ChemWiki - User contributions [en]

Rep:Mod:TS0606

2012-11-02T16:28:54Z

Ts3610: /* Reaction Pathway */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's or Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two terminal Carbons of the transition state were frozen at 2.2Å apart.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure. This is expected as the original guess for the chair transition structure was close to the actual transition structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken. It is very similar to the chair conformation Cope Rearrangement and therefore occurs at a similar frequency. The bond breaking/forming distance is 2.14Å.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state. The bond breaking/forming distance is 1.97Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* basis set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation. The bond breaking/forming distance is 2.20Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain and being the lower energy conformation.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:800px-Cis- butadiene HOMO ts3610.png|200px]] || [[File:800px-Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. yz plane || Symmetric wrt. yz plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the two new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the two new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Similarly to the exo conformation, the imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond, this gives a positive orbital overlap, known as the secondary orbital overlap effect. This additional orbital effect stabilises the molecule and is the main reason why the endo conformation is the lowest energy conformation.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T16:28:34Z

Ts3610: /* Molecular Orbitals */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's or Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two terminal Carbons of the transition state were frozen at 2.2Å apart.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure. This is expected as the original guess for the chair transition structure was close to the actual transition structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken. It is very similar to the chair conformation Cope Rearrangement and therefore occurs at a similar frequency. The bond breaking/forming distance is 2.14Å.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state. The bond breaking/forming distance is 1.97Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* basis set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation. The bond breaking/forming distance is 2.20Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain and being the lower energy conformation.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:800px-Cis- butadiene HOMO ts3610.png|200px]] || [[File:800px-Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. yz plane || Symmetric wrt. yz plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the two new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Similarly to the exo conformation, the imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond, this gives a positive orbital overlap, known as the secondary orbital overlap effect. This additional orbital effect stabilises the molecule and is the main reason why the endo conformation is the lowest energy conformation.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T16:26:29Z

Ts3610: /* Gauche Conformation */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's or Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two terminal Carbons of the transition state were frozen at 2.2Å apart.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure. This is expected as the original guess for the chair transition structure was close to the actual transition structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken. It is very similar to the chair conformation Cope Rearrangement and therefore occurs at a similar frequency. The bond breaking/forming distance is 2.14Å.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state. The bond breaking/forming distance is 1.97Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* basis set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation. The bond breaking/forming distance is 2.20Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain and being the lower energy conformation.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:800px-Cis- butadiene HOMO ts3610.png|200px]] || [[File:800px-Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. yz plane || Symmetric wrt. yz plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Similarly to the exo conformation, the imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond, this gives a positive orbital overlap, known as the secondary orbital overlap effect. This additional orbital effect stabilises the molecule and is the main reason why the endo conformation is the lowest energy conformation.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T14:57:11Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two terminal Carbons of the transition state were frozen at 2.2Å apart.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure. This is expected as the original guess for the chair transition structure was close to the actual transition structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken. It is very similar to the chair conformation Cope Rearrangement and therefore occurs at a similar frequency. The bond breaking/forming distance is 2.14Å.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state. The bond breaking/forming distance is 1.97Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* basis set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation. The bond breaking/forming distance is 2.20Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain and being the lower energy conformation.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:800px-Cis- butadiene HOMO ts3610.png|200px]] || [[File:800px-Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. yz plane || Symmetric wrt. yz plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Similarly to the exo conformation, the imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond, this gives a positive orbital overlap, known as the secondary orbital overlap effect. This additional orbital effect stabilises the molecule and is the main reason why the endo conformation is the lowest energy conformation.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T13:43:02Z

Ts3610: /* Endo Transition State */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two terminal Carbons of the transition state were frozen at 2.2Å apart.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure. This is expected as the original guess for the chair transition structure was close to the actual transition structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken. It is very similar to the chair conformation Cope Rearrangement and therefore occurs at a similar frequency. The bond breaking/forming distance is 2.14Å.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state. The bond breaking/forming distance is 1.97Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* basis set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation. The bond breaking/forming distance is 2.20Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain and being the lower energy conformation.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:800px-Cis- butadiene HOMO ts3610.png|200px]] || [[File:800px-Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. yz plane || Symmetric wrt. yz plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Similarly to the exo conformation, the imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T13:41:38Z

Ts3610: /* Exo Transition State */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two terminal Carbons of the transition state were frozen at 2.2Å apart.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure. This is expected as the original guess for the chair transition structure was close to the actual transition structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken. It is very similar to the chair conformation Cope Rearrangement and therefore occurs at a similar frequency. The bond breaking/forming distance is 2.14Å.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state. The bond breaking/forming distance is 1.97Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* basis set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation. The bond breaking/forming distance is 2.20Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain and being the lower energy conformation.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:800px-Cis- butadiene HOMO ts3610.png|200px]] || [[File:800px-Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. yz plane || Symmetric wrt. yz plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T13:39:24Z

Ts3610: /* Cis-Butadiene */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two terminal Carbons of the transition state were frozen at 2.2Å apart.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure. This is expected as the original guess for the chair transition structure was close to the actual transition structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken. It is very similar to the chair conformation Cope Rearrangement and therefore occurs at a similar frequency. The bond breaking/forming distance is 2.14Å.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state. The bond breaking/forming distance is 1.97Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* basis set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation. The bond breaking/forming distance is 2.20Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain and being the lower energy conformation.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:800px-Cis- butadiene HOMO ts3610.png|200px]] || [[File:800px-Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. yz plane || Symmetric wrt. yz plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

File:800px-Cis-butadiene LUMO ts3610.png

2012-11-02T13:38:42Z

Ts3610:

Rep:Mod:TS0606

2012-11-02T13:37:39Z

Ts3610: /* Cis-Butadiene */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two terminal Carbons of the transition state were frozen at 2.2Å apart.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure. This is expected as the original guess for the chair transition structure was close to the actual transition structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken. It is very similar to the chair conformation Cope Rearrangement and therefore occurs at a similar frequency. The bond breaking/forming distance is 2.14Å.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state. The bond breaking/forming distance is 1.97Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* basis set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation. The bond breaking/forming distance is 2.20Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain and being the lower energy conformation.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:800px-Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

File:800px-Cis- butadiene HOMO ts3610.png

2012-11-02T13:36:57Z

Ts3610:

Rep:Mod:TS0606

2012-11-02T13:32:19Z

Ts3610: /* Activation Energies */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two terminal Carbons of the transition state were frozen at 2.2Å apart.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure. This is expected as the original guess for the chair transition structure was close to the actual transition structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken. It is very similar to the chair conformation Cope Rearrangement and therefore occurs at a similar frequency. The bond breaking/forming distance is 2.14Å.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state. The bond breaking/forming distance is 1.97Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* basis set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation. The bond breaking/forming distance is 2.20Å.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain and being the lower energy conformation.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T13:22:02Z

Ts3610: /* Boat Transition State Optimisation */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two terminal Carbons of the transition state were frozen at 2.2Å apart.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure. This is expected as the original guess for the chair transition structure was close to the actual transition structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken. It is very similar to the chair conformation Cope Rearrangement and therefore occurs at a similar frequency. The bond breaking/forming distance is 2.14Å.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T13:20:37Z

Ts3610: /* Boat Transition State Optimisation */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two terminal Carbons of the transition state were frozen at 2.2Å apart.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure. This is expected as the original guess for the chair transition structure was close to the actual transition structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken. It is very similar to the chair conformation Cope Rearrangement and therefore occurs at a similar frequency. The bond breaking/forming distance is

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T13:16:34Z

Ts3610: /* Frozen Coordinates */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two terminal Carbons of the transition state were frozen at 2.2Å apart.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure. This is expected as the original guess for the chair transition structure was close to the actual transition structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T13:12:14Z

Ts3610: /* Force Constant Matrix */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

The chair transition state was optimised by calculating the force constant matrix. For this method to work the guess of the transition structure has to be similar to the optimised output.

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T13:05:52Z

Ts3610: /* Optimising the "Chair" and "Boat" Transition Structures */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build both the chair and the boat transition states when the two fragments are correctly orientated with a distance of approximately 2.2Å between the terminal Carbon atoms..

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T13:04:24Z

Ts3610: /* Anti 2 Conformation */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

A frequency calculation was run to view the vibrations of the molecule and produce an IR spectrum.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T13:03:32Z

Ts3610: /* Anti 2 Conformation */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule. An imaginary, or negative, frequency is expected in a molecule that is a transition state.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T13:00:13Z

Ts3610: /* Gauche Conformation */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy because there is much less steric repulsions than in the gauche conformations. There is also very little Van der Waal's of Pauli effects. In this case it was found that the lowest energy conformation was a gauche conformation due to the π orbital overlap between the alkene group and the vinyl proton.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T12:54:56Z

Ts3610: /* Energy Comparisons */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (a.u.) !! B3LYP/6-31G* (a.u.) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T12:54:33Z

Ts3610: /* Gauche Conformation */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy conformation. It is normally expected that one of the anti conformations will have the lowest energy

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T12:51:53Z

Ts3610: /* Gauche Conformation */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn in an attempt to find the lowest energy conformation and then was optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-02T12:48:29Z

Ts3610: /* Antiperiplanar Conformation */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation due to the steric effects, the two alkene groups are antiperiplanar to each other which reduces the steric interaction between them and hence stabilises the molecule.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T15:23:06Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The steric repulsions between the anhydride and the -CH2-CH2- group is greater than that of the -CH=CH- group this means that the exo product has greater steric repulsions and is therefore more strained. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 a.u. and 0.00000 a.u. respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T15:15:37Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The exo product is more strained because there will be greater steric repulsion's between the -CH2-CH2- group and the anhydride than the -CH=CH- group in the endo form. The geometry of the endo form also ensures that the steric repulsions have much less of an effect. The steric effects combined with the secondary orbital overlap between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment show that the endo form is expected to be the thermodynamically most stable product. The relative energies for the exo and endo form, 0.00409 and 0.00000 respectively, show that this is in fact the case.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T15:10:11Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

The exo product is more strained because there will be greater steric repulsion's between the -CH2-CH2- group and the anhydride than the -CH=CH- group in the endo form. The geometry of the endo form also ensures that the steric repulsions have much less of an effect.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T14:43:59Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved. The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies away from the alkene bond of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the alkene bond.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T14:42:43Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved. The σ C-C bond forming distance for both the exo and endo form is 2.3Å, all of the other bond lengths are the same in both conformations. The main structural difference between the exo and endo form is that in the exo form the anhydride section of the diene lies beneath the -CH2-CH2- part of the dienophile whereas in the endo form the anhydride section of the diene lies beneath the -CH=CH- bond.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

File:Exo bond lengths ts3610.png

2012-11-01T14:36:21Z

Ts3610: uploaded a new version of "File:Exo bond lengths ts3610.png"

File:Endo bond lengths ts3610.png

2012-11-01T14:36:05Z

Ts3610: uploaded a new version of "File:Endo bond lengths ts3610.png"

Rep:Mod:TS0606

2012-11-01T14:24:26Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths !! Through Space Distance
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]] || [[File:Exo secondary ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]] || [[File:Endo secondary ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

File:Endo secondary ts3610.png

2012-11-01T14:24:05Z

Ts3610:

File:Exo secondary ts3610.png

2012-11-01T14:23:32Z

Ts3610:

Rep:Mod:TS0606

2012-11-01T14:15:46Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy !! Bond Lengths
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u. || [[File:Exo bond lengths ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u. || [[File:Endo bond lengths ts3610.png|200px]]
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

File:Endo bond lengths ts3610.png

2012-11-01T14:15:28Z

Ts3610:

File:Exo bond lengths ts3610.png

2012-11-01T14:13:35Z

Ts3610:

Rep:Mod:TS0606

2012-11-01T13:42:02Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u.
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u.
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile combine to give an asymmetric orbital with only one nodal plane. The overall symmetry is expected to be asymmetric due to the HOMO of the diene and the LUMO of the dienophile both being asymmetric, therefore the symmetry of the reaction is conserved.

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T13:40:15Z

Ts3610: /* Intrinsic Reaction Coordinate */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC using the 6-31G* basis set, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u.
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u.
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T13:39:29Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u.
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u.
|}

For both the Exo and the Endo form the HOMO of the diene and the LUMO of the dienophile

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T13:30:25Z

Ts3610: /* Regioselectivity of the Diels Alder Reaction */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

Both the endo and exo transition state were calculated using the 6-31G* basis set and the TS Berny method for finding a transition state. The "Opt=NoEigen" term was added as the force constants were being calculated once.

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u.
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u.
|}

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T13:13:03Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Exo vs Endo
! !! HOMO !! LUMO !! Energy
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u.
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u.
|}

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T13:12:45Z

Ts3610: /* Endo Transition State */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

==== Comparison ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! !! HOMO !! LUMO !! Energy
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u.
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u.
|}

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T13:12:36Z

Ts3610: /* Exo Transition State */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Energy of endo transition state: -612.68339678 a.u.

==== Comparison ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! !! HOMO !! LUMO !! Energy
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u.
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u.
|}

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T13:12:23Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

Energy of exo transition state: -612.67931096 a.u.

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Energy of endo transition state: -612.68339678 a.u.

==== Comparison ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! !! HOMO !! LUMO !! Energy
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u.
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]] || -612.68339678 a.u.
|}

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T13:12:08Z

Ts3610: /* Comparison */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

Energy of exo transition state: -612.67931096 a.u.

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Energy of endo transition state: -612.68339678 a.u.

==== Comparison ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! !! HOMO !! LUMO !! Energy
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]] || -612.67931096 a.u.
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]]
|}

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T13:11:27Z

Ts3610: /* Endo Transition State */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

Energy of exo transition state: -612.67931096 a.u.

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Energy of endo transition state: -612.68339678 a.u.

==== Comparison ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! !! HOMO !! LUMO
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]]
|}

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T13:11:05Z

Ts3610: /* Regioselectivity of the Diels Alder Reaction */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

Energy of exo transition state: -612.67931096 a.u.

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Energy of endo transition state: -612.68339678 a.u.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! !! HOMO !! LUMO
|-
| Exo || [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]]
|-
| Endo || [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]]
|}

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T13:06:24Z

Ts3610: /* Energy Comparisons */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K. The activation energy of the boat conformation is much higher than that of the chair conformation, this is as expected due to the chair conformation having less steric strain.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

Energy of exo transition state: -612.67931096 a.u.

{|class="wikitable" border="1"
|+ Molecular Orbitals of the Exo Transition State
! HOMO !! LUMO
|-
| [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]]
|}

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Energy of endo transition state: -612.68339678 a.u.

{|class="wikitable" border="1"
|+ Molecular Orbitals of the Endo Transition State
! HOMO !! LUMO
|-
| [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]]
|}

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T13:01:34Z

Ts3610: /* Cis-Butadiene */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

The AM1 semi-empirical molecular orbital method was used to optimise the molecule of cis-butadiene.

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

Energy of exo transition state: -612.67931096 a.u.

{|class="wikitable" border="1"
|+ Molecular Orbitals of the Exo Transition State
! HOMO !! LUMO
|-
| [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]]
|}

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Energy of endo transition state: -612.68339678 a.u.

{|class="wikitable" border="1"
|+ Molecular Orbitals of the Endo Transition State
! HOMO !! LUMO
|-
| [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]]
|}

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>

Rep:Mod:TS0606

2012-11-01T11:59:15Z

Ts3610: /* Energy Comparisons */

== The Cope Rearrangement ==

==== Antiperiplanar Conformation ====

1,5-hexadiene was originally optimised with the intention of finding an antiperiplanar conformation. A molecule was drawn in what was expected to be a low energy antiperiplanar conformation and this molecule was optimised.

[http://hdl.handle.net/10042/20999 1,5-Hexadiene Optimisation DOI:10042/20999]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene app opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69260235 a.u.
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000435 0.000450 YES
RMS Force 0.000069 0.000300 YES
Maximum Displacement 0.001576 0.001800 YES
RMS Displacement 0.000522 0.001200 YES
Predicted change in Energy=-9.100268D-07
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene app ts3610.png|200px]]

The energy and point group match that of the anti 1 conformation<ref name="Appendix1" />. This is the lowest energy antiperiplanar conformation.

==== Gauche Conformation ====

Similar to the anitperiplanar conformation a guess structure of the lowest energy gauche conformation was drawn and then optimised.

[http://hdl.handle.net/10042/21001 1,5-Hexadiene Gauche Optimisation DOI:10042/21001]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene gauche opt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.68771613 a.u.
|-
| RMS Gradient Norm || 0.00001461 a.u.
|-
| Dipole Moment || 0.4550 Debye
|-
| Point Group || C2
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000040 0.000450 YES
RMS Force 0.000007 0.000300 YES
Maximum Displacement 0.000612 0.001800 YES
RMS Displacement 0.000151 0.001200 YES
Predicted change in Energy=-2.473769D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

[[File:1,5 hexadiene gauche ts3610.png|200px]]

The energy and point group match the gauche 1 conformation<ref name="Appendix1" />. This is actually the highest energy gauche conformation, due to the two terminal carbon atoms being close together there is likely to be steric strain.

The gauche conformation was redrawn to find the lowest energy conformation and then optimised.

[http://hdl.handle.net/10042/21355 Gauche 3 Optimisation DOI:10042/21355]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || gauche3
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69266122 a.u.
|-
| RMS Gradient Norm || 0.00000357 a.u.
|-
| Dipole Moment || 0.3405 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000012 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.001094 0.001800 YES
RMS Displacement 0.000281 0.001200 YES
Predicted change in Energy=-3.416150D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was correctly optimised.

[[File:Gauche 3 ts3610.png|200px]]

The energy and point group of the newly optimised molecule matches that of the gauche 3 conformation<ref name="Appendix1" /> and is the lowest energy gauche conformation.

==== Anti 2 Conformation ====

The anti 2 conformation had not been found in the original calculation, so an additional calculation was run.

[http://hdl.handle.net/10042/21005 Anti2 Optimisation DOI:10042/21005]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -231.69253521 a.u.
|-
| RMS Gradient Norm || 0.00003619 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000074 0.000450 YES
RMS Force 0.000019 0.000300 YES
Maximum Displacement 0.001453 0.001800 YES
RMS Displacement 0.000540 0.001200 YES
Predicted change in Energy=-7.570432D-08
Optimization completed.
-- Stationary point found. </pre>

[[File:1,5 hexadiene anti 2 ts3610.png|200px]]

The point group and energy match the anti 2 conformation.

The molecule was then reoptimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21008 Anti 2 Reoptimisation DOI:10042/21008]

{| class="wikitable" border="1"
|+ 1,5-Hexadiene Summary
|-
| File Name || 1,5 hexadiene anti 2 reopt
|-
| File Type || .fch
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || -234.61171856 a.u.
|-
| RMS Gradient Norm || 0.00001217 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000014 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000190 0.001800 YES
RMS Displacement 0.000077 0.001200 YES
Predicted change in Energy=-1.458658D-08
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

The geometry changes very little when changing to a higher basis set, but the energy differs quite significantly. The change in energy from using the 3-21G basis set to the 6-31G basis set is -231.69253521 a.u. to -234.61171856 a.u. This gives a difference of 2.91918335 a.u. or 1831 kcal mol-1. This is a very large difference in energy due to a better basis set being used, giving a more accurate representation of the energy of the conformation.

[http://hdl.handle.net/10042/21012 Anti 2 Frequency DOI:10042/21012]

''Low frequencies --- -10.6330 -0.0007 -0.0006 -0.0006 8.1799 35.7303''

''Low frequencies --- 71.0594 81.2098 121.4252''

No negative frequencies are observed which shows that all of the vibrations are real, as expected for a ground state molecule.

[[File:1,5 hexadiene anti 2 IR spectrum ts3610.png]]

The thermochemistry data can be seen below:

''Sum of electronic and zero-point Energies= -234.469198''

''Sum of electronic and thermal Energies= -234.461846''

''Sum of electronic and thermal Enthalpies= -234.460901''

''Sum of electronic and thermal Free Energies= -234.500807''

== Optimising the "Chair" and "Boat" Transition Structures ==

An allyl fragment was optimised as it is essentially one half of the transition states and can therefore be used to build each of the transition states.

[http://hdl.handle.net/10042/21018 Allyl Fragment Optimisation DOI:10042/21018]

<pre> Item Value Threshold Converged?
Maximum Force 0.000048 0.000450 YES
RMS Force 0.000018 0.000300 YES
Maximum Displacement 0.000142 0.001800 YES
RMS Displacement 0.000070 0.001200 YES
Predicted change in Energy=-1.277268D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was successfully optimised.

=== Chair Transition State ===

The chair transition state was built using two allyl fragments, with the terminal Carbon atoms approximately 2.2Å apart.

==== Force Constant Matrix ====

[http://hdl.handle.net/10042/21021 Chair TS Guess Optimisation TS (Berny) DOI:10042/21021]

<pre> Item Value Threshold Converged?
Maximum Force 0.000008 0.000450 YES
RMS Force 0.000001 0.000300 YES
Maximum Displacement 0.000167 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-2.513811D-09
Optimization completed.
-- Stationary point found. </pre>

The item table shows that molecule was optimised successfully.

''Low frequencies --- -817.9475 -1.3131 -1.0053 -0.0007 0.0006 0.0007''

''Low frequencies --- 1.1224 209.5401 396.0042''

****** 1 imaginary frequencies (negative Signs) ******

[[File:Chair ts animation ts3610.gif|400px]]

1 imaginary frequency was found at 818cm-1 and from the above animation it can be seen that this corresponds to the Cope Rearrangement where one bond is being formed whilst at the opposite end of the fragment a bond is being broken. The bond breaking/forming distance is approximately 2.02Å.

==== Frozen Coordinates ====

The transition state was also computed using the frozen coordinate method, where the two ends of the transition state were frozen at 2.2Å.

[http://hdl.handle.net/10042/21032 Chair Optimisation Frozen Coordinate DOI:10042/21032]

<pre> Item Value Threshold Converged?
Maximum Force 0.000037 0.000450 YES
RMS Force 0.000013 0.000300 YES
Maximum Displacement 0.001492 0.001800 YES
RMS Displacement 0.000267 0.001200 YES
Predicted change in Energy=-4.668955D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows that the molecule was optimised successfully.

The frozen bonds were then changed to derivative and the transition state was reoptimised using the force constant method.

[http://hdl.handle.net/10042/21043 Chair Optimisation Frozen Coordinate 2 DOI:10042/21043]

<pre> Item Value Threshold Converged?
Maximum Force 0.000026 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001110 0.001800 YES
RMS Displacement 0.000197 0.001200 YES
Predicted change in Energy=-3.494538D-07
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was optimised successfully.

''Low frequencies --- -817.9060 -4.4511 -3.8316 0.0002 0.0002 0.0006''

''Low frequencies --- 3.0445 209.5148 396.2539''

''****** 1 imaginary frequencies (negative Signs) ******''

1 imaginary frequency was found and it is clear that is corresponds to the same Cope Rearrangement as the optimised structure using the force constant matrix method. The bond breaking/forming distance is approximately 2.02Å which again is the same as for the force constant matrix method. There is no obvious change in geometry between the frozen coordinates structure and the force constant structure.

=== Boat Transition State Optimisation ===

The boat transition structure was optimised using the QST2 method. The reactant and product were both the anti 2 conformer that had been previously optimised.

[http://hdl.handle.net/10042/21297 QST2 Optimisation Fail DOI:10042/21297]

The calculation fails as the reactant and product are not close enough to the transition state for the calculation to find the boat structure. The molecule was then edited to closer resemble the transition state and the calculation was re-run.

[http://hdl.handle.net/10042/21313 Boat QST2 Optimisation DOI:10042/21313]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -840.1357 -0.0023 0.0001 0.0005 1.3923 6.4842''

''Low frequencies --- 8.4751 155.5449 381.9632''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat animation ts3610.gif|400px]]

1 imaginary frequency was found and this corresponds to the Cope Rearrangement, where one bond is being formed whilst another is being broken.

=== Intrinsic Reaction Coordinate ===

The IRC was originally calculated using 50 points, however the output file shows that only 26 points were computed. There also appears to be an error on the second step as it does not fit the pattern.

[http://hdl.handle.net/10042/21110 Chair IRC DOI:10042/21110]

{|class="wikitable" border="1"
|+ IRC 50 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC ts3610.png|600px|centre]] || [[File:IRC animation ts3610.gif|400px|centre]]
|}

An optimisation was completed on the last point of the IRC, as this is expected to be close to the minimum.

[http://hdl.handle.net/10042/21138 Chair IRC Optimisation DOI:10042/21138]

The IRC was re-run using 100 points and again the output file shows less points were computed, only 46. The anomaly in the IRC calculate using only 50 points is not carried through to the IRC with 100 points.

[http://hdl.handle.net/10042/21307 Chair IRC 2 DOI:10042/21307]

{|class="wikitable" border="1"
|+ IRC 100 Points
! IRC !! Molecule Vibrations
|-
| [[File:IRC 100 ts3610.png|600px|centre]] || [[File:IRC 100 animation ts3610.gif|400px|centre]]
|}

By comparing the IRC with 50 points to the IRC with 100 points it is clear to see that is was appropriate to optimise the last point form the IRC with 50 points as this is close enough to the minium value shown in the IRC with 100 points.

=== Activation Energies ===

The transition state optimised using the frozen coordinate method was re-optimised using the 6-31G* basis set.

[http://hdl.handle.net/10042/21190 Chair Activation Energy Optimisation DOI:10042/21190]

<pre> Item Value Threshold Converged?
Maximum Force 0.000016 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.000894 0.001800 YES
RMS Displacement 0.000158 0.001200 YES
Predicted change in Energy=-4.049382D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule has been successfully optimised.

''Low frequencies --- -569.0992 -21.5412 -7.1927 -0.0008 -0.0006 -0.0005''

''Low frequencies --- 25.5105 195.1486 262.5505''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Chair opt and freq animation ts3610.gif|400px]]

The imaginary frequency corresponds to the bond breaking/forming vibration of the transition state.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.414881''

''Sum of electronic and thermal Energies= -234.408953''

''Sum of electronic and thermal Enthalpies= -234.408009''

''Sum of electronic and thermal Free Energies= -234.443121''

The boat that had originally been optimised using the QST2 method with the 3-21G basis set was reoptimiesd using the 6-31G* baisi set.

[http://hdl.handle.net/10042/21203 Boat Activation Energy Optimisation DOI:10042/21203]

<pre> Item Value Threshold Converged?
Maximum Force 0.000145 0.000450 YES
RMS Force 0.000035 0.000300 YES
Maximum Displacement 0.001201 0.001800 YES
RMS Displacement 0.000283 0.001200 YES
Predicted change in Energy= 3.644196D-08
Optimization completed.
-- Stationary point found.</pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -533.9244 -16.2044 0.0005 0.0007 0.0008 1.3065''

''Low frequencies --- 10.0658 133.6793 258.4286''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Boat opt and freq animation ts3610.gif|400px]]

The imaginary frequency again corresponds to the bond breaking/forming vibration of the transition state as it does for the chair conformation.

Thermochemistry data:

''Sum of electronic and zero-point Energies= -234.402331''

''Sum of electronic and thermal Energies= -234.395985''

''Sum of electronic and thermal Enthalpies= -234.395041''

''Sum of electronic and thermal Free Energies= -234.431760''

==== Energy Comparisons ====

{|class="wikitable" border="1"
|+ Activation Energies for the Reaction
! Structure !! HF/321-G (hartree) !! B3LYP/6-31G* (hartree) !! Activation Energy, ΔE (kcal/mol)
|-
| Boat TS || -231.60280242 || -234.54307810 || 43.072(5750546)
|-
| Chair TS|| -231.61932180 || -234.55693214 || 34.379(0264142)
|-
| Anti2 (reactant)|| -231.69253521 || -234.61171856 || ---
|}

Although the geometries change very little when optimised with a higher basis set, it is clear to see that the energies vary quite a lot. The energies calculated using the higher basis are more accurate and for that reason the activation energies were calculated using the values from the 6-31G* basis set. The calculated activation energies fall within the error range given for the experimental energies<ref name="Appendix1" /> at 0K.

== The Diels Alder Cycloaddition ==

=== Cis-Butadiene ===

[http://hdl.handle.net/10042/21143 Cis-Butadiene Optimisation DOI:10042/21143]

{| class="wikitable" border="1"
|+ Cis-Butandiene Summary
|-
| File Name || cis-butandiene opt
|-
| File Type || .log
|-
| Calculation Type || FOPT
|-
| Calculation Method || RAM1
|-
| Basis Set || ZDO
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy || 0.04879719 a.u.
|-
| RMS Gradient Norm || 0.00001745 a.u.
|-
| Dipole Moment || 0.0414 Debye
|-
| Point Group || C1
|}

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000011 0.000300 YES
Maximum Displacement 0.000382 0.001800 YES
RMS Displacement 0.000162 0.001200 YES
Predicted change in Energy=-9.691123D-09
Optimization completed.
-- Stationary point found. </pre>

The summary and item table show that the molecule was successfully optimised.

{|class="wikitable" border="1"
|+ Molecular Orbitals of Cis-Butadiene
! HOMO !! LUMO
|-
| [[File:Cis- butadiene HOMO ts3610.png|200px]] || [[File:Cis-butadiene LUMO ts3610.png|200px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

=== Transition Structure ===

The transition structure was calculated using the force constant matrix method as was used for the first determination of the transition state for the chair conformation. The estimated bond distance was originally set at 2.25Å as this is close to the expected bond distance.

[http://hdl.handle.net/10042/21205 Transition State Optimisation DOI:10042/21205]

<pre> Item Value Threshold Converged?
Maximum Force 0.000044 0.000450 YES
RMS Force 0.000006 0.000300 YES
Maximum Displacement 0.001167 0.001800 YES
RMS Displacement 0.000227 0.001200 YES
Predicted change in Energy=-1.936170D-08
Optimization completed.
-- Stationary point found. </pre>

The item table shows the molecule was successfully optimised.

''Low frequencies --- -818.6295 0.0007 0.0008 0.0008 2.4128 3.9865''

''Low frequencies --- 4.3768 166.5547 284.3601''

''****** 1 imaginary frequencies (negative Signs) ******''

The imaginary frequency corresponds to the bond breaking/forming vibration of the Diels-Alder reaction.

==== Structure ====

[[File:Diels alder TS ts3610.png|300px]]

The ethene molecule can be seen to be approaching the face of the butadiene molecule from above or below the plane. This allows the two new sigma bonds to be formed at either end of the ethene molecule in a concerted fashion.

The partially formed σ C-C bond length is: 2.209Å

Typical C-C sp3-sp3 bond length<ref name="C Bond Lengths" />: 1.53Å

Typical C-C sp2-sp2 bond length<ref name="C Bond Lengths" />: 1.46Å

Van der Waals radius of Carbon<ref name="C VdW Radii" />: 1.70Å

The partially formed σ C-C bond is less than 3.4Å which is the value for two Carbon Van der Waals radii, this means there is some orbital overlap between the two molecules in the transition state and therefore displays some bonding character. The partially formed σ C-C bond is however much longer than both the typical C-C sp3-sp3 and C-C sp2-sp2 bond lengths for a stable molecule. This shows that in the transition state the bond formation is not yet complete, which is as expected.

==== Molecular Orbitals ====

{|class="wikitable" border="1"
|+ Molecular Orbitals of Diels-Alder Reaction TS
! HOMO !! LUMO
|-
| [[File:Envelope HOMO ts3610.png|300px]] || [[File:Envelope LUMO ts3610.png|300px]]
|-
| Anti-symmetric wrt. plane || Symmetric wrt. plane
|}

The HOMO at the transition structure is antisymmetric with respect to the the plane. The MO's that have formed the HOMO of the transition state are the the HOMO of the butadiene molecule and the LUMO of the ethene molecule. Both of these orbitals are antisymmetric and give the antisymmetric HOMO. The reaction is allowed because during the formation of the 2 new σ C-C bonds by the interaction of the butadiene HOMO with the ethene LUMO the symmetry is conserved.

==== Reaction Pathway ====

{|class="wikitable" border="1"
|+ Lowest Energy Vibrations of the Diels Alder Transition State
! Imaginary Frquency (-818.63 cm-1) !! Lowest Positive Frequency (166.55 cm-1)
|-
| [[File:TS imaginary vibration ts3610.gif|400px]] || [[File:TS lowest real ts3610.gif|400px]]
|}

The imaginary frequency corresponds to the bond forming reaction in the transition state and therefore the reaction pathway, it is clear to see form the animation that the forming of the 2 new σ C-C bonds is a synchronous process. The lowest positive frequency shows the butadiene molecule and the ethene molecule vibrate as individual molecules rather interacting to form the transition state. There is no interaction between the two fragments in the the lowest positive frequency as there was in the imaginary frequency.

=== Regioselectivity of the Diels Alder Reaction ===

==== Exo Transition State ====

[http://hdl.handle.net/10042/21345 Exo Transition State DOI:10042/21345]

<pre> Item Value Threshold Converged?
Maximum Force 0.000030 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000383 0.001800 YES
RMS Displacement 0.000101 0.001200 YES
Predicted change in Energy=-2.439873D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -448.4756 -13.9333 -11.7572 0.0007 0.0009 0.0014''

''Low frequencies --- 3.0477 53.2974 109.0854''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Exo animation ts3610.gif|400px]]

Energy of exo transition state: -612.67931096 a.u.

{|class="wikitable" border="1"
|+ Molecular Orbitals of the Exo Transition State
! HOMO !! LUMO
|-
| [[File:Exo HOMO ts3610.png|200px]] || [[File:Exo LUMO ts3610.png|200px]]
|}

==== Endo Transition State ====

[http://hdl.handle.net/10042/21347 Endo Transtition State DOI:10042/21347]

<pre> Item Value Threshold Converged?
Maximum Force 0.000020 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000843 0.001800 YES
RMS Displacement 0.000213 0.001200 YES
Predicted change in Energy=-6.289740D-09
Optimization completed.
-- Stationary point found. </pre>

''Low frequencies --- -446.9904 -14.3290 -0.0010 -0.0009 -0.0006 4.2809''

''Low frequencies --- 11.2860 59.6264 118.3320''

''****** 1 imaginary frequencies (negative Signs) ******''

[[File:Endo animation ts3610.gif|400px]]

Energy of endo transition state: -612.68339678 a.u.

{|class="wikitable" border="1"
|+ Molecular Orbitals of the Endo Transition State
! HOMO !! LUMO
|-
| [[File:Endo HOMO ts3610.png|200px]] || [[File:Endo LUMO ts3610.png|200px]]
|}

== References ==

<references>
<ref name="Appendix1"> M.Bearpark., (2012)., Moecule 3, Physical (Computational Lab)., [Lab Script]., Imperial College London., Autumn 2012., [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_/ Lab Script]</ref>
<ref name="C Bond Lengths">F.H. Allen, O. Kennard, D.G. Watson, et.al., Tables of Bond Lengths determined by X-Ray and Neutron Diffraction. Part 1. Bond Lengths in Organic Compounds., J. Chem. Soc. Perkin Trans. 2., 1987, S1-S19., {{DOI|10.1039/P298700000S1}}</ref>
<ref name="C VdW Radii">A. Bondi., Van der Waals Volumes and Radii., J. Phys. Chem., 1964, 68 (3) pp 441-451., {{DOI|10.1021/j100785a001}}</ref>
</references>