone to 'clumping' - the entire system is not in the same spin state, but large regions of spin-aligned atoms form. These are metastable states: spins have aligned within the clump to lower the energy, but there is not enough thermal energy in the system to break up neighbouring clumps of opposing spins. A local minimum of the PES has been found, but the global minimum requires crossing an activation barrier. A good example of this is the 32x32 lattice above: using exactly the same conditions, the system can either reach the lowest possible energy state or show clumping.

For the effect of temperature, consider the 8x8 lattice shown above. Well below the critical temperature, there is not sufficient thermal energy to cause large fluctuations in the spin states of the system and spontaneous magnetisation occurs quickly. Approaching the critical temperature, more fluctuations are possible, and more steps are required for the system to finally reach the lowest possible energy state. Above the critical temperature, spontaneous magnetisation no longer occurs, and entropic effects start to dominate - from the first step the entire system is fluctuating rapidly.

'''<big>TASK</big>: Use ILtemperaturerange.py to plot the average energy and magnetisation for each temperature, ''with error bars'', for an <math>8\times 8</math> lattice. Use your initution and results from the script ILfinalframe.py to estimate how many cycles each simulation should be. The temperature range 0.25 to 5.0 is sufficient. Use as many temperature points as you feel necessary to illustrate the trend, but do not use a temperature spacing larger than 0.5. T NumPy function savetxt() stores your array of output data on disk — you will need it later. Save the file as ''8x8.dat'' so that you know which lattice size it came from.'''

The experiment runs for 500000 steps, with the first 250000 steps discarded from the calculation of the average energies. This is a compromise between allowing the larger lattice systems a better chance of converging to the final state, and running the experiment on a reasonable timescale. The results from the 8x8 lattice, with error bars:

{| align=center
|-
|[[File:wm1415_New8x8energyplot.png|400px]]
|[[File:wm1415_New8x8magplot.png|400px]]
|}

==The effect of system size==

'''<big>TASK</big>: Repeat the final task of the previous section for the following lattice sizes: 2x2, 4x4, 8x8, 16x16, 32x32. Make sure that you name each datafile that your produce after the corresponding lattice size! Write a Python script to make a plot showing the energy ''per spin'' versus temperature for each of your lattice sizes. Hint: the NumPy loadtxt function is the reverse of the savetxt function, and can be used to read your previously saved files into the script. Repeat this for the magnetisation. As before, use the plot controls to save your a PNG image of your plot and attach this to the report. How big a lattice do you think is big enough to capture the long range fluctuations?'''

{| class="wikitable"
|-
!Lattice Size
!Energy plot
!Magnetisation plot
|-
|2x2
|[[File:wm1415_New2x2energyplot.png|400px|center]]
|[[File:wm1415_New2x2magplot.png|400px|center]]
|-
|4x4
|[[File:wm1415_New4x4energyplot.png|400px|center]]
|[[File:wm1415_New4x4magplot.png|400px|center]]
|-
|8x8
|[[File:wm1415_New8x8energyplot.png|400px|center]]
|[[File:wm1415_New8x8magplot.png|400px|center]]
|-
|16x16
|[[File:wm1415_New16x16energyplot.png|400px|center]]
|[[File:wm1415_New16x16magplot.png|400px|center]]
|-
|32x32
|[[File:wm1415_New32x32energyplot.png|400px|center]]
|[[File:wm1415_New32x32magplot.png|400px|center]]
|-
|64x64
|[[File:wm1415_New64x64energyplot.png|400px|center]]
|[[File:wm1415_New64x64magplot.png|400px|center]]
|-
|All
|[[File:wm1415_Newener_comp.png|400px|center]]
|[[File:wm1415_Newmag_comp.png|400px|center]]
|}

For each finite lattice size plot, there is a noticeable trend in the size of the error bars. These represent the number of energy and magnetisation states available at each temperature. Far below <math>T_C</math>, the error bars are almost negligible: there is spontaneous magnetisation and there is not sufficient thermal energy in the system to overcome the activation barrier in flipping the spins. From around T = 1.5, more states become accessible to the system and the error bars steadily increase in size until the critical temperature is reached, where the system undergoes a secondary phase transition and the error bars are at their largest: the states available at this temperature span the largest range of energies and magnetisations. At <math> T > T_C</math>, the system has enough thermal energy to undergo fluctuations and the error bars remain considerably large.

As the size of the lattice increases, the size of the error bars decrease. For a 2x2 lattice, a single spin flip represents a huge change in the energy and magnetisation of the whole system. For a 64x64 lattice, the difference in energy and magnetisation for the whole system when a single spin flips is almost negligible. The fluctuations result in smaller fractional changes in the system as the lattice size increases, hence the error bars decrease.

When plotted on the same graph, the energy and magnetisation plots appear to converge for the higher lattice sizes. The 16x16 and higher lattices are big enough to capture the long-range fluctuations: a single spin flip brings about a finer change in the whole system's energy.

==Determining the heat capacity==
'''<big>TASK</big>: By definition,'''

<math>C = \frac{\partial \left\langle E\right\rangle}{\partial T}</math>

'''From this, show that'''

<math>C = \frac{\mathrm{Var}[E]}{k_B T^2}</math>

(Where <math>\mathrm{Var}[E]</math> is the variance in <math>E</math>.)

The average energy can be related to the partition function<sup>[3]</sup>, <math>Z=\sum\exp\left(-\beta\varepsilon_{i}\right)</math>, with <math>\beta=\frac{1}{k_{B}T}</math> and <math>\varepsilon_{i}</math> being the energy of each state <math>i</math>:

<center><math>\langle E\rangle=\frac{E}{N}=\frac{1}{N}\sum N_{i}\varepsilon_{i}=\frac{1}{Z}\sum\varepsilon_{i}\exp\left(-\beta\varepsilon_{i}\right)=-\frac{1}{Z}\frac{\partial Z}{\partial\beta}=-\frac{\partial\ln Z}{\partial\beta}</math></center>

Similarly, the expectation value for <math>\langle E^{2}\rangle</math> can be calculated from the partition function:

<center><math>\langle E^{2}\rangle=\sum\varepsilon_{i}^{2}p_{i}=\frac{\sum\varepsilon_{i}^{2}\exp\left(-\beta\varepsilon_{i}\right)}{\sum\exp\left(-\beta\varepsilon_{i}\right)}=\frac{1}{Z}\frac{\partial^{2}}{\partial\beta^{2}}\left[\sum\exp\left(-\beta\varepsilon_{i}\right)\right]=\frac{1}{Z}\frac{\partial^{2}Z}{\partial\beta^{2}}</math></center>

Variance, <math>Var\left[E\right]</math>, can be expressed as:

<center><math>Var\left[E\right]=\langle E^{2}\rangle-\langle E\rangle^{2}</math></center>

<center><math>Var\left[E\right]=\frac{1}{Z}\frac{\partial^{2}Z}{\partial\beta^{2}}-\left(-\frac{1}{Z}\frac{\partial Z}{\partial\beta}\right)^{2}</math></center>

Let us show that this expression for variance is equivalent to <math>\frac{\partial}{\partial\beta}\left(\frac{1}{Z}\frac{\partial Z}{\partial\beta}\right)</math>:

<center><math>\frac{\partial}{\partial\beta}\left(\frac{1}{Z}\frac{\partial Z}{\partial\beta}\right)=\frac{1}{Z}\frac{\partial^{2}Z}{\partial\beta^{2}}+\frac{\partial\left(\frac{1}{Z}\right)}{\partial\beta}\frac{\partial Z}{\partial\beta}=\frac{1}{Z}\frac{\partial^{2}Z}{\partial\beta^{2}}+\left(\frac{\partial\left(\frac{1}{Z}\right)}{\partial Z}\frac{\partial Z}{\partial\beta}\right)\frac{\partial Z}{\partial\beta}=\frac{1}{Z}\frac{\partial^{2}Z}{\partial\beta^{2}}-\frac{1}{Z^{2}}\left(\frac{\partial Z}{\partial\beta}\right)^{2}=Var\left[E\right]</math></center>

Hence:

<center><math>Var\left[E\right]=\frac{\partial}{\partial\beta}\left(\frac{1}{Z}\frac{\partial Z}{\partial\beta}\right)=\frac{\partial^{2}\ln Z}{\partial\beta^{2}}</math></center>

But, since <math>\langle E\rangle=-\frac{\partial\ln Z}{\partial\beta}</math>, and <math>\frac{\partial}{\partial\beta}=\frac{\partial T}{\partial\beta}\frac{\partial}{\partial T}=\left(\frac{1}{\frac{\partial\beta}{\partial T}}\right)\frac{\partial}{\partial T}=\left(-k_{B}T^{2}\right)\frac{\partial}{\partial T}</math>, the above can be rewritten as<sup>[3]</sup>:

<center><math>Var\left[E\right]=\frac{\partial^{2}\ln Z}{\partial\beta^{2}}=-\frac{\partial}{\partial\beta}\left(-\frac{\partial\ln Z}{\partial\beta}\right)=-\frac{\partial}{\partial\beta}\langle E\rangle=-\left(-k_{B}T^{2}\frac{\partial}{\partial T}\right)\langle E\rangle=k_{B}T^{2}C_{V}</math></center>

Therefore:

<center><math>C_{V}=\frac{Var\left[E\right]}{k_{B}T^{2}}</math></center>

'''<big>TASK</big>: Write a Python script to make a plot showing the heat capacity versus temperature for each of your lattice sizes from the previous section. You may need to do some research to recall the connection between the variance of a variable, <math>\mathrm{Var}[X]</math>, the mean of its square <math>\left\langle X^2\right\rangle</math>, and its squared mean <math>\left\langle X\right\rangle^2</math>. You may find that the data around the peak is very noisy — this is normal, and is a result of being in the critical region. As before, use the plot controls to save your a PNG image of your plot and attach this to the report.'''

{| class="wikitable"
|-
!2x2 Heat Capacity
!4x4 Heat Capacity
!8x8 Heat Capacity
!16x16 Heat Capacity
!32x32 Heat Capacity
!64x64 Heat Capacity
!Combined
|-
|[[File:wm1415_Newheatcap_2x2.png|200px|center]]
|[[File:wm1415_Newheatcap_4x4.png|200px|center]]
|[[File:wm1415_Newheatcap_8x8.png|200px|center]]
|[[File:wm1415_Newheatcap_16x16.png|200px|center]]
|[[File:wm1415_Newheatcap_32x32.png|200px|center]]
|[[File:wm1415_Newheatcap_64x64.png|200px|center]]
|[[File:wm1415_Newheatcap_comp.png|200px|center]]
|}

These plots show the heat capacities for each lattice size, using the same data from the previous section. In all cases, the regions <math>T << T_C</math> and <math>T >> T_C</math> are well-resolved, and in the next section it will be shown that they map to the reference C++ data perfectly. However, in the region of the critical temperature, larger lattices give a very noisy plot. Further repeats of larger lattices at the critical temperature will be explored later in the report.

The last plot, showing the heat capacity curves for all lattice sizes, somewhat demonstrates an equation stated in the lab script, relating the temperature at which heat capacity is maximised with lattice size:

<center><math>T_{C, L} = \frac{A}{L} + T_{C,\infty}</math></center>

From this, one would expect the peak in the heat capacity to shift to lower temperatures as the lattice size increases. This can be seen up to the 8x8 lattice: above this, only one repeat of the experiment is not sufficient to accurately map the critical region and the trend is not clear for this example.

==Locating the Curie temperature==

'''<big>TASK</big>: A C++ program has been used to run some much longer simulations than would be possible on the college computers in Python. You can view its source code [https://github.com/niallj/ducking-avenger/tree/master/Ising here] if you are interested. Each file contains six columns: <math>T, E, E^2, M, M^2, C</math> (the final five quantities are per spin), and you can read them with the NumPy loadtxt function as before. For each lattice size, plot the C++ data against your data. For ''one'' lattice size, save a PNG of this comparison and add it to your report — add a legend to the graph to label which is which. To do this, you will need to pass the label="..." keyword to the plot function, then call the legend() function of the axis object (documentation [http://matplotlib.org/api/axes_api.html#matplotlib.axes.Axes.legend here]).'''

{| class="wikitable"
|-
!Lattice Size
!Energy: Python vs C++
!Magnetisation: Python vs C++
!Heat Capacity: Python vs C++
|-
|2x2
|[[File:wm1415_New2_energy_ref_comp.png|400px|center]]
|[[File:wm1415_New2_magnetisation_ref_comp.png|400px|center]]
|[[File:wm1415_New2_capacity_ref_comp.png|400px|center]]
|-
|4x4
|[[File:wm1415_New4_energy_ref_comp.png|400px|center]]
|[[File:wm1415_New4_magnetisation_ref_comp.png|400px|center]]
|[[File:wm1415_New4_capacity_ref_comp.png|400px|center]]
|-
|8x8
|[[File:wm1415_New8_energy_ref_comp.png|400px|center]]
|[[File:wm1415_New8_magnetisation_ref_comp.png|400px|center]]
|[[File:wm1415_New8_capacity_ref_comp.png|400px|center]]
|-
|16x16
|[[File:wm1415_New16_energy_ref_comp.png|400px|center]]
|[[File:wm1415_New16_magnetisation_ref_comp.png|400px|center]]
|[[File:wm1415_New16_capacity_ref_comp.png|400px|center]]
|-
|32x32
|[[File:wm1415_New32_energy_ref_comp.png|400px|center]]
|[[File:wm1415_New32_magnetisation_ref_comp.png|400px|center]]
|[[File:wm1415_New32_capacity_ref_comp.png|400px|center]]
|-
|64x64
|[[File:wm1415_New64_energy_ref_comp.png|400px]]
|[[File:wm1415_New64_magnetisation_ref_comp.png|400px|center]]
|[[File:wm1415_New64_capacity_ref_comp.png|400px]]
|}

For all lattice sizes, the energy and magnetisation are in agreement with the reference C++ data. The heat capacity plots also match the reference for smaller lattice sizes, producing a smooth curve with almost no noise: the system is small enough to probe all the possible configurations. However, there is substantial noise in the heat capacity for larger lattices (16x16 and above) in the region of the critical temperature: one repeat of the experiment is not sufficient to accurately model the many possible mechanisms of phase transition.

The relaxation process for a random lattice configuration can be described by a free energy surface. The path whereby the system relaxes depends on the initial configuration - which, in our model, is random. Thus, for each run of the experiment, a different section of the free energy surface is explored: the local minimum where the system stabilises will have a different energy in the different paths taken. The activation barriers between local minima are mediated by 'clumping': regions of spin-aligned lattice sites form, which are unfavourable to disrupt. In the critical temperature region, there is not sufficient thermal energy to eliminate clumping: as a result, there are many metastable states the system can relax to, each with different energies. This results in a large number of possible configurations, and hence a noisy plot of energy, magnetisation and heat capacity.

To reduce the noise, more repeats must be performed and the results averaged to sample more relaxation paths on the free energy surface. The figures below show the results of further simulations run in the critical temperature range for larger lattices, with a comparison between a single run of the experiment and an averaged plot from 10 repeats. Error bars in the heat capacity plot were obtained by calculating the standard deviation across repeats.

{| class="wikitable"
|-
!Lattice Size
!Heat Capacity: Single run
!Heat Capacity: 10 repeats
|-
|16x16
|[[File:wm1415_16x16_heatcap_1run.png|400px]]
|[[File:wm1415_16x16_heatcap_rpts.png|400px]]
|-
|32x32
|[[File:wm1415_32x32_heatcap_1run.png|400px]]
|[[File:wm1415_32x32_heatcap_rpts.png|400px]]
|-
|64x64
|[[File:wm1415_64x64_heatcap_1run.png|400px]]
|[[File:wm1415_64x64_heatcap_rpts.png|400px]]
|}

Even with these repeat experiments, the values for the heat capacity for the larger 32x32 and 64x64 lattices seem to be consistently smaller than the reference data at the critical temperature. Perhaps this is because the system is not being given enough steps to fully explore the potential energy surface. All these plots have been using the same method for this report - discounting the first 250,000 steps and recording the next 250,000. To test this, an additional run of the 32x32 lattice was performed: discounting the first 250,000 steps and recording the next 1,250,000 steps:

<center>[[File:wm1415_Large_32x32_heatcap_1run.png|400px]]</center>

As this was computationally very expensive, only one repeat was manageable for the timescale of this experiment. While it may not be scientifically rigorous to draw any conclusions from one repeat of an experiment which depends on chance, this figure is certainly promising - the extra steps seem to have alleviated the underestimation of the heat capacity - and a closer match to the C++ data may be obtained with further repeats. Unfortunately, this was not investigated further due to time constraints of the experiment.

To extract the Curie temperature, the reference data - with clearly resolved peaks in the heat capacity - will be examined first, and then compared to the value obtained from the data generated in Python.

===Locating the Curie temperature - Reference C++ data===

'''<big>TASK</big>: write a script to read the data from a particular file, and plot C vs T, as well as a fitted polynomial. Try changing the degree of the polynomial to improve the fit — in general, it might be difficult to get a good fit! Attach a PNG of an example fit to your report.'''

<center>[[File:wm1415_Initial_polyfits.png|400px|thumb|In general, a higher order polyfit, which includes higher order terms, gives the best fit. However, without restricting the range of the fit, the peak cannot be located with this approach.]]</center>

'''<big>TASK</big>: Modify your script from the previous section. You should still plot the whole temperature range, but fit the polynomial only to the peak of the heat capacity! You should find it easier to get a good fit when restricted to this region.'''

{| align=center
|-
|[[File:wm1415_Polyfit_max.png|400px]]
|[[File:wm1415_Polyfit_max_zoom.png|400px]]
|}

For the remainder of the report, a polyfit order of 8 has been chosen: high enough to obtain a good fit, but sufficiently low to avoid poorly optimised high-order term fits.

'''<big>TASK</big>: find the temperature at which the maximum in C occurs for each datafile that you were given. Make a text file containing two colums: the lattice side length (2,4,8, etc.), and the temperature at which C is a maximum. This is your estimate of <math>T_C</math> for that side length. Make a plot that uses the scaling relation given above to determine <math>T_{C,\infty}</math>. By doing a little research online, you should be able to find the theoretical exact Curie temperature for the infinite 2D Ising lattice. How does your value compare to this? Are you surprised by how good/bad the agreement is? Attach a PNG of this final graph to your report, and discuss briefly what you think the major sources of error are in your estimate.'''
{| class="wikitable"
|-
!2x2 Polyfit
!4x4 Polyfit
!8x8 Polyfit
!16x16 Polyfit
!32x32 Polyfit
!64x64 Polyfit
|-
|[[File:wm1415_Polyfit_2x2.png|200px|center]]
|[[File:wm1415_Polyfit_4x4.png|200px|center]]
|[[File:wm1415_Polyfit_8x8.png|200px|center]]
|[[File:wm1415_Polyfit_16x16.png|200px|center]]
|[[File:wm1415_Polyfit_32x32.png|200px|center]]
|[[File:wm1415_Polyfit_64x64.png|200px]]
|}

<center>[[File:wm1415_Curie_t_inf.png|500px]]
<math>T_{C, L} = \frac{A}{L} + T_{C,\infty}</math></center>

A plot of <math>T_{C, L}</math> against <math>\frac{1}{L}</math> yields a straight line, with intercept <math>T_{C, \infty}</math> and gradient <math>A</math> (a constant). Fitting the C++ data resulted in a value for <math>T_{C,\infty} = 2.277 \pm 0.017 \frac{J}{k_B}</math>, with the uncertainty calculated from the error in the linear fit. This is in good agreement with the reference value<sup>[1]</sup> of <math>T_{C,\infty} \thickapprox 2.269 \frac{J}{k_B}</math>. The main source of error is the fitting of the straight line, which is quantifiable and represents a 1% uncertainty. There is also an uncertainty in the extraction of <math>T_{C,L}</math> values from the polynomial fitting of the peaks in each heat capacity plot, as the fits are not exact. The simulations themselves also introduce an uncertainty: for example, if the lattice did not equilibrate within the limit of the number of simulation steps.

However, this experiment has reproduced the theoretical value within the error of the straight line fit, demonstrating the power of the Monte Carlo methods used.

===Locating the Curie temperature - Python data===

Below are the fitted functions for the Python heat capacity data. For the 16x16, 32x32 and 64x64 lattices, the averaged data points over 10 repeats were used (see above).

{| class="wikitable"
|-
!2x2 Polyfit
!4x4 Polyfit
!8x8 Polyfit
!16x16 Polyfit
!32x32 Polyfit
!64x64 Polyfit
|-
|[[File:wm1415_Pypolyfit_2x2.png|200px|center]]
|[[File:wm1415_Pypolyfit_4x4.png|200px|center]]
|[[File:wm1415_Pypolyfit_8x8.png|200px|center]]
|[[File:wm1415_Pypolyfit_16x16.png|200px|center]]
|[[File:wm1415_Pypolyfit_32x32.png|200px|center]]
|[[File:wm1415_Pypolyfit_64x64.png|200px|center]]
|}

Compiling this data to plot the straight line, as outlined previously, yielded very similar results to the C++ data:

<center>[[File:wm1415_PyCurie_t_inf.png|400px]]</center>

There is a larger uncertainty in the fitting of the straight line but the resulting curie temperature, <math>T_{C,\infty} = 2.28 \pm 0.03 \frac{J}{k_B}</math>, is nearly identical to the result from the C++ data and is in good agreement with the reference value<sup>[1]</sup>. Due to the noise in the heat capacity plots in the critical temperature region, the polynomial fits of the peaks were not as convincing as the fits for the C++ data. The <math>T_{C,L}</math> values that were subsequently extracted from these polynomial fits were shifted from the C++ data, resulting in the greater spread of data points in the straight line plot above. Furthermore, there are contributing errors from the simulations themselves - for example, the lattice not reaching equilibrium or getting stuck in a local minimum (the 'clumping' effect discussed earlier) affecting the heat capacity.

==References==

1 B. Liu, M. Gitterman, ''American Journal of Physics'', '''71''', 806 (2003), pp. 1-4

2 R. Fitzpatrick, 2006, The Ising Model, ''Computational Physics'', retrieved from http://farside.ph.utexas.edu/teaching/329/lectures/node110.html in November 2017.

3 P. Atkins and J. de Paula, ''Atkins' Physical Chemistry'', Oxford University Press, UK, 8th edn, 2006, pp. 564-573

Rep:Wm1415CMP

2025-09-01T09:50:00Z

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Rep:Winston

2025-09-01T09:49:57Z

Move page script: Move page script moved page Winston to Rep:Winston: Move to report namespace

= Physical Computational Laboratory =

= Cope Rearrangement =

==Introduction==

Proposed transition structures for the [3,3] Cope Rearrangement of ''1,5''-hexadiene has been the topic of debate by theoriticians for an extensive period of time. Initially three transition states for the reaction were proposed, however the Bis-Allyl mechanism was dismissed on experimental grounds. Thus two mechanisms for the reaction pathway remained open for discussion, one involving the concerted sigma bond shift aromatic transition state and the other being the formation of a stable 1,4-Diyl diradical intermediate<ref name="http://pubs.acs.org/doi/pdf/10.1021/ja044734d">Balancing Dynamic and Nondynamic Correlation for Diradical
and Aromatic Transition States: A Renormalized Coupled-Cluster Study of the Cope Rearrangement of 1,5-Hexadiene {(http://pubs.acs.org/doi/pdf/10.1021/ja044734d )}</ref>:
[[image:Gaucheanti_confomers.JPG|thumb]]
[[image:Chem_draw_cope_mech.JPG|left|400px]]

Computational chemistry was used to investigate this further, and it was found that the transition states concluded from calculations largely depended upon the methods employed. Such that semi-empirical methods supported the Diyl pathway and DFT quantum mechanical methods supported the concerted aromatic transition state.

In this investigation the density functional theory was used with B3LYP/6-31G method and basis sets. Activation and enthalpy energies for the 'boat' and 'chair' confomers were then analysed.

''1,5''-hexadiene exists in various forms, they can be classified as either ''anti-'' (roughly anti-periplanar) or ''gauche-'' (roughly eclipsed) based upon the orientation of the molecule about the central carbon atoms (C3 and C4).

Prior to calculation using the DFT/6-31G method, various confomers of 1,5-hexadiene were optimised using the less complex HF/3-21G method. This enabled quicker and more accurate calculation by the DFT method since the optimisations were almost step-wise in complexity.

==Optimising 1,5-hexadiene==
[[image:Anti_2_hexadiene.JPG|thumb|''anti-2'' hexadiene]]
[[image:Gauche_3_hexadiene.JPG|thumb|''gauche-3''hexadiene]]
1,5-hexadiene structures were optimised in Gaussian using a HF/3-21G method and values for the energies were compared to that of literature:

{|
|[[image:Results_summary_anti_2.JPG|left|''anti-2'' hexadiene]]
|[[image:Results_summary_gauche_3.JPG|left|''gauche-3''hexadiene]]
|}
Summary of the results from the optimisation calculations could be obtained from gaussian. From the tables it can be deduced that the HF/3-21G minimised energy for ''anti-2'' hexadiene is -231.69254 a.u. and for ''gauche-3'' hexadiene it is -231.69266. These energies are consistent with that of the values given in the appendix. Where 1 Hartree a.u. is equivalent to 627.50947 kcal/mol<ref name="http://www.chem.arizona.edu/courseweb/074/CHEM5181/conversions.htm">Common Conversion Factors in Computational Chemistry {(http://www.chem.arizona.edu/courseweb/074/CHEM5181/conversions.htm )}</ref>. Thus, it can be concluded that the gauche ''1,5''-hexadiene conformation is thermodynamically more stable than that of the anti ''1,5''-hexadiene conformation by 0.0753 kcal/mol.

This is unexpected, since basic theory would suggest that the ''anti'' confomer shows greater stability, as it is less sterically strained. Dipole moments for the confomers also indicate that the anti molecule should exhibit greater stability. However, one reason that could be used to explain why the ''gauche'' conformation is preffered, is due to possible ''pi''-overlap between the two ''pi''-systems. As the gauche conformation enables the ''pi''-systems to be orientated close to each other, this factor could in fact play a major role in stabilisation and thus overcome all the other factors which favour the ''anti'' conformation.

Symmetry point groups of the molecules were deduced by Gaussian, these were found to be C''i'' for the ''anti-2'' hexadiene and C''1'' for the ''gauche-3'' hexadiene. Again, these properties were consistent with those in the appendix.

Optimisation of the ''anti-2'' hexadiene was increased to the B3LYP/6-31G method. Given the increased complexity of this calculation method, it is expected that values for energies differ such that a more stable conformation is provided by the B3LYP/6-31G method.

[[image:Results_summary_DFT_HF_anti_2.JPG|left]]
As expected the energy for the B3LYP/6-31G method provides a significantly better minimum for the ''anti-2'' hexadiene {(https://www.ch.imperial.ac.uk/wiki/images/e/ea/REACT_COPE_ANTI_2_DFT.LOG )}. The calculation provides a structure that is 1831.81 kcal/mol more stable than the previous HF/3-21G method.

The geometry of the molecule remains the same, such that both calculations produce a structure that has a C''i'' symmetry point group. This indicates that the energy difference is largely down to manipulation of bond lengths rather than modification of bond angles and overall geometry.

Further detailed analysis of the bond lengths in each of the molecules supports this assumption:
HF/3-21G ''anti-2'' ''1,5''-hexadiene - (C1 to C2/C5 to C6) ''pi''-bond length = 1.316Ǻ, (C3 to C4) ''sigma''-bond length = 1.552Ǻ.
B3LYP/6-31G ''anti-2'' ''1,5''-hexadiene - (C1 to C2/C5 to C6) ''pi''-bond length = 1.333Ǻ, (C3 to C4) ''sigma''-bond length = 1.548Ǻ.
Typical Alkene (C=C)bond length = 1.33Ǻ, and typical (C-C) bond length = 1.54Ǻ.

Thus, it is ability of the B3LYP/6-31G method to more accurately predict the length of the C=C bond within the molecule which results in a more stable geometry.

Frequency analysis of the ''anti-2'' ''1,5''-hexadiene was then conducted in order to determine the calculation had reached a minimum value, this was confirmed by the presence of fully positive vibrational frequencies. - Since the frequency calculation is essentially a second derivative of the potential energy curve for optimisation, subsequently a minima is indicated by positive values.

[[image:IR_hexadiene_anti_2.JPG|left]]

{| border='1px'
|+ IR Analysis of ''Anti-2'' ''1,5''-hexadiene
! Bond Type !! Frequency (cm-1) !! Lit.<ref name="http://www.cem.msu.edu/~reusch/VirtualText/Spectrpy/InfraRed/infrared.htm">Infrared Spectrscopy {(http://www.cem.msu.edu/~reusch/VirtualText/Spectrpy/InfraRed/infrared.htm )}</ref> Frequency (cm-1)
|-
| =CH2 (bend)
| 940
| 880-995
|-
| C=C (asymmetric stretch)
| 1734
| 1900-200
|-
| CH2 (alkane)
| 3031,3080
| 2850-3000
|-
|}

Tabulated frequencies for the molecule was formed by analysis of the animated frequencies, these values were then compared to that of literature. It can be noted that the frequencies for the B3LYP/6-31G are consistent to that of literture, thus reinforcing the use of this method.

By examination of the .log file for the frequency calculation, the sum of electronic and zero-point energies, the sum of electronic and thermal energies, the sum of electronic and thermal enthalpies, and the sum of electronic and thermal free energies could be determined:

'''sum of electronic and zero-point energies''' = -234.469204 a.u.
'''sum of electronic and thermal energies''' = -234.461857 a.u.
'''sum of electronic and thermal enthalpies''' = -234.460913 a.u.
'''sum of electronic and thermal free energies''' = -234.500777 a.u.

==Optimising the Chair Transition Structure==

Both chair and boat transition states can exist during the Cope rearrangement. The chair transition structure consists of two C3H5 allyl fragments positioned 2.2Ǻ apart with a symmetry point group of C''2h'', and the boat transisiotn structure consists of the same allyl fragments positioned 2.2Ǻ apart with a different symmetry point group of C''2v''.
[[image:Chair_ts_guess_.bmp|thumb|chair transition state - '''''guess''''']]
Given the properties associated with both of the transition structures, it seemed feasible to first optimise the basic fragment that is common in both structures. Thus, prior to further investigation into the overall optimisation of the full transition states, the HF/3-21G method was used to optimise the allyl fragment (CH2CHCH2). Once optimised, this was copied and pasted such that two fragments could be orientated about each other to vaguely represent the chair transisiton state - (terminal carbons were positioned 2.2Ǻ apart).

This '''''guess''''' provided a viable basis structure that could be optimised further and used to analyse the chair transisiton structure for the Cope rearrangement. Transition state optimisation calculations require a '''''guess''''' structure that is as close as possible to the proposed structure, such that Hessian calculations performed throughout the optimisaton follow that of the true optimistion PES curve.

HF/3-21G method was used to provide an optmisation and frequency analysis of the '''''guess''''' Transition state, the key words '''opt=noeigen''' were inserted in order to prevent the calculation aborting if more than one imaginary frequency was detected.This occurs when the '''''guess''''' structure is not close enough to that of the real transition structure. - (Force constants were calculated once throughout the optimisation.)
[[image:Results_summary_ts_optfreq_chair.JPG|left]]
[[image:Chair_ts_optandfreq.JPG|thumb|
<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair_opt+freq_ts_.mol</uploadedFileContents>
<text>chair transition state</text>
</jmolAppletButton>
</jmol>]]
Results for the HF/3-21G Optimisation and Frequency analysis of the Chair transisiton structure are summarised in the Table (https://www.ch.imperial.ac.uk/wiki/images/e/e6/CHAIR_TS_OPT%2BFREQ.LOG }. It can be noted that the energy for the transition state is 45.94kcal/mol higher than that of the reactant - ('''''anti-2''''' ''1,5''-hexadiene). This is consistent with theory, and is known as the activation energy for the reaction.

Completion of a successful calculation leads to the presence of one imaginary vibration in the frequency analysis. This has a value of -818cm-1. The vibration is shown below:

[[image:Freq analysis chair ts.JPG|centre|550px]]

This vibration is consistent with that for the proposed aromatic mechanism of the Cope rearrangement, whereby there is a concerted sigma bond shift between C1,C6 and C3,C4 in the reactant.
[[image:Cope_aromatic_mechanism.JPG|centre]]

Although this calculation proved successful, and subsequently it can be deduced the '''''guess''''' structure was close enough to that of the real transition structure for accurate hessian calculations, extra steps can be encorporated to provide a more precise and therefore reliable calculation. It is knwon as the frozen co-ordinate method. Whereby the reaction coordinate is frozen, the molecule is then optimised and once the molecule is fully relaxed, the reaction coordinate can then be unfrozen and a transition state optimisation performed.

The additional key words of '''opt=modredundant''' were needed for this calculation to proceed.
[[image:Chair_ts_optandfreqfrozen.JPG|thumb|frozen co-ordinate optimisation]]
First the '''''guess''''' structure was optimsed to a minimum, using the HF/3-21G method, with a fixed (frozen) bond length of 2.2Ǻ between the two allyl fragments. The next stage is to run a transition state optimisation of the structure with the distance between the two allyl fragments unfrozen. Force constant were not calculated in the optimisation, instead a normal guess Hessian was used:
[[image:Results_summary_freeze_comparison.JPG|left]]
A comparison between the HF/3-21G Optimisation and Frequency analysis for the normal and frozen methods is shown in the table on the left. It is clear to see that the energy difference between the two structures is negligible, therefore suggesting that the initial HF/3-21G method with one force constant calculation was more than satisfactory.

Comparison of bond lengths between the two transition structures also supports the use of the HF/3-21G calculation without frozen parameters, since all exhibit roughly the same value - differences are negligible.

As for the previous calculation, the vibrations of the molecule were analysed (https://www.ch.imperial.ac.uk/wiki/images/9/9c/CHAIR_TS_OPT%2BFREQ%2CFREEZE.LOG). This showed one imaginery vibration (as expected) corresponding to the aromatic concerted sigma-tropic rearrangement, the frequency of this vibration was at -818cm-1.

==Optimising the Boat Transition Structure==

When caculating the boat transition structure, the QST2 method was used upon the '''''anti-2''''' ''1,5''-hexadiene. This required a reactant and product molecule to be constructed, as the calculation interpolates between the two structures to find the optimum transition structure for the reaction.

Thus the first stage was to construct both the product and reactant such that they were numbered as follows:
[[image:QTS2_anti-2_reactandprod.JPG|centre]]
Numbering of molecules in such a way was critical for the successful outcome of the calculation. QTS2 Transition optimisation and frquency analysis was then carried out, and this was found to fail.[[image:Fail_boat.JPG|thumb|failed QTS2 calculation]]
Reasons for this failed calculation arise from the geometry of the reactant. The C''2h'' '''''anti-2''''' geometry adopted by the reatant is clearly far from that of the boat transition structure, thus the calculation cannot cleary see how best to optimise/ rearrange the molecules in a sigma tropic rearrangemnt such that the aromatic boat transition structure is formed.

Alterations to the reactant clearly had to be made such that the calculation method would form and optimise the boat transition structure. This was achieved by orientating the C3,C4 sigma bond into an eclipsed form - (C2,C3,C4,C5 dihedral angle of 0 degrees), by doing so this orientated C1 and C6 closer together, much like in the diagrammatic reaction mechanism shown previously. Furthermore, the C1,C2,C3 and C4,C5,C6 bond angles were altered to 100 degrees:

[[image:Boat_proper_conf.JPG|centre]]
As before, the two molecules were subjected to a QTS2 transition optimisation and frequency analysis. As expected the alterations that were made to both the product and reactant resulted in a successful calculation.
[[image:Boat_ts_QTS2_optandfreq.JPG|thumb|Boat Transition State]]
[[image:321G_comparison_boat_chair.JPG|left]]
Comparison of the 3-21G level calculations for the chair and boat transition states shows that the chair structure is more energetically favourable. With an energy difference of 10.37 kcal/mol between the two structures.

Previous calculations showed that the 3-21G activation energy for the Cope rearrangement with the Chair transition state was 45.94kcal/mol, therefore it can be concluded that the 3-21G activation energy for the Cope rearrangement with the Boat transition state is 56.31 kcal/mol.

==Intrinsic Reaction Coordinate Method for Chair Transition State==

IRC can be used to follow the minimum energy path from a transition state down to its minimum on a PES. To investigate this further the chair transition structure calculated not using the frozen co-ordinate method was used. The number of points along the IRC optimisation and frequency analysis was changed from its default setting of 6 to a more thorough 50:

[[Image:IRC 50.JPG|centre]]

The summary attatched to the IRC plot shows the data associated with the minima point of the calculation with 50 points.
In order to determine whether the PES had infact come to a local minima, other comparisons were made. First of which was to increase the number of points in the IRC to 100. This did not alter the minima obtained and the IRC plots were exactly the same. Next came a standard minimisation calculation, this resulted in a local minima structure with 3.04 kcal/mol less energy than that of the IRC 50/100 plots:

[[image:Irc50_minimisation.JPG|centre]]

Next came the most reliable and expensive method of locating the minima of the PES curve, this involved computing the force constants at every step along the IRC optimisation and frequency analysis.

[[Image:Force everystep IRC.JPG|centre]]

However, this method provided a local minima structure with energy negligibly higher than that of the standard minimisation calculation. Suggesting the minimisation calculation has over compensated in areas.

[[image:Summary_forceeverystep_IRC.JPG|centre]]

==Optimisation Using B3LYP/6-31G* and Comparisons==
[[image:B3LYP_boat.JPG|(https://www.ch.imperial.ac.uk/wiki/images/a/ac/BOAT_OPT_TS_B3LYP.LOG )|thumb]]
[[image:B3LYP_chair.JPG|(https://www.ch.imperial.ac.uk/wiki/index.php/Image:CHAIR_TS_B3LYP.LOG )|thumb]]
Structures for the Boat and Chair transition staes were subject to more complex optimisation using the B3LYP/6-31G* method. This provided more accurate representation for the structures and data from these calculation could then be compared to that of literature.[[image:631G_321G_comp_boat.JPG|631G_321G_comp_boat.JPG]]
[[image:631G_321G_comp_chair.JPG|631G_321G_comp_chair.JPG]]

Cleary the B3LYP/6-31G* method provides far greater accuracy and from the previuos calculation used to optimise the reactant ('''''anti-2''''' ''1,5''-hexadiene) at B3LYP/6-31G* method, the activation energies for the reactions can be decided.

'''Activation Energy for Boat transition structure''' = 74.55 kcal/mol

'''Activation energy for Chair transition structure''' = 66.67 kcal/mol

'''Energy difference between transition structures''' = 7.88 kcal/mol

= The Diels Alder Cycloaddition =

The [4+2] cycloaddition reaction of cis-butadiene with ethene was studied. To do this, cis-butadiene was initially optmiised using the AM1 semi-empirical method of analysis, this was compared to analysis of increasing colpexity. Energies and bond angles were compared.
[[image:Cis_butadiene_AM1.JPG|'''AM1''' optmiised ''cis''-butadiene (https://www.ch.imperial.ac.uk/wiki/images/b/be/CIS_BUTADIENE.LOG )|thumb]]
[[image:Cis_butadiene_321G.JPG|'''3-21G''' optimised ''cis''-butadiene (https://www.ch.imperial.ac.uk/wiki/images/a/a3/CIS_BUTADIENE_31G.LOG )|thumb]]
[[image:Cis_butadiene_DFT.JPG|'''6-31G''' optimised ''cis''-butadiene (https://www.ch.imperial.ac.uk/wiki/images/0/07/CIS_BUTADIENE_DFT_6-31G.LOG )|thumb]]

[[Image:Summary dielsalder cisbutadiene.JPG|850px]]

By analysis of the summary of results for the optimisation, it is evident that solutions for optimisation become progressively more accurate as the method is changed from semi-empirical AM1 to the quantum mechanical B3LYP/6-31G. It can also be noted that the largest difference in energy is experienced from changing the semi-empirical calcultion to the quantum mechanical approach, after which the energy difference is relatively minimal.

{| border='1px'
|+ Cis-Butadiene Poroperties of Optmisation
! !! C=C Bond Length !! C-C Bond Length !! < C=C-C (degrees)
|-
| Semi-Empirical AM1
| 1.39
| 1.36
| 128.12
|-
| Quantum Mechanical 6-31G
| 1.43
| 1.40
| 134.33
|-
| Literature
| 1.35
| 1.47
| 124.4
|-
|}

Suprisingly the literature values for the bond lengths and angles are not biased toward the more complex quantum mechanical calculation, but instead the bond lengths and angles sway between both methods. Suggesting the moethods do not cater for other forces such as conjugation which effect the structure.

During a diels-alder reaction the HOMO/LUMO are involved depending upon which type of mechanism the reaction follows. This depends upon the reaction being following the inverse electron demand are normal electron demand. Given this, the MO for the HOMO and LUMO of cis-butadiene were investigated and symmetry with respect to the plane determined.

{| border='1px'
|+ MO of cis-butadiene
! !! LUMO !! HOMO !! Symmetry with respect to plane
|-
| Semi-Empirical AM1
| [[image:LUMO_AM1.JPG]]
| [[image:HOMO_AM1.JPG]]
| '''LUMO''' - yes '''HOMO''' - no
|-
| Quantum Mechanical 6-31G
| [[image:LUMO_631G.JPG]]
| [[image:HOMO_631G.JPG]]
| '''LUMO''' - yes '''HOMO''' - no
|-
|}

= References and Citations =
<references />

Rep:Wilee5534.IIII

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[[Image:boatTSimaginaryfreq.jpg]]

Rep:Wilee5534.III

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==Module 3:Transistion Structures on Potential Energy Surfaces==
==The Cope Rearrangement Tutorial==
===Optimizing the ''anti'' and ''gauche'' conformers===
[[Image:anti1_5lowlevel.jpg]]
[[Image:anti1_5highlevel.jpg]]
[[chair_imaginaryfreq.gif]]
[[chair_imaginaryfreq(imaginaryfreq).gif]]
[[Image:BoatTSimaginaryfreq.jpg]]

[[Image:gauche1_5hexadiene3_21G.jpg]]
[[Image:gauche1_5hexdiene6_31G]]

[[IR_anti2_sg.jpg]]

Rep:Wilee5534.3

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==Module 3: Transistion Structures on Potential Energy Surfaces==
===Objectives===
==The Cope Rearrangement Tutorial==
===Optimisation of ''anti'' and ''gauche'' conformers===
ANTI
anti optimisation
File Name anti1_5_hexadiene_optimisation
File Type .chk
Calculation Type FOPT
Calculation Method RHF
Basis Set 3-21G
Charge 0
Spin Singlet
Total Energy -231.69253528 a.u.
RMS Gradient Norm 0.00001891 a.u.
Imaginary Freq
Dipole Moment 0.0000 Debye
Point Group

Symmetrize? Point Group = C<sub>i</sub>

Energy given in table -231.69254, in agreement with my energy Total Energy -231.69253528

GAUCHE

HF/3-21G

File Name gauche1_5hexadiene_optimisation
File Type .chk
Calculation Type FOPT
Calculation Method RHF
Basis Set 3-21G
Charge 0
Spin Singlet
Total Energy -231.69266118 a.u.
RMS Gradient Norm 0.00001742 a.u.
Imaginary Freq
Dipole Moment 0.3404 Debye
Point Group

Symmetrized.. Point Group = C1

[[Image:gauche1_5hexadiene3_21g.jpg|thumb|centre|200px|''gauche'' conformer optimised with low level basis set HF/3-21G.jpg]]

Run anti2 again with B3LYP/6-31G*

anti2 high level optimisation
File Name anti1_5_hexadiene_optimisation_highlevel
File Type .chk
Calculation Type FOPT
Calculation Method RB3LYP
Basis Set 6-31G(D)
Charge 0
Spin Singlet
Total Energy -234.61171027 a.u.
RMS Gradient Norm 0.00000985 a.u.
Imaginary Freq
Dipole Moment 0.0000 Debye
Point Group
{|
|[[Image:anti1_5_hexadiene.jpg|thumb|centre|200px|''anti'' conformer optimised with low level basis set HF/3-21G.jpg]]
|[[Image:anti1_5_highlevel.jpg|thumb|centre|200px|''anti'' conformer optimised with high level basis set B3LYP/6-31G*.jpg]]
|}

Torsion Angles (high level)
C 1,2,3,4 118.54991
C 2,3,4,5 180.0
C 3,4,5,6 118.54991

Torsion Angles (low level)
C 1,2,3,4 114.66878
C 2,3,4,5 180.0
C 3,4,5,6 114.66878

===Vibrational analysis of ''anti2''===
B3LYP/6-31G*

Modes all positive and real

[[Image:IR_anti2_sg.jpg|thumb|400px|center]]

Sum of electronic and zero-point Energies= -234.469203
Sum of electronic and thermal Energies= -234.461856
Sum of electronic and thermal Enthalpies= -234.460912
Sum of electronic and thermal Free Energies= -234.500777

Freq analysis ran at lower basis set for activation energies [later].

Sum of electronic and zero-point Energies= -231.539539
Sum of electronic and thermal Energies= -231.532565
Sum of electronic and thermal Enthalpies= -231.531621
Sum of electronic and thermal Free Energies= -231.570916

===Optimising the Chair and Boat Transistion Structures===
'''''Chair'''''
Optimised to a TS (Berny)
[[Image:chair_ts_sg.jpg|thumb|left|200px|''Chair'' optimised to TS (Berny)]]
C-C [terminal] = 2.02055

Imaginary vibration at 817.9cm<sup>-1</sup>
[[Image:chair_imaginaryfreq.gif|left|thumb|400px|imaginary freq 817.9cm<sup>-1</sup>]]

Optimised using frozen method
[[Image:chair_imaginaryfreq(frozen).gif|left|400px|thumb|imaginary freq 818cm<sup>-1</sup>]]

Rep:Wdmn 3rd Year Liquid Simulations

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== Report ==

===Abstract===

Molecular dynamic simulations are a well developed technique for probing the qualities of liquids. Here, the temperature dependency of number density and heat capacity was determined for a monoatomic fluid through a Lennard-Jones force field. This gave a comparison between the simulated model and rationalisations from theory. The radial distribution function for a monoatomic solid, liquid and vapour elucidated their different dynamic and structural properties. The diffusion characteristics of each phase was studied from by simulating the mean displacement of an atom from a reference position and the correlation of its future velocities. The results are discussed in relation to the degree of freedom for each phase. The limitation of the model was assessed by simulating with 1000 and 100000 atoms.

===Introduction===

High-performance computers have made molecular dynamics (MD) simulations a standard lab technique, allowing chemists to probe the behaviour of atoms in extreme reaction conditions. The ability to change thermodynamic parameters such as pressure, temperature and density easily and cheaply has led to its proliferation.

One of the most interesting applications of MD is simulating the conditions found in space. NASA needs to test the stability of metallic alloys and simulates the response of polymeric adhesives in the presence of moisture by MD <ref>M. Polanco, S. Kellas and K. Jackson, ''65th Annu. Forum Proc. - AHS Int.'', 2009, '''2''', 1513–1524.</ref>. Additionally, due to the vast temperature differences between the inside and outside of a spacecraft, being able to assess the thermodynamic properties of electrical conductors is essential for heat management <ref>S. V Garimella, T. Persoons, J. A. Weibel and V. Gektin, ''IEEE Trans. Components, Packag. Manuf. Technol.'', 2016, '''PP''', 1191–1205.</ref>. MD can also be used to determine flow rates, ratios and temperatures for mixing rocket fuel (liquid hydrogen and liquid oxygen).

Whilst MD simulations have aided the development of new technologies, it has also informed on earth's most abundant and yet enigmatic liquid: water. The evolution of modelling water has progressed from the pairwise force fields implemented in this report to considering many body and quantum effects <ref>G. A. Cisneros, K. T. Wikfeldt, L. Ojamäe, J. Lu, Y. Xu, H. Torabifard, A. P. Bartók, G. Csányi, V. Molinero and F. Paesani, ''Chem. Rev.'', 2016, '''116''', 7501–7528.</ref>.

Due to the ability of water to form four strong hydrogen bonds per molecule, which are constantly being broken and reformed because of water's dynamic nature, it displays many anomalous features. Water reaches its most dense state at 4 <sup>o</sup>C and displays extraordinarily high surface tension and low compressibility <ref>E. Cartlidge, ''New Sci.''', 2010.</ref>. This is a result of the competing structural effects of ordered hydrogen bonded tetrahedrons minimising lone pair repulsions at low temperatures and molecules with disordered thermal motion at high temperatures. Water's high polarity gives it an affinity for charged surfaces and makes it the ubiquitous ionic solvent. Water also heavily influences the dynamic and reaction characteristics of proteins. Hydrophobic interactions with carbon chains and hydrogen bonding with permanent dipoles are the driving force for protein folding <ref>M. C. Bellissent-Funel, A. Hassanali, M. Havenith, R. Henchman, P. Pohl, F. Sterpone, D. Van Der Spoel, Y. Xu and A. E. Garcia, ''Chem. Rev.'', 2016, '''116''', 7673–7697.</ref>. Furthermore, the phase diagram water at low temperature and pressure has attracted much attention for its complexity and the presence of a liquid-liquid critical point <ref>P. Gallo, K. Amann-Winkel, C. A. Angell, M. A. Anisimov, F. Caupin, C. Chakravarty, E. Lascaris, T. Loerting, A. Z. Panagiotopoulos, J. Russo, J. A. Sellberg, H. E. Stanley, H. Tanaka, C. Vega, L. Xu and L. G. M. Pettersson, ''Chem. Rev.'', 2016, '''116''', 7463–7500.</ref>.

===Model and Methodology===

Simulations were performed in LAMMPS with pairwise interactions modelled by the truncated Lennard-Jones potential with parameters under Lennard-Jones reduced units [http://lammps.sandia.gov/doc/99/units.html] including σ=ε=1 :

<math>u(r)=\begin{cases} 4\epsilon\left(\frac{\sigma^{12}}{r^{12}} - \frac{\sigma^6}{r^6} \right), & r \le 3 \\ 0, & r > 3 \end{cases} </math>

Atoms were considered spheres of mass 1. A monoatomic crystal of 15x15x15 unit cells was initially generated and then brought to the required state to prevent random and potentially repulsive atomic positioning. All melting was performed under the NVE ensemble with the particles assigned velocities according to Maxwell-Boltzmann statistics.

The effect of temperature on fluid number density at high and low pressure was studied under the NpT ensemble to allow its variation. Fluid heat capacity was calculated at a range of temperatures from energy fluctuations at a fixed density in the NVT ensemble. The diffusive behaviour for a solid (ρ*=1.2, T=0.5, fcc), liquid (ρ*=0.8, T=1.2, sc) and vapour(ρ*=0.02, T=1.2, sc) was examined by the radial distribution function processed by the University of Ilionois' VMD software [http://www.ks.uiuc.edu/Research/vmd/] using conditions determined by Hansen and Verlet<ref>J.-P. Hansen and L. Verlet, ''Phys. Rev.'', 1969, '''184''', 151–161.</ref>. The diffusion coefficient for each phase was estimated from two functions of time: the mean square displacement and the velocity autocorrelation function.

===Results and Discussion===

'''Variation of Number Density with Temperature'''

The relationship between fluid number density and temperature was at examined at two fixed pressure levels. The LJ reduced ideal gas law <math>\frac{N}{V} =\frac{p}{T}</math> was fitted to the data to give a comparison between the Lennard-Jones force field and ideality.

{|style="margin: 0 auto;"
|[[Image:WdmnNV18.png|thumb|upright=2|Number density against temperature for p*=1.8.]]
|[[Image:WdmnNV26.png|thumb|upright=2|Number density against temperature for p*=2.6.]]
|}

Whilst both models give a decrease in number density as temperature increases and an inverse proportionality, the ideal gas law predicts greater values than the simulation. Increasing temperature, increases the kinetic energy, which means that the atoms collide with the box at a greater frequency and velocity, expanding it As number density is defined as <math>\left(\frac{N}{V}\right)</math>, it decreases. The ideal gas model only considers perfectly elastic collisions between atoms and the container walls and ignores electrostatic interactions. The simulated conditions includes pairwise Lennard-Jones potential that leads to repulsion between atoms at short distances. Hence the number of atoms per unit volume is less for the simulation than the ideal gas law. As the temperature increases, the discrepancy between the number densities decreases. At higher temperatures, the collision velocities are sufficient to overcome the intermolecular repulsive forces that dominate at lower temperatures as v<sup>2</sup>αT.

'''Heat Capacity'''

The heat capacity was determined using the following relationship by recording the variance in energy around equilibrium at a fixed temperature.

<math>C_V = \frac{\partial E}{\partial T} = \frac{\mathrm{Var}\left[E\right]}{k_B T^2} = N^2\frac{\left\langle E^2\right\rangle - \left\langle E\right\rangle^2}{k_B T^2}</math>

[[Image:Wdmnheatcapacity.png|thumb|upright=2|centre|Volume reduced isochoric heat capacity against temperature for ρ*=0.2 and 0.8.]]

The plot indicates that the isochoric heat capacity decreases with increasing temperatures for ρ*=0.8. In a system of finite energy levels, there is a higher density of levels towards the energy maximum. At higher temperatures, electrons populate higher energy levels according to the Boltzmann distribution. Therefore, as energy level gap decreases, the thermal energy required to raise the internal energy by populating higher levels subsequently decreases. As <math>C_V=(\frac{{\partial U}}{{\partial T}})_V</math>, the heat capacity decreases with increasing temperatures until all energy levels are equally occupied and it tends to infinity.

Whilst this trend is observed initially in the case of ρ*=0.2, the atoms go through a phase change, indicated by the rapid increase. At T*=2.4 all energy is converted to latent heat. After the phase change, the fluid can access further degrees of freedom such as vibration and rotation which would raise the heat capacity as they are an avenue for thermal energy. These extra degrees of freedom were unavailable at ρ*=0.8.

Furthermore, the heat capacity is consistently higher in the more dense system of ρ*=0.8 than ρ*=0.2 while obeying the same trend for temperature because the thermal energy must be distributed over a greater number of atomic energy levels per unit cell, requiring more energy.

'''Radial Distribution Function'''

The radial distribution function (RDF) relates density to the radial distance from a reference atom <ref>M. I. Barker and T. Gaskell, ''J. Phys. C Solid State Phys.'', 2001, '''5''', 353–365.</ref>. The RDF was determined for the solid, liquid and vapour phases. At short distances, <math>r<\sigma, g(r)=0</math> due interatomic repulsion, according to the Lennard-Jones potential. Eventually, the RDF for each of the three phases tended to 1, the bulk value. After reaching a maximum for the single coordination shell of a gas, <math>g(r)</math> decays to 1. Although liquids are significantly more ordered than a gas, the radial distribution peaks for approximately the first three coordination shells and goes below 1 due to collisions before decaying. As liquids are dynamic, the peaks are broad and coordination shells are not correlated to the reference particle over long distances. The radial distribution function for the solid, displays regular maxima for the coordination spheres at <math>\sqrt{n}</math> multiples of <math>\sigma</math>. The peak broadness results from atomic vibrations.

{|style="margin: 0 auto;"
|[[Image:WdmnG(r).png|thumb|upright=2|The radial distribution function.]]
|[[Image:WdmnIntg(r).png|thumb|upright=2|The running integral of the radial distribution function]]
|}

The running integral, <math>\int(g(r)</math> corresponds to the number of atoms at that radial distance by <math>n(r)=4\pi\rho \int_0^{r'}g(r)r^2dr </math> <ref>J. M. Seddon, ''Thermodynamics and statistical mechanics'', Royal Society of Chemistry, Cambridge, 2001.</ref> The coordination and number match the theoretical fcc model.

{|style="margin: 0 auto;"
|[[Image:WdmnCoordination.png|thumb|upright=2|right|This region of the running integral plot shows the number of atoms in the first three coordination shells.]]
|}

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+''' Data reported for lattice site coordination'''
|-
! Peak !!="col" | Latice spacing !!="col" | Intg(r) !! ="col" | Number of atoms !! ="col" | Coordination number
|-
! scope="row" | 1
| σ || 12 || 12 || 12
|-
! scope="row" | 2
|√2 σ ||18 || 6 || 12
|-
!scope="row" | 3
| √3 σ || 42 || 24 || 12
|}

{|style="margin: 0 auto;"
|[[Image:WdmnFcc.png|thumb|upright=2|right|fcc lattice illustrating atoms corresponding to peaks and radial diagram showing coordination shells for solid]]
|}

'''Diffusion Coefficient: Mean Square Displacement'''

The mean squared displacement (MSD) gives the average deviation of a particle from a reference position. A logarithmic plot of MSD against step was generated to allow diagnosis of the ballistic and diffusion regions. After ln(step)=4, the particle motion changes from Brownian ballistic to Brownian diffusive <ref>R. Huang, I. Chavez, K. M. Taute, B. Lukić, S. Jeney, M. G. Raizen and E.-L. Florin, ''Nat. Phys.'', 2011, '''7''', 576–580.</ref>. The three phases diverge as their degrees of freedom determine the extent to which the atoms can diffuse.

{|style="margin: 0 auto;"
|[[Image:WdmnMSD1000.png|thumb|upright=2|The total MSD for 1000 atoms against steps.]]
|[[Image:WdmnMSD1000000.png|thumb|upright=2|The total MSD for 1000000 atoms against steps.]]
|}

The gradient in the diffusive region of the normal plot, the point at which lines diverge in the exponential plot, was determined and the diffusion coefficient calculated using the relationship <math>D = \frac{1}{6}\frac{\partial\left\langle r^2\left(t\right)\right\rangle}{\partial t}</math>.[See Tasks for compete plots] These values can be rationalised by theory. Atoms in a solid are restricted to one degree freedom, vibration, and so are not able to diffuse throughout the box. Liquid atoms are dynamic and experience limited translational motion allowing diffusion. Atoms in the vapour state, however, possess full translational freedom and so diffuse throughout the box. The difference between the 1000 and 100000 atom diffusion coefficients is caused by significantly greater averaging which reduces the effects of fluctuations on the simulation and considers a greater number of scenarios.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+''' Data reported for diffusion coefficient determined from MSD'''
|-
! Number of Atoms !!="col" | Solid !!="col" | Liquid !! ="col" | Gas
|-
! scope="row" | 1000
| 1.06x10<sup>-6</sup> || 0.106 || 7.25
|-
! scope="row" | 1000000
|8.27x10<sup>-6</sup> ||0.0873 || 3.01
|}

'''Diffusion Coefficient: Velocity Auto-correlation Function'''

The diffusion coefficient was also determined from the velocity auto-correlation function (VACF). The VACF describes the extent to which an atoms' current velocity is correlated with its future velocity <ref>M. I. Barker and T. Gaskell, ''J. Phys. C Solid State Phys.'', 2001, '''5''', 353–365.</ref>. It was evaluated analytically by considering the atoms as 1D harmonic oscillators and then compared to simulated VACF data.

<math>C\left(\tau\right) = \left\langle \mathbf{v}\left(t\right) \cdot \mathbf{v}\left(t+\tau\right)\right\rangle</math>

<math>C\left(\tau\right)=\frac{\int_{-\infty}^{\infty} (-A\omega\sin{(\omega t+\phi)} (-A\omega\sin{(\omega(t+\tau)+\phi)})\mathrm{d}t}{\int_{-\infty}^{\infty} (-A\omega\sin{(\omega t+\phi}))^2\mathrm{d}t}= \cos{(\omega\tau)}</math>

{|style="margin: 0 auto;"
|[[Image:WdmnVACF1000.png|thumb|upright=2|The VACF for the three states with the correlation function plotted. 1000 atoms.]]
|[[Image:WdmnVACF1000000.png|thumb|upright=2|The VACF for the three states with the correlation function plotted. 1000000 atoms]]
|}

The analytic correlation predicted by the 1D harmonic oscillator maps similarly to the simulated VACF for solid and liquid initially but as they decay to zero it retains the sinusoidal pattern. This reflects the extent to which the atoms in the solid and liquid phases are able to oscillate around a fixed point like the 1D harmonic oscillator. Moreover, the simulated VACFs resemble the shape of the Lennard-Jones potential. The regularly structured solid vibrates around a fixed point but this reduced as the system tend to equilibrium. The translational motion of the liquid initially follows the same path as the 1D harmonic oscillator in the ballistic region but as diffusion begins to dominate its motion the VACF tends to 1. It also demonstrates the difference between the motion of liquid and solid: the solid undergoes vibrations and liquid undergoes diffusion. Introducing damping to the 1D harmonic oscillator would improve the fit. The minima in the VACF simulations are due to the change in direction of the atoms resulting from interatomic collisions.

The diffusion coefficient values estimated from the VACF by determining the area under the curve using the trapezium rule and applying the relationship <math>D = \frac{1}{3}\int_0^\infty \mathrm{d}\tau \left\langle\mathbf{v}\left(0\right)\cdot\mathbf{v}\left(\tau\right)\right\rangle</math>

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+''' Data reported for diffusion coefficient determined from VACF'''
|-
! Number of Atoms !!="col" | Solid !!="col" | Liquid !! ="col" | Gas
|-
! scope="row" | 1000
| 6.11x10<sup>-5</sup> || 0.0979 || 8.45
|-
! scope="row" | 1000000
|4.55x10<sup>-5</sup> ||0.0901 || 3.27
|}

The results follow the same trends described for the calculation by MSD. The largest errors in the estimation of the diffusion coefficient from the area under the VACF is the trapezium rule and that the running integrals for vapour and solid do not converge to a limit [See Tasks for running integrals]. The trapezium rule estimates the area under a straight line connecting two points, ignoring the path taken between them. As this simulation produced a curve, an error results in measuring the area under it. Additionally, as the running integrals continued to increase for the liquid and vapour, the VACF has not converged to 0 and the diffusion coefficient is larger than that stated. This is less pronounced for 1000000 atoms, which suggests better averaging.

===Conclusion===

Imperial College's high performance computer was leveraged to perform molecular dynamic simulations in LAMMPS of atoms governed by Lennard-Jones interactions. It was found that number density was inversely proportional to temperature, although lower than that predicted by the ideal gas law. Heat capacity generally decreased with increasing temperature as higher atomic energy levels became occupied unless the atoms went through a phase change, which accessed further degrees of freedom. The impact of the dynamic capabilities of the atoms was also evident in the radial distribution function for solid, liquid and gas phases. The diffusion coefficients calculated for the three phases from the MSD and VACF data were in good agreement, which reflects the inherent link velocity and displacement measurements. Moreover, the MSD highlighted the transition from ballistic to diffusive motion and the VACF conveyed the rate of decay of velocity correlation over time, what was dependent on the atoms' dynamism.

== Tasks ==

===Theory===

* '''Open the file HO.xls. In it, the velocity-Verlet algorithm is used to model the behaviour of a classical harmonic oscillator. Complete the three columns "ANALYTICAL", "ERROR", and "ENERGY": "ANALYTICAL" should contain the value of the classical solution for the position at time <math>t</math>, "ERROR" should contain the ''absolute'' difference between "ANALYTICAL" and the velocity-Verlet solution (i.e. ERROR should always be positive -- make sure you leave the half step rows blank!), and "ENERGY" should contain the total energy of the oscillator for the velocity-Verlet solution. Remember that the position of a classical harmonic oscillator is given by <math> x\left(t\right) = A\cos\left(\omega t + \phi\right)</math> (the values of <math>A</math>, <math>\omega</math>, and <math>\phi</math> are worked out for you in the sheet).'''
* '''For the default timestep value, 0.1, estimate the positions of the maxima in the ERROR column as a function of time. Make a plot showing these values as a function of time, and fit an appropriate function to the data.'''
* '''Experiment with different values of the timestep. What sort of a timestep do you need to use to ensure that the total energy does not change by more than 1% over the course of your "simulation"? Why do you think it is important to monitor the total energy of a physical system when modelling its behaviour numerically?'''
[[Image:WdmnError.png|thumb|upright=3|centre|Absolute error between experimental and analytical values.]]
[[Image:WdmnEnergyerror.png|thumb|upright=2.1|centre|Percentage oscillation in energy for different timesteps.]]

* '''For a single Lennard-Jones interaction, <math>\phi\left(r\right) = 4\epsilon \left( \frac{\sigma^{12}}{r^{12}} - \frac{\sigma^6}{r^6} \right)</math>, find the separation, <math>r_0</math>, at which the potential energy is zero. What is the force at this separation? Find the equilibrium separation, <math>r_{eq}</math>, and work out the well depth (<math>\phi\left(r_{eq}\right)</math>). Evaluate the integrals <math>\int_{2\sigma}^\infty \phi\left(r\right)\mathrm{d}r</math>, <math>\int_{2.5\sigma}^\infty \phi\left(r\right)\mathrm{d}r</math>, and <math>\int_{3\sigma}^\infty \phi\left(r\right)\mathrm{d}r</math> when <math>\sigma = \epsilon = 1.0</math>.'''

<math>\phi\left(r\right) = 4\epsilon \left( \frac{\sigma^{12}}{r_0^{12}} - \frac{\sigma^6}{r_0^6} \right)=0</math> when the potential energy is zero.

<math>\frac{\sigma^{12}}{r^{12}} =\frac{\sigma^6}{r^6}</math>

<math>r_0=\sigma</math>

To calculate the force at this separation <math>F=-\frac{d\phi(r)}{dr}</math>

<math>F=-4\epsilon \left( -\frac{12\sigma^{12}}{r_0^{13}} + \frac{6\sigma^{6}}{r_0^7} \right)</math>

At <math>r=r_0=\sigma</math>, <math>F=-4\epsilon \left( -\frac{12\sigma^{12}}{\sigma^{13}} + \frac{6\sigma^{6}}{\sigma^7} \right)</math>

<math>F=\frac{24\epsilon}{\sigma}</math>

Equilibrium occurs when the potential gradient is 0. <math>\frac{d\phi(r)}{dr}=4\epsilon \left( -\frac{12\sigma^{12}}{r_{eq}^{13}} + \frac{6\sigma^{6}}{r_{eq}^7} \right)=0</math>

<math>\frac{12\sigma^{12}}{r_{eq}^{13}} = \frac{6\sigma^{6}}{r_{eq}^7} </math>

<math>r_{eq}=\sqrt[6]{2}\sigma</math>

The well depth at equilibrium is given by <math>\phi\left(r_{eq}\right) = 4\epsilon \left( \frac{\sigma^{12}}{\sqrt[6]{2}\sigma^{12}} - \frac{\sigma^6}{\sqrt[6]{2}\sigma^6} \right)</math>

<math>\phi\left(r_{eq}\right)=4\epsilon \left( \frac{1}{4} - \frac{1}{2} \right)</math>

<math>\phi\left(r_{eq}\right)=-\epsilon</math>

Integrals. <math>\sigma = \epsilon = 1.0</math>

<math>\int_{2\sigma}^\infty \phi\left(r\right)\mathrm{d}r=\int_2^\infty 4\left( \frac{1}{r^{12}} - \frac{1}{r^{6}} \right)\mathrm{d}r</math>

<math>=4 \left[ -\frac{1}{11r^{11}} + \frac{1}{5r^{5}}\right]_{2}^{\infty} =-0.0248</math>

<math>\int_{2.5\sigma}^\infty \phi\left(r\right)\mathrm{d}r=\int_2.5^\infty 4\left( \frac{1}{r^{12}} - \frac{1}{r^{6}} \right)\mathrm{d}r</math>

<math>=4 \left[ -\frac{1}{11r^{11}} + \frac{1}{5r^{5}}\right]_{2.5}^{\infty} =-0.00818</math>

<math>\int_{3\sigma}^\infty \phi\left(r\right)\mathrm{d}r=\int_3^\infty 4\left( \frac{1}{r^{12}} - \frac{1}{r^{6}} \right)\mathrm{d}r</math>

<math>=4 \left[ -\frac{1}{11r^{11}} + \frac{1}{5r^{5}}\right]_{3}^{\infty} =-0.00329</math>

* '''Estimate the number of water molecules in 1ml of water under standard conditions. Estimate the volume of 10000 water molecules under standard conditions.'''

Water has a density of 1 g/mL and a molar mass of 18 g/mol. Thus there are 0.055 moles of water in 1 mL and 3.34x10<sup>22</sup> molecules, found by multiplying the number of moles with Avagrado's constant (6.022x10<sup>23</sup>). 1000 water molecules is equivalent 31.66x10<sup>-22</sup> moles, found by dividing by Avagrado's number. This is 2.99x10<sup>-19</sup> g and consequently 2.99x10<sup>-19</sup> mL.

* '''Consider an atom at position (0.5, 0.5, 0.5) in a cubic simulation box which runs from (0, 0, 0) to (1, 1, 1). In a single timestep, it moves along the vector (0.7, 0.6, 0.2). At what point does it end up, after the periodic boundary conditions have been applied?'''

The particle ends up at (0.2, 0.1, 0.7) after applying periodic boundary conditions to the position (1.2, 1.1, 0.7) achieved by vector addition.

* '''The Lennard-Jones parameters for argon are <math>\sigma = 0.34\mathrm{nm}, \epsilon\ /\ k_B= 120 \mathrm{K}</math>. If the LJ cutoff is <math>r^* = 3.2</math>, what is it in real units? What is the well depth in <math>\mathrm{kJ\ mol}^{-1}</math>? What is the reduced temperature <math>T^* = 1.5</math> in real units?'''

<math>r=r^*\sigma=3.2\times0.32x10^{-9}= 1.09 nm</math>

<math>T=\frac{T^*k_b}{\epsilon}=1.5\times120=180K</math>

<math>\epsilon=120k_b=120\times 1.38\times 10^{22}=1.66\times 10^{-21}=0.996 kJmol^{-1} </math>

===Equilibration===

*'''Why do you think giving atoms random starting coordinates causes problems in simulations? Hint: what happens if two atoms happen to be generated close together?'''

Random starting coordinates could generate atoms too close together, leading the atoms to unrealistically large interatomic repulsion and so velocities which would distort the simulation unless a very small timestep was used.

*'''Satisfy yourself that this lattice spacing corresponds to a number density of lattice points of 0.8. Consider instead a face-centred cubic lattice with a lattice point number density of 1.2. What is the side length of the cubic unit cell?'''

A face-centred cubic lattice contains has 4 atoms per unit cell.

<math>\rho_N=\frac{N}{V} </math>

<math>1.2=\frac{4}{l^3}</math>

<math>l=1.49 </math>

*'''TASK: Consider again the face-centred cubic lattice from the previous task. How many atoms would be created by the create_atoms command if you had defined that lattice instead?'''

1000 unit cells containing 4 atoms each would generate 4000 atoms.

*'''Using the [http://lammps.sandia.gov/doc/Section_commands.html#cmd_5 LAMMPS manual], find the purpose of the following commands in the input script:'''

<pre>
mass 1 1.0
pair_style lj/cut 3.0
pair_coeff * * 1.0 1.0
</pre>

mass: atom type 1 with mass of 1.0

pair_style: Lennard-Jones potential between atom pairs ignoring coulombic interactions with the interaction cutoff at a separation of 3.0 A.

pair_coeff: for interactions between all atoms 1 to N, sigma and epsilon are set to 1.0.

*''' Given that we are specifying <math>\mathbf{x}_i\left(0\right)</math> and <math>\mathbf{v}_i\left(0\right)</math>, which integration algorithm are we going to use?'''

Velocity-verlet

*'''Look at the lines below.'''
<pre>
### SPECIFY TIMESTEP ###
variable timestep equal 0.001
variable n_steps equal floor(100/${timestep})
timestep ${timestep}

### RUN SIMULATION ###
run ${n_steps}
run 100000
</pre>
'''The second line (starting "variable timestep...") tells LAMMPS that if it encounters the text ${timestep} on a subsequent line, it should replace it by the value given. In this case, the value ${timestep} is always replaced by 0.001. In light of this, what do you think the purpose of these lines is? Why not just write:'''
<pre>
timestep 0.001
run 100000
</pre>

Defining timestep as a variable allows the number of steps required to reach the set time for each timestep to be calculated by LAMMPS. This standardises the input file.

*'''Make plots of the energy, temperature, and pressure, against time for the 0.001 timestep experiment (attach a picture to your report). Does the simulation reach equilibrium? How long does this take? When you have done this, make a single plot which shows the energy versus time for all of the timesteps (again, attach a picture to your report). Choosing a timestep is a balancing act: the shorter the timestep, the more accurately the results of your simulation will reflect the physical reality; short timesteps, however, mean that the same number of simulation steps cover a shorter amount of actual time, and this is very unhelpful if the process you want to study requires observation over a long time. Of the five timesteps that you used, which is the largest to give acceptable results? Which one of the five is a ''particularly'' bad choice? Why?'''

{|style="margin: 0 auto;"
|[[Image:WdmnEnergy.png|thumb|upright=1.5|A plot of total energy against time for timestep 0.001.]]
|[[Image:WdmnTemperature.png|thumb|upright=1.5|A plot of temperature against time for timestep 0.001.]]
|[[Image:WdmnPressure.png|thumb|upright=1.5|A plot of pressure against time for timestep 0.001.]]
|}

The system reaches equilibrium at 0.4 τ*, indicated by the convergence of energy.

[[Image:WdmnTimestep.png|thumb|upright=2.5|centre|A comparison on the effect of timestep on energy convergence.]]

Timestep 0.015 fails to converge and increases constantly throughout the simulation. Additionally, the timesteps 0.01 and 0.075 are unsuitable as they converge to an incorrect energy value. Thus the only practicable timesteps are 0.0025 and 0.001. The largest acceptable timestep, 0.0025, is more computationally efficient than 0.001 without compromising significant accuracy.

===Running Simulation over Specific Conditions===

*'''Choose 5 temperatures (above the critical temperature <math>T^* = 1.5</math>), and two pressures (you can get a good idea of what a reasonable pressure is in Lennard-Jones units by looking at the average pressure of your simulations from the last section). This gives ten phase points — five temperatures at each pressure. Create 10 copies of npt.in, and modify each to run a simulation at one of your chosen <math>\left(p, T\right)</math> points. You should be able to use the results of the previous section to choose a timestep. Submit these ten jobs to the HPC portal. While you wait for them to finish, you should read the next section.'''

*''' We need to choose <math>\gamma</math> so that the temperature is correct <math>T = \mathfrak{T}</math> if we multiply every velocity <math>\gamma</math>. We can write two equations:
<math>\frac{1}{2}\sum_i m_i v_i^2 = \frac{3}{2} N k_B T</math>

<math>\frac{1}{2}\sum_i m_i \left(\gamma v_i\right)^2 = \frac{3}{2} N k_B \mathfrak{T}</math>

'''Solve these to determine <math>\gamma</math>.'''

<math>\frac{1}{2}\sum_i m_i \left(\gamma v_i\right)^2 = \frac{1}{2}\gamma^2\sum_i m_i \left( v_i\right)^2 </math>

Dividing <math>\frac{1}{2}\gamma^2\sum_i m_i \left( v_i\right)^2= \frac{3}{2} N k_B \mathfrak{T} </math> by <math>\frac{1}{2}\sum_i m_i v_i^2 = \frac{3}{2} N k_B T</math> gives

<math>\gamma^2=\frac{\mathfrak{T}}{T} </math>

<math>\gamma=\sqrt{\frac{\mathfrak{T}}{T}} </math>

*'''Use the [http://lammps.sandia.gov/doc/fix_ave_time.html manual page] to find out the importance of the three numbers ''100 1000 100000''. How often will values of the temperature, etc., be sampled for the average? How many measurements contribute to the average? Looking to the following line, how much time will you simulate?'''

fix_aves specifies Nevery, Nrepeat and Nfreq, in this case as the three numbers ''100 1000 100000''. An average is calculated every on timestep that is a multiple of Nfreq, using the directly preceding Nrepeat values that are multiples of Nevery and averaging over Nrepeat. If 1000000 timesteps are simulated in this experiment, 10 averages are calculated using the preceding 1000 numbers that are a multiple of 100 and dividing by 1000.

*'''When your simulations have finished, download the log files as before. At the end of the log file, LAMMPS will output the values and errors for the pressure, temperature, and density <math>\left(\frac{N}{V}\right)</math>. Use software of your choice to plot the density as a function of temperature for both of the pressures that you simulated. Your graph(s) should include error bars in both the x and y directions. You should also include a line corresponding to the density predicted by the ideal gas law at that pressure. Is your simulated density lower or higher? Justify this. Does the discrepancy increase or decrease with pressure?'''

{|style="margin: 0 auto;"
|[[Image:WdmnNV18.png|thumb|upright=2|The typeface is Myriad Pro.]]
|[[Image:WdmnNV26.png|thumb|upright=2|The typeface is Myriad Pro.]]
|}

The simulated number density for the fluid is higher than that predicted by the ideal gas law. The ideal gas model only considers perfectly elastic collisions between atoms and ignores electrostatic interactions. The simulated conditions includes pairwise Lennard-Jones potential that leads to repulsion between atoms at short distances. Hence the number of atoms per unit volume is less for the simulation than the ideal gas law. As the temperature increases, the discrepancy between the number densities decreases. At higher temperatures, the collision velocities are sufficient to overcome the intermolecular repulsive forces that dominate at lower temperatures as <math>v^2\alpha T</math>.

===Calculating Heat Capacities using Statistical Physics===

''' As in the last section, you need to run simulations at ten phase points. In this section, we will be in density-temperature <math>\left(\rho^*, T^*\right)</math> phase space, rather than pressure-temperature phase space. The two densities required at <math>0.2</math> and <math>0.8</math>, and the temperature range is <math>2.0, 2.2, 2.4, 2.6, 2.8</math>. Plot <math>C_V/V</math> as a function of temperature, where <math>V</math> is the volume of the simulation cell, for both of your densities (on the same graph). Is the trend the one you would expect? Attach an example of one of your input scripts to your report.'''

[[Image:Wdmnheatcapacity.png|thumb|upright=2.5|centre|Volume reduced isochoric heat capacity against temperature for ρ*=0.2 and 0.8.]]

The plot indicates that the isochoric heat capacity decreases with increasing temperatures for ρ*=0.8. In a system of finite energy levels, there is a higher density of levels towards the energy maximum. At higher temperatures, electrons populate higher energy levels according to the Boltzmann distribution. Therefore, as energy level gap decreases, the thermal energy required to raise the internal energy by populating higher levels subsequently decreases. As <math>C_V=(\frac{{\partial U}}{{\partial T}})_V</math>, the heat capacity decreases with increasing temperatures until all energy levels are equally occupied and it tends to infinity.

Whilst this trend is observed initially in the case of ρ*=0.2, the atoms go through a phase change, indicated by the rapid increase. At T*=2.4 all energy is converted to latent heat. After the phase change, the fluid can access further degrees of freedom such as vibration and rotation which would raise the heat capacity as they are an avenue for thermal energy. These extra degrees of freedom were unavailable at ρ*=0.8.

Furthermore, the heat capacity is consistently higher in the more dense system of ρ*=0.8 than ρ*=0.2 while obeying the same trend for temperature because the thermal energy must be distributed over a greater number of atomic energy levels per unit cell, requiring more energy.

Heat Capacity Input Script[[File:wdmnNvt_0.8_2.8.txt]]

===Structural Properties and the Radial Distribution Function===

*'''Perform simulations of the Lennard-Jones system in the three phases. When each is complete, download the trajectory and calculate <math>g(r)</math> and <math>\int g(r)\mathrm{d}r</math>. Plot the RDFs for the three systems on the same axes, and attach a copy to your report. Discuss qualitatively the differences between the three RDFs, and what this tells you about the structure of the system in each phase. In the solid case, illustrate which lattice sites the first three peaks correspond to. What is the lattice spacing? What is the coordination number for each of the first three peaks?'''

{|style="margin: 0 auto;"
|[[Image:WdmnG(r).png|thumb|upright=2|The radial distribution function.]]
|[[Image:WdmnIntg(r).png|thumb|upright=2|The running integral of the radial distribution function]]
|}

The radial distribution function relates density to the radial distance from a reference atom. At short distances, <math>r<\sigma, g(r)=0</math> due interatomic repulsion, according to the Lennard-Jones potential. After reaching a maximum for the single coordination shell of a gas, <math>g(r)</math> decays to 1. Although liquids are significantly more ordered than a gas, radial distribution peaks for approximately the first three coordination shells before tending to 1. As liquids are dynamic, the peaks are broad and coordination shells are not correlated to the reference particle over longer distances. The radial distribution function for the solid, displays regular maxima for the coordination spheres at <math>\sqrt{n}</math> multiples of <math>\sigma</math>. The peak broadness results from atomic vibrations. The running integral, <math>\int(g(r)</math> corresponds to the number of atoms at that radial distance by <math>n(r)=4\pi\rho \int_0^{r'}g(r)r^2dr </math>

{|style="margin: 0 auto;"
|[[Image:WdmnCoordination.png|thumb|upright=2|right|This region of the running integral plot shows the number of atoms in the first three coordination shells.]]
|}

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+''' Data reported for lattice site coordination'''
|-
! Peak !!="col" | Latice spacing !!="col" | Intg(r) !! ="col" | Number of atoms !! ="col" | Coordination number
|-
! scope="row" | 1
| σ || 12 || 12 || 12
|-
! scope="row" | 2
|√2 σ ||18 || 6 || 12
|-
!scope="row" | 3
| √3 σ || 42 || 24 || 12
|}

{|style="margin: 0 auto;"
|[[Image:WdmnFcc.png|thumb|upright=2|right|fcc lattice illustrating atoms corresponding to peaks and radial diagram showing coordination shells for solid]]
|}

===Dynamical Properties and the Diffusion Coefficient===

'''Make a plot for each of your simulations (solid, liquid, and gas), showing the mean squared displacement (the "total" MSD) as a function of timestep. Are these as you would expect? Estimate D in each case. Be careful with the units! Repeat this procedure for the MSD data that you were given from the one million atom simulations.'''

{| class="wikitable" style="text-align: center"
|+ Mean Square Displacement against Time
|-
! Number of Atoms
! Solid
! Liquid
! Vapour
|-
! 1000
| [[Image:WdmnMSDsolid1000.png|thumb|upright=1.4]] || [[Image:WdmnMSDliquid1000.png|thumb|upright=1.4]] || [[Image:WdmnMSDgas1000.png|thumb|upright=1.4]]
|-
! 1000000
| [[Image:WdmnMSDsolid1000000.png|thumb|upright=1.4]] || [[Image:WdmnMSDliquid1000000.png|thumb|upright=1.4]] || [[Image:WdmnMSDgas1000000.png|thumb|upright=1.4]]
|}

The gradient in the diffusive region (straight line) was determined and the diffusion coefficient calculated using the relationship <math>D = \frac{1}{6}\frac{\partial\left\langle r^2\left(t\right)\right\rangle}{\partial t}</math>. These values can be rationalised by theory. Atoms in a solid are restricted to one degree freedom, vibration, and so are not able to diffuse throughout the box. Liquid atoms are dynamic and experience limited translational motion allowing diffusion. Atoms in the vapour state, however, possess full translational freedom and so diffuse throughout the box. The difference between the 1000 and 100000 atom diffusion coefficients is caused by significantly greater averaging which reduces the effects of fluctuations on the simulation and considers a greater number of scenarios.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+''' Data reported for diffusion coefficient determined from MSD'''
|-
! Number of Atoms !!="col" | Solid !!="col" | Liquid !! ="col" | Gas
|-
! scope="row" | 1000
| 1.06x10<sup>-6</sup> || 0.106 || 7.25
|-
! scope="row" | 1000000
|8.27x10<sup>-6</sup> ||0.0873 || 3.01
|}

'''In the theoretical section at the beginning, the equation for the evolution of the position of a 1D harmonic oscillator as a function of time was given. Using this, evaluate the normalised velocity autocorrelation function for a 1D harmonic oscillator (it is analytic!):'''

<math>C\left(\tau\right) = \frac{\int_{-\infty}^{\infty} v\left(t\right)v\left(t + \tau\right)\mathrm{d}t}{\int_{-\infty}^{\infty} v^2\left(t\right)\mathrm{d}t}</math>

'''Be sure to show your working in your writeup. On the same graph, with x range 0 to 500, plot <math>C\left(\tau\right)</math> with <math>\omega = 1/2\pi</math> and the VACFs from your liquid and solid simulations. What do the minima in the VACFs for the liquid and solid system represent? Discuss the origin of the differences between the liquid and solid VACFs. The harmonic oscillator VACF is very different to the Lennard Jones solid and liquid. Why is this? Attach a copy of your plot to your writeup.'''

<math>=\frac{\int_{-\infty}^{\infty} (-A\omega\sin{(\omega t+\phi)} (-A\omega\sin{(\omega(t+\tau)+\phi)})\mathrm{d}t}{\int_{-\infty}^{\infty} (-A\omega\sin{(\omega t+\phi}))^2\mathrm{d}t}</math>

<math>=\frac{\int_{-\infty}^{\infty} \frac{1}{2}( \cos{(-\omega\tau)}-cos{(2\omega t+\omega\tau+2\phi)})\mathrm{d}t}{\int_{-\infty}^{\infty} \frac{1}{2} (1-\cos{(2\omega t+2\phi)})\mathrm{d}t}</math>

<math>= \left[\frac{t\cos{(-\omega\tau)}-\sin{(2\omega t+\omega\tau+2\phi)}\times \frac{1}{2\omega}} {t-\sin{(2\omega t+2\phi)}\times\frac{1}{2\omega} } \right]_{-\infty}^{\infty} </math>

<math>= \left[\frac{\cos{(-\omega\tau)}-\frac{1}{2\omega t}\sin{(2\omega t+\omega\tau+2\phi)}} {1-\frac{1}{2\omega t}\sin{(2\omega t+2\phi)} } \right]_{-\infty}^{\infty} </math>

<math>= \frac{\cos{(-\omega\tau)}-0}{1-0} </math>

<math>= \cos{(\omega\tau)}</math>

{|style="margin: 0 auto;"
|[[Image:WdmnVACF1000.png|thumb|upright=2|The VACF for the three states with the correlation function plotted. 1000 atoms.]]
|[[Image:WdmnVACF1000000.png|thumb|upright=2|The VACF for the three states with the correlation function plotted. 1000000 atoms]]
|}

The analytic correlation predicted by the 1D harmonic oscillator maps similarly to the simulated VACF for solid and liquid initially but as they decay to zero it retains the sinusoidal pattern. This reflects the extent to which the atoms in the solid and liquid phases are able to oscillate around a fixed point like the 1D harmonic oscillator. Moreover, the simulated VACFs resemble the shape of the Lennard-Jones potential. The regularly structured solid vibrates around a fixed point but this reduced as the system tend to equilibrium. The translational motion of the liquid initially follows the same path as the 1D harmonic oscillator in the ballistic region but as diffusion begins to dominate its motion the VACF tends to 1. It also demonstrates the difference between the motion of the liquid and solid: the solid undergoes vibrations and liquid undergoes diffusion. Introducing damping to the 1D harmonic oscillator would improve the fit. The minima in the VACF simulations are due to the change in direction of the atoms resulting from interatomic collisions.

''' Use the trapezium rule to approximate the integral under the velocity autocorrelation function for the solid, liquid, and gas, and use these values to estimate <math>D</math> in each case. You should make a plot of the running integral in each case. Are they as you expect? Repeat this procedure for the VACF data that you were given from the one million atom simulations. What do you think is the largest source of error in your estimates of D from the VACF?'''

{| class="wikitable" style="text-align: center"
|+ Diffusion Coefficient against Time for the Velocity Correlation Function
|-
! Number of Atoms
! Solid
! Liquid
! Vapour
|-
! 1000
| [[Image:WdmnVACFRI1000solid.png|thumb|upright=1.4]] || [[Image:WdmnVACFRI1000liquid.png|thumb|upright=1.4]] || [[Image:WdmnVACFRI1000gas.png|thumb|upright=1.4]]
|-
! 1000000
| [[Image:WdmnVACFRI1000000solid.png|thumb|upright=1.4]] || [[Image:WdmnVACFRI1000000liquid.png|thumb|upright=1.4]] || [[Image:WdmnVACFRI1000000gas.png|thumb|upright=1.4]]
|}

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+''' Data reported for diffusion coefficient determined from VACF'''
|-
! Number of Atoms !!="col" | Solid !!="col" | Liquid !! ="col" | Gas
|-
! scope="row" | 1000
| 6.11x10<sup>-5</sup> || 0.0979 || 8.45
|-
! scope="row" | 1000000
|4.55x10<sup>-5</sup> ||0.0901 || 3.27
|}

The diffusion coefficient values estimated from the VACF by determining the area under the curve and applying the relationship <math>D = \frac{1}{3}\int_0^\infty \mathrm{d}\tau \left\langle\mathbf{v}\left(0\right)\cdot\mathbf{v}\left(\tau\right)\right\rangle</math> follow the same trends as the VACF. The results follow the same trends described for the calculation by MSD. The largest errors in the estimation of the diffusion coefficient from the area under the VACF is the trapezium rule and that the running integrals for vapour and solid do not converge to a limit. The trapezium rule estimates the area under a straight line connecting two points, ignoring the path taken between them. As this simulation produced a curve, an error results in measuring the area under it. Additionally, as the running integrals continued to increase for the liquid and vapour, the VACF has not converged to 0 and the diffusion coefficient is larger than that stated. This is less pronounced for 1000000 atoms, which suggests better averaging.

==References==

<references />

Rep:WM1415MgO

2025-09-01T09:49:49Z

Move page script: Move page script moved page WM1415MgO to Rep:WM1415MgO: Move to report namespace

'''<big>This is William Micou's report on the 3rd year computational MgO experiment, starting Monday 27/11/2017</big>'''

==Introduction==

===Thermal Expansion===

Thermal expansion is an important property of materials to consider in engineering. Demonstrative examples include rails and bridges, which are specifically designed to expand and contract with temperature. The thermal expansion coefficient, <math>\alpha</math>, can be used to calculate thermodynamic properties of crystals, such as entropy<sup>[1]</sup>. It measures the fractional change in volume as temperature varies (at constant pressure):

<center><math>\alpha=\frac{1}{V_{0}}\left(\frac{\partial V}{\partial T}\right)_{p}</math></center>

In this experiment, the thermal expansion coefficient of the ''fcc'' crystal of MgO (or periclase<sup>[2]</sup>), will be determined using computational methods. Two models will be investigated and compared: the quasi-harmonic approximation (QHA), and molecular dynamics (MD) calculations. As one of the components of the Earth's mantle, studies of periclase are important to better understand seismic velocities at the Earth's interior<sup>[2]</sup>. Since a wide range of studies have been performed on the thermal expansion of MgO, this is a useful exercise to appreciate the level of the approximations used in these calculations and to explore the theory of vibrations in crystals.

===Vibrational band theory===

In order to calculate thermodynamic properties for the MgO lattice, the vibrational band structure must be calculated. The electronic band structure of metals is based on very similar theory: it is simplest to consider the vibrations within a crystal as a series of H 1s wavefunctions<sup>[3]</sup>, with their phases arranged in accordance with the vibrational mode of interest. The interactions between these wavefunctions gives rise to the continuous series of vibrational energy levels - a band. A linear combination of these orbitals yields the Bloch function<sup>[3]</sup>:

<center><math>\psi_{\boldsymbol{k}}=\sum\limits_{n}\phi_{n}\exp\left(i\boldsymbol{k}n\boldsymbol{a}\right)</math></center>

The lattice parameter is given by '''a'''. The wavevector '''k''' is given by <math>\boldsymbol{k}=\frac{2\pi}{\lambda}</math>. As the value of '''k''' increases, the wavelength of the vibration decreases and hence the energy of vibration increases. The First Brillouin Zone (FBZ) is comprised of all the unique values of '''k''': it is delimited by the region in '''k''' space given by <math>-\frac{\pi}{\boldsymbol{a}}\leq\boldsymbol{k}<\frac{\pi}{\boldsymbol{a}}</math> <sup>[3]</sup>. The lattice vibration modes exhibit wave-particle duality: ''phonons'' are the particle associated with a lattice vibration at a particular frequency. In the 1D case, the frequency associated with a particular value of '''k''' is given by:

<center><math>\omega_{\boldsymbol{k}}=\sqrt{\frac{4J}{M}}|\sin\left(\frac{\boldsymbol{ka}}{2}\right)|</math></center>

The density of states (DOS) is a useful construct to appreciate the number of energy levels associated with a particular frequency. It is defined by<sup>[3]</sup>:

<center><math>DOS(E) dE = </math>number of levels between <math>E</math> and <math>E+dE</math></center>

===The Quasi-Harmonic Approximation===

In order to calculate thermodynamic properties, the phonon modes over the First Brillouin Zone (FBZ) must be known<sup>[4]</sup>. In this experiment, a grid of '''k'''-points under the shrinking factor 32 was used to sample the phonon modes of the MgO crystal. Under the harmonic approximation, 3M (M - number of atoms in the cell) harmonic oscillators are associated with every point in '''k'''-space in the FBZ, with energies<sup>[4]</sup>:

<center><math>\varepsilon_{v}^{i,\boldsymbol{k}}=\left(v+\frac{1}{2}\right)\omega_{\boldsymbol{k}i}</math> where <math>i</math> is the phonon index, taking values from 1 to 3M.</center>

Due to the symmetry of the crystal lattice, the vibrational canonical partition function can be explicitly calculated<sup>[4]</sup>:

<center><math>Q_{vib}(T)=\sum\limits_{\boldsymbol{k}=\boldsymbol{0}}^{L-1}\sum\limits_{i=1}^{3M}\sum\limits_{v=0}^{\infty}\exp\left[-\frac{\varepsilon_{v}^{i,\boldsymbol{k}}}{k_{B}T}\right]</math></center>

Explicitly evaluating the vibrational partition function allows for the calculation of the temperature dependence of entropy, <math>S\left(T\right)</math> and internal energy, <math>E\left(T\right)</math> from statistical mechanics relations<sup>[4,5]</sup>. However, under the harmonic approximation, the thermodynamics of the system are calculated independently of volume: as a result, zero thermal expansion is predicted<sup>[4]</sup>, among other properties which cannot be modelled at this simple level<sup>[6,7]</sup>.

To calculate the thermal expansion a dependence of phonon frequency on on volume <sup>[4]</sup> is introduced to the free energy. This is the basis of the Quasi-Harmonic Approximation (QHA)<sup>[4]</sup>:

<center><math>F^{QHA}(T,V)=U_{0}(V)+F_{vib}^{QHA}(T,V)</math></center>

<center><math>F_{vib}^{QHA}(T,V)=E_{0}^{ZP}(V)+k_{B}T\sum_{\boldsymbol{k}i}\left[\ln\left(1-e^{-\frac{\hbar\omega_{\boldsymbol{k}i}(V)}{k_{B}T}}\right)\right]</math></center>

<math>U_{0}(V)</math> is the internal energy of the crystal at 0 K, with zero vibrational contribution<sup>[4]</sup> and <math>E_{0}^{ZP}</math> is the zero point energy contribution of the phonons: <math>E_{0}^{ZP}(V)=\sum_{\boldsymbol{k}i}\frac{1}{2}\hbar\omega_{\boldsymbol{k}i}(V)</math>. To probe the temperature dependence of the volume, <math>V(T)</math>, the geometry of the MgO crystal is optimised by minimising the free energy, <math>F^{QHA}(T,V)</math>, at constant T<sup>[4]</sup>.

===Molecular Dynamics===

The molecular dynamics method is a purely classical model. The physical properties of MgO are estimated by numerical integration of Newtonian equations of motion<sup>[8]</sup>. The algorithm used in this experiment, lifted directly from the lab script:
*Generate an ideal lattice of MgO, with individual atoms having random velocities assigned by the Boltzmann distribution
*Compute the forces on the atoms
*Compute the accelerations from Newton's Law <math>F = ma</math>
*Update the velocities: <math>\boldsymbol{v}_{new}=\boldsymbol{v}_{old}+\boldsymbol{a}\times dt</math>
*Update the positions of the atoms: <math>\boldsymbol{r}_{new}=\boldsymbol{r}_{old}+\boldsymbol{v}_{new}\times dt</math>

MD calculations are typically executed on a ''supercell'' of a crystal lattice; that is, a large cell containing many MgO units. The larger the supercell, the better the representation of the collective motion of the lattice - the phonons<sup>[9]</sup>.

==Methodology==

Input calculation files were constructed in DLVisualize and were executed using GULP (General Utility Lattice Program) version 1.4.43 (last modified March 2003).
All calculations were run under an ionic potential, excluding shells. Non-Coulombic interatomic interactions were modelled with Buckingham potentials. Long-range interactions were accounted for by Ewald summation.

===Phonon Dispersion and DOS===

A single-point phonon dispersion calculation was run on a primitive cell of MgO. The phonon eigenvectors were calculated at 50 points along the W-L-G-X-W-K direction in '''k'''-space. To calculate the DOS, a grid of points in '''k'''-space were chosen to sample the possible vibrations: the number of points in the grid depends on the shrinking factors chosen in the experiment.

===Thermal Expansion in QHA===

From temperatures 0-2900 K (in steps of 100 K), an optimisation of the free energy was computed on a primitive cell of MgO, keeping T constant and allowing the volume to vary. A shrinking factor of 32 (to give a grid size 32x32x32) was used to calculate the phonons, as this was deemed sufficient to obtain a satisfactory approximation of the free energy (further detailed later in the report).

===Thermal Expansion in MD===

From temperatures 100-3000K (in steps of 100 K), a single-point MD calculation was performed on a supercell of MgO, containing 32 MgO units, in the NPT ensemble. This allowed for the volume to vary while keeping the temperature constant. Time intervals <math>dt</math> were chosen to be 1.0 fs. The system was allowed to equilibrate for 500 steps; the volumes of the next 500 steps were recorded and averaged.

==Results and Discussion==

===Phonon Modes===

====MgO Experiment====

[[File:wm1415_Combined_DOS.PNG|600 px|thumb|right|Figure 1: The phonon dispersion of MgO and the DOS calculated for a variety of shrinking factors. (Click for expanded view)]]

''How does the DOS related to the dispersion curves?''

Figure 1 shows the phonon dispersion of MgO, calculated at 50 points along the conventional path in '''k'''-space: W-L-G-X-W-K, as well as the density of states calculated for a selection of shrinking factors. The density of states represents the number of states available to the system at a particular frequency (energy)<sup>[3]</sup>. The exact density of states is obtained by integration of the dispersion curve across the Brillouin Zone:

<center><math>DOS(E) dE = </math>number of levels between <math>E</math> and <math>E+dE</math></center>

Since analytical integration is computationally costly, the DOS is instead approximated via a numerical integration: a finite grid of points in '''k'''-space is chosen. At each point, the phonons are calculated and weighted (by the inverse of the total number of grid points). It is worth noting that not all points in the grid are evaluated, as some are related by symmetry and hence are identical.

''Which '''k'''-point was computed in the DOS 1x1x1 grid calculation?''

For a 1x1x1 grid, only ony point in '''k'''-space is sampled: this corresponds to the point L on the phonon dispersion curve shown in Figure 1 (coordinates in '''k'''-space: [0.5, 0.5, 0.5]). Four distinct peaks are obtained at 286, 351, 676 and 806 cm<sup>-1</sup>: these agree with the intersections of the phonon dispersion curve with the vertical line at L. The intensities of the peaks at 281 and 351 cm<sup>-1</sup> are exactly double the intensities of the other two peaks, as there are doubly degenerate states at these frequencies (it can be seen that the dispersion curve splits up into two lines).

''How does the density of states vary with grid size? What grid size is the minimum for a reasonable approximation of the DOS?''

As the grid becomes more and more subdivided, more points in '''k''' space are sampled for the numerical integration of the phonon dispersion, and the increased number of peaks in the DOS plot begins to approach a continuous function. If the grid were further subdivided into infinitesimally small separations, the analytical integration of the phonon dispersion would be obtained. With an 8x8x8 grid, a reasonable approximation of the DOS is obtained, but there remains a considerable amount of noise, and smaller features of the plot cannot be seen. The resolution continues to improve until the 32x32x32 grid, which is almost identical to the plot obtained from the 64x64x64 grid. However, since the 64x64x64 calculation is so much more intensive, further calculations in this experiment involving MgO will use shrinking factors of 32x32x32.

====Extension to other systems====
CaO: Ca is in the same group as Mg: one would expect a similar lattice parameter and hence the periodic repeat unit in reciprocal space, '''a*''' with <math>\boldsymbol{a^{*}}=\frac{2\pi}{\boldsymbol{a}}</math>, will also be similar. A reasonable approximation of the DOS would require sampling over a similar number of points in '''k'''-space, so a 32x32x32 grid would most likely be optimal for CaO as well. A study<sup>[4]</sup> has confirmed that the shrinking parameters from MgO also led to convergence for CaO.

Zeolites: The periodic repeat dimension in real space, lattice parameter '''a''' for zeolites is much larger than for MgO. Therefore, the repeat unit in reciprocal space '''a*''' would be much smaller: the first Brillouin zone would span a smaller range of '''k'''-values. One would expect a greater extent of 'folding'<sup>[3]</sup> in the branch structure for a zeolite. Hence, a single '''k''' point would sample many more frequencies in a zeolite compared to MgO. Therefore, in total, fewer '''k'''-points would need to be sampled in order to obtain an approximation for the DOS plot, and a smaller grid size could be used.

Metals (e.g. Lithium): The lattice parameter for a metal such as lithium is expected to be much smaller. Therefore, in reciprocal space, the Brillouin zone would span a larger range of '''k''' values. A higher number of '''k'''-points must be sampled in order to obtain a good representation of the phonon dispersion; hence a larger grid size would be required compared to MgO.

===Computing the Free Energy===

====MgO Experiment====

Table 1 shows the calculated free energies in meV obtained from various grid sizes. All significant figures output from the calculation have been included to demonstrate the convergence of the free energy optimisations. A 2x2x2 grid is sufficient to obtain an approximation for the free energy to within 1 meV; 3x3x3 and 4x4x4 grids are sufficient for approximations to within 0.5 and 0.1 meV respectively. This table of data is further evidence that a 32x32x32 grid of '''k''' points yields a numerical integration that is sufficiently accurate: there is no distinction between the values from the 32x32x32 and 64x64x64 grids at the level of precision of this experiment. In the next section, when free energies of MgO will be calculated at different temperatures, a shrinking factor of 32 will be chosen (grids 32x32x32).
[[File:wm1415_New_free_energies_shrink.PNG|400 px|thumb|left|Figure 2: The graph shows how the free energy of the optimised structure varies with the grid size used to sample the phonons.]]
{| class="wikitable"
|+Table 1: Free energies calculated at different grid sizes.
|-
! Lattice Size
! Free Energy (meV)
|-
| 1x1x1
| -40930.301
|-
| 2x2x2
| -40926.609
|-
| 3x3x3
| -40926.432
|-
| 4x4x4
| -40926.450
|-
| 5x5x5
| -40926.463
|-
| 6x6x6
| -40926.471
|-
| 8x8x8
| -40926.478
|-
| 16x16x16
| -40926.482
|-
| 32x32x32
| -40926.483
|-
| 64x64x64
| -40926.483

|}

====Extension to other systems====

The calculation of the free energy requires the evaulation of the phonons over the FBZ<sup>[4]</sup>. Therefore, the required grid sizes for CaO, zeolites and metals will be the same as required for an accurate depiction of the DOS. As outlined in the prior section: CaO can use approximately the same grid size as MgO, zeolites a smaller grid size, and metals a larger grid size.

It is worth noting that the increased polarisability<sup>[4]</sup> of Ca with respect to Mg leads to the breakdown of the QHA at lower temperatures: around 100 K for CaO <sup>[4]</sup> compared to around 1000 K for MgO<sup>[4]</sup>.

===Thermal Expansion of MgO===

====Quasi-Harmonic Approximation====

=====Free energy and lattice constant relationship with temperature=====

[[File:wm1415_Free_energy_lat_param_T.png|400 px|thumb|right|Figure 3: The dependence of the free energy <math>F(V,T)</math> and the primitive lattice parameter '''a<sub>p</sub>''' on the temperature., under the QHA model]]

A grid of '''k'''-points under the shrinking factor 32 was used to sample the phonon modes of the MgO crystal. An optimisation of the free energy was run at constant temperature, allowing only the volume to minimise F(V,T).

Figure 3 shows how the free energy and lattice parameter of MgO varies with temperature under the QHA model. Below 500 K, there is a non-linear decrease in the free energy as temperature increases. Above 500 K and below the breakdown of the QHA at around 1400 K, there is a linear dependence of the free energy on temperature: however, there is deviation from this straight line at higher temperatures.

The equation for the free energy under the QHA, as outlined in the introduction, can be generalised as:

<center><math>F^{QHA}(T,V)=U_{0}(V)+E_{0}^{ZP}(V)-TS(T,V)</math></center>

As higher vibrational levels are populated, the entropy increases as there are more states accessible to the system. Plotting the entropy term of the free energy against temperature, <math>k_{B}T\sum_{\boldsymbol{k}i}\left[\ln\left(1-e^{-\frac{\hbar\omega_{\boldsymbol{k}i}(V)}{k_{B}T}}\right)\right]</math>, one finds a linear relationship above the vibrational temperature <math>\theta_v</math> - the temperature above which excited vibrational states begin to be appreciably populated. The curved region at low temperatures is a manifestation of the quantised, discrete energy levels in the harmonic oscillator model: there is insufficient thermal energy to fully populate the excited, <math>v > 0</math>, states. The decrease in free energy with temperature is dominated by the entropy term: the variations in <math>U_{0}(V)</math> and <math>E_{0}^{ZP}</math> are comparatively minor (since the volume has a temperature dependence, these terms will also depend on T). At 0 K, the free energy is the sum of the internal energy with zero vibrational contribution and the sum of the zero point energies of all the phonons.

''As the temperature approaches the melting point of MgO, how well do the phonon modes represent the actual motions of the ions?''

At high temperatures, over 1500 K, there is a deviation away from the linear relationship: the gradients of the plots for both the free energy and the lattice parameter increase. These deviations are more pronounced in the volume-temperature plot, which will be examined in the next section. This is a result of the breakdown of the quasi-harmonic approximation. At such elevated temperatures, the anharmonic terms of the phonons that are ignored in the QHA become increasingly important. The infinite number of equally spaced vibrational energy levels in the harmonic approximation is no longer an accurate picture and the QHA is no longer valid. As the melting point is approached, an anharmonic model would predict bond dissociation: however, no such transition occurs in the QHA. In fact, attempts to run an optimisation calculation at 3000 K did not yield a converged structure within a reasonable time frame. The optimisation of the free energy for such an unrealistic system results in the deviation from linearity at high temperatures.

=====Computing the coefficient of thermal expansion, <math>\alpha</math>=====

[[File:wm1415_Quasi_linear_alpha.png|400 px|thumb|right|Figure 4: Determining <math>\alpha</math> from the QHA calculations via a linear fit in the region 500 K < T < 1400K.]]

The volumetric thermal expansion coefficient at constant pressure, <math>\alpha</math>, expresses the fractional change in volume of MgO as the temperature changes:

<center><math>\alpha=\frac{1}{V_{0}}\left(\frac{\partial V}{\partial T}\right)_{p}</math> with units K<sup>-1</sup></center>

For temperatures 500 K < T < 1400 K, the cell volume of MgO is an approximately linear function of T. By performing a linear fit in this restricted region, <math>\alpha</math> can be calculated from the gradient, with <math>V_{0}</math> as the cell volume at 500 K: <math>\alpha=\left(3.13\pm0.06\right)\times10^{-5}</math> <math>K^{-1}</math>. It is worth noting that this uncertainty is only a reflection of the quality of the linear fit: the error in the gradient was extracted from the Numpy polyfit() function to calculate the error in <math>\alpha</math>.

The main approximations in this calculation are the quasi-harmonic approximation itself - assuming that the phonons exhibit a harmonic potential at all volumes - and the assumption that the thermal expansion coefficient is independent of temperature in the region 500 K < T < 1400 K. Evidently smaller approximations have been made, such as in the methods for evaluating interatomic potentials, the exclusion of shells and the polarisability of the ions, and the assumption that the MgO crystal is perfect - with no defects or impurities. In the literature, some attempts have been made to account for quantum effects<sup>[8]</sup>, polarisability <sup>[4]</sup>, and defects<sup>[17]</sup>. Later in this report, an attempt will be made to determine the temperature dependence of <math>\alpha</math> from the data obtained from the QHA calculations, and the values for the thermal expansion coefficient will be compared to those found from the MD simulations and literature values.

''What is the physical origin of thermal expansion?''

In a classical sense, thermal expansion is caused by the increased kinetic energy of the atoms within the lattice. As the temperature increases, the atoms vibrate more strongly and there is increased average separation of the atoms, leading to thermal expansion. The origins of thermal expansion within the QHA will now be discussed.

''In a diatomic molecule with an exactly harmonic potential, would the bond length increase with temperature? Why does the bond length increase in the solid under the quasi-harmonic approximation?''

Under an exactly harmonic potential, the bond length is independent of temperature and will thus remain constant for all T. In the QHA, the phonons follow a harmonic potential at every volume: however, the phonon modes themselves have a volume-dependence. It is the inclusion of this anharmonic effect<sup>[10]</sup> that results in an increase in the bond length at higher temperatures and hence a thermal expansion is predicted<sup>[4,9]</sup>. To properly account for anharmonic effects, phonon-phonon interaction coefficients for all modes in the Brillouin Zone would have to be calculated<sup>[9]</sup>. However, this is not a practical exercise, and the other anharmonic terms are ignored in the QHA. In the literature, attempts have been made to include these higher terms within the approximation, giving rise to the Modified-QHA<sup>[17]</sup>.

Since the pair potentials as a function of interatomic distance are known for our system, the volume-dependence of the phonon modes can be obtained by performing harmonic-lattice-dynamics calculations for a large number of selected volumes<sup>[10]</sup>. Hence the minimisation of the free energies by varying the volume is made possible, and thermal expansion results.

====Molecular Dynamics Simulation====

[[File:wm1415_MD_linear_alpha_fullrange.png|400 px|thumb|right|Figure 5: The averaged conventional cell volume across 500 MD steps against temperature.]]

[[File:wm1415_Linear_MD_QH_comp_2.png|400 px|thumb|right|Figure 6: Comparison between the conventional cell volume-temperature relationships for the MD and QHA calculations.]]

The volume-temperature relationship for the MD calculations is plotted in Figure 5. A comparison between the volume-temperature curves in the MD and QHA models is shown in Figure 6.

Because of the classical nature of the calculation, the volume has a linear relationship with temperature for all T in the molecular dynamics simulation. Since the energy states are continuous rather than discrete, this linear relationship holds for even low temperatures (which do not excite the first vibrational state in the QHA). A linear fit over the full temperature range yields the thermal expansion coefficient <math>\alpha=\left(3.12\pm0.05\right)\times10^{-5}</math> <math>K^{-1}</math>. Once again, the uncertainty stated refers only to the error in the gradient of the linear fit calculated in Python. Over the same temperature range as used in the QHA (500 K < T < 1400 K), the MD simulation gives <math>\alpha=3.29\times10^{-5}</math> (with the uncertainty in the gradient being negligible).

''How large a cell should be used to reliably perform MD for MgO?''

In this experiment, 32 MgO units were used in the MD calculations. A prior study of MgO with MD techniques utilised a cell containing 256 MgO units<sup>[8]</sup>. With larger supercells, the phonons are better represented<sup>[9]</sup>: this comes at the cost of higher computational effort. The lack of ergodicity<sup>[9]</sup> limits the numerical efficiency of MD at low temperatures. For the purposes of this experiment, a 32 MgO unit supercell was sufficient. However, for more precise predictions, a larger cell should be used.

''How does the thermal expansion predicted by MD compare to that from QHA?''

Despite the very different approaches used in each model, the predicted thermal expansion coefficients are in remarkable agreement - at least, in the range where the QHA gives a linear relationship for V(T). In the MD simulation, where V(T) is linear for all T, the calculated thermal expansion coefficient is independent of temperature. Further in this report, the temperature dependence of <math>\alpha</math> in the QHA will be examined and compared to reference values.

''Why do the two methods produce different answers - and how does the difference depend on temperature? What would happen to the cell volume at high temperature in both models?''

Figure 6 shows the MD and QHA V(T) plots line up almost exactly from 500 K < T < 1400 K; the same region where a linear dependence was found in the QHA calculation. The values for the thermal expansion in this region match up almost exactly. At low temperatures, the QHA predicts a slightly larger volume: this is because the zero point energy of the phonons is taken into account; the interatomic spacings cannot be further reduced. The energy spacings for the harmonic oscillator (phonons) are quantised: at low temperatures, the first excited state is only partially accessible. A molecular dynamics calculation was not performed at 0 K since the system would not change from its initial state. At high temperatures, the QHA deviates significantly from the MD calculations as the anharmonicity of the phonons becomes more pronounced, and a deviation from linearity is observed as previously discussed. Near the melting point, such is the breakdown of the QHA that the simulation no longer converges (or at least, not within a reasonable timescale) as the free energy cannot be optimised for T > 3000K. At high temperatures, large fluctuations are seen in the MD calculations: the ions have much higher velocities and move much further in each timestep. It was observed in the animations produced from the trajectories of the MD simulations that, as a result of very high temperature, some quite unusual vibrations were calculated - the atoms move so far in each timestep that this is made possible. Using a smaller timescale and a greater number of equilibration steps may reduce the noise at high temperatures.

By calculating the forces on each individual atom at every step of the calculation, the MD simulation can account for the anharmonicity of the bonds in the lattice, unlike the QHA model.

====Comparison with reference data====

The thermal expansion coefficient is temperature-dependent. In the literature, <math>\alpha</math> is usually reported at a large selection of temperatures. The thermal expansion coefficients calculated so far have assumed a linear relationship between the lattice volume and the temperature. Table 2 gives some reference values for <math>\alpha</math> under standard conditions (T = 300 K), collated by Scanavino<sup>[2]</sup>.

The thermal expansion coefficients obtained in this experiment from the linear region of QHA, and MD calculations, are in remarkable agreement with literature values under standard conditions. In the literature<sup>[4,17]</sup>, <math>\alpha</math> is sometimes determined via a polynomial fit of the volume-temperature relationship. A similar approach will be attempted for this experiment, under the QHA model.

Polynomial fits for V(T) were calculated using the polyfit() function in Matplotlib. For a fit of degree <math>n</math>, the function returns:

<center><math>V(T)=a+bT+cT^{2}+...+mT^{n}</math></center>

To find <math>\alpha</math>, the gradient at any point can be explicitly calculated:

<center><math>V'(T)=b+2cT+...+nmT^{n-1}</math></center>

<center><math>\alpha=\frac{V'(T)}{V(T)}</math></center>

Figure 7 shows the computed <math>\alpha</math> values plotted against reference MgO data from Suzuki and Reebers. The polynomial fit with 16 terms is a better match than the 4 term fit to the general shape of the reference data from 0 < T < 1500 K. However, both produce curves which significantly underestimate the thermal expansion coefficient of MgO. Perhaps this is due to the more advanced models used in the literature, such as the MQHA, which account for higher anharmonic terms. The inclusion of shells and atom polarisability in this experiment may have improved the fit compared to the reference data. However, the shape of the function produced by the 16 term fit is similar to the <math>\alpha(T)</math> dependencies calculated in the literature. As a further extension, it would be beneficial to repeat this experiment using the more developed models from the literature.

[[File:wm1415_Poly_reference.png|400 px|thumb|right|Figure 7: Temperature dependence of the thermal expansion calculated for the QHA data and compared to reference values from Suzuki<sup>[16]</sup> and Reeber<sup>[17]</sup>.]]
{| class = "wikitable" align="left"
|+Table 2: Reference thermal expansion coefficients.
|-
!Source
!<math>\alpha</math> / <math>10^{-5} K^{-1}</math>
|-
|Saxena et al.<sup>[11]</sup>
|3.12
|-
|Fei<sup>[12]</sup>
|3.16
|-
|Chopelas<sup>[13]</sup>
|3.11
|-
|Isaak et al.<sup>[14]</sup>
|3.12
|-
|Karki et al. (computational)<sup>[15]</sup>
|3.11
|-
|This experiment - QHA
|3.13 ± 0.06
|-
|This experiment - MD
|3.12 ± 0.05
|-
|QHA Polynomial Fit
|3.08 ± 0.24
|}

==Conclusion==
In this experiment, the phonon dispersion of MgO was calculated and analysed. For a number of shrinking factors, DOS plots were drawn and optimised free energies were determined in order to select an appropriate level of calculations for further parts of the investigation. The thermal expansion coefficient was determined using a quasi-harmonic approximation and molecular dynamics methods, to within remarkable accuracy of reference values recorded under standard conditions. As an extension, the temperature dependence of the thermal expansion coefficient was evaluated for the QHA simulations: a similar shape was obtained to the QHA fits from literature, but the thermal expansion coefficient was significantly underestimated compared to reference values. A further investigation into the improved models used in the literature could explain this difference.

==References==
1 O. L. Anderson and K. Zou, J. Phys. Chem. Ref. Data, 1990, '''19''', 69–83.

2 I. Scanavino, R. Belousov and M. Prencipe, Phys. Chem. Miner., 2012, '''39''', 649–663.

3 R. Hoffmann, Angew. Chemie-International Ed. English, 1987, '''26''', 846–878.

4 A. Erba, M. Shahrokhi, R. Moradian and R. Dovesi, J. Chem. Phys., 2015, '''142'''.

5 P. Atkins and J. de Paula, ''Atkins' Physical Chemistry'', Oxford University Press, UK, 8th edn, 2006, pp. 560-600

6 N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, USA, 1976).

7 S. Baroni, P. Giannozzi, and E. Isaev, Rev. Mineral. Geochem. '''71''', 39 (2010).

8 M. Matsui, J. Chem. Phys., 1989, '''91''', 489.

9 S. Baroni, P. Giannozzi and E. Isaev, Rev. Mineral. Geochemistry, 2009, '''71''', 1–19.

10 L. L. Boyer, Phys. Rev. Lett., 1979, '''42''', 584–587.

11 S. Saxena, N. Chatterjee, Y. Fei, G. Shen (1993) Assessment of Data on Some Oxides and Silicates: Calorimetry and High-Pressure Phase Equilibrium Experiments. In: Thermodynamic Data on Oxides and Silicates. Springer, Berlin, Heidelberg

12 Y. Fei, L. Zhang, A. Corgne, H. Watson, A. Ricolleau, Y. Meng and V. Prakapenka, Geophys. Res. Lett., 2007, '''34''', 1–5.

13 A. Chopelas, Phys. Chem. Miner., 1990, '''17''', 142–148.

14 D. G. Isaak, O. L. Anderson and T. Goto, Phys. Chem. Miner., 1989, '''16''', 704–713.

15 B. B. Karki, Am. Mineral., 2000, '''85''', 1447–1451.

16 I. Suzuki, J. Earth Sci., 1975, '''23''', 145–159.

17 R. R. Reeber, K. Goessel and K. Wang, Eur. J. Mineral., 1995, '''7''', 1039–1047.

Rep:Transition states (st4215): Exercise 3

2025-09-01T09:49:48Z

Move page script: Move page script moved page Transition states (st4215): Exercise 3 to Rep:Transition states (st4215): Exercise 3: Move to report namespace

== Exercise 3: Diels-Alder vs Cheletropic ==
In this exercise, we examine the possible reactions between o-xylylene and SO<sub>2</sub> - the Diels-Alder reaction and the cheletropic reaction, as illustrated in the reaction scheme below.
[[File:st4215_Ex3_reactionscheme.png|thumb|center|600px| Reaction of o-xylylene and SO<sub>2</sub>
[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_exercise (Source)]]]

Again, in the Diels-Alder reaction, the dienophile SO<sub>2</sub> can approach the diene o-xylylene at different orientations, resulting in the endo- and exo- DA products. The cheletropic reaction, which is a pericyclic reaction in which the new bonds formed are made to the same atom, results in a single product.

In this exercise, we compare the three different reaction pathways - both Diels-Alder reactions and the cheletropic reaction - in terms of their reaction coordinates, as well as which is the most thermodynamically or kinetically favoured. We also investigate the possibility of a Diels-Alder reaction at a second site in xylylene.

=== Calculations ===
Calculations were performed at the PM6 level. Method 3 (see [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_tutorial#Method_3 tutorial]) was used to locate the transition state.

The optimised molecules can be seen here:

{| class="wikitable"
! colspan="2" | Reactants
|-
| style="text-align: center;" | o-xylylene
| style="text-align: center;" | SO<sub>2</sub>
|-
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 16</script>
<uploadedFileContents>st4215_XYLYLENE_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 16</script>
<uploadedFileContents>st4215_SO2_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

{| class="wikitable"
! rowspan="2" |
! colspan="2" | Diels-Alder
! rowspan="2" | Cheletropic
|-
| style="text-align: center;" | Endo || style="text-align: center;" | Exo
|-
| style="text-align: center;" | Transition state
| style="text-align: center;" | <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14</script>
<uploadedFileContents>St4215_ex3_ENDO_TS_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
[[Media:st4215_ex3_ENDO_TS_IRC2_PM6.LOG | IRC file]]
| style="text-align: center;" | <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14</script>
<uploadedFileContents>st4215_ex3_Exo_TS_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
[[Media:st4215_ex3_Exo_TS_IRC_PM6.LOG | IRC file]]
| style="text-align: center;" | <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 12</script>
<uploadedFileContents>st4215_ex3_CHEL_TS_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
[[Media:st4215_ex3_CHEL_TS_IRC_PM6.LOG | IRC file]]
|-
| style="text-align: center;" | Product
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38</script>
<uploadedFileContents>St4215_ex3_ENDO_PDT_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38</script>
<uploadedFileContents>st4215_ex3_EXO_PDT_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38</script>
<uploadedFileContents>st4215_ex3_CHEL_PDT_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Reaction coordinate ===
<small><i>Visualise the reaction coordinate with an IRC calculation for each path. Include a .gif file in the wiki of these IRCs.</i></small>

The formation of the products for each of the endo- and exo- Diels-Alder reactions, as well as the cheletropic reaction, can be visualised below.

{| class="wikitable"
! colspan="2" | Diels-Alder
! rowspan="2" | Cheletropic
|-
| style="text-align: center;" | Endo
| style="text-align: center;" | Exo
|-
| [[File:st4215_ENDO_TS_IRCmovie.gif]]
| [[File:st4215_Exo_TS_IRCmovie.gif]]
| [[File:st4215_Chel_TS_IRCmovie.gif]]
|}

We can see that SO<sub>2</sub> approaches xylylene at a different orientation and angle for each of these reactions. This difference in approach trajectories hence lead to the formation of different products.

It can also be observed that in each of the reactions, as the product is formed, the 6-membered ring in xylylene becomes aromatic. This means that there is a strong driving force for the formation of the products. In contrast, the reactant xylylene is highly unstable as it is non-aromatic, and as can be seen in these reactions, can readily react with unsaturated bonds to form a more stable aromatic product.

=== Energy analysis ===
<small><i>Calculate the activation and reaction energies (converting to kJ/mol) for each step as in Exercise 2 to determine which route is preferred. 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.</i></small>

==== Free energies ====
The free energies of the reactants, transition states and products for each of the 3 reactions are tabulated below. The reaction barriers and energies are also calculated.

{| class="wikitable"
! rowspan="2" scope="col" style="width: 150px;" |
! colspan="2" scope="col" style="width: 300px;" | Energy at 298 K (PM6)
|-
| style="text-align: center;" | Hartree
| style="text-align: center;" | kJmol<sup>-1</sup>
|-
| colspan="3" style="background: #B7C3D0" | <b>Reactants</b>
|-
| Xylylene || style="text-align: center;" | 0.17876 || style="text-align: center;" | 469.34
|-
| SO<sub>2</sub> || style="text-align: center;" | -0.11927 || style="text-align: center;" | -313.14
|-
| Total reactants || style="text-align: center;" | 0.059496 || style="text-align: center;" | 156.21
|-
| colspan="3" style="background: #B7C3D0" | <b>Endo- Diels-Alder</b>
|-
| Transition state || style="text-align: center;" | 0.090560 || style="text-align: center;" | 237.77
|-
| Product || style="text-align: center;" | 0.021704 || style="text-align: center;" | 56.984
|-
| Reaction barrier (E<sub>a</sub>) || style="text-align: center;" | 0.0311 || style="text-align: center;" | 81.6
|-
| Reaction energy (ΔG) || style="text-align: center;" | -0.0378 || style="text-align: center;" | -99.2
|-
| colspan="3" style="background: #B7C3D0" | <b>Exo- Diels-Alder</b>
|-
| Transition state || style="text-align: center;" | 0.092078 || style="text-align: center;" | 241.75
|-
| Product || style="text-align: center;" | 0.021455 || style="text-align: center;" | 56.330
|-
| Reaction barrier (E<sub>a</sub>) || style="text-align: center;" | 0.0326 || style="text-align: center;" | 85.5
|-
| Reaction energy (ΔG) || style="text-align: center;" | -0.0380 || style="text-align: center;" | -99.9
|-
| colspan="3" style="background: #B7C3D0" | <b>Cheletropic</b>
|-
| Transition state || style="text-align: center;" | 0.099058 || style="text-align: center;" | 260.08
|-
| Product || style="text-align: center;" | -0.000002 || style="text-align: center;" | -0.0052510
|-
| Reaction barrier (E<sub>a</sub>) || style="text-align: center;" | 0.0396 || style="text-align: center;" | 104
|-
| Reaction energy (ΔG) || style="text-align: center;" | -0.0595 || style="text-align: center;" | -156
|}
<small><i>^Note that all values from the calculations are reported to 5 s.f. unless more decimal places are needed to differentiate between each reaction type. Derived quantities (reaction barrier and reaction energy) are reported to 3 s.f.</i></small>

==== Thermodynamic and kinetic products ====
From the values above, a combined reaction profile showing the reaction barriers and reaction energies can also be plotted as seen below.
[[File:st4215_ex3_Reaction_profile2.PNG|600px|thumb|center|Reaction profile diagram for the 3 Diels-Alder and cheletropic reactions, setting the energy of the reactants to zero]]

We can clearly see that the endo- Diels-Alder product is the kinetic product as it has the lowest reaction barrier of +81.6 kJmol<sup>-1</sup>, while the cheletropic product is the thermodynamic product as it is most stable as has the lowest energy of -156 kJmol<sup>-1</sup>. This is consistent with experimental and computational data.<ref name="chel_3"/><ref name="chel_1"/>

The high stability of the cheletropic product is likely due to its high symmetry, which minimises steric clash, as well as the preference of S to adopt a five-membered compared to 6-membered ring structure. This is in contrast to saturated carbon rings, where 6-membered cyclohexane is more stable than 5-membered cyclopentane due to deviation from tetrahedral bond angles of 109° in cyclopentane, resulting in angle strain. However, due to the larger size of the S atom, C-C<sub>tet</sub>-S bond angles in a 5-membered unsaturated sulfur ring (like in the cheletropic product) are much closer to the tetrahedral bond angle, at 105°. In comparison, the 6-membered heterocycle in the Diels-Alder products have C-C<sub>tet</sub>-S bond angles of around 114°, which are comparatively larger than 109°. This is illustrated below:

{| class="wikitable"
! rowspan="2" |
! colspan="2" | Diels-Alder
! rowspan="2" | Cheletropic
|-
| style="text-align: center;" | Endo
| style="text-align: center;" | Exo
|-
| Products
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38; measure 3 16 11</script>
<uploadedFileContents>St4215_ex3_ENDO_PDT_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38; rotate y 180; measure 3 16 11</script>
<uploadedFileContents>st4215_ex3_EXO_PDT_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38; measure 1 7 15</script>
<uploadedFileContents>st4215_ex3_CHEL_PDT_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

(Nice analysis. I would think that it's mostly because the Diels-Alder products are highly distorted due to the size of the sulfur atoms. This creates ring strain, which is largely from the inefficient sp3 hybridisation that results from having to twist the structure. The cheletropic product doesn't suffer from this [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:22, 30 October 2017 (UTC))

It is also expected that one of the Diels-Alder products (the endo- product) is the kinetic product instead of the cheletropic product.<ref name="chel_3"/> While the reaction barriers for both endo- and exo- Diels-Alder reactions are similar, the reaction barrier for the cheletropic reaction is much higher. Examining their transition states (see below) allows us to infer why.

{| class="wikitable"
! rowspan="2" |
! colspan="2" | Diels-Alder
! rowspan="2" | Cheletropic
|-
| style="text-align: center;" | Endo
| style="text-align: center;" | Exo
|-
| Transition state HOMO
| style="text-align: center;" | [[File:st4215_ex3_Endo_TS_HOMO.PNG]]
| style="text-align: center;" | [[File:st4215_ex3_Exo_TS_HOMO.PNG]]
| style="text-align: center;" | [[File:st4215_ex3_Chel_TS_HOMO.PNG]]
|-
| Discussion
| colspan="2" | Some stabilising secondary orbital interactions can be seen in the HOMOs of the endo- and exo- transition states, hence lowering its energy and reducing the reaction barriers for these reactions.
| No stabilising secondary orbital interactions are present here - in fact, the HOMO of the transition state is highly antisymmetric with many nodes, and hence likely to be high in energy.
|}

Like most Diels-Alder reactions, the endo and not the exo- product is the kinetic product due to more or greater stabilising secondary orbital interactions in the transition state. However, in this reaction, both the endo- and exo- transition states seem to have an approximately equal but not significant amount of stabilising orbital interactions in the HOMO. This explains why the endo- and exo- transition states are relatively close in energy.

(Very good - not many people try to rationalise the kinetics with MOs. Here it's less obvious than exercise 2. [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:22, 30 October 2017 (UTC))

== Extra: Diels-Alder reaction at a different site ==
O-xylylene also contains a cis-butadiene fragment that can act as the diene in another Diels-Alder reaction. However, this reaction is very kinetically and thermodynamically favourable, as proven by computational methods.

=== Calculations ===
Calculations were performed at the PM6 level. Method 3 (see [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_tutorial#Method_3 tutorial]) was used to locate the transition state.

The optimised molecules can be seen here. Optimised reactants (o-xylylene and SO<sub>2</sub>) from the earlier reaction were used.

{| class="wikitable"
! rowspan="2" |
! colspan="2" | Diels-Alder (2)
|-
| style="text-align: center;" | Endo || style="text-align: center;" | Exo
|-
| style="text-align: center;" | Transition state
| style="text-align: center;" | <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 20</script>
<uploadedFileContents>st4215_BAD_ENDO_TS_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
[[Media:st4215_BAD_ENDO_IRC_PM6.LOG | IRC file]]
| style="text-align: center;" | <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14</script>
<uploadedFileContents>st4215_BAD_EXO_TS_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
[[Media:st4215_BAD_EXO_IRC_PM6.LOG | IRC file]]
|-
| style="text-align: center;" | Product
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 64</script>
<uploadedFileContents>st4215_BAD_ENDO_PDT_OPTFREQ_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6</script>
<uploadedFileContents>st4215_BAD_EXO_PDT_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Energy analysis ===
The free energies of the reactants, transition states and products for both the endo- and exo- Diels Alder reactions are tabulated below. The reaction barriers and energies are also calculated.

{| class="wikitable"
! rowspan="2" scope="col" style="width: 150px;" |
! colspan="2" scope="col" style="width: 300px;" | Energy at 298 K (PM6)
|-
| style="text-align: center;" | Hartree
| style="text-align: center;" | kJmol<sup>-1</sup>
|-
| colspan="3" style="background: #B7C3D0" | <b>Reactants</b>
|-
| Xylylene || style="text-align: center;" | 0.17876 || style="text-align: center;" | 469.34
|-
| SO<sub>2</sub> || style="text-align: center;" | -0.11927 || style="text-align: center;" | -313.14
|-
| Total reactants || style="text-align: center;" | 0.059496 || style="text-align: center;" | 156.21
|-
| colspan="3" style="background: #B7C3D0" | <b>Endo- Diels-Alder (2)</b>
|-
| Transition state || style="text-align: center;" | 0.10207 || style="text-align: center;" | 267.99
|-
| Product || style="text-align: center;" | 0.065613 || style="text-align: center;" | 172.27
|-
| Reaction barrier (E<sub>a</sub>) || style="text-align: center;" | 0.0426 || style="text-align: center;" | 112
|-
| Reaction energy (ΔG) || style="text-align: center;" | +0.00612 || style="text-align: center;" | +16.1
|-
| colspan="3" style="background: #B7C3D0" | <b>Exo- Diels-Alder (2)</b>
|-
| Transition state || style="text-align: center;" | 0.10505 || style="text-align: center;" | 275.82
|-
| Product || style="text-align: center;" | 0.067305 || style="text-align: center;" | 176.71
|-
| Reaction barrier (E<sub>a</sub>) || style="text-align: center;" | 0.0456 || style="text-align: center;" | 120
|-
| Reaction energy (ΔG) || style="text-align: center;" | +0.00781 || style="text-align: center;" | +20.5
|}
<small><i>^Note that all values from the calculations are reported to 5 s.f. unless more decimal places are needed to differentiate between each reaction type. Derived quantities (reaction barrier and reaction energy) are reported to 3 s.f.</i></small>

We can hence see that this reaction is both kinetically and thermodynamically unfavourable compared to the other Diels-Alder reaction (previously discussed) and the cheletropic reaction.

Comparing the reaction barriers, the activation energies of 112 kJmol<sup>-1</sup> (endo) and 120 kJmol<sup>-1</sup> (exo) are much higher than that of the other DA reaction, at 81.6 kJmol<sup>-1</sup> (endo) and 85.5 kJmol<sup>-1</sup> (exo), as well as the cheletropic reaction, at 104 kJmol<sup>-1</sup>. Hence this Diels-Alder reaction is the most kinetically unfavourable reaction.

Comparing the reaction energies, we can see that both the endo- and exo- reactions are endothermic - compared to the other two DA reactions as well as the cheletropic reaction, which are exothermic. This indicates that this Diels-Alder reaction is also highly thermodynamically unfavourable as ΔG<0 and it is not spontaneous.

(Very good! [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 11:27, 30 October 2017 (UTC))

=== Conclusion ===
While there are several possible reactions between o-xylylene and SO<sub>2</sub>, including a Diels-Alder reaction at two different sites and a cheletropic reaction, not all are comparable in terms of kinetic and thermodynamic favourability. Calculations show that the cheletropic reaction is the most thermodynamically favoured due to the higher stability of a 5-membered S ring, and the Diels-Alder reaction between the two terminal alkene carbons of xylylene is much more kinetically and thermodynamically favourable compared to the Diels-Alder reaction involving the cis-butadiene fragment in the aromatic 6-membered ring. This is because the aromaticity of the ring is not broken in the former reaction.

== References ==

<references>
<ref name="chel_1">F. Monnat, P. Vogel, J. A. Sordo. [http://onlinelibrary.wiley.com/doi/10.1002/1522-2675(200203)85:3%3C712::AID-HLCA712%3E3.0.CO;2-5/epdf <i>Hetero-Diels-Alder and Cheletropic Additions of Sulfur Dioxide to 1,2-Dimethylidenecycloalkanes. Determination of Thermochemical and Kinetics Parameters for Reactions in Solution and Comparison with Estimates From Quantum Calculations</i>] Helv. Chim. Acta. <b>85</b> (3), 712–732 (2002).</ref>
<ref name="chel_3">D. Suarez, T. L. Sordo, J. A. Sordo. [http://pubs.acs.org/doi/pdf/10.1021/jo00114a039 <i>A Comparative Analysis of the Mechanisms of Cheletropic and Diels-Alder Reactions of 1,3-Dienes with Sulfur Dioxide: Kinetic and Thermodynamic Controls</i>] J. Org. Chem., <b>60</b> (9), 2848–2852 (1995).</ref>

</references>

Rep:Transition states (st4215): Exercise 2

2025-09-01T09:49:48Z

Move page script: Move page script moved page Transition states (st4215): Exercise 2 to Rep:Transition states (st4215): Exercise 2: Move to report namespace

== Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole ==
The reaction of cyclohexadiene and 1,3-dioxole is also a Diels-Alder reaction, in which cyclohexadiene is the diene and 1,3-dioxole is the dienophile. However, unlike the simple Diels-Alder reaction between butadiene and ethylene, this reaction is slightly more complex - 1,3-dioxole can approach the diene at different orientations, leading to the formation of the endo- and exo- Diels-Alder products. This can be seen in the reaction scheme below:
[[File:st4215_Ex2_reactionscheme.png|thumb|center|400px| Reaction of cyclohexadiene and 1,3-dioxole [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_exercise (Source)]]]

The endo- product is formed from a transition state where the substituents on the dienophile point towards the π system of the diene, while the exo- product is formed from a transition state where the substituents are pointing away.

In this exercise, we take a closer look at a different aspect of the Diels-Alder reaction - whether it is normal or inverse electron demand. We also consider the Diels-Alder reaction when an unsymmetrical diene is involved, and compare the endo- and exo- DA reactions in terms of which is more kinetically or thermodynamically favourable. In the process, we confirm that the orbital symmetry requirements as elucidated in Exercise 1 still apply, and illustrate that there are differences between the PM6 and B3LYP calculation methods, with one being more suitable than the other in this case.

=== Calculations ===
Calculations were performed at both the PM6 and B3LYP/6-31G(d) level. Method 2 (see [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_tutorial#Method_2 tutorial]) was used to locate the transition state.

The optimised molecules can be seen here:

{| class="wikitable"
! rowspan="2" | Calculation method
! colspan="2" | Reactants
! colspan="2" | Endo
! colspan="2" | Exo
|-
| style="text-align: center;" | Cyclohexadiene
| style="text-align: center;" | 1,3-dioxole
| style="text-align: center;" | Transition state
| style="text-align: center;" | Product
| style="text-align: center;" | Transition state
| style="text-align: center;" | Product
|-
| style="text-align: center;" | B3LYP
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 18</script>
<uploadedFileContents>st4215_CYCLOHEXADIENE_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 26</script>
<uploadedFileContents>st4215_DIOXOLE_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38</script>
<uploadedFileContents>st4215_ENDO_TS_OPTFREQ2_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 18</script>
<uploadedFileContents>st4215_ENDO_PDT_OPTFREQ2_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 20</script>
<uploadedFileContents>st4215_EXO_TS_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14</script>
<uploadedFileContents>st4215_EXO_PDT_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| style="text-align: center;" | PM6
| style="text-align: center;" | [[Media:st4215_CYCLOHEXADIENE_OPTFREQ_PM6.LOG| Cyclohexadiene (PM6)]]
| style="text-align: center;" | [[Media:st4215_DIOXOLE_OPTFREQ_PM6.LOG| 1,3-dioxole (PM6)]]
| style="text-align: center;" | [[Media:st4215_ENDO_TS_OPTFREQ_PM6.LOG| Endo-TS (PM6)]] <br> [[Media:st4215_ENDO_TS_IRC2_PM6ONPM6.LOG| IRC file]]
| style="text-align: center;" | [[Media:st4215_ENDO_PDT_OPTFREQ_PM6.LOG| Endo-product (PM6)]]
| style="text-align: center;" | [[Media:st4215_EXO_TS_OPTFREQ_PM6.LOG| Exo-TS (PM6)]] <br> [[Media:st4215_EXO_TS_IRC_PM6ONPM6.LOG| IRC file]]
| style="text-align: center;" | [[Media:st4215_EXO_PDT_OPTFREQ_PM6.LOG| Exo-product(PM6)]]

|}

=== Molecular orbitals ===

==== Computed MOs ====
<small><i>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).</i></small>

The MOs involved in this reaction - the HOMOs and LUMOs for the reactants (cyclohexadiene and 1,3-dioxole), as well as the 4 MOs that these produce in each of the endo and exo transition states - are shown below. As determined in Exercise 1, which is also a Diels-Alder reaction, only orbitals of the same symmetry are able to interact. This is illustrated in the table below, where the symmetry or antisymmetry of each orbital can be clearly seen.

{| class="wikitable"
! rowspan="2"| Symmetry
! colspan="2" | Reactants
! colspan="2" rowspan="2"| Transition state (Endo)
! colspan="2" rowspan="2"| Transition state (Exo)
|-
| style="text-align: center;" | Cyclohexadiene
| style="text-align: center;" | 1,3-Dioxole
|-
| Antisymmmetric
| <b>MO 22</b> (HOMO) <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 18; mo 22; rotate x 90; rotate y 90; rotate z -20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_CYCLOHEXADIENE_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 20</b> (LUMO)<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 26; mo 20; rotate x 90; rotate y 90; rotate z -20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_DIOXOLE_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 40</b><jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38; mo 40; rotate x 90; rotate y 90; rotate z -20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_ENDO_TS_OPTFREQ2_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 43</b><jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38; mo 43; rotate x 90; rotate y 90; rotate z -20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_ENDO_TS_OPTFREQ2_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 40</b><jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 20; mo 40; rotate x 90; rotate y 90; rotate z -20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_EXO_TS_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 43</b><jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 20; mo 43; rotate x 90; rotate y 90; rotate z -20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_EXO_TS_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Symmetric
| <b>MO 23</b> (LUMO)<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 18; mo 23; rotate x 90; rotate y 90; rotate z -20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_CYCLOHEXADIENE_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 19</b> (HOMO) <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 26; mo 19; rotate x 90; rotate y 90; rotate z -20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_DIOXOLE_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 41</b><jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38; mo 41; rotate x 90; rotate y 90; rotate z -20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_ENDO_TS_OPTFREQ2_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 42</b><jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38; mo 42; rotate x 90; rotate y 90; rotate z -20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_ENDO_TS_OPTFREQ2_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 41</b><jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 20; mo 41; rotate x 90; rotate y 90; rotate z -20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_EXO_TS_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 42</b><jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 20; mo 42; rotate x 90; rotate y 90; rotate z -20; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_EXO_TS_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

It is important to note that these MOs were generated from the B3LYP and not the PM6 calculation. While both calculation methods were used, the B3LYP calculation proved to be a better choice in visualising the MOs as it allowed us to see the symmetry of the orbitals more clearly, and corresponded more closely to the graphical representation of the orbitals in the MO diagram (see next section).

==== MO diagrams ====
<i><small>Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. Is this a normal or inverse demand DA reaction?</small></i>

MO diagrams for both the endo- and exo- Diels-Alder reactions were constructed as shown below. Energy levels of the MOs have been adjusted to reflect their actual values, as calculated in the B3LYP calculation.

{| class="wikitable"
! Endo- Diels-Alder
! Exo- Diels-Alder
|-
| [[File:st4215_Endo_MO_diagram.PNG|500px]]
| [[File:st4215_Exo_MO_diagram.PNG|500px]]
|}

From the MO diagrams above, we can conclude that both the endo- and exo- reactions are inverse demand DA reactions. In contrast to a normal electron demand DA reaction, where the diene is electron rich with a high LUMO and the dienophile is electron poor with a low HOMO, cyclohexadiene (the diene) is electron poor with a low HOMO and 1,3-dioxole (the dienophile) is electron rich with a high LUMO. Hence we can classify these reactions as inverse demand.

This is likely due to the nature of the dienophile, 1-3-dioxole, which has an electron-donating O atom. The lone pairs on the two adjacent O atoms are able to donate electron density into the double bond, making the dienophile more electron rich and raising the energy of its MOs, including its LUMO. In contrast, cyclohexadiene does not have any electron-donating substituents and is relatively electron poor. Hence, considering the Frontier Molecular Orbitals (FMOs), the LUMO<sub>dienophile</sub> is relatively high-energy and closer in energy to the low-energy HOMO<sub>diene</sub>, as compared to HOMO<sub>dienophile</sub> and LUMO<sub>diene</sub>. Thus the interaction between the LUMO<sub>dienophile</sub> and HOMO<sub>diene</sub> is the strongest and dominates in this reaction, causing it to be inverse demand.<ref name="inverse"/>

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 00:01, 8 November 2017 (UTC) Good understanding of the demand of the electron of a DA reaction

=== Energy analysis ===
<small><i>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). Which are the kinetically and thermodynamically favourable products? Look at the HOMO of the TSs. Are there any secondary orbital interactions or sterics that might affect the reaction barrier energy?</i></small>

==== Free energies ====
The free energy of the reactants as well as the endo- and exo- Diels-Alder transition states and products are tabulated below. The reaction barriers and energies are also calculated.

{| class="wikitable"
! rowspan="3" scope="col" style="width: 150px;" |
! colspan="4" scope="col" style="width: 500px;" | Energy at 298 K
|-
| colspan="2" style="text-align: center;" | B3LYP
| colspan="2" style="text-align: center;" | PM6
|-
| style="text-align: center;" | Hartree
| style="text-align: center;" | kJmol<sup>-1</sup>
| style="text-align: center;" | Hartree
| style="text-align: center;" | kJmol<sup>-1</sup>
|-
| colspan="5" style="background: #B7C3D0" | <b>Reactants</b>
|-
| Cyclohexadiene || style="text-align: center;" | -233.32 || style="text-align: center;" | -6.1259 x 10<sup>5</sup> || style="text-align: center;" | 0.11688 || style="text-align: center;" | 306.86
|-
| 1,3-dioxole || style="text-align: center;" | -267.07 || style="text-align: center;" | -7.0119 x 10<sup>5</sup> || style="text-align: center;" | -0.052279 || style="text-align: center;" | -137.26
|-
| Total reactants || style="text-align: center;" | -500.39 || style="text-align: center;" | -1.3138 x 10<sup>6</sup> || style="text-align: center;" | 0.064598 || style="text-align: center;" | 169.60
|-
| colspan="5" style="background: #B7C3D0" | <b>Endo- Diels-Alder</b>
|-
| Transition state || style="text-align: center;" | -500.332 || style="text-align: center;" | -1.3136 x 10<sup>6</sup> || style="text-align: center;" | 0.13794 ||style="text-align: center;" | 362.17
|-
| Product || style="text-align: center;" | -500.419 || style="text-align: center;" | -1.3138 x 10<sup>6</sup> || style="text-align: center;" | 0.037807 || style="text-align: center;" | 99.26
|-
| Reaction barrier (E<sub>a</sub>) || style="text-align: center;" | 0.0609 || style="text-align: center;" | 160 || style="text-align: center;" | 0.0733 || style="text-align: center;" | 193
|-
| Reaction energy (ΔG) || style="text-align: center;" | -0.0257 || style="text-align: center;" | -67.4 || style="text-align: center;" | -0.0268 || style="text-align: center;" | -70.3
|-
| colspan="5" style="background: #B7C3D0" | <b>Exo- Diels-Alder</b>
|-
| Transition state || style="text-align: center;" | -500.329 || style="text-align: center;" | -1.3136 x 10<sup>6</sup> || style="text-align: center;" | 0.13890 || style="text-align: center;" | 364.68
|-
| Product || style="text-align: center;" | -500.417 || style="text-align: center;" | -1.3138 x 10<sup>6</sup> || style="text-align: center;" | 0.037977 || style="text-align: center;" | 99.71
|-
| Reaction barrier (E<sub>a</sub>) || style="text-align: center;" | 0.0639 || style="text-align: center;" | 168 || style="text-align: center;" | 0.0743 || style="text-align: center;" | 195
|-
| Reaction energy (ΔG) || style="text-align: center;" | -0.0243 || style="text-align: center;" | -63.8 || style="text-align: center;" | -0.0266 || style="text-align: center;" | -69.9
|}
<small><i>^Note that all values from the calculations are reported to 5 s.f. unless more decimal places are needed to differentiate between endo- and exo- geometries. Derived quantities (reaction barrier and reaction energy) are reported to 3 s.f.</i></small>

==== Thermodynamic and kinetic products ====
From the table above, we can conclude that the endo- product is both the kinetic and thermodynamic product. This is because it has a lower reaction barrier, meaning that less activation energy is required for the reaction, making it the kinetic product. The endo- product also has a lower reaction energy (ΔG), meaning that the endo- product is lower in energy and more stable than the exo- product, hence making it the thermodynamic product.

This is in contrast to most Diels-Alder reactions, in which the exo- product is the thermodynamic product as it is less sterically hindered and more stable, while the endo- product is the kinetic product due to stabilising orbital interactions in the transition state.<ref name="fmo"/> However, the difference for this reaction can be explained.

As seen in the jmols below, the endo- TS indeed has stabilising secondary orbital interactions between the p orbitals of O in 1,3-dioxole and the p orbitals of C in cyclohexadiene, which are absent in the exo- TS. The presence of these interactions lower the energy of the transition state, hence lowering the reaction barrier and making the endo-product the kinetic product.

Comparing the endo- and exo- products, we can also see that these stabilising orbital interactions are also in present in the endo- product. Additionally, the endo- product is less sterically hindered compared to the exo-product, in which there is steric clash between the hydrogens which are pointing towards each other. These two factors cause the energy of the endo- product to be lower than that of the exo- product, hence lowering the reaction energy and making the endo- product the thermodynamic product as well.

{| class="wikitable"
!
! Endo
! Exo
|-
| Transition state
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 38; mo 41; rotate x 90; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on; mo cutoff 0.01</script>
<uploadedFileContents>st4215_ENDO_TS_OPTFREQ2_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 20; mo 41; rotate x 90; rotate y 180; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on; mo cutoff 0.01</script>
<uploadedFileContents>st4215_EXO_TS_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Product
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 18; mo 41; rotate x 90; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on; mo cutoff 0.01</script>
<uploadedFileContents>st4215_ENDO_PDT_OPTFREQ2_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 14; mo 41; rotate x 90; rotate y 180; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on; mo cutoff 0.01</script>
<uploadedFileContents>st4215_EXO_PDT_OPTFREQ_631G(D).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 00:05, 8 November 2017 (UTC) This is an excellent understanding of the thermodynamics and kenetics of the reaction. You could have improved your answer by addding a drawing showing the steric clashes.

=== Conclusion ===
As per Exercise 1, MO calculations and visualisation for this Diels-Alder reaction prove that only orbitals with the same symmetry can overlap and interact, hence showing that this is not unique to a singular reaction. A closer examination of the MO diagram also allows us to determine that this reaction is an inverse electron demand DA reaction, due to the presence of electron-donating groups on the dienophile (1,3-dioxole) which raises the energy of its LUMO. Analysing the energy levels of the reactants, TSs and products also reveal that the endo- product is both the kinetic and thermodynamic product, due to greater stability of the transition state (due to stabilising secondary orbital interactions) and the final product (due to lesser steric clash).

== References ==
<references>
<ref name="inverse">D. Boger. <i>Progress in heterocyclic chemistry. (1st ed.).</i> (1989).</ref>
<ref name="fmo">I. Fleming. <i>Frontier Orbitals and Organic Chemical Reactions.</i> (1976).</ref>
</references>

Rep:Transition states (st4215): Exercise 1

2025-09-01T09:49:43Z

Move page script: Move page script moved page Transition states (st4215): Exercise 1 to Rep:Transition states (st4215): Exercise 1: Move to report namespace

== Exercise 1: Reaction of Butadiene with Ethylene ==
The reaction between butadiene and ethylene is a [4+2] cycloaddition reaction, or more specifically, a Diels-Alder reaction - a reaction involving a conjugated diene in the s-cis conformation, and a dienophile containing a double bond. In this simple Diels-Alder reaction, butadiene is the diene and ethylene is the dienophile. The reaction scheme for this reaction is shown below:

[[File:st4215_Ex1_reactionscheme.png|thumb|center|300px| Reaction of butadiene with ethylene [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_exercise (Source)]]]

In this exercise, we use this Diels-Alder reaction to determine the orbital symmetry requirements for a reaction to take place, and to show the effect of hybridisation and bond order on the length of C-C bonds. We also probe into the mechanism of the reaction, and determine if bond formation is synchronous or asynchronous.

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) 17:17, 1 November 2017 (UTC) Very well done on the exercise as a whole!)
=== Calculations ===
Calculations were performed at the PM6 level. Method 1 (see [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_tutorial#Method_1 tutorial]) was used to locate the transition state.

The optimised molecules can be seen here:

{| class="wikitable"
! colspan="2" | Reactants
! Transition state
! Product
|-
|<jmol>
<jmolApplet>
<title>Butadiene</title>
<color>white</color>
<size>200</size>
<script>frame 6</script>
<uploadedFileContents>st4215_BUTADIENE_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<title>Ethylene</title>
<color>white</color>
<size>200</size>
<script>frame 6</script>
<uploadedFileContents>st4215_ETHYLENE_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| style="text-align: center;" | <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6</script>
<uploadedFileContents>st4215_CYCLOHEXENE_TS_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
[[Media:st4215_CYCLOHEXENE_TS_IRC3_PM6.LOG | IRC file]]
| <jmol>
<jmolApplet>
<title>Cyclohexene</title>
<color>white</color>
<size>200</size>
<script>frame 6</script>
<uploadedFileContents>st4215_CYCLOHEXENE_PDT_OPT_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

=== Molecular orbitals ===

==== MO diagram ====
<small><i>Construct an MO diagram for the formation of the butadiene/ethene TS, including basic symmetry labels (symmetric/antisymmetric or s/a).</i></small><br>

The MO diagram for this cycloaddition reaction was constructed as shown below. We first assume based on prior knowledge that only orbitals of the same symmetry can combine; however, this is proven in a later section.

The MOs have been labelled accordingly with their numbers according to the calculations, and can be visualised in the section below.<br>
[[File:st4215_ex1_MO_diagram.PNG]]

Considering the actual energy levels of the MOs computed from the calculations, a more accurate MO diagram can be constructed, as seen [[Media:st4215_ex1_MO_diagram_actual.PNG|here]]. However, this does not affect the relative energy levels of the MOs (within each molecule) or the interactions between these MOs.

==== Computed MOs ====
<small><i>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.</i></small><br>

The HOMOs and LUMOs for each reactant, butadiene and ethylene, as well as the 4 MOs that these produce for the TS, were calculated and can be visualised below. The MO numbers correspond to that labelled on the MO diagram, allowing us to see which orbitals interact. We can also observe that these computed MOs bear physical resemblance to that drawn in the MO diagram, in terms of the bonding and antibonding phases in each MO.

{| class="wikitable"
! rowspan="2" | Symmetry
! colspan="2" | Reactants
! rowspan="2" colspan="2"| Transition state
|-
| style="text-align: center;" | Butadiene
| style="text-align: center;" | Ethylene
|-
| Antisymmetric
| <b>MO 11</b> (HOMO) <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_BUTADIENE_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 7</b> (LUMO)<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>st4215_ETHYLENE_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 16</b><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>st4215_CYCLOHEXENE_TS_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 19</b><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>st4215_CYCLOHEXENE_TS_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Symmetric
| <b>MO 12</b> (LUMO)<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>st4215_BUTADIENE_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 6</b> (HOMO) <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>st4215_ETHYLENE_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 17</b><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>st4215_CYCLOHEXENE_TS_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <b>MO 18</b><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>st4215_CYCLOHEXENE_TS_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

==== Symmetry requirements ====
<small><i>What can you conclude about the requirements for symmetry for a reaction (when is a reaction 'allowed' and when is it 'forbidden')? 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.</i></small><br>

From the table above as well as the MO diagram, we can see that the HOMO/LUMO of butadiene interact with the corresponding LUMO/HOMO in ethylene respectively. These interactions can occur as each pair of orbitals share the same symmetry. The HOMO of butadiene (MO 11) and the LUMO of ethylene (MO 7) are both antisymmetric, and interact to give two antisymmetric MOs in the TS (MOs 16 and 19). Similarly, the LUMO of ethylene (MO 12) and the HOMO of butadiene (MO 6) are both symmetric, and interact to give two symmetric MOs in the TS (MOs 17 and 18). ([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) 17:17, 1 November 2017 (UTC) You mean the LUMO of butadiene and the HOMO of ethylene ;) ) Hence we can conclude that a reaction is only allowed when the relevant orbitals share the same symmetry, as this would allow orbitals to overlap and thus interact. This is illustrated in the following table:

{| class="wikitable"
! Symmetry of interaction
! Orbital overlap integral
|-
| style="text-align: center;" | Symmetric-antisymmetric
| style="text-align: center;" | Zero
|-
| style="text-align: center;" | Symmetric-symmetric
| style="text-align: center;" | Non-zero
|-
| style="text-align: center;" | Antisymmetric-antisymmetric
| style="text-align: center;" | Non-zero
|}

=== C-C bond lengths ===
<small><i>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.</i></small>

The C-C bond lengths in the reactants, TS and products are tabulated below.

{| class="wikitable"
! rowspan="2" colspan="2" |
! colspan="2" | Reactants
! rowspan="2" | Transition state
! rowspan="2" | Product
|-
| style="text-align: center;" | Butadiene
| style="text-align: center;" | Ethylene
|-
| colspan="2" style="text-align: center;"| Labelled C atoms
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>select atomno=1, atomno=4, atomno=6, atomno=7; label display</script>
<uploadedFileContents>st4215_BUTADIENE_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>select atomno=1, atomno=4; label display</script>
<uploadedFileContents>st4215_ETHYLENE_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>select atomno=1, atomno=4, atomno=6, atomno=7, atomno=11, atomno=14; label display</script>
<uploadedFileContents>st4215_CYCLOHEXENE_TS_OPTFREQ2_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>select atomno=1, atomno=4, atomno=6, atomno=7, atomno=11, atomno=14; label display</script>
<uploadedFileContents>st4215_CYCLOHEXENE_PDT_OPT_PM6.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| rowspan="6" style="text-align: center;" | C-C bond<br> length / Å
| style="width: 50px;" | Bond A || <center> C1-C4: 1.34 </center>|| || <center> C1-C4: 1.38 </center> || <center> C1-C4: 1.50 </center>
|-
| Bond B || <center> C4-C6: 1.47 </center> || || <center> C4-C6: 1.41 </center> || <center> C4-C6: 1.34 </center>
|-
| Bond C || <center> C6-C7: 1.34 </center> || || <center> C6-C7: 1.38 </center> || <center> C6-C7: 1.50 </center>
|-
| Bond D || || <center> C1-C4: 1.33 </center> || <center> C11-C14: 1.38 </center> || <center> C11-C14: 1.54 </center>
|-
| Bond E || || || <center> C7-C14: 2.11 </center> || <center> C7-C14: 1.54 </center>
|-
| Bond F || || || <center> C1-C11: 2.11 </center> || <center> C1-C11: 1.54 </center>
|}

From the values above, we can observe the following changes in bond length as the reaction progresses.

{| class="wikitable" style="width: 750px;"
! Bond
! Trend in bond length as reaction progresses
! Graph
|-
| A || As the reaction progresses, the length of Bond A increases. This is because Bond A is initially a C=C double bond in butadiene, which becomes a sp<sup>2</sup>-sp<sup>3</sup> C-C single bond in cyclohexene. These lengths are similar to typical values of 1.34 Å for alkenes and 1.50 Å sp<sup>2</sup>-sp<sup>3</sup> C-C bonds.<ref name="bondlength"/>
| [[File:st4215_BondA.JPG|200px]]
|-
| B || As the reaction progresses, the length of Bond B decreases. This is because Bond B is initially an sp<sup>2</sup>-sp<sup>2</sup> C-C single bond in butadiene, which becomes a C=C double bond in cyclohexene. These lengths are similar to typical values of 1.47 Å for sp<sup>2</sup>-sp<sup>2</sup> C-C bonds and 1.34 Å for alkenes.<ref name="bondlength"/>
| [[File:st4215_BondB.JPG|200px]]
|-
| C || Similar to Bond A, as the reaction progresses, the length of Bond C increases. This is due to the same reason as explained for Bond A.
| [[File:st4215_BondC.JPG|200px]]
|-
| D || As the reaction progresses, the length of Bond D increases. This is because Bond B is initially an C=C single bond in ethylene, which becomes a sp<sup>3</sup>-sp<sup>3</sup> C-C single bond in cyclohexene. These lengths are similar to typical values of 1.34 Å for alkenes and 1.54 Å for sp<sup>3</sup>-sp<sup>3</sup> C-C bonds.<ref name="bondlength"/>
| [[File:st4215_BondD.JPG|200px]]
|-
| E || Bond E is formed in this reaction. Hence as the reaction progresses from the transition state to the product, the length of Bond E increases. The length of the fully formed Bond E in the product, an sp<sup>3</sup>-sp<sup>3</sup> C-C single bond, corresponds to the typical value of 1.54 Å.<ref name="bondlength"/> The distance between the C atoms in the TS (2.11 Å) falls between twice the van der Waal radius of C (2 x 1.70 Å<ref name="vdw_c"/> = 3.40 Å) and the length of the fully formed Bond E, indicating that the atoms are approaching each other.
| [[File:st4215_BondE.JPG|200px]]
|-
| F || Like Bond E, Bond F is also formed in this reaction. Hence as the reaction progresses from the transition state to the product, the length of Bond F increases, due to the same reasons as explained above.
| [[File:st4215_BondF.JPG|200px]]
|}

We can hence conclude that bond lengths in the reactants and product do not deviate from typical values. Additionally, we can also show that an increase in bond order from 1 to 2 indeed shortens the C-C bond, while an increase in hybridisation from sp<sup>2</sup> to sp<sup>3</sup> does lengthen the C-C bond, as seen in the comparison of theoretical values.

=== Transition state ===
<i><small>Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous?</small></i>
<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<script>frame 7; rotate x -40; select atomno=1, atomno=7, atomno=11, atomno=14; label display </script>
<uploadedFileContents>st4215_CYCLOHEXENE_TS_OPTFREQ2_PM6.LOG</uploadedFileContents>
<name>cyclohexeneTS</name>
</jmolApplet>
<jmolbutton>
<script>if(vibrating==0) vibrating=1; vibration 2; else; vibrating=0; vibration off; endif</script>
<text>Vibration</text>
<target>cyclohexeneTS</target>
</jmolbutton>
<jmolbutton>
<script>measure 1 11; measure 7 14</script>
<text>C-C Distances</text>
<target>cyclohexeneTS</target>
</jmolbutton>
</jmol>
<br><br>
As illustrated above, the formation of the two bonds are synchronous as the relevant pairs of atoms (C1/C11 and C7/C14) are moving towards each other at the same time, with the distance between the two atoms being approximately equal in both pairs. This is to be expected as evidence shows that the Diels-Alder reaction can proceed via a concerted, synchronous mechanism.<ref name="DA_1"/> However, it is important to note that this may also be the result of the calculation method used, as studies with other calculation methods (DFT/B3LYP) have also shown that other mechanisms are feasible.<ref name="DA_2"/>

=== Conclusion ===
By examining this simple Diels-Alder reaction between butadiene and ethylene, we can conclude from computational data that only orbitals need to share the same symmetry to be able to overlap and hence interact. This knowledge will be useful in Exercise 2, where we again construct a MO diagram for a another Diels-Alder reaction. By following the change in C-C bond lengths over the course of the reaction, it can also be concluded that as per theoretical data, bond lengths shorten when bond order increases and hybridisation decreases. Lastly, based on this calculation, we can conclude that the Diels-Alder reaction between butadiene and ethylene proceeds via synchronous bond formation in the transition state.

== References ==
<references>
<ref name="bondlength">M. A. Fox, J. K. Whitesell, E. Buchholz. <i>Organische Chemie: Grundlagen, Mechanismen, bioorganische Anwendungen</i> (1995).</ref>
<ref name="vdw_c">A. Bondi. [http://pubs.acs.org/doi/pdf/10.1021/j100785a001 <i>van der Waals Volumes and Radii </i>] J. Phys. Chem. <b>68</b>(3), 441–451 (1996).</ref>
<ref name="DA_1">K. N. Houk, Y. T Lin & F. K. Brown. [http://pubs.acs.org/doi/pdf/10.1021/ja00263a059 <i>Evidence for the Concerted Mechanism of the Diels-Alder Reaction of Butadiene with Ethylene.</i>] J. Am. Chem. SOC. <b>108</b>, 554-556 (1986).</ref>
<ref name="DA_2">E. Goldstein, B. Beno & K. N. Houk. [http://pubs.acs.org/doi/pdf/10.1021/ja9601494 <i>Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene</i>] J. Am. Chem. Soc. <b>118</b>, 6036-6043 (1996).
</ref>
</references>

Rep:Transition states (st4215)

2025-09-01T09:49:43Z

Move page script: Move page script moved page Transition states (st4215) to Rep:Transition states (st4215): Move to report namespace

== Introduction ==
In a chemical reaction, reactants can change into the products via a multitude of different configurations of atoms, each with a different energy. These energies corresponds to a energy profile (when there is 1 degree of freedom), or a Potential Energy Surface (PES) (when there are two degrees of freedom). An example of the PES for a reaction can be seen below.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 23:47, 7 November 2017 (UTC) A PES can be N dimensions it can just be visualised in 2.

[[File:st4215_Pes.jpeg|thumb|250px|right| A possible potential energy surface (PES) for a reaction. (Source: [http://www.kf.elf.stuba.sk/~ballo/piatok/prezentacia/hartree-fock/hf_2.html Department of Physics, Slovak University of Technology in Bratislava])]]

As illustrated in the PES, the reactants and products correspond to energy minima - where energy increases in all directions, meaning that the reactants and products sit in a well. At this point, the gradient of the PES curve is zero, and the second derivative yields a positive value, indicating it is a minimum.

The reaction coordinate is hence the minimum energy pathway that leads from the reactants to the products, where the maximum energy along this pathway corresponds to the transition state. In terms of the PES, this corresponds to a first-order saddle point - where energy increases in all directions but one. At this point, the gradient of the PES curve is also zero - it is both a maximum and a minimum. However, the second derivative leads a negative value, indicating that energy is a maximum in one direction but a minimum in all other directions. This value is a negative force constant which corresponds to the imaginary frequency in the vibration spectrum of the transition state. The normal mode corresponding to this imaginary frequency usually show the relevant atoms of the reactants moving towards each other to form the product.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 23:51, 7 November 2017 (UTC) A few points, You haven't defined what you meant by direction. The second derivative is negative in the reaction coored and all other are positive.

Hence the presence of a negative imaginary frequency in the transition state calculation confirms that the structure at a saddle point and hence corresponds to the transition state. However, energy minima where reactants and products lie should not contain any imaginary frequencies, as the second derivative only yields positive force constant values. The lack of imaginary frequencies in a frequency calculation can confirm that the structures lie in energy minima, and correspond to either the reactants or products.

In this module, computational methods were used to locate and characterise transition states for various pericyclic reactions, including Diels-Alder and cheletropic reactions. An overview of the various methods that were used in the calculations are detailed [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_tutorial here].

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 23:51, 7 November 2017 (UTC) A good explanation of a TS however, you could have spoke in abit moe detail about the Quantum mechanical methods used.

== Exercises ==
The instructions for the exercises can be found [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_exercise here].

=== Exercise 1: Reaction of Butadiene with Ethylene ===
The reaction between butadiene and ethylene is a [4+2] cycloaddition reaction, or more specifically, a Diels-Alder reaction - a reaction involving a conjugated diene in the s-cis conformation, and a dienophile containing a double bond. In this simple Diels-Alder reaction, butadiene is the diene and ethylene is the dienophile. The reaction scheme for this reaction is shown below:
[[File:st4215_Ex1_reactionscheme.png|thumb|center|300px| Reaction of butadiene with ethylene [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_exercise (Source)]]]

In this exercise, we use this Diels-Alder reaction to determine the orbital symmetry requirements for a reaction to take place, and to show the effect of hybridisation and bond order on the length of C-C bonds. We also probe into the mechanism of the reaction, and determine if bond formation is synchronous or asynchronous.

The link to this completed exercise can be found [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Transition_states_(st4215):_Exercise_1 here].

=== Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole ===
The reaction of cyclohexadiene and 1,3-dioxole is also a Diels-Alder reaction, in which cyclohexadiene is the diene and 1,3-dioxole is the dienophile. However, unlike the simple Diels-Alder reaction between butadiene and ethylene, this reaction is slightly more complex - 1,3-dioxole can approach the diene at different orientations, leading to the formation of the endo- and exo- Diels-Alder products. This can be seen in the reaction scheme below:
[[File:st4215_Ex2_reactionscheme.png|thumb|center|400px| Reaction of butadiene with ethylene [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_exercise (Source)]]]

The endo- product is formed from a transition state where the substituents on the dienophile point towards the π system of the diene, while the exo- product is formed from a transition state where the substituents are pointing away.

In this exercise, we take a closer look at a different aspect of the Diels-Alder reaction - whether it is normal or inverse electron demand. We also consider the Diels-Alder reaction when an unsymmetrical diene is involved, and compare the endo- and exo- DA reactions in terms of which is more kinetically or thermodynamically favourable. In the process, we confirm that the orbital symmetry requirements as elucidated in Exercise 1 still apply, and illustrate that there are differences between the PM6 and B3LYP calculation methods, with one being more suitable than the other in this case.

The link to this completed exercise can be found [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Transition_states_(st4215):_Exercise_2 here].

=== Exercise 3: Diels-Alder vs Cheletropic ===
In this exercise, we examine the possible reactions between o-xylylene and SO<sub>2</sub> - the Diels-Alder reaction and the cheletropic reaction, as illustrated in the reaction scheme below.
[[File:st4215_Ex3_reactionscheme.png|thumb|center|600px| Reaction of o-xylylene and SO<sub>2</sub>
[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ts_exercise (Source)]]]

Again, in the Diels-Alder reaction, the dienophile SO<sub>2</sub> can approach the diene o-xylylene at different orientations, resulting in the endo- and exo- DA products. The cheletropic reaction, which is a pericyclic reaction in which the new bonds formed are made to the same atom, results in a single product.

In this exercise, we compare the three different reaction pathways - both Diels-Alder reactions and the cheletropic reaction - in terms of their reaction coordinates, as well as which is the most thermodynamically or kinetically favoured. We also investigate the possibility of a Diels-Alder reaction at a second site in xylylene.

The link to this completed exercise can be found [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Transition_states_(st4215):_Exercise_3 here].

== Conclusion ==
<small><i>Note: Please see each exercise for conclusions pertaining to that exercise</i></small><br>Computational methods allowed us to successfully examine the various pericyclic reactions, by locating and characterising their transition states via a variety of methods. In the process, we were able to determine the kinetic and thermodynamic products of each reaction, prove that orbitals must share the same symmetry in order to interact, and construct MO diagrams based on calculations, among others.

Rep:Transition States and Reactivity ZWL115

2025-09-01T09:49:42Z

Move page script: Move page script moved page Transition States and Reactivity ZWL115 to Rep:Transition States and Reactivity ZWL115: Move to report namespace

==Introduction==

===What is a potential energy surface?===

A potential energy surface (PES) describes the potential energy of a chemical system in terms of certain parameters such as time and extent of reactions. Each geometry of the atoms in a system has its own unique potential energy due to the varying distances of the different atoms. Two atoms which are very far apart will have no interactions while 2 atoms which are very close together will have strong repulsive forces between them. This results in the creation of a smooth energy landscape such as the one shown in Figure 1. In Figure 1, the vertical coordinate gives the potential energy which varies with the 2 horizontal coordinates which are the bond lengths of atoms AB and atoms BC

[[File:PES_zwl115.jpg|centre|frame|Figure 1. Example of a PES<sup>1</sup>]]

===What is a transition state?===

Minima in a PES are assiociated with the structures of the reactants, products or intermediates of a reaction system. The reaction path is the lowest energy pathway between the reactant minimum and the product minimum which is denoted as a red line in Figure 1. The highest point on the reaction path corresponds to the transition state. A minimum and a transition state are both stationary points as they have zero gradient on the PES. For a stationary point, the Hessian Index, which is the number of negative eigenvalues of a force constant matrix, corresponds to the number of internal degrees of freedom along which that point is a potential energy maximum<sup>2</sup>. The Hessian Index is also equal to the number of imaginary vibrational frequencies. A minimum and a transition state can be distinguished via this Hessian Index. A minimum will have only positive eigenvalues of the Hessian and have an Hessian Index of 0. Hence, there will be no imaginary frequencies. However, since a transition state is the maximum of the lowest energy pathway, it will have a single negative Hessian eigenvalue and have a Hessian Index of 0. Hence, it will have a single imaginary vibrational frequency.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 10:58, 9 March 2018 (UTC) Good understanding of the Hessian Matrix. The eigenvectors are the normal modes which are linear combination of the degrees of freedom and hne you move along these vectors this is what looks like a vibration.

===Quantum Chemical methods used===
The semi-empirical PM6 method was used for initial calculations, followed by the more accurate DFT (Density Functional Theory) method B3LYP. The PM6 method was used to speed up initial calculations due to the method replacing some of the two-electron integrals, the Coulomb and the exchange integrals (which are difficult to calculate), used in the Hartree Fock, with empirical parameters.<sup>3</sup> The DFT method is a computational method that derives the properties of a molecular system based on the determination of the electron density of the molecule.<sup>4</sup> Unlike a wavefunction, which becomes increasingly complex as the number of electrons increase, the determination of electron density is independent of the number of electrons. The B3LYP method is a hybrid method which utilises the useful features from <i>ab initio</i> methods and improves the accuracy of the calculation by DFT mathematics. The 6-31G basis set was chosen as a larger basis set gave a better approximation to the atomic orbitals as they placed fewer restrictions on the wavefunction while the split basis sets allowed for size changes during bonding.<sup>5</sup>

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 10:58, 9 March 2018 (UTC) Good understanding here you have clearly read beyond the script here. Some equations would have been nice to back up your arguements.

==Exercise 1: Reaction of Butadiene and Ethene==
([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) Very good work across the whole exercise. Only hiccups is your discussion of the bond lengths: the VdW distance between two atoms is the sum of their VdW radius, so at the TS the terminal carbons are starting to bond.)

===Reaction Scheme===
[[File:Reaction_Scheme_Ex1_zwl115.PNG]]
===Jmol Files===
{| class="wikitable" style="margin-left: auto; margin-right: auto; border: 1;"
|-
! Butadiene !! Ethene !! Transition State !! Cyclohexene
|-
| <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENE_FRAG_OPT_PM6_zwl115.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>ETHYLENE_OPT_PM6_zwl115.LOG</uploadedFileContents>
</jmolApplet>
</jmol> || <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>CYCLOHEXENE_OPT_POSTFREEZE_PM6_zwl115.LOG</uploadedFileContents>
</jmolApplet>
</jmol> || <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>CYCLOHEXENE_PDT_OPT_1_PM6_zwl115.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|}

===MO Diagram===

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: 1;"
|-
! Butadiene !! MO diagram for the formation of the Butadiene/Ethene transition state !! Ethene
|-
|<jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>BUTADIENE_FRAG_OPT_PM6_zwl115.LOG</uploadedFileContents>
</jmolApplet>
</jmol> LUMO of butadiene||rowspan="2"|[[File:MO_Diagram_Ex_1_3_zwl115.PNG|centre|frame]] || <jmol>
<jmolApplet>
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<uploadedFileContents>ETHYLENE_OPT_PM6_zwl115.LOG</uploadedFileContents>
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</jmol> LUMO of Ethene
|-
| <jmol>
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<uploadedFileContents>BUTADIENE_FRAG_OPT_PM6_zwl115.LOG</uploadedFileContents>
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</jmol> HOMO of Butadiene|| <jmol>
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<uploadedFileContents>ETHYLENE_OPT_PM6_zwl115.LOG</uploadedFileContents>
</jmolApplet>
</jmol> HOMO of Ethene
|}

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: 1;"
|-
! MO 1 !! MO 2 !! MO 3 !! MO 4
|-
| <jmol>
<jmolApplet>
<color>white</color>
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<uploadedFileContents>CYCLOHEXENE_OPT_POSTFREEZE_PM6_zwl115.LOG</uploadedFileContents>
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</jmol> || <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>CYCLOHEXENE_OPT_POSTFREEZE_PM6_zwl115.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|}

Referring to the MO diagram above, the HOMO of butadiene is higher in energy than that of ethene, indicating that this is a normal electron demand cycloaddition. Bonding MOs (MOs 1 and 2) of the transition state formed from the MOs of the reactants are much higher in energy than what is expected for a product. Similarly, the anti-bonding MOs of the transition state are lower in energy than the corresponding MOs of the product. This is due to the fact that the transition state lies at the maximum of the lowest energy pathway and the overlap of orbitals are not as strong as what would be seen in the products of the reaction. This leads to an overall destabilisation of the structure as the electrons are occupying MOs which are higher in energy than that of the reactants.

===Symmetry===

[[File:WH_analysis__2_EX1_zwl115.PNG|frame|100px|Figure 2. Woodward-Hoffmann analysis of cycloaddition of butadiene and ethene.]]

For a thermal pericyclic reaction, whether a reaction is 'allowed' or 'forbidden' depends on the Woodward-Hoffmann rules which states that the total number of (4q+2)<sub>s</sub> and (4r)<sub>a</sub> components must be odd.<sup>6</sup> The suffix 's' represents suprafacial where the new bonds are formed on the same side at both ends of the component. The suffix 'a' represents antarafacial where the new bonds formed are on opposite sides of both ends of the component. As seen from Figure 2, when both components are suprafacial, it would allow for good orbital overlap and the Woodward-Hoffmann rules are obeyed. This is in agreement with the results obtained for the combination of the Frontier Orbitals of butadiene and ethene in the MO diagram. Reactions are only allowed when the symmetry of the FO of butadiene is the same as that for the FO of ethene as this would lead to a non-zero orbital overlap. Conversely, an overlap between a symmetric FO and an anti-symmetric FO would lead to an orbital overlap integral of 0 and hence, no reaction will occur.

===Bond Length Analysis===

[[File:Bond_lengths_Ex1.PNG|centre|frame| Figure 3: Reaction Scheme showing the bond lengths of the reactants, transition state and the products]]
The C-C bond lengths for the reactants, transition state and the products are summarised in the Figure 3. Butadiene has shorter C=C terminal bonds of 1.34 Å and a longer C-C internal bond of 1.47 Å while ethene has a bond length of 1.33 Å. The shorter bond lengths observed in butadiene are due to the overlap between sp<sup>2</sup> C atoms which have a typical value of 1.34 Å<sup>7</sup> while the longer C-C bonds observed are due to the overlap between sp<sup>3</sup> C atoms which have a typical value of 1.54 Å.<sup>7</sup> At the transition state, the C=C terminal bonds of butadiene and the C=C bond in ethene lengthen to 1.38 Å, moving towards the value of a bond length observed between sp<sup>3</sup> C atoms. This indicates that the bond order has decreased from 2 to a value between 1 and 2. In contrast, the internal bond shortens to 1.41 Å, moving towards the value of a bond length observed between sp<sup>2</sup> C atoms and the bond order has increased from 1 to a value between 1 and 2, similar to the bond lengths of the terminal bonds. Finally in the products, the bond lengths continue to change in the same way. C5-C6 which was previously the internal bond of the butadiene has the shortest bond length of 1.34 Å due to the overlap between sp<sup>2</sup> C atoms. C1-C2, C2-C3 and C3-C4 have very similar bond lengths of 1.53-1.54 Å which reflects the overlap between sp<sup>3</sup> C atoms. C1-C6 and C4-C5 which were previously the terminal bonds of the butadiene have a bond length of 1.50 Å which is slightly shorter than the other single bonds in the molecule. This is due to the sp<sup>2</sup>-sp<sup>3</sup> overlap of the C atoms. The greater s character of the sp<sup>2</sup> C atom results in a stronger and shorter sigma bond. The length of the partially formed C-C bonds in the transition state is 2.11 Å which is significantly greater than the Van Der Waals radius of C (1.70 Å).<sup>8</sup> This indicates that at the transition state, no new C-C bonds were formed.

===Reaction path towards the products from the transition state===

[[File:Vibration_TS_Ex1_zwl115.gif|centre|frame| Figure 3. Illustration of vibration of reaction path at the Transition state]]
As seen from the illustration showing the vibration corresponding to the transition state, the formation of the 2 bonds is synchronous. The ends of the butadiene and ethene fragments move towards each other in a symmetric fashion showing a concerted mechanism.

==Link to Exercise 2==
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Transition_States_and_Reactivity_Exercise2_ZWL115

==Link to Exercise 3==
https://wiki.ch.ic.ac.uk/wiki/index.php?title=Transition_States_and_Reactivity_Exercise3_ZWL115

==Conclusion==
The use of the semi-empirical method PM6, followed by the DFT method B3LYP has provided the tools required to locate the transition states for several reactions, allowing great insight into the nature of the reactions. Following a successful vibration calculation, an IRC can be carried out to ensure that the minima of the reactants and the products were reached and were connected. In Exercise 1, the energy levels of the individual MOs of the reactants, transition states and the products were obtained, allowing for the construction of the MO diagram which was useful in studying the symmetry of the 'allowed' and 'forbidden' reactions. Exercise 2 involved an inverse electron demand Diels-Alder reaction due to the different ordering of energies of the HOMO and LUMO of the dienophile and the diene as compared to Exercise 1. This allowed for the comparison between the MO diagrams of a normal versus an inverse electron demand Diels-Alder. It was found that the MO diagrams for the 2 reactions had similar relative energy levels for their respective transition states. The study of the reaction thermodynamics in Exercises 2 and 3 confirmed the role of steric clashes and secondary orbital interactions in determining which transition states and products are favourable and vice versa. Steric clashes raised the energy of the compound while secondary orbital interactions helped to lower the energy of the chemical system. Exercise 3 also showed the role of aromaticity in providing stability to a molecule and lowering its energy while anti-aromatic compounds are highly unstable.

==References==
1. Potential Energy Surface. Encyclopedia of Human thermodynamics.

2. Potential Energy Surfaces. C, David Sherrill. School of Chemistry and Biochemistry Georgia Institute of Technology.

3. Computational Quantum Chemistry. C, David Sherrill.

4. Introduction to Computational Quantum Chemistry: Theory. A, Gilbert. The Australian National University.

5. Introduction to Computational Chemistry. V H¨anninen. University of Helsinki

6. Woodward, R. B.; Hoffmann, R. J. Am. Chem. Soc. 1965, 87(2), 395-397.

7. Bond Length. Wikipedia.

8. Van der Waals radius. Wikipedia.

([[User:Fv611|Fv611]] ([[User talk:Fv611|talk]]) Wikipedia IS NOT an acceptable primary source. Any inorganic textbook would have done here.)