ChemWiki - User contributions [en]

Rep:Mod:96499348

2015-11-05T18:23:57Z

Yo113: /* References */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and electronic energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has a C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 a.u which corresponds to the appendix. This structure was then optimised twice, separately, once with DFT 6-31G and another time at Hartree-Fock 3-21G level.

The table below shows the structure of the antiperiplanar 1,5-hexadiene antiperiplanar of Ci symmetry, optimised at HF/3-21G level (left) and optimised at DFT 6-31G level (right). The diagrams in the table show that the arrangement of the hydrogen atoms on the middle two carbons differ between the structures optimised at different levels. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies were found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they resemble the chair structure, and the distance between the carbons involved in bond formation were set to 2.2 Å. To get the chair transition state, the first the force constant matrix was computed and the structure constructed was optimised to a transition state (optimising to a TS (Berny)). This was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. The animation of this vibration also corresponds to the Cope rearrangement, giving confidence that this is the transition state we are looking for. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, opt+freq QST2 TS optimisation at HF/3-21G level was employed. This method attemptsto find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki> This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), both the geometries (point groups) and energies differ before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*).
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
(Point group = C1)
|Boat TS
(Point group = CS)
|Reactant (''anti''2)
(Point group = C1)
|Chair TS
(Point group = C2h)
|Boat TS
(Point group = CS)
|Reactant (''anti''2)
(Point group = C2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5 kcal/mol
|44.7 ± 2.0 kcal/mol
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the 'boat' transition state is higher in energy (less stable) and hence the chair transition state is favoured.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. Typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>, while the ''van der'' ''Waals'' radius of a C atom is 1.85 Å<ref>Chemwiki.ucdavis.edu, (2013). ''Covalent Bond Distance, Radius and van der Waals Radius - Chemwiki''. [online] Available at: <nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki> [Accessed 5 Nov. 2015].</ref>. The distance between the carbons of the partially formed C-C sigma bonds are larger than typical sp3 bondlengths, implying the approach of the two components in the transition state the two components as the bond forms. This distance is also shorter than twice the van der Waals radius of a C atom, which reflects the interaction between the two carbons towards bond formation.

For this obtained transition state, two vibrations, the one with the imaginary frequency and the one with the lowest positive frequency, were visualised. These are shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle.
|}
This cycloaddition reaction involves 6π-electrons. Systems with [4n+2]π electrons are thermally allowed only if the symmetry of both substrates involved in the cycloaddition are the same (symmetric or antisymmetric).<ref>Clayden, J. (2001). ''Organic chemistry''. Oxford: Oxford University Press.</ref> Based on this, since the HOMO of the transition state is antisymmetric with respect to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene (both antisymmetric).

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimisation using the Frozen Coordinate Method. (All calculations were done using the AM1 semi-empircal molecular orbital method.) The table below gives the information for the two transition states found.
{| class="wikitable"
!
!
!Endo Transition State
!Exo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartrees, is 2.85 kJ mol-1.The ''exo'' transition state is higher in energy than the ''endo'' transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured, and are shown in the table below:
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|2.94
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|2.89
|}
Without considering electronic effects, based on the structure of the ''exo'' and ''endo'' products, this distance is expected to be greater in the ''endo'' transition state (see diagram below).

[[File:Exoendostructures.png|500px]]

However, the measured distances prove otherwise, suggesting that there is a greater extent of favourable interaction between the two substrates (maleic anhydride and hexadiene) in the ''endo'' transition state. This explanation is supported by the lower energy of the ''endo ''transition as calculated above (which corresponds to a lower activation energy for the cycloaddition reaction ''via ''this transition state) compared to the ''exo'' transition state. The secondary orbital interaction between the pi orbitals of the maleic anhydride and cyclohexadiene substrates that is only possible in the ''endo'' transition state accounts for this.

== References ==

Rep:Mod:96499348

2015-11-05T18:20:46Z

Yo113: /* References */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and electronic energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has a C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 a.u which corresponds to the appendix. This structure was then optimised twice, separately, once with DFT 6-31G and another time at Hartree-Fock 3-21G level.

The table below shows the structure of the antiperiplanar 1,5-hexadiene antiperiplanar of Ci symmetry, optimised at HF/3-21G level (left) and optimised at DFT 6-31G level (right). The diagrams in the table show that the arrangement of the hydrogen atoms on the middle two carbons differ between the structures optimised at different levels. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies were found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they resemble the chair structure, and the distance between the carbons involved in bond formation were set to 2.2 Å. To get the chair transition state, the first the force constant matrix was computed and the structure constructed was optimised to a transition state (optimising to a TS (Berny)). This was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. The animation of this vibration also corresponds to the Cope rearrangement, giving confidence that this is the transition state we are looking for. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, opt+freq QST2 TS optimisation at HF/3-21G level was employed. This method attemptsto find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki> This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), both the geometries (point groups) and energies differ before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*).
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
(Point group = C1)
|Boat TS
(Point group = CS)
|Reactant (''anti''2)
(Point group = C1)
|Chair TS
(Point group = C2h)
|Boat TS
(Point group = CS)
|Reactant (''anti''2)
(Point group = C2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5 kcal/mol
|44.7 ± 2.0 kcal/mol
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the 'boat' transition state is higher in energy (less stable) and hence the chair transition state is favoured.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. Typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref>Wiredchemist.com, (2015). ''Common Bond Energies (D''. [online] Available at: <nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki> [Accessed 3 Nov. 2015].</ref>, while the ''van der'' ''Waals'' radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are larger than typical sp3 bondlengths, implying the approach of the two components in the transition state the two components as the bond forms. This distance is also shorter than twice the van der Waals radius of a C atom, which reflects the interaction between the two carbons towards bond formation.

For this obtained transition state, two vibrations, the one with the imaginary frequency and the one with the lowest positive frequency, were visualised. These are shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle.
|}
This cycloaddition reaction involves 6π-electrons. Systems with [4n+2]π electrons are thermally allowed only if the symmetry of both substrates involved in the cycloaddition are the same (symmetric or antisymmetric).<ref>Organic Chemistry. By J. P. Clayden, N. Greeves, S. Warren, and P. D. Wothers; Oxford University Press, 2001</ref> Based on this, since the HOMO of the transition state is antisymmetric with respect to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene (both antisymmetric).

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimisation using the Frozen Coordinate Method. (All calculations were done using the AM1 semi-empircal molecular orbital method.) The table below gives the information for the two transition states found.
{| class="wikitable"
!
!
!Endo Transition State
!Exo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartrees, is 2.85 kJ mol-1.The ''exo'' transition state is higher in energy than the ''endo'' transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured, and are shown in the table below:
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|2.94
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|2.89
|}
Without considering electronic effects, based on the structure of the ''exo'' and ''endo'' products, this distance is expected to be greater in the ''endo'' transition state (see diagram below).

[[File:Exoendostructures.png|500px]]

However, the measured distances prove otherwise, suggesting that there is a greater extent of favourable interaction between the two substrates (maleic anhydride and hexadiene) in the ''endo'' transition state. This explanation is supported by the lower energy of the ''endo ''transition as calculated above (which corresponds to a lower activation energy for the cycloaddition reaction ''via ''this transition state) compared to the ''exo'' transition state. The secondary orbital interaction between the pi orbitals of the maleic anhydride and cyclohexadiene substrates that is only possible in the ''endo'' transition state accounts for this.

== References ==

Rep:Mod:96499348

2015-11-05T18:13:09Z

Yo113: /* Transition state of the Diels-Alder Cycloaddition Reaction between cis-butadiene and ethene */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and electronic energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has a C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 a.u which corresponds to the appendix. This structure was then optimised twice, separately, once with DFT 6-31G and another time at Hartree-Fock 3-21G level.

The table below shows the structure of the antiperiplanar 1,5-hexadiene antiperiplanar of Ci symmetry, optimised at HF/3-21G level (left) and optimised at DFT 6-31G level (right). The diagrams in the table show that the arrangement of the hydrogen atoms on the middle two carbons differ between the structures optimised at different levels. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies were found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they resemble the chair structure, and the distance between the carbons involved in bond formation were set to 2.2 Å. To get the chair transition state, the first the force constant matrix was computed and the structure constructed was optimised to a transition state (optimising to a TS (Berny)). This was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. The animation of this vibration also corresponds to the Cope rearrangement, giving confidence that this is the transition state we are looking for. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, opt+freq QST2 TS optimisation at HF/3-21G level was employed. This method attemptsto find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki> This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), both the geometries (point groups) and energies differ before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*).
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
(Point group = C1)
|Boat TS
(Point group = CS)
|Reactant (''anti''2)
(Point group = C1)
|Chair TS
(Point group = C2h)
|Boat TS
(Point group = CS)
|Reactant (''anti''2)
(Point group = C2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5 kcal/mol
|44.7 ± 2.0 kcal/mol
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the 'boat' transition state is higher in energy (less stable) and hence the chair transition state is favoured.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. Typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>, while the ''van der'' ''Waals'' radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are larger than typical sp3 bondlengths, implying the approach of the two components in the transition state the two components as the bond forms. This distance is also shorter than twice the van der Waals radius of a C atom, which reflects the interaction between the two carbons towards bond formation.

For this obtained transition state, two vibrations, the one with the imaginary frequency and the one with the lowest positive frequency, were visualised. These are shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle.
|}
This cycloaddition reaction involves 6π-electrons. Systems with [4n+2]π electrons are thermally allowed only if the symmetry of both substrates involved in the cycloaddition are the same (symmetric or antisymmetric).<ref>Organic Chemistry. By J. P. Clayden, N. Greeves, S. Warren, and P. D. Wothers; Oxford University Press, 2001</ref> Based on this, since the HOMO of the transition state is antisymmetric with respect to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene (both antisymmetric).

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimisation using the Frozen Coordinate Method. (All calculations were done using the AM1 semi-empircal molecular orbital method.) The table below gives the information for the two transition states found.
{| class="wikitable"
!
!
!Endo Transition State
!Exo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartrees, is 2.85 kJ mol-1.The ''exo'' transition state is higher in energy than the ''endo'' transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured, and are shown in the table below:
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|2.94
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|2.89
|}
Without considering electronic effects, based on the structure of the ''exo'' and ''endo'' products, this distance is expected to be greater in the ''endo'' transition state (see diagram below).

[[File:Exoendostructures.png|500px]]

However, the measured distances prove otherwise, suggesting that there is a greater extent of favourable interaction between the two substrates (maleic anhydride and hexadiene) in the ''endo'' transition state. This explanation is supported by the lower energy of the ''endo ''transition as calculated above (which corresponds to a lower activation energy for the cycloaddition reaction ''via ''this transition state) compared to the ''exo'' transition state. The secondary orbital interaction between the pi orbitals of the maleic anhydride and cyclohexadiene substrates that is only possible in the ''endo'' transition state accounts for this.

== References ==

Rep:Mod:96499348

2015-11-05T18:10:50Z

Yo113: /* Tutorial:The Cope Rearrangement */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and electronic energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has a C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 a.u which corresponds to the appendix. This structure was then optimised twice, separately, once with DFT 6-31G and another time at Hartree-Fock 3-21G level.

The table below shows the structure of the antiperiplanar 1,5-hexadiene antiperiplanar of Ci symmetry, optimised at HF/3-21G level (left) and optimised at DFT 6-31G level (right). The diagrams in the table show that the arrangement of the hydrogen atoms on the middle two carbons differ between the structures optimised at different levels. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies were found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they resemble the chair structure, and the distance between the carbons involved in bond formation were set to 2.2 Å. To get the chair transition state, the first the force constant matrix was computed and the structure constructed was optimised to a transition state (optimising to a TS (Berny)). This was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. The animation of this vibration also corresponds to the Cope rearrangement, giving confidence that this is the transition state we are looking for. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, opt+freq QST2 TS optimisation at HF/3-21G level was employed. This method attemptsto find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki> This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), both the geometries (point groups) and energies differ before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*).
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
(Point group = C1)
|Boat TS
(Point group = CS)
|Reactant (''anti''2)
(Point group = C1)
|Chair TS
(Point group = C2h)
|Boat TS
(Point group = CS)
|Reactant (''anti''2)
(Point group = C2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5 kcal/mol
|44.7 ± 2.0 kcal/mol
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the 'boat' transition state is higher in energy (less stable) and hence the chair transition state is favoured.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. Typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>, while the ''van der'' ''Waals'' radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are larger than typical sp3 bondlengths, implying the approach of the two components in the transition state the two components as the bond forms. This distance is also shorter than twice the van der Waals radius of a C atom, which reflects the interaction between the two carbons towards bond formation.

For this obtained transition state, two vibrations, the one with the imaginary frequency and the one with the lowest positive frequency, were visualised. These are shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle.
|}
This cycloaddition reaction involves 6π-electrons. Systems with [4n+2]π electrons are thermally allowed only if the symmetry of both substrates involved in the cycloaddition are the same (symmetric or antisymmetric). Based on this, since the HOMO of the transition state is antisymmetric with respect to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene (both antisymmetric).

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimisation using the Frozen Coordinate Method. (All calculations were done using the AM1 semi-empircal molecular orbital method.) The table below gives the information for the two transition states found.
{| class="wikitable"
!
!
!Endo Transition State
!Exo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartrees, is 2.85 kJ mol-1.The ''exo'' transition state is higher in energy than the ''endo'' transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured, and are shown in the table below:
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|2.94
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|2.89
|}
Without considering electronic effects, based on the structure of the ''exo'' and ''endo'' products, this distance is expected to be greater in the ''endo'' transition state (see diagram below).

[[File:Exoendostructures.png|500px]]

However, the measured distances prove otherwise, suggesting that there is a greater extent of favourable interaction between the two substrates (maleic anhydride and hexadiene) in the ''endo'' transition state. This explanation is supported by the lower energy of the ''endo ''transition as calculated above (which corresponds to a lower activation energy for the cycloaddition reaction ''via ''this transition state) compared to the ''exo'' transition state. The secondary orbital interaction between the pi orbitals of the maleic anhydride and cyclohexadiene substrates that is only possible in the ''endo'' transition state accounts for this.

== References ==

Rep:Mod:96499348

2015-11-05T17:32:57Z

Yo113: /* Optimising the Boat Transition State */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5 kcal/mol
|44.7 ± 2.0 kcal/mol
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are clearly larger than typical sp3 bondlengths, clearly in the transition state the two components are approaching each other and the bond is not fully formed yet. However, this distance is shorter than twice the van der Waals radius of a C atom, which reflects that the two carbons are interacting and the bond is forming.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}
This cycloaddition reaction involves 6π-electrons. Systems with [4n+2]π electrons are thermally allowed only if the symmetry of both substrates involved in the cycloaddition are the same (symmetric or antisymmetric). Based on this, since the HOMO of the transition state is antisymmetric with respect to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene (both antisymmetric).

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Endo Transition State
!Exo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartrees, is 2.85 kJ mol-1.The ''exo'' transition state is higher in energy than the ''endo'' transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|2.94
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|2.89
|}
Without considering electronic effects, based on the structure of the ''exo'' and ''endo'' products, this distance is expected to be greater in the ''endo'' transition state (see diagram below).

[[File:Exoendostructures.png|500px]]

However, the measured distances prove otherwise, suggesting that there is a greater extent of favourable interaction between the two substrates (maleic anhydride and hexadiene) in the ''endo'' transition state. This explanation is supported by the lower energy of the ''endo ''transition as calculated above (which corresponds to a lower activation energy for the cycloaddition reaction ''via ''this transition state) compared to the ''exo'' transition state. The secondary orbital interaction between the pi orbitals of the maleic anhydride and cyclohexadiene substrates that is only possible in the ''endo'' transition state accounts for this.

== References ==

Rep:Mod:96499348

2015-11-05T17:31:31Z

Yo113: /* Transition state of the Diels-Alder Cycloaddition Reaction between cis-butadiene and ethene */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5
|44.7 ± 2.0
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are clearly larger than typical sp3 bondlengths, clearly in the transition state the two components are approaching each other and the bond is not fully formed yet. However, this distance is shorter than twice the van der Waals radius of a C atom, which reflects that the two carbons are interacting and the bond is forming.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}
This cycloaddition reaction involves 6π-electrons. Systems with [4n+2]π electrons are thermally allowed only if the symmetry of both substrates involved in the cycloaddition are the same (symmetric or antisymmetric). Based on this, since the HOMO of the transition state is antisymmetric with respect to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene (both antisymmetric).

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Endo Transition State
!Exo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartrees, is 2.85 kJ mol-1.The ''exo'' transition state is higher in energy than the ''endo'' transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|2.94
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|2.89
|}
Without considering electronic effects, based on the structure of the ''exo'' and ''endo'' products, this distance is expected to be greater in the ''endo'' transition state (see diagram below).

[[File:Exoendostructures.png|500px]]

However, the measured distances prove otherwise, suggesting that there is a greater extent of favourable interaction between the two substrates (maleic anhydride and hexadiene) in the ''endo'' transition state. This explanation is supported by the lower energy of the ''endo ''transition as calculated above (which corresponds to a lower activation energy for the cycloaddition reaction ''via ''this transition state) compared to the ''exo'' transition state. The secondary orbital interaction between the pi orbitals of the maleic anhydride and cyclohexadiene substrates that is only possible in the ''endo'' transition state accounts for this.

== References ==

Rep:Mod:96499348

2015-11-05T17:17:42Z

Yo113: /* Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5
|44.7 ± 2.0
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are clearly larger than typical sp3 bondlengths, clearly in the transition state the two components are approaching each other and the bond is not fully formed yet. However, this distance is shorter than twice the van der Waals radius of a C atom, which reflects that the two carbons are interacting and the bond is forming.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}
Since the HOMO is antisymmetric to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene.

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Endo Transition State
!Exo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartrees, is 2.85 kJ mol-1.The ''exo'' transition state is higher in energy than the ''endo'' transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|2.94
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|2.89
|}
Without considering electronic effects, based on the structure of the ''exo'' and ''endo'' products, this distance is expected to be greater in the ''endo'' transition state (see diagram below).

[[File:Exoendostructures.png|500px]]

However, the measured distances prove otherwise, suggesting that there is a greater extent of favourable interaction between the two substrates (maleic anhydride and hexadiene) in the ''endo'' transition state. This explanation is supported by the lower energy of the ''endo ''transition as calculated above (which corresponds to a lower activation energy for the cycloaddition reaction ''via ''this transition state) compared to the ''exo'' transition state. The secondary orbital interaction between the pi orbitals of the maleic anhydride and cyclohexadiene substrates that is only possible in the ''endo'' transition state accounts for this.

== References ==

File:Exoendostructures.png

2015-11-05T17:17:13Z

Yo113:

Rep:Mod:96499348

2015-11-05T17:13:27Z

Yo113: /* Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5
|44.7 ± 2.0
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are clearly larger than typical sp3 bondlengths, clearly in the transition state the two components are approaching each other and the bond is not fully formed yet. However, this distance is shorter than twice the van der Waals radius of a C atom, which reflects that the two carbons are interacting and the bond is forming.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}
Since the HOMO is antisymmetric to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene.

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Endo Transition State
!Exo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartrees, is 2.85 kJ mol-1.The ''exo'' transition state is higher in energy than the ''endo'' transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|2.94
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|2.89
|}
Without considering electronic effects, based on the structure of the ''exo'' and ''endo'' products, this distance is expected to be greater in the ''endo'' transition state (see diagram below).

[[File:Endo and exo.png|500px]]

However, the measured distances prove otherwise, suggesting that there is a greater extent of favourable interaction between the two substrates (maleic anhydride and hexadiene) in the ''endo'' transition state. This explanation is supported by the lower energy of the ''endo ''transition as calculated above (which corresponds to a lower activation energy for the cycloaddition reaction ''via ''this transition state) compared to the ''exo'' transition state. The secondary orbital interaction between the pi orbitals of the maleic anhydride and cyclohexadiene substrates that is only possible in the ''endo'' transition state accounts for this.

== References ==

Rep:Mod:96499348

2015-11-05T17:12:16Z

Yo113: /* Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5
|44.7 ± 2.0
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are clearly larger than typical sp3 bondlengths, clearly in the transition state the two components are approaching each other and the bond is not fully formed yet. However, this distance is shorter than twice the van der Waals radius of a C atom, which reflects that the two carbons are interacting and the bond is forming.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}
Since the HOMO is antisymmetric to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene.

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Endo Transition State
!Exo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartrees, is 2.85 kJ mol-1.The ''exo'' transition state is higher in energy than the ''endo'' transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|2.94
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|2.89
|}
Without considering electronic effects, based on the structure of the ''exo'' and ''endo'' products, this distance is expected to be greater in the ''endo'' transition state (see diagram below). <gallery>
[[File:Endo and exo.png|500px]]

However, the measured distances prove otherwise, suggesting that there is a greater extent of favourable interaction between the two substrates (maleic anhydride and hexadiene) in the ''endo'' transition state. This explanation is supported by the lower energy of the ''endo ''transition as calculated above (which corresponds to a lower activation energy for the cycloaddition reaction ''via ''this transition state) compared to the ''exo'' transition state. The secondary orbital interaction between the pi orbitals of the maleic anhydride and cyclohexadiene substrates that is only possible in the ''endo'' transition state accounts for this.

== References ==

Rep:Mod:96499348

2015-11-05T17:11:37Z

Yo113: /* Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5
|44.7 ± 2.0
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are clearly larger than typical sp3 bondlengths, clearly in the transition state the two components are approaching each other and the bond is not fully formed yet. However, this distance is shorter than twice the van der Waals radius of a C atom, which reflects that the two carbons are interacting and the bond is forming.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}
Since the HOMO is antisymmetric to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene.

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Endo Transition State
!Exo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartrees, is 2.85 kJ mol-1.The ''exo'' transition state is higher in energy than the ''endo'' transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|2.94
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|2.89
|}
Without considering electronic effects, based on the structure of the ''exo'' and ''endo'' products, this distance is expected to be greater in the ''endo'' transition state (see diagram below). <gallery>
File:Endo and exo.png
</gallery>

However, the measured distances prove otherwise, suggesting that there is a greater extent of favourable interaction between the two substrates (maleic anhydride and hexadiene) in the ''endo'' transition state. This explanation is supported by the lower energy of the ''endo ''transition as calculated above (which corresponds to a lower activation energy for the cycloaddition reaction ''via ''this transition state) compared to the ''exo'' transition state. The secondary orbital interaction between the pi orbitals of the maleic anhydride and cyclohexadiene substrates that is only possible in the ''endo'' transition state accounts for this.

== References ==

Rep:Mod:96499348

2015-11-05T17:09:50Z

Yo113: /* Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5
|44.7 ± 2.0
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are clearly larger than typical sp3 bondlengths, clearly in the transition state the two components are approaching each other and the bond is not fully formed yet. However, this distance is shorter than twice the van der Waals radius of a C atom, which reflects that the two carbons are interacting and the bond is forming.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}
Since the HOMO is antisymmetric to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene.

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Endo Transition State
!Exo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1.The ''exo'' transition state is higher in energy than the ''endo'' transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|2.94
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|2.89
|}
Without considering electronic effects, based on the structure of the ''exo'' and ''endo'' products, this distance is expected to be greater in the ''endo'' transition state (see diagram below). <gallery>
File:Endo and exo.png
</gallery>

However, the measured distances prove otherwise, suggesting that there is a greater extent of favourable interaction between the two substrates (maleic anhydride and hexadiene) in the ''endo'' transition state. This explanation is supported by the lower energy of the ''endo ''transition as calculated above (which corresponds to a lower activation energy for the cycloaddition reaction ''via ''this transition state) compared to the ''exo'' transition state. The secondary orbital interaction between the pi orbitals of the maleic anhydride and cyclohexadiene substrates that is only possible in the ''endo'' transition state accounts for this.

== References ==

Rep:Mod:96499348

2015-11-05T16:52:01Z

Yo113: /* Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5
|44.7 ± 2.0
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are clearly larger than typical sp3 bondlengths, clearly in the transition state the two components are approaching each other and the bond is not fully formed yet. However, this distance is shorter than twice the van der Waals radius of a C atom, which reflects that the two carbons are interacting and the bond is forming.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}
Since the HOMO is antisymmetric to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene.

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Endo Transition State
!Exo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1.The ''exo'' transition state is higher in energy than the exo transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|2.94
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|2.89
|}
Without considering electronic effects, one would expect greater steric hinderence in the ''exo'' transition state and hence a greater through space distance between the two parts of the molecule on the same face. (See diagram below)<gallery>
File:Endo and exo.png
</gallery>

However, this distance measured for the ''endo'' transition state appears to be shorter. This can be explained by the secondary orbital interaction between the carbonyl groups in the maleic anhydride substrate and the

== References ==

File:Endo and exo.png

2015-11-05T16:31:33Z

Yo113:

Rep:Mod:96499348

2015-11-05T16:31:09Z

Yo113: /* Optimising the Chair Transition State */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|33.5 ± 0.5
|44.7 ± 2.0
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|N/A
|N/A
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

Comparing the activation energies computed to the experimental activation energies for both the chair and boat transition states, it is clear that the values computed from the DFT - B3LYP/6-31G* level is much closer to the experimental values and hence more accurate.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are clearly larger than typical sp3 bondlengths, clearly in the transition state the two components are approaching each other and the bond is not fully formed yet. However, this distance is shorter than twice the van der Waals radius of a C atom, which reflects that the two carbons are interacting and the bond is forming.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}
Since the HOMO is antisymmetric to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene.

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Exo Transition State
!Endo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1.The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

To compare the sterics of both transition states, the through-space distance between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the nearest carbon on the same face of the molecule were measured
{| class="wikitable"
!Measurement
!Distance (Å)
|-
|EXO TS:
Through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2-
|3.90
|-
|ENDO TS:
through-space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH-
|3.78
|}
Without considering electronic effects, one would expect greater steric hinderence in the ''exo'' transition state and hence a greater through space distance between the two parts of the molecule on the same face. (See diagram below)

However, this distance measured for the ''endo'' transition state appears to be shorter. This can be explained by the secondary orbital interaction between the carbonyl groups in the maleic anhydride substrate and the

== References ==

Rep:Mod:96499348

2015-11-05T15:36:37Z

Yo113: /* Transition state of the Diels-Alder Cycloaddition Reaction between cis-butadiene and ethene */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|<nowiki>-231.466700</nowiki>
|<nowiki>-231.450931</nowiki>
|<nowiki>-231.539601</nowiki>
|<nowiki>-234.414910</nowiki>
|<nowiki>-234.402352</nowiki>
|<nowiki>-234.469309</nowiki>
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|0.072901 Hartrees

= 45.7 kcal/mol
|0.08867 Hartrees

= 55.6 kcal/mol
|N/A
|0.054399 Hartrees

= 34.1 kcal/mol
|0.066957 Hartrees

= 42.0 kcal/mol
|N/A
|
|
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|<nowiki>-231.461340</nowiki>
|<nowiki>-231.445302</nowiki>
|<nowiki>-231.532645</nowiki>
|<nowiki>-234.408982</nowiki>
|<nowiki>-234.396009</nowiki>
|<nowiki>-234.461976</nowiki>
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|<nowiki>-231.460396</nowiki>
|<nowiki>-231.444358</nowiki>
|<nowiki>-231.531701</nowiki>
|<nowiki>-234.408038</nowiki>
|<nowiki>-234.395065</nowiki>
|<nowiki>-234.461032</nowiki>
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|0.071305 Hartrees

= 44.7 kcal/mol
|0.087343 Hartrees

= 54.8 kcal/mol
|N/A
|0.052994 Hartrees

= 33.3 kcal/mol
|0.065967 Hartrees

= 41.4 kcal/mol
|N/A
|
|
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are clearly larger than typical sp3 bondlengths, clearly in the transition state the two components are approaching each other and the bond is not fully formed yet. However, this distance is shorter than twice the van der Waals radius of a C atom, which reflects that the two carbons are interacting and the bond is forming.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}
Since the HOMO is antisymmetric to the plane of reflection, the MOs involved in bond formation in this reaction should also be antisymmetric. This would correspond to the HOMO of cis-butadiene and the LUMO of ethene.

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Exo Transition State
!Endo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1. The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

== References ==

Rep:Mod:96499348

2015-11-05T14:45:25Z

Yo113: /* Transition state of the Diels-Alder Cycloaddition Reaction between cis-butadiene and ethene */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|
|
|N/A
|
|
|N/A
|
|
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|
|
|N/A
|
|
|N/A
|
|
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the computed transition state with a point group of Cs:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are clearly larger than typical sp3 bondlengths, clearly in the transition state the two components are approaching each other and the bond is not fully formed yet. However, this distance is shorter than twice the van der Waals radius of a C atom, which reflects that the two carbons are interacting and the bond is forming.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Exo Transition State
!Endo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1. The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

== References ==

Rep:Mod:96499348

2015-11-05T14:26:52Z

Yo113: /* Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|
|
|N/A
|
|
|N/A
|
|
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|
|
|N/A
|
|
|N/A
|
|
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the geometry of the computed transition state:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>. The distance between the carbons of the partially formed C-C sigma bonds are clearly larger than typical sp3 bondlengths, clearly in the transition state the two components are approaching each other and the bond is not fully formed yet. However, this distance is shorter than twice the van der Waals radius of a C atom, which reflects that the two carbons are interacting and the bond is forming.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Exo Transition State
!Endo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1. The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

== References ==

Rep:Mod:96499348

2015-11-04T21:35:31Z

Yo113: /* Optimising the Chair Transition State */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!HF/3-21G
!DFT/6-31G
|-
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The transition state from both optimisations look similar but have different point groups (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]Point Group = Ci

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]Point Group = C2

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|
|
|N/A
|
|
|N/A
|
|
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|
|
|N/A
|
|
|N/A
|
|
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the geometry of the computed transition state:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Exo Transition State
!Endo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1. The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

== References ==

Rep:Mod:96499348

2015-11-04T13:05:49Z

Yo113: /* Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!Level of Optimisation
!HF/3-21G
!DFT/6-31G
|-
|Geometry of optimised structures
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The geometry of the transition state from both optimisations appear to be the same (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|
|
|N/A
|
|
|N/A
|
|
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|
|
|N/A
|
|
|N/A
|
|
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the geometry of the computed transition state:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is antisymmetric with respect to the plane cutting through the middle.

|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]

This MO is symmetric with respect to the plane cutting through the middle
|}

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Exo Transition State
!Endo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1. The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

== References ==

Rep:Mod:96499348

2015-11-04T12:39:04Z

Yo113: /* Transition state of the Diels-Alder Cycloaddition Reactoin between cis-butadiene and ethene */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!Level of Optimisation
!HF/3-21G
!DFT/6-31G
|-
|Geometry of optimised structures
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The geometry of the transition state from both optimisations appear to be the same (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|
|
|N/A
|
|
|N/A
|
|
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|
|
|N/A
|
|
|N/A
|
|
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the geometry of the computed transition state:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is antisymmetric with respect to the plane cutting through the middle
|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is symmetric with respect to the plane cutting through the middle
|}

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Exo Transition State
!Endo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1. The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

== References ==

Rep:Mod:96499348

2015-11-04T12:37:00Z

Yo113:

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!Level of Optimisation
!HF/3-21G
!DFT/6-31G
|-
|Geometry of optimised structures
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The geometry of the transition state from both optimisations appear to be the same (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|
|
|N/A
|
|
|N/A
|
|
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|
|
|N/A
|
|
|N/A
|
|
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reactoin between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the geometry of the computed transition state:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is antisymmetric with respect to the plane cutting through the middle
|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is symmetric with respect to the plane cutting through the middle
|}

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!
!Exo Transition State
!Endo Transition State
|-
|
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|
|Geometry
|CS
|CS
|-
| rowspan="4" |Internuclear distances (Å)
|C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
|2.16
|2.17
|-
|π-bond to be formed (in the cyclohexadiene substrate)
|1.39
|1.40
|-
|π-bond to be broken (in maleic anhydride substrate)
|1.41
|1.41
|-
|π-bond to be broken (in cyclohexadiene substrate)
|1.39
|1.40
|-
|
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1. The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

== References ==

Rep:Mod:96499348

2015-11-04T12:28:41Z

Yo113: /* Optimising the Boat Transition State */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!Level of Optimisation
!HF/3-21G
!DFT/6-31G
|-
|Geometry of optimised structures
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The geometry of the transition state from both optimisations appear to be the same (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|
|
|N/A
|
|
|N/A
|
|
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|
|
|N/A
|
|
|N/A
|
|
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reactoin between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the geometry of the computed transition state:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is antisymmetric with respect to the plane cutting through the middle
|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is symmetric with respect to the plane cutting through the middle
|}

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!Exo Transition State
!Endo Transition State
|-
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|Geometry
|CS
|CS
|-
|Internuclear distances (Å):
* C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
* π-bond to be formed (in the cyclohexadiene substrate)
* π-bond to be broken (in maleic anhydride substrate)
* π-bond to be broken (in cyclohexadiene substrate)
|
* 2.16
* 1.39
* 1.41
* 1.39
|
* 2.17
* 1.40
* 1.41
* 1.40
|-
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1. The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

== References ==

Rep:Mod:96499348

2015-11-04T12:26:36Z

Yo113: /* Optimising the Boat Transition State */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!Level of Optimisation
!HF/3-21G
!DFT/6-31G
|-
|Geometry of optimised structures
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The geometry of the transition state from both optimisations appear to be the same (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|
|
|N/A
|
|
|N/A
|
|
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|
|
|N/A
|
|
|N/A
|
|
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reactoin between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the geometry of the computed transition state:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is antisymmetric with respect to the plane cutting through the middle
|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is symmetric with respect to the plane cutting through the middle
|}

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!Exo Transition State
!Endo Transition State
|-
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|Geometry
|CS
|CS
|-
|Internuclear distances (Å):
* C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
* π-bond to be formed (in the cyclohexadiene substrate)
* π-bond to be broken (in maleic anhydride substrate)
* π-bond to be broken (in cyclohexadiene substrate)
|
* 2.16
* 1.39
* 1.41
* 1.39
|
* 2.17
* 1.40
* 1.41
* 1.40
|-
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1. The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

== References ==

Rep:Mod:96499348

2015-11-04T12:24:20Z

Yo113: /* Optimising the Boat Transition State */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!Level of Optimisation
!HF/3-21G
!DFT/6-31G
|-
|Geometry of optimised structures
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The geometry of the transition state from both optimisations appear to be the same (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
|+Computed Energies for the Reactant and Transition States for the Cope Rearrangement Reaction
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|
|
|N/A
|
|
|N/A
|
|
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|
|
|N/A
|
|
|N/A
|
|
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reactoin between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the geometry of the computed transition state:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is antisymmetric with respect to the plane cutting through the middle
|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is symmetric with respect to the plane cutting through the middle
|}

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!Exo Transition State
!Endo Transition State
|-
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|Geometry
|CS
|CS
|-
|Internuclear distances (Å):
* C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
* π-bond to be formed (in the cyclohexadiene substrate)
* π-bond to be broken (in maleic anhydride substrate)
* π-bond to be broken (in cyclohexadiene substrate)
|
* 2.16
* 1.39
* 1.41
* 1.39
|
* 2.17
* 1.40
* 1.41
* 1.40
|-
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1. The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

== References ==

Rep:Mod:96499348

2015-11-04T12:22:28Z

Yo113: /* Optimising the Boat Transition State */

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!Level of Optimisation
!HF/3-21G
!DFT/6-31G
|-
|Geometry of optimised structures
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The geometry of the transition state from both optimisations appear to be the same (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

<nowiki> </nowiki>This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|
|
|N/A
|
|
|N/A
|
|
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|
|
|N/A
|
|
|N/A
|
|
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reactoin between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the geometry of the computed transition state:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is antisymmetric with respect to the plane cutting through the middle
|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is symmetric with respect to the plane cutting through the middle
|}

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!Exo Transition State
!Endo Transition State
|-
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|Geometry
|CS
|CS
|-
|Internuclear distances (Å):
* C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
* π-bond to be formed (in the cyclohexadiene substrate)
* π-bond to be broken (in maleic anhydride substrate)
* π-bond to be broken (in cyclohexadiene substrate)
|
* 2.16
* 1.39
* 1.41
* 1.39
|
* 2.17
* 1.40
* 1.41
* 1.40
|-
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1. The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

== References ==

Rep:Mod:96499348

2015-11-04T12:19:58Z

Yo113:

= Transition Structures =

=== Tutorial:The Cope Rearrangement ===

==== Part A: Optimising the reactants and products in an example with 1,5-hexadiene ====
Using the HF/3-21G level of theory, the antiperiplanar conformation of 1,5-hexadiene was computed to have C2 symmetry and an energy of -231.69260235 a.u. This is expected to be lower than that of the gauche conformer that is subjected to higher steric clash between the alkyl groups (at 60 degrees dihedral angle) - this steric clash increases torsional strain and increases the energy of the gauche conformer. Still using HF/3-21G, the energy of the gauche conformer was calculated to be The calculated energy of the gauche conformer -231.69166702 a.u, which is higher than that of the antiperiplanar conformer, as we would expect. This structure also has the C2 point group.

The antiperiplanar conformer made corresponds to the ''anti''1 structure in the table in Appendix 1 of the [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 lab manual]. They both have C2 point groups and exactly the same energy of -231.69260 a.u. This, according to the appendix, is also the lowest energy ''anti conformation. ''For the gauche conformer, the molecule made corresponds to ''gauche''2 in the same table, with matching energies of -231.69167 a.u. and point group C2.

The construction of the Ci ''anti''2 conformer of 1,5-hexadiene was then attempted. The resulting conformer's final energy was -231.69254 which corresponds to the table. This structure was then optimised twice, separately, once with DFT 6-31G and another time with Hartree-Fock 3-21G.

The table below shows the geometries of the 1,5-hexadiene antiperiplanar structure with Ci symmetry, optimised at HF/3-21G level (left) and optimised with DFT 6-31G (right). The figure below shows a different arrangement of the hydrogen atoms on the middle two carbons. In terms of the geometry, the point group of the re-optimised (DFT/6-31G) structure is closer to a C2h structure than the Ci structure (which is the case for the structure optimised at HF/3-21G).
{| class="wikitable"
!Level of Optimisation
!HF/3-21G
!DFT/6-31G
|-
|Geometry of optimised structures
|
[[File:APPCi-unoptimised.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at HF/3-21G|200px]]

|
[[File:APPCi-optimised-DFT-6-31G.png|1,5-hexadiene antiperiplanar structure with Ci symmetry - optimised at DFT/6-31G level|200px]]

|}

After a frequency calculation at DFT/6-31G, all vibrational frequencies are found to be real and positive. Results showed that:
# the sum electronic and zero-point energies computed = 234.416255 a.u. (This is the potential energy of the molecule at 0 K plus the zero-point vibrational energy.)
# the sum of electronic and thermal energies = 234.408963 a.u (This is the potential energy of the molecule at 298.15 K and 1 atm - this energy comes from the translational, rotational and vibrational energy modes under these conditions.)
# the sum of electronic and thermal enthalpies = 234.408019 a.u (this contains an additional correction for RT (H = E + RT) - crucial component when investigating dissociation reactions; also includes entropy in free energy calculations)
# the sum of electronic and thermal free energies = 234.447872 a.u

==== Part B: Optimizing the "Chair" and "Boat" Transition Structures ====
To understand the Cope reaction, the 'Chair' and 'Boat' transition states were investigated.

====== Optimising the Chair Transition State ======
A guess structure for the chair transition state was constructed by orienting two allyl fragments such that they look like the transition state, and the distance between the carbons involved in bond formation was set to 2.2 Å. To get the chair transition state, the first method used involved the computation of the force constant matrix and optimising the guess chair structure to a transition state (optimising to a TS (Berny)) - this was done using Hartree Fock calculation at the 3-21G level. The resulting structure had one imaginary vibration frequency of magnitude 818 cm-1, which confirms that the structure is indeed a transition state. Furthermore, the animation of this vibration also corresponds to the Cope rearrangement. This structure was then further optimised using the frozen coordinate method. After this second optimisation, the bond forming/bond breaking bond lengths are 2.0 Å, which is slightly shorter than that of the structure after the first optimisation (bond forming/bond breaking bond lengths = 2.2 Å). The geometry of the transition state from both optimisations appear to be the same (see table below).
{| class="wikitable"
! colspan="2" |Level of Optimisation
|-
!HTF/3-21G
!Frozen Coordinate Method
|-
|[[File:ChairTS321gopt.png|centre|200x200px]]

|[[File:ChairTSoptmodredundant.png|centre|200x200px]]

|}

====== Optimising the Boat Transition State ======
To find the boat transition state, optimisation was done on 1,5-hexadiene using the opt+freq QST2 TS optimisation at method HF/3-21G level, an attempt was made to find the transition state from the reactant and product.This was unsuccessful because bond rotation was not automatically considered in the computation - the figure below shows two images of the erroneous transition structures computed.

{| class="wikitable"
!Top View
!Side View
|-
|[[File:Qst2topview.png|centre|121x121px]]

|[[File:SideviewoferroneousTS.png|centre|150x150px]]

|}
As a result, the optimisation was repeated, but with the dihedral angle for the central 4 carbon atoms set to 0o and the two innermost angles in the molecule set to 100o for both the reactant and product. The resulting structure is shown below:
[[File:BoatTSyo.png|left|250x250px]]

This structure has only one imaginary frequency at -840.04 cm-1, which confirms that it is indeed a transition state.

The Intrinsic Reaction Coordinate (IRC) method is then used to find the minimum energy path from a transition state to its local minimum on a potential energy surface.

In order to find the activation energy (to be compared with experimental data), the boat and chair transition structures are reoptimised at a higher level (DFT - B3LYP/6-31G*) for greater accuracy, and frequency calculations were also run. For each transition state (chair and boat), the geometries of the structures were similar before and after higher level optimisation (i.e HF/3-21G vs DFT - B3LYP/6-31G*), however differences were apparent in the energies of the computed structures.
{| class="wikitable"
!
!
! colspan="3" |HF/3-21G
! colspan="3" |DFT - B3LYP/6-31G*
! colspan="2" |Experimental
|-
|
|Structure
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|Reactant (''anti''2)
|Chair TS
|Boat TS
|-
|
|Electronic Energy (a.u)
|-231.61932247
|-231.60280241
|-231.69260236
|-234.55693103
|-234.54307800
|-234.61179137
|N/A
|N/A
|-
| rowspan="2" |At 0 K
|Electronic Energy + Zero-Point Energy (a.u)
(i.e Potential Energy at 0 K)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Potential Energy of TS at 0 K - Potential Energy of Reactant (''anti''2) at 0 K)
(i.e the activation energy via the chair or boat transition state of this reaction at 0 K)
|
|
|N/A
|
|
|N/A
|
|
|-
| rowspan="3" |At 298.15 K
|Sum of Electronic and Thermal Energies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|Sum of Electronic and Thermal Enthalpies (a.u)
|
|
|
|
|
|
|N/A
|N/A
|-
|ΔE (Sum of electronic and thermal enthalpies of TS - sum of electronic and thermal enthalpies of reactant, at 298.15 K)
(i.e Activation energy of the reaction ''via ''either the chair or boat transition state, at 298.15 K)
|
|
|N/A
|
|
|N/A
|
|
|}
These energies show that the Boat Transition State is higher in energy (less stable) and hence the chair transition state is favoured at lower temperatures where there is lower thermal energy.

== Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) ==

=== Diels-Alder Cycloaddition Reaction between Cis-butadiene and ethene ===

==== Cis-butadene ====
After constructing ''cis-''butadiene on Gaussview, the structure was optimised using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the molecule were then visualised (see below).
{| class="wikitable"
|+Visualisations of the HOMO and LUMO of ''cis''-butadiene
!HOMO
!LUMO
|-
|[[File:CisbutadieneHOMO.jpg|centre|357x357px]]

The HOMO is antisymmetric with

respect to the plane through the middle.
|[[File:CisbutadieneLUMOyo.jpg|centre|357x357px]]

The LUMO is symmetric with respect

to the plane through the middle.
|}

==== Transition state of the Diels-Alder Cycloaddition Reactoin between ''cis-''butadiene and ethene ====
A guess for transition state of the Diels-Alder Cycloaddition Reaction between ''cis-''butadiene and ethene was built and optimised using the AM1 semi-empirical molecular orbital method (opt+freq). The result from this computation was a structure that has just one imaginary frequency - this corresponds a transition state, which is what we are looking for. The diagram below shows the geometry of the computed transition state:
[[File:GeometryofbutadienetheneTS.jpg|centre|550x550px]]

For each (of the two) partially formed C-C sigma bonds in this transition state. the 'bond' length is 2.12 Å. The typical bond lengths for sp3 and sp2 C-C bonds are 1.54 Å and 1.34 Å respectively<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The van der Waals radius of a C atom is 1.85 Å<ref><nowiki>http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Covalent_Bond_Distance,_Radius_and_van_der_Waals_Radius</nowiki></ref>.

Two vibrations, the one with the imaginary frequency and one with the lowest positive frequency, were visualised, as shown below.
{| class="wikitable"
!Animation corresponding to imaginary frequency
!Animation corresponding to lowest positive frequency
|-
|[[File:TSimaginaryfreqanimation.gif|centre|600x600px]]

This vibration shows that the formation of both C-C bonds are synchronous.
|[[File:TSlowestpositivefreqanimation.gif|centre|600x600px]]

<nowiki> </nowiki>This vibration does not favour bond formation between the ''cis''-butadiene and ethene fragment,

hence is neither synchronous nor asynchronous
|}

The HOMO and LUMO of this optimised transition state structure were also visualised. They are shown in the table below.
{| class="wikitable"
!HOMO
!LUMO
|-
|
[[File:HOMO-butadienetheneTS.jpg|HOMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is antisymmetric with respect to the plane cutting through the middle
|
[[File:LUMO-butadienetheneTS.jpg|LUMO of the transition state for the Diels-Alder reaction between cis-butadiene and ethene|200px]]This MO is symmetric with respect to the plane cutting through the middle
|}

=== Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride ===
The exo and endo transition states for the Diels-Alder Cycloaddition Reaction involving cyclo-1,3-hexadiene and maleic anhydride were found by optimising cyclo-1,3-hexadiene and maleic anhydride separately and then orienting them to look like the exo and endo transition states, followed by final optimising using the Frozen Coordinate Method (all calculated using the AM1 semi-empircal molecular orbital method). The table below gives the information obtained for the two transition states found.
{| class="wikitable"
!
!Exo Transition State
!Endo Transition State
|-
|Structure
|
[[File:ExoTSstructure.jpg|500px]]

|
[[File:EndoTSstructure.jpg|500px]]

|-
|Geometry
|CS
|CS
|-
|Internuclear distances (Å):
* C-C σ bond to be formed (between maleic anhydride substrate and cyclohexadiene substrate)
* π-bond to be formed (in the cyclohexadiene substrate)
* π-bond to be broken (in maleic anhydride substrate)
* π-bond to be broken (in cyclohexadiene substrate)
|
* 2.16
* 1.39
* 1.41
* 1.39
|
* 2.17
* 1.40
* 1.41
* 1.40
|-
|Magnitude of imaginary frequency (cm-1)
|806.39
|812.45
|-
|Energy (a.u)
|<nowiki>-0.05150480</nowiki>
|<nowiki>-0.05041968 </nowiki>
|-
|HOMO
|[[File:ExoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|[[File:EndoTSHOMO.jpg|centre|446x446px]]The HOMO is antisymmetric with respect to the plane through the middle of the structure.

|}
The difference in energy of the two transition states, converted from Hartree units, is 2.85 kJ mol-1. The endo transition state is higher in energy than the exo transition state, as expected due to the favourable secondary orbital interaction available in the exo transition state but not the endo transition state.

== References ==

Rep:Mod:96499348

2015-11-04T10:41:50Z

Yo113:

Rep:Mod:96499348

2015-11-03T18:45:08Z

Yo113: /* Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride */

Rep:Mod:96499348

2015-11-03T18:38:52Z

Yo113: /* Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride */

Rep:Mod:96499348

2015-11-03T18:37:11Z

Yo113: /* Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride */

Rep:Mod:96499348

2015-11-03T18:36:23Z

Yo113:

Rep:Mod:96499348

2015-11-03T18:33:16Z

Yo113:

File:EndoTSHOMO.jpg

2015-11-03T18:17:34Z

Yo113: Yo113 uploaded a new version of File:EndoTSHOMO.jpg

File:ExoTSHOMO.jpg

2015-11-03T18:17:04Z

Yo113: Yo113 uploaded a new version of File:ExoTSHOMO.jpg

File:EndoTSstructure.jpg

2015-11-03T18:14:28Z

Yo113:

File:ExoTSstructure.jpg

2015-11-03T18:13:41Z

Yo113:

Rep:Mod:96499348

2015-11-03T17:53:32Z

Yo113: /* Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) */

Rep:Mod:96499348

2015-11-03T17:33:41Z

Yo113: /* Building the TS and Visualising the MOs (HOMO and LUMO) */

Rep:Mod:96499348

2015-11-03T17:32:50Z

Yo113: /* Exercise: Diels-Alder Cycloaddition Reaction (using AM1 semi-empirical molecular orbital method) */

Rep:Mod:96499348

2015-11-03T17:31:50Z

Yo113: /* Building the TS and Visualising the MOs (HOMO and LUMO) */

File:TSlowestpositivefreqanimation.gif

2015-11-03T17:30:18Z

Yo113:

Rep:Mod:96499348

2015-11-03T17:25:06Z

Yo113: /* Part B: Optimizing the "Chair" and "Boat" Transition Structures */

Rep:Mod:96499348

2015-11-03T17:23:05Z

Yo113: /* Building the TS and Visualising the MOs (HOMO and LUMO) */

Rep:Mod:96499348

2015-11-03T17:21:12Z

Yo113:

Rep:Mod:96499348

2015-11-03T17:20:24Z

Yo113:

Rep:Mod:96499348

2015-11-03T17:18:42Z

Yo113: /* Diels-Alder Cycloaddition Reaction between Cyclo-1,3-hexadiene and Maleic Anhydride */

File:TSimaginaryfreqanimation.gif

2015-11-03T17:17:56Z

Yo113:

Rep:Mod:96499348

2015-11-03T17:12:24Z

Yo113:

Rep:Mod:96499348

2015-11-03T17:10:56Z

Yo113: