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Rep:Mod:PhysBE411

2014-03-21T16:36:39Z

Be411: /* Cis-butadiene */

'''Physical Module: Transition States and Reactivity'''

Computational methods can be used to study reaction pathways and more specifically transition states. In this experiment the Cope rearrangement reaction and Diels-Alder cycloaddition reaction are used as examples of this. Using various methods the lowest energy conformation of 1,5-hexadiene can be found as well as the conformation that exists as a reactant in the Cope rearrangement reaction. The two possible transition states of this reaction can be studied and their activation energies calculated to gain some understanding of the mechanism of this reaction. The Diels-Alder cycloaddition will also be studied. The origin of the symmetry allowed reaction between ethene and cis-butadiene will be visualized and the reaction between maleic anhydride and hexa-1,5-diene studied in terms of the regioselectivity of its two conformational different transition states.

=='''The Cope Rearrangement Tutorial'''==

The Cope rearrangement is a specific [3,3] sigmatropic rearrangement in which a sigma bond moves and the pi system is redistributed across the molecule. In this case the Cope rearrangement of hexa-1,5-diene is considered with regard to its conformations and the boat and chair forms of its reaction's transition state.

==='''Optimizing the Reactants and Products'''===

===='''Conformation of 1,5-hexadiene'''====
[[File:BE_1,5-HEXADIENE_CONFORTMATIONS.jpg|thumb|The Conformations of 1,5-Hexadiene]]

1,5-hexadiene is one of the simplest precursors in the Cope rearrangement. The three rotating carbon-carbon bonds within the center of the molecule cause there to be 10 energetically different conformations, when taking into account enantiomers and symmetry.

Within these conformations you would expect that where two carbons are in the eclipsed form (F-J) the conformer will be of higher energy than conformers with CH-eclipsed bonds (A-E)<ref name="Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols">B. W. Gung, Z. Zhu and R. A. Fouch, ''J. Am. Chem. Soc'', '''1995''', ''117'', 1783-1788.</ref>. This is demonstrated in the case of butane where the difference in energy between the CC eclipsed form and the CH eclipsed form is 1.5kcal/mol<ref name="Website - Butane Conformational Analysis">[http://research.cm.utexas.edu/nbauld/teach/butane.html]N. Bauld, '''"Butane Conformational Analysis"''', ''University of Texas'', Retreived 20 March 2014.</ref>. The energies of these forms can be optimized by computational methods in order to find the most suitable candidates for the reactants and products in the Cope Rearrangement reaction.

The table below shows the total energy for all energetically distinct conformers of the molecule, it shows that in the case of this molecule the anti conformer is generally of higher energy. Relating these structures to the literature<ref name="Instructional Wiki">[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3]M. Bearpark, Imperial College, Physical Module: Transition states and reactivity</ref>, the energies found are all in line with the energies expected, as is also the case with the point groups.

The lowest energy conformation is the Gauche Conformer D (-231.69266Hartrees) this maybe due to an attractive interaction between the pi orbital of the double bond and the neighboring proton, whose orbital would interact with that of the pi bond<ref name="An Analysis of the Conformers of 1,5-hexadiene">B. G. Rocque, J. M. Gonzales, H. F. Schaeffer III, ''Mol. Phys'', '''2002''', ''100'', 441-446.</ref>. This conjugation will stabilize the molecule making it lower in energy.
The highest energy conformer of the CH eclipsed forms is C (-231.69153Hartrees) this maybe explained by interactions between the two alkene protons which are 2.51A apart. This proximity will have a repulsive effect increasing the overall energy of the molecule.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimized Energies for the Anti and Gauche Conformations of 1,5-Hexadiene (Relating to the Instructional Wiki<ref name="Instructional Wiki"/>)
!
! align="center" | '''<jmolFile text="Conformer A - Gauche2">BE_A.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer B - Anti1 ">BE_B.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer C - Gauche4">BE_C.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer D - Gauche3">BE_D.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer E - Anti2">BE_E.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer F - Gauche6">BE_F.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer G - Anti4">BE_G.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer H - Gauche5">BE_H.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer I - Gauche1">BE_I.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer J - Anti3">BE_J.mol</jmolFile>'''
|-
|'''Total Energy/Hartrees'''
| -231.69167
| -231.69260
| -231.69153
| -231.69266
| -231.69254
| -231.68916
| -231.69097
| -231.68962
| -231.68772
| -231.68907
|-
|'''Point Group'''
|C2
|C2
|C2
|C1
|Ci
|C1
|C1
|C1
|C2
|C2h
|-
|'''Dihedral Angle (C2-C3-C4-C5)'''
|64.17
| -176.91
| -63.65
|67.70
| -179.996
|70.23
| -178.32
| -71.09
|75.76
| -179.98
|}

===='''Analysis of Conformer E'''====

Further Analysis of Conformer E was done at a higher basis set in order to get an even better approximation for its lowest energy conformation. An optimization calculation was run as before at the B3LYP/6-31G(d)level of theory.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimised Conformations of Conformer E using the HF/3-21G and B3LYP/6-31G(d) Basis Sets
!<jmol><jmolApplet><title>Anti 2 Conformation at HF level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_E.mol</uploadedFileContents> </jmolApplet></jmol>
!<jmol><jmolApplet><title>Anti 2 Conformation at B3LYP level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_DFT_optmisation.mol</uploadedFileContents> </jmolApplet></jmol>
|}

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Geometrical Features of Conformer E Optimized Under Different Basis Sets
! rowspan ="2" | '''Basis Set'''
! colspan ="3" | '''Bond Length'''
! colspan ="3" | '''Angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3
|C2-C3-C4
|C2-C3-C4-C5
|-
|HF/3-21G
|1.32
|1.51
|1.55
|179.996
|124.81
|111.35
|-
|B3LYP/6-31G(d)
|1.33
|1.50
|1.55
|180
|125.29
|112.68
|}

[[image:BE_IR_Spectrum_of_Anti2.jpg|thumb|200px|IR Spectrum of Conformer E]]

This shows that the geometries of the results of the two different approaches don't differ wildly, the bond lengths especially are very similar, however there is some deviation between their angles.

Following on from this a frequency calculation was run using the same higher level of theory (B3LYP/6-31G(d)). This calculation can be used to confirm that this conformation is of a minimum energy by the absence of an imaginary vibrational frequency. The infrared spectrum of the molecule can be generated from the molecular vibrations and compared to literature.

The thermochemical energies for this molecule can be found in the output file of the calculation:

Sum of electronic and zero-point Energies= -234.46921 Hartrees
Sum of electronic and thermal Energies= -234.46186 Hartrees
Sum of electronic and thermal Enthalpies= -234.46091 Hartrees
Sum of electronic and thermal Free Energies= -234.50082 Hartrees

The various information above can be used to consider the potential and free energy of the system. The sum of the electronic and zero-point energies (E= Eelec + ZPE) show the potential energy at 0K and the sum of electronic and thermal energies (E = E + Evib + Erot + Etrans) considers it at 298.15K and 1atm. The sum of electronic and thermal enthalpies (H = E + RT) shows the enthalpy of the system containing a correction for thermal energy and the sum of electronic and thermal free energies ( G= H - TS) suggests the entropic contribution to the free energy of the system.

This data shows that the energy at 0K is lower than that at room temperature, this is as expected because at 0K there will be no internal movement of the molecule and hence no contribution from vibrational, rotational or translational energies.

==='''Optimizing the "Chair" and "Boat" Transition Structures'''===

Conformer E was considered for this reaction and used to model both the chair and boat transition states for the Cope rearrangement reaction. A number of different ways of optimizing the geometry of the transition states were considered including using redundant coordinates and QTS2(TS)optimization and frequency calculation.

===='''Chair Transition State'''====

The chair transition state was constructed from two allyl fragments. Firstly the geometry of a single allyl fragment was optimized using the HF/3-21G level of theory giving an energy of -117.61330Hartrees. Two allyl fragments are placed 2.2A apart and then using this as a guess for the transition state, the molecule is then optimized in two different ways.

[[File:BE_B_vibration.gif|thumb|Cope Rearrangement Chair Transition State Vibration]]

Firstly it can be optimized using a opt + freq calculation to a TS using the Berny Method. By calculating the force constant once an imaginary vibration can be found. The imaginary vibration was found at 817.95cm-1 (expected value of 818cm-1<ref name="Instructional Wiki"/>) on animation it can be seen that this corresponds to the coming together of the two terminal carbons on each allyl molecule, the movement expected during this reaction. In this case the total energy is -231.61932Hartrees.

[[File:BE_D_ChairTS.jpg|thumb|left|Chair Transition State by Redundant Coordinate Method]]
Secondly the transition state was optimized by freezing the reaction coordinate in order to minimize the rest of the molecule first and then unfreezing it in order to find the minimum for the transition state. This means that the force constant Hessian matrix doesn't need to be calculated and therefore is a faster method of computing the transition state.

By looking at the atom distances between the bond that will be formed the suitability of both of these methods to find the true transition state can be studied. All the measured geometric parameters were identical in both versions of the transition state, namely the intranuclear bond lengths was 1.39A, the internuclear distance between the two allyl molecules is 2.02A and the angle within the each molecule is 120.05o. This suggests that both methods for determining the transition state are equally effect from the same guess input file. It also confirms that the guess structure was close to the exact structure as the first method was as successful in calculating the force constants as the second which doesn't need to calculate them.

===='''The Boat Transition State'''====

The boat transition state was constructed in much the same way as the chair using two pre-optimized allyl fragments, placing them together in the correct geometry and setting them to 2.2A apart. In the case of the boat transition state the QTS2 method was used to optimize the energy. By ensuring that the same labeling system is used in the reactants and products, an optimization and frequency calculation was run that could help define the geometry of the boat transition state. This calculation was run with a predicted transition state with a dihedral angle (C2-C3-C4-C5) of 0o and inside C2-C3-C4 and C3-C4-C5 angles of 100o.

[[File:BE_E_Vibration.gif|thumb|200px|Cope Rearrangement Boat Transition State Vibration]]

The result of this calculation shows the boat transition state with a single imaginary vibration of -840nm and energy of -231.60280Hartrees. Visualizing this vibration shows that it corresponds to the coming together of the two terminal carbons and the separation of the two central carbons. This is the expected vibration seen during the Cope rearrangement.
The QTS2 method is advantageous because it is fully automated however it required a good starting guess for the transition state and therefore a lot of permutations of structure, in the input file, were needed for a successful calculation.

{| class="wikitable" style="margin: auto;"
|+ Summary Table of the Energies of the Two Transition States
! rowspan="2" | Transition State
! colspan="3" | Energy
|-
|<jmolFile text="Berny TS">BE_ChairB.mol</jmolFile>
|<jmolFile text="Redunant Coordinates">BE_chairD.mol</jmolFile>
|<jmolFile text="QTS2 (TS)">BE_Boat_TS_Again2.mol</jmolFile>
|-
!Chair
| -231.61932
| -231.61932
|
|-
!Boat
|
|
| -231.60280
|}

===='''The Reaction Pathway'''====

The reaction pathway can be studied through both the minimum energy position on its potential energy surface and the activation energies of the two transition states.

====='''IRC'''=====

[[File:BE_IRC_Plot.jpg|thumb|400px|IRC Energy Plot of the Chair Transition State]]
An intrinsic reaction coordinate (IRC) was run as it can be used to determine the minimum energy pathway from the transition state to its local minimum by using many points along the pathway to the point where the energy plateaus. The IRC in this case has been run in the forward direction only and the force constants calculated at every step.

The lowest energy conformation is achieved after 44 steps. The structure of the first and final conformations can be compared by considering their different geometry and energy. The IRC determines the minimum energy conformation and therefore the energy of step 44 (-231.69158Hartrees) is, as expected, lower than that of the first step (-231.61932Hartrees), this confirms that the calculation has found a minimum. The most noticeable different between the two conformations is the bond symmetry, in step 1 the bond lengths within each molecule were identical (1.39A) and the distance between the two molecules is the same (2.02A) for both sets of terminal carbons. However in step 44 the bond lengths within each molecule are different in preparation for the formation of two different types of bonds - single and double - this means one bond is shorter than the other (1.32 and 1.51A). As well as this difference in intranuclear distances there is also a difference in the internuclear distances as the one end approaches much closer (1.55 and 4.23A) almost within the van der Waals radius of the carbon atom as it prepares to form a new bond whereas the other distance is much larger as there is no bond formation at those carbons.

{| class="wikitable" style="margin: auto;"
|+ Structures of Steps 1 and 44 in the Determination of the IRC of the Chair Transition State
! [[File:BE_F1_Image.jpg|400px]]
! [[File:BE_F44_Image.jpg|400px]]
|}

====='''Activation Energies'''=====

In order to determine the activation energy of the reaction via the two different transition states a higher level of theory needed to be used. The B3LYP/6-31G(d) was used to optimize the two transition states to a minimum. Comparing this to the activation energies obtained at the lower level of theory and literature values<ref name="Instructional Wiki"/> shows the large difference in the energies obtained at each level of theory. The Energies at 0 and 298.15K were then found by running a frequency calculation, using the same basis set, on the optimised structure and checking the sum of the electronic and zero-point (0K) and sum of electronic and thermal energies (298.15K).

{| class="wikitable" style="margin: auto;"
|+ Geometrical Features of the Reactant, Boat and Chair Transition States
!
! colspan="2" | Reactants
! colspan="2" | Chair Transition State
! colspan="2" | Boat Transition State
|-
|
|<jmol><jmolApplet><title>Reactants Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Reactants Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Transition State Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Transition State Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Transition State Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again2.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Transition State Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again3_-_EA.mol</uploadedFileContents> </jmolApplet></jmol>
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
!Intramolecular Bond Lengths/A
|1.32, 1.51
|1.33, 1.50
|1.39
|1.41
|1.38
|1.39
|-
!Intermolecular Bond Lengths/A
|1.55
|1.55
|2.02
|1.97
|2.14
|2.21
|-
!Dihedral Angle/o
|179.996
|180
|54.96
|54
|0
|0
|}

In the reactants the higher level of theory shows a tightening of the dihedral angle to exactly 180o as well as a shortening of the single bonds and lengthening of the double bonds within the molecule. This acts to bring the bond lengths within the molecule closer to the expected bond lengths of 1.47 and 1.54A<ref name="Wiki Book">M. A. Fox, J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen.'', '''1995''', Springer.</ref>. Within chair transition state the same effect can be seen, increasing the dihedral angle and double bond length and decreasing the single and intermolecular bond distances.

{| class="wikitable" style="margin: auto;"
|+ Electronic Energies and Energies of the Boat and Chair Transition States at 0K
!
! colspan="2" | '''Electronic Energy/Hartrees'''
! colspan="2" | '''Sum of Electronic and Zero-Point Energies'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.60280
| -234.54309
| -231.45093
| -234.40234
| 0.08861
| 0.06687
| 55.60
| 41.96
| 44.7 ± 2.0
|-
|'''Chair Transition State'''
| -231.69167
| -234.55698
| -231.46669
| -234.41488
| 0.07286
| 0.05433
| 45.72
| 34.09
| 33.5 ± 0.5
|}

The activation energies were calculated with reference to the conformer E energies as the reactant ( At 0K HF/3-21G = -231.53954 and B2LYP/6-31G(d) = -234.46921Hartrees, At 298.15K HF/3-21G = -231.53257 and B2LYP/6-31G(d) = -234.46186Hartrees ).

{| class="wikitable" style="margin: auto;"
|+ Energies of the Boat and Chair Transition States at 298.15K
!
! colspan="2" | '''Sum of Electronic and Thermal Energies'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.44530
| -234.39599
| 0.08727
| 0.06587
| 54.76
| 41.33
| 41.32
|-
|'''Chair Transition State'''
| -231.46133
| -234.40895
| 0.07124
| 0.05291
| 44.70
| 33.20
| 33.17
|}

From this it is clear why a higher level of theory is needed. The energies calculated at the B3LYP/6-31G(d) level yield much more accurate activation energies that those at HF/3-21G. In both cases this increase in accuracy has brought the activation energies significantly closer to the expected values.

The activation energy of the Chair transition state is a good match for that in the literature and is very close to the boundaries of error given. When comparing this to the energy obtained at the lower level of theory there is a much greater difference and therefore illustrates the ability of the higher level of energy to calculate a much more accurate transition state.

=='''Diels-Alder Cycloaddition Reaction'''==

The Diels-Alder cycloaddition reaction is a pericyclic reaction that occurs between two pi systems, a diene and a dienophile, in order to form a cyclohexene. It offers good regio- and stereo-selective control that can be extended to other conjugated systems<ref name="The Diels-Alder Reaction in Total Synthesis">K. C. Nicolaou, S. A. Snyder, T. Montagnon and G. Vassilikogiannakis ''Angewandte Chemie International Edition'','''2002''', ''41'', 1668-1698.</ref>.

In this experiment the reaction between cis butadiene and ethene will be studied in order to determine the symmetry of both the reactant and product molecular orbitals, as well as the geometry of the reaction's transition state. The regioselectivity of the reaction will also be looked at into order to determine the preferred product of the reaction, endo or exo.

==='''Cis-butadiene'''===

Cis-butadiene will be used as the diene in this studied reaction with ethene. Its geometry was first optimized using the HF/3-21G level of theory. The final energy of the product was -154.05514Hartrees. The molecular orbitals of the optimized structure can be modeled and used further to look at the electronic contribution of the diene to the final molecule.

{| class="wikitable" style="margin: auto;"
|+ Images of the Molecular Orbitals of Cis-Butadiene
! [[File:BE_I_HOMO.jpg|200px]]
! rowspan="3" | [[File:BE_Diene.jpg|150px]]
! [[File:BE_I_LUMO.jpg|200px]]
|-
![[File:BE_I_HOMO2.jpg|200px]]
![[File:BE_I_LUMO2.jpg|200px]]
|-
| align="center" | The '''HOMO''' orbital of the molecule is asymmetric. The nodes in the molecule fall between the red and green lobes of the HOMO. This shows the two separate pi systems within the molecule, but the proximity of the two systems suggests that the molecule could be fully conjugated.
| align="center" | The '''LUMO''' orbital of the molecule is symmetric. It shows the pi* orbitals of the two pi systems and the conjugation of these across the central single bond. The nodes fall within the plane of the molecule and between the double bonds of the molecule suggesting that for this orbital there is no conjugation over the double bonds.
|}

==='''The Transition State'''===

A number of methods, that had previously been discussed above for the Cope rearrangement reaction, were considered to optimize the transition state of this reaction. However it was decided to use a method previously used in optimizing the chair transition state of 1,5-hexadiene due to it's simplicity and that a good guess for the transition state could be made initially.

Firstly the reactants were optimized separately under the HF/3-21G level of theory. Then the two molecules were combined in an envelope like geometry and the internuclear distances between the two molecules set to 2.2A in order to give a best guess for the transition state. An optimization and frequency calculation was then run, optimizing to a transition state using the Berny method, calculating the force constants only once.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Structure and Molecular Orbitals of the Diels-Alder Transition State
![[File:BE_TSHOMO.jpg|center|200px]][[File:BE_TSHOMO2.jpg|center|200px]]
!<jmol><jmolApplet><title>1,5-hexdiene formation transition state</title><color>white</color> <size>300</size><uploadedFileContents>BE_Reactants.mol</uploadedFileContents> </jmolApplet></jmol>
|[[File:BE_TSLUMO.jpg|center|200px]][[File:BE_TSLUMO2.jpg|center|200px]]
|-
! align="center" | HOMO
| align="center" | The transition state for the Diels-Alder reaction between ethene and cis-butadiene has an energy of -231.60321Hartrees
! align="center" | LUMO
|}

[[File:BE_TSVibration.gif|thumb|left|Imaginary Vibration of the Diels-Alder Transition State]][[File:BE_Positive_Vibration.gif|thumb|right|Lowest Energy Positive Vibration of the Diels-Alder Transition State]]

This gave a good approximation for the transition state with the imaginary vibration, of -817.71cm-1, showing the two molecules approaching each other in a synchronous motion. Comparatively the lowest positive vibration at 166.54cm-1 is an asynchronous rocking motion that does not contribute to the coming together of the two molecules.

[[File:BE_TSDIELSALDERIR_Spectrum.png|thumb|center|300px|Predicted IR Spectrum of the Diels-Alder Transition State]]

The justification of calling this the transition state is also shown in the final bond lengths between the two molecules, the length of approach is 2.21A. Typically an sp3-sp3 carbon bond length is 1.54A and for a sp2-sp2 carbon it is 1.47A<ref name="Wiki Book"/>this suggests that a bond has not been fully formed in the transition state. However it is within the van der Waals radius of the carbon atom, which is 1.7A<ref name="Van der Waals Volumes and Radii">A. Bondi, ''J. Phys. Chem.'','''1964''', ''68 '', 441-51.</ref> so a partial bond may have started to form at this point in the reaction.

Looking at the HOMO and LUMO orbitals for the reactants there is a clear contribution from both to the molecular orbitals of the transition state, this is especially the case for the ethene molecule. The visualization of the HOMO orbital throws some doubt onto the final structure obtained however considering the imaginary vibration is as expected, it can be assumed that this is a form of the transition state, if not the most minimal energy one. Further calculations were run in order to find the transition state with the correct HOMO orbital however it could not be found with the correct geometry and imaginary vibration.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Molecular Orbitals of Ethene and Cis-Butadiene
! colspan="4" | Ethene
! colspan="4" | cis-butadiene
|-
|[[File:BE_Ethene_HOMO.jpg|center|100px]]
|[[File:BE_Ethene_HOMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO.jpg|center|100px]]
|[[File:BE_I_HOMO.jpg|center|100px]]
|[[File:BE_I_HOMO2.jpg|center|100px]]
|[[File:BE_I_LUMO.jpg|center|100px]]
|[[File:BE_I_LUMO2.jpg|center|100px]]
|-
! colspan="2" | HOMO - Symmetric
! colspan="2" | LUMO - Asymmetric
! colspan="2" | HOMO - Asymmetric
! colspan="2" | LUMO - Symmetric
|}

The reactants orbitals above display frontier molecular orbital theory. The HOMO of cis-butadiene and the LUMO of ethene are of correct symmetry to interact to form a bonding molecular orbital in the product. This means that the reaction is symmetry allowed and therefore no other conditions are needed, such as light, in order for the reaction to proceed<ref>[http://www.ch.ic.ac.uk/rzepa/] Henry S. Rzepa, Imperial College, Organic Pericyclic Reactions: Categories</ref>.

==='''Regioselectivity'''===

The reaction can form two different products depending on the transition state. The exo product is formed when the approach of the maleic anhydride occurs with the oxygens pointing away from the double bond, and the endo product is formed when the oxygens point towards it.
In order to determine the energies of the transition states first the reactants were optimized separately and then together using the HF/3-21G level of theory. Then the endo and exo products were optimized the same way. The transition state was found using the QTS2 method, an Opt+Freq calculation under the HF/3-21G level of theory, this gave an optimized version of each transition state. This was used in order to utilise the reactants and products to find the transition state as the initial guess for the transition state was not a confident enough one.

The imaginary vibrations of these transition at -647.54 and -643.32cm-1 for the exo and endo transition states respectively. These vibrations show the coming together of the reactant molecules to form the product as is expected. These Transition state structures were then optimized under a higher level of theory in order to obtain a more accurate energy, and the HOMO and LUMO orbitals of each visualized.

[[File:BE_Exo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Exo Transition State]][[File:BE_Endo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Endo Transition State]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Energies of the Exo and Endo Transition States under the B3LYP/6-31G(d) Level of Theory
!
! Exo Transition State
! Endo Transition State
|-
|
|<jmol><jmolApplet><title>EXO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_Exo_TS_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>ENDO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_ENDO_TS_3_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|-
!Total Energy/Hartrees
| -612.66122
| -612.68340
|-
!Sum of Electronic and Zero-Point Energies/Hartrees
| -612.47669
| -612.50214
|-
!Sum of Electronic and Thermal Energies/Hartrees
| -612.46612
| -612.49179
|-
!Activation Energy at 0K/kcalmol-1
| 34.90
| 18.93
|-
!Activation Energy at 298.15K/kcalmol-1
| 33.58
| 17.47
|-
!Dihedral Angle/o
| -169.56
| -175.67
|-
!Intermolecular Bond Lengths/A
|1.5
|2.27
|}

Activation energies were calculated by comparing the energy of the transition state to the energy of the reactants (At 0K B3LYP/6-31G(d) = -612.53231, At 298.15K B3LYP/6-31G(d) = -612.51964). The endo product should be the kinetic product, this is shown by its much lower activation energy. The exo product has a tighter dihedral angle than the endo, this is due to the repulsion between the two reactants as seen in the HOMO of the exo transition state below<ref name="Steric Effects vs. Secondary Orbital Overlap in Diels-Alder Reactions. MNDO and AM1 Studies">, M. A. Fox, R. Cardons and N. J. Kiwiet, ''J. Org. Chem.'', '''1987''', ''52'',1469-1474</ref>.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The HOMO and LUMO Molecular Orbitals of the Exo and Endo Transition States
! colspan="2" | Exo Transition State
! colspan="2" | Endo Transition State
|-
!HOMO
!LUMO
!HOMO
!LUMO
|-
|[[File:BE_HOMO_exo.jpg|center|200px]]
|[[File:BE_LUMO_Exo.jpg|center|200px]]
|[[File:BE_HOMO_Endo.jpg|center|200px]]
|[[File:BE_LUMO_endo.jpg|center|200px]]
|}

The endo and exo products will be stabilised by secondary orbital overlap (SOO). This is the mixing of the HOMO orbital of one reactant with the LUMO orbital of the other<ref name="Orbital symmetries and endo-exo relationships in concerted cycloaddition reactions">, R. Hoffmann, R. B. Woodward, ''J. Am. Chem. Soc.'', '''1965''', ''87'', 4388-4389</ref>. This can be seen in both transition state HOMO orbitals above. In the case of the endo transition state the orbitals from the central oxygen shows a node around it, the orbitals on that oxygen are of the same symmetry and hence will interact with the orbitals from the double bond that is formed in the product above it. For the exo transition state the position of the oxygen and double bond are much separated and therefore there is no SOO. This SOO effect in the endo transition state, and hence the greater stabilisation, is a reason that the energy of the endo transition state is lower than that of the endo.

==Conclusion==

These reactions have shown the usefulness of molecular modelling in the visualization and analysis of transition states. A wide range of methods for optimizing the transition states have been considered and tested for their accuracy and ease of use. Despite initial difficulties the QTS2 method proved to be the most effective in determining the transition state of both the boat conformer in the Cope rearrangement and the endo and exo forms in the Diels-Alder reaction. It gave a good prediction for the geometries and energies of the transition states and allowed comparison between the reactants and products of the reactions. Further examples of Diels-Alder reactions could be looked at to determine the stereo and regio-selective implications of the mechanism and further prove the effectiveness of calculating activation energies by this method as compared to experimentally determined values. Also calculations from different basis sets could be run to determine the accuracy of the different levels of theory and to improve the calculated geometries and energies of the transition states studied in this experiment.

==References==

<references />

Rep:Mod:PhysBE411

2014-03-21T16:34:11Z

Be411: /* The Boat Transition State */

'''Physical Module: Transition States and Reactivity'''

Computational methods can be used to study reaction pathways and more specifically transition states. In this experiment the Cope rearrangement reaction and Diels-Alder cycloaddition reaction are used as examples of this. Using various methods the lowest energy conformation of 1,5-hexadiene can be found as well as the conformation that exists as a reactant in the Cope rearrangement reaction. The two possible transition states of this reaction can be studied and their activation energies calculated to gain some understanding of the mechanism of this reaction. The Diels-Alder cycloaddition will also be studied. The origin of the symmetry allowed reaction between ethene and cis-butadiene will be visualized and the reaction between maleic anhydride and hexa-1,5-diene studied in terms of the regioselectivity of its two conformational different transition states.

=='''The Cope Rearrangement Tutorial'''==

The Cope rearrangement is a specific [3,3] sigmatropic rearrangement in which a sigma bond moves and the pi system is redistributed across the molecule. In this case the Cope rearrangement of hexa-1,5-diene is considered with regard to its conformations and the boat and chair forms of its reaction's transition state.

==='''Optimizing the Reactants and Products'''===

===='''Conformation of 1,5-hexadiene'''====
[[File:BE_1,5-HEXADIENE_CONFORTMATIONS.jpg|thumb|The Conformations of 1,5-Hexadiene]]

1,5-hexadiene is one of the simplest precursors in the Cope rearrangement. The three rotating carbon-carbon bonds within the center of the molecule cause there to be 10 energetically different conformations, when taking into account enantiomers and symmetry.

Within these conformations you would expect that where two carbons are in the eclipsed form (F-J) the conformer will be of higher energy than conformers with CH-eclipsed bonds (A-E)<ref name="Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols">B. W. Gung, Z. Zhu and R. A. Fouch, ''J. Am. Chem. Soc'', '''1995''', ''117'', 1783-1788.</ref>. This is demonstrated in the case of butane where the difference in energy between the CC eclipsed form and the CH eclipsed form is 1.5kcal/mol<ref name="Website - Butane Conformational Analysis">[http://research.cm.utexas.edu/nbauld/teach/butane.html]N. Bauld, '''"Butane Conformational Analysis"''', ''University of Texas'', Retreived 20 March 2014.</ref>. The energies of these forms can be optimized by computational methods in order to find the most suitable candidates for the reactants and products in the Cope Rearrangement reaction.

The table below shows the total energy for all energetically distinct conformers of the molecule, it shows that in the case of this molecule the anti conformer is generally of higher energy. Relating these structures to the literature<ref name="Instructional Wiki">[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3]M. Bearpark, Imperial College, Physical Module: Transition states and reactivity</ref>, the energies found are all in line with the energies expected, as is also the case with the point groups.

The lowest energy conformation is the Gauche Conformer D (-231.69266Hartrees) this maybe due to an attractive interaction between the pi orbital of the double bond and the neighboring proton, whose orbital would interact with that of the pi bond<ref name="An Analysis of the Conformers of 1,5-hexadiene">B. G. Rocque, J. M. Gonzales, H. F. Schaeffer III, ''Mol. Phys'', '''2002''', ''100'', 441-446.</ref>. This conjugation will stabilize the molecule making it lower in energy.
The highest energy conformer of the CH eclipsed forms is C (-231.69153Hartrees) this maybe explained by interactions between the two alkene protons which are 2.51A apart. This proximity will have a repulsive effect increasing the overall energy of the molecule.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimized Energies for the Anti and Gauche Conformations of 1,5-Hexadiene (Relating to the Instructional Wiki<ref name="Instructional Wiki"/>)
!
! align="center" | '''<jmolFile text="Conformer A - Gauche2">BE_A.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer B - Anti1 ">BE_B.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer C - Gauche4">BE_C.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer D - Gauche3">BE_D.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer E - Anti2">BE_E.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer F - Gauche6">BE_F.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer G - Anti4">BE_G.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer H - Gauche5">BE_H.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer I - Gauche1">BE_I.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer J - Anti3">BE_J.mol</jmolFile>'''
|-
|'''Total Energy/Hartrees'''
| -231.69167
| -231.69260
| -231.69153
| -231.69266
| -231.69254
| -231.68916
| -231.69097
| -231.68962
| -231.68772
| -231.68907
|-
|'''Point Group'''
|C2
|C2
|C2
|C1
|Ci
|C1
|C1
|C1
|C2
|C2h
|-
|'''Dihedral Angle (C2-C3-C4-C5)'''
|64.17
| -176.91
| -63.65
|67.70
| -179.996
|70.23
| -178.32
| -71.09
|75.76
| -179.98
|}

===='''Analysis of Conformer E'''====

Further Analysis of Conformer E was done at a higher basis set in order to get an even better approximation for its lowest energy conformation. An optimization calculation was run as before at the B3LYP/6-31G(d)level of theory.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimised Conformations of Conformer E using the HF/3-21G and B3LYP/6-31G(d) Basis Sets
!<jmol><jmolApplet><title>Anti 2 Conformation at HF level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_E.mol</uploadedFileContents> </jmolApplet></jmol>
!<jmol><jmolApplet><title>Anti 2 Conformation at B3LYP level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_DFT_optmisation.mol</uploadedFileContents> </jmolApplet></jmol>
|}

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Geometrical Features of Conformer E Optimized Under Different Basis Sets
! rowspan ="2" | '''Basis Set'''
! colspan ="3" | '''Bond Length'''
! colspan ="3" | '''Angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3
|C2-C3-C4
|C2-C3-C4-C5
|-
|HF/3-21G
|1.32
|1.51
|1.55
|179.996
|124.81
|111.35
|-
|B3LYP/6-31G(d)
|1.33
|1.50
|1.55
|180
|125.29
|112.68
|}

[[image:BE_IR_Spectrum_of_Anti2.jpg|thumb|200px|IR Spectrum of Conformer E]]

This shows that the geometries of the results of the two different approaches don't differ wildly, the bond lengths especially are very similar, however there is some deviation between their angles.

Following on from this a frequency calculation was run using the same higher level of theory (B3LYP/6-31G(d)). This calculation can be used to confirm that this conformation is of a minimum energy by the absence of an imaginary vibrational frequency. The infrared spectrum of the molecule can be generated from the molecular vibrations and compared to literature.

The thermochemical energies for this molecule can be found in the output file of the calculation:

Sum of electronic and zero-point Energies= -234.46921 Hartrees
Sum of electronic and thermal Energies= -234.46186 Hartrees
Sum of electronic and thermal Enthalpies= -234.46091 Hartrees
Sum of electronic and thermal Free Energies= -234.50082 Hartrees

The various information above can be used to consider the potential and free energy of the system. The sum of the electronic and zero-point energies (E= Eelec + ZPE) show the potential energy at 0K and the sum of electronic and thermal energies (E = E + Evib + Erot + Etrans) considers it at 298.15K and 1atm. The sum of electronic and thermal enthalpies (H = E + RT) shows the enthalpy of the system containing a correction for thermal energy and the sum of electronic and thermal free energies ( G= H - TS) suggests the entropic contribution to the free energy of the system.

This data shows that the energy at 0K is lower than that at room temperature, this is as expected because at 0K there will be no internal movement of the molecule and hence no contribution from vibrational, rotational or translational energies.

==='''Optimizing the "Chair" and "Boat" Transition Structures'''===

Conformer E was considered for this reaction and used to model both the chair and boat transition states for the Cope rearrangement reaction. A number of different ways of optimizing the geometry of the transition states were considered including using redundant coordinates and QTS2(TS)optimization and frequency calculation.

===='''Chair Transition State'''====

The chair transition state was constructed from two allyl fragments. Firstly the geometry of a single allyl fragment was optimized using the HF/3-21G level of theory giving an energy of -117.61330Hartrees. Two allyl fragments are placed 2.2A apart and then using this as a guess for the transition state, the molecule is then optimized in two different ways.

[[File:BE_B_vibration.gif|thumb|Cope Rearrangement Chair Transition State Vibration]]

Firstly it can be optimized using a opt + freq calculation to a TS using the Berny Method. By calculating the force constant once an imaginary vibration can be found. The imaginary vibration was found at 817.95cm-1 (expected value of 818cm-1<ref name="Instructional Wiki"/>) on animation it can be seen that this corresponds to the coming together of the two terminal carbons on each allyl molecule, the movement expected during this reaction. In this case the total energy is -231.61932Hartrees.

[[File:BE_D_ChairTS.jpg|thumb|left|Chair Transition State by Redundant Coordinate Method]]
Secondly the transition state was optimized by freezing the reaction coordinate in order to minimize the rest of the molecule first and then unfreezing it in order to find the minimum for the transition state. This means that the force constant Hessian matrix doesn't need to be calculated and therefore is a faster method of computing the transition state.

By looking at the atom distances between the bond that will be formed the suitability of both of these methods to find the true transition state can be studied. All the measured geometric parameters were identical in both versions of the transition state, namely the intranuclear bond lengths was 1.39A, the internuclear distance between the two allyl molecules is 2.02A and the angle within the each molecule is 120.05o. This suggests that both methods for determining the transition state are equally effect from the same guess input file. It also confirms that the guess structure was close to the exact structure as the first method was as successful in calculating the force constants as the second which doesn't need to calculate them.

===='''The Boat Transition State'''====

The boat transition state was constructed in much the same way as the chair using two pre-optimized allyl fragments, placing them together in the correct geometry and setting them to 2.2A apart. In the case of the boat transition state the QTS2 method was used to optimize the energy. By ensuring that the same labeling system is used in the reactants and products, an optimization and frequency calculation was run that could help define the geometry of the boat transition state. This calculation was run with a predicted transition state with a dihedral angle (C2-C3-C4-C5) of 0o and inside C2-C3-C4 and C3-C4-C5 angles of 100o.

[[File:BE_E_Vibration.gif|thumb|200px|Cope Rearrangement Boat Transition State Vibration]]

The result of this calculation shows the boat transition state with a single imaginary vibration of -840nm and energy of -231.60280Hartrees. Visualizing this vibration shows that it corresponds to the coming together of the two terminal carbons and the separation of the two central carbons. This is the expected vibration seen during the Cope rearrangement.
The QTS2 method is advantageous because it is fully automated however it required a good starting guess for the transition state and therefore a lot of permutations of structure, in the input file, were needed for a successful calculation.

{| class="wikitable" style="margin: auto;"
|+ Summary Table of the Energies of the Two Transition States
! rowspan="2" | Transition State
! colspan="3" | Energy
|-
|<jmolFile text="Berny TS">BE_ChairB.mol</jmolFile>
|<jmolFile text="Redunant Coordinates">BE_chairD.mol</jmolFile>
|<jmolFile text="QTS2 (TS)">BE_Boat_TS_Again2.mol</jmolFile>
|-
!Chair
| -231.61932
| -231.61932
|
|-
!Boat
|
|
| -231.60280
|}

===='''The Reaction Pathway'''====

The reaction pathway can be studied through both the minimum energy position on its potential energy surface and the activation energies of the two transition states.

====='''IRC'''=====

[[File:BE_IRC_Plot.jpg|thumb|400px|IRC Energy Plot of the Chair Transition State]]
An intrinsic reaction coordinate (IRC) was run as it can be used to determine the minimum energy pathway from the transition state to its local minimum by using many points along the pathway to the point where the energy plateaus. The IRC in this case has been run in the forward direction only and the force constants calculated at every step.

The lowest energy conformation is achieved after 44 steps. The structure of the first and final conformations can be compared by considering their different geometry and energy. The IRC determines the minimum energy conformation and therefore the energy of step 44 (-231.69158Hartrees) is, as expected, lower than that of the first step (-231.61932Hartrees), this confirms that the calculation has found a minimum. The most noticeable different between the two conformations is the bond symmetry, in step 1 the bond lengths within each molecule were identical (1.39A) and the distance between the two molecules is the same (2.02A) for both sets of terminal carbons. However in step 44 the bond lengths within each molecule are different in preparation for the formation of two different types of bonds - single and double - this means one bond is shorter than the other (1.32 and 1.51A). As well as this difference in intranuclear distances there is also a difference in the internuclear distances as the one end approaches much closer (1.55 and 4.23A) almost within the van der Waals radius of the carbon atom as it prepares to form a new bond whereas the other distance is much larger as there is no bond formation at those carbons.

{| class="wikitable" style="margin: auto;"
|+ Structures of Steps 1 and 44 in the Determination of the IRC of the Chair Transition State
! [[File:BE_F1_Image.jpg|400px]]
! [[File:BE_F44_Image.jpg|400px]]
|}

====='''Activation Energies'''=====

In order to determine the activation energy of the reaction via the two different transition states a higher level of theory needed to be used. The B3LYP/6-31G(d) was used to optimize the two transition states to a minimum. Comparing this to the activation energies obtained at the lower level of theory and literature values<ref name="Instructional Wiki"/> shows the large difference in the energies obtained at each level of theory. The Energies at 0 and 298.15K were then found by running a frequency calculation, using the same basis set, on the optimised structure and checking the sum of the electronic and zero-point (0K) and sum of electronic and thermal energies (298.15K).

{| class="wikitable" style="margin: auto;"
|+ Geometrical Features of the Reactant, Boat and Chair Transition States
!
! colspan="2" | Reactants
! colspan="2" | Chair Transition State
! colspan="2" | Boat Transition State
|-
|
|<jmol><jmolApplet><title>Reactants Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Reactants Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Transition State Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Transition State Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Transition State Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again2.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Transition State Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again3_-_EA.mol</uploadedFileContents> </jmolApplet></jmol>
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
!Intramolecular Bond Lengths/A
|1.32, 1.51
|1.33, 1.50
|1.39
|1.41
|1.38
|1.39
|-
!Intermolecular Bond Lengths/A
|1.55
|1.55
|2.02
|1.97
|2.14
|2.21
|-
!Dihedral Angle/o
|179.996
|180
|54.96
|54
|0
|0
|}

In the reactants the higher level of theory shows a tightening of the dihedral angle to exactly 180o as well as a shortening of the single bonds and lengthening of the double bonds within the molecule. This acts to bring the bond lengths within the molecule closer to the expected bond lengths of 1.47 and 1.54A<ref name="Wiki Book">M. A. Fox, J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen.'', '''1995''', Springer.</ref>. Within chair transition state the same effect can be seen, increasing the dihedral angle and double bond length and decreasing the single and intermolecular bond distances.

{| class="wikitable" style="margin: auto;"
|+ Electronic Energies and Energies of the Boat and Chair Transition States at 0K
!
! colspan="2" | '''Electronic Energy/Hartrees'''
! colspan="2" | '''Sum of Electronic and Zero-Point Energies'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.60280
| -234.54309
| -231.45093
| -234.40234
| 0.08861
| 0.06687
| 55.60
| 41.96
| 44.7 ± 2.0
|-
|'''Chair Transition State'''
| -231.69167
| -234.55698
| -231.46669
| -234.41488
| 0.07286
| 0.05433
| 45.72
| 34.09
| 33.5 ± 0.5
|}

The activation energies were calculated with reference to the conformer E energies as the reactant ( At 0K HF/3-21G = -231.53954 and B2LYP/6-31G(d) = -234.46921Hartrees, At 298.15K HF/3-21G = -231.53257 and B2LYP/6-31G(d) = -234.46186Hartrees ).

{| class="wikitable" style="margin: auto;"
|+ Energies of the Boat and Chair Transition States at 298.15K
!
! colspan="2" | '''Sum of Electronic and Thermal Energies'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.44530
| -234.39599
| 0.08727
| 0.06587
| 54.76
| 41.33
| 41.32
|-
|'''Chair Transition State'''
| -231.46133
| -234.40895
| 0.07124
| 0.05291
| 44.70
| 33.20
| 33.17
|}

From this it is clear why a higher level of theory is needed. The energies calculated at the B3LYP/6-31G(d) level yield much more accurate activation energies that those at HF/3-21G. In both cases this increase in accuracy has brought the activation energies significantly closer to the expected values.

The activation energy of the Chair transition state is a good match for that in the literature and is very close to the boundaries of error given. When comparing this to the energy obtained at the lower level of theory there is a much greater difference and therefore illustrates the ability of the higher level of energy to calculate a much more accurate transition state.

=='''Diels-Alder Cycloaddition Reaction'''==

The Diels-Alder cycloaddition reaction is a pericyclic reaction that occurs between two pi systems, a diene and a dienophile, in order to form a cyclohexene. It offers good regio- and stereo-selective control that can be extended to other conjugated systems<ref name="The Diels-Alder Reaction in Total Synthesis">K. C. Nicolaou, S. A. Snyder, T. Montagnon and G. Vassilikogiannakis ''Angewandte Chemie International Edition'','''2002''', ''41'', 1668-1698.</ref>.

In this experiment the reaction between cis butadiene and ethene will be studied in order to determine the symmetry of both the reactant and product molecular orbitals, as well as the geometry of the reaction's transition state. The regioselectivity of the reaction will also be looked at into order to determine the preferred product of the reaction, endo or exo.

==='''Cis-butadiene'''===

Cis-butadiene will be used as the diene in this studied reaction with ethene. Its geometry was first optimized using the HF/3-21G level of theory. The final energy of the product was -154.05514Hartrees. The molecular orbitals of the optimized structure can be modeled and used further to look at the electronic contribution of the diene to the final molecule.

{| class="wikitable" style="margin: auto;"
|+ Images of the Molecular Orbitals of Cis-Butadiene
! [[File:BE_I_HOMO.jpg|200px]]
! rowspan="3" | [[File:BE_Diene.jpg|150px]]
! [[File:BE_I_LUMO.jpg|200px]]
|-
![[File:BE_I_HOMO2.jpg|200px]]
![[File:BE_I_LUMO2.jpg|200px]]
|-
| align="center" | The HOMO orbital of the molecule is asymmetric. The nodes in the molecule fall between the red and green lobes of the HOMO. This shows the two separate pi systems within the molecule, but the proximity of the two systems suggests that the molecule could be fully conjugated.
| align="center" | The LUMO orbital of the molecule is symmetric. It shows the pi* orbitals of the two pi systems and the conjugation of these across the central single bond. The nodes fall within the plane of the molecule and between the double bonds of the molecule suggesting that for this orbital there is no conjugation over the double bonds.
|}

==='''The Transition State'''===

A number of methods, that had previously been discussed above for the Cope rearrangement reaction, were considered to optimize the transition state of this reaction. However it was decided to use a method previously used in optimizing the chair transition state of 1,5-hexadiene due to it's simplicity and that a good guess for the transition state could be made initially.

Firstly the reactants were optimized separately under the HF/3-21G level of theory. Then the two molecules were combined in an envelope like geometry and the internuclear distances between the two molecules set to 2.2A in order to give a best guess for the transition state. An optimization and frequency calculation was then run, optimizing to a transition state using the Berny method, calculating the force constants only once.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Structure and Molecular Orbitals of the Diels-Alder Transition State
![[File:BE_TSHOMO.jpg|center|200px]][[File:BE_TSHOMO2.jpg|center|200px]]
!<jmol><jmolApplet><title>1,5-hexdiene formation transition state</title><color>white</color> <size>300</size><uploadedFileContents>BE_Reactants.mol</uploadedFileContents> </jmolApplet></jmol>
|[[File:BE_TSLUMO.jpg|center|200px]][[File:BE_TSLUMO2.jpg|center|200px]]
|-
! align="center" | HOMO
| align="center" | The transition state for the Diels-Alder reaction between ethene and cis-butadiene has an energy of -231.60321Hartrees
! align="center" | LUMO
|}

[[File:BE_TSVibration.gif|thumb|left|Imaginary Vibration of the Diels-Alder Transition State]][[File:BE_Positive_Vibration.gif|thumb|right|Lowest Energy Positive Vibration of the Diels-Alder Transition State]]

This gave a good approximation for the transition state with the imaginary vibration, of -817.71cm-1, showing the two molecules approaching each other in a synchronous motion. Comparatively the lowest positive vibration at 166.54cm-1 is an asynchronous rocking motion that does not contribute to the coming together of the two molecules.

[[File:BE_TSDIELSALDERIR_Spectrum.png|thumb|center|300px|Predicted IR Spectrum of the Diels-Alder Transition State]]

The justification of calling this the transition state is also shown in the final bond lengths between the two molecules, the length of approach is 2.21A. Typically an sp3-sp3 carbon bond length is 1.54A and for a sp2-sp2 carbon it is 1.47A<ref name="Wiki Book"/>this suggests that a bond has not been fully formed in the transition state. However it is within the van der Waals radius of the carbon atom, which is 1.7A<ref name="Van der Waals Volumes and Radii">A. Bondi, ''J. Phys. Chem.'','''1964''', ''68 '', 441-51.</ref> so a partial bond may have started to form at this point in the reaction.

Looking at the HOMO and LUMO orbitals for the reactants there is a clear contribution from both to the molecular orbitals of the transition state, this is especially the case for the ethene molecule. The visualization of the HOMO orbital throws some doubt onto the final structure obtained however considering the imaginary vibration is as expected, it can be assumed that this is a form of the transition state, if not the most minimal energy one. Further calculations were run in order to find the transition state with the correct HOMO orbital however it could not be found with the correct geometry and imaginary vibration.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Molecular Orbitals of Ethene and Cis-Butadiene
! colspan="4" | Ethene
! colspan="4" | cis-butadiene
|-
|[[File:BE_Ethene_HOMO.jpg|center|100px]]
|[[File:BE_Ethene_HOMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO.jpg|center|100px]]
|[[File:BE_I_HOMO.jpg|center|100px]]
|[[File:BE_I_HOMO2.jpg|center|100px]]
|[[File:BE_I_LUMO.jpg|center|100px]]
|[[File:BE_I_LUMO2.jpg|center|100px]]
|-
! colspan="2" | HOMO - Symmetric
! colspan="2" | LUMO - Asymmetric
! colspan="2" | HOMO - Asymmetric
! colspan="2" | LUMO - Symmetric
|}

The reactants orbitals above display frontier molecular orbital theory. The HOMO of cis-butadiene and the LUMO of ethene are of correct symmetry to interact to form a bonding molecular orbital in the product. This means that the reaction is symmetry allowed and therefore no other conditions are needed, such as light, in order for the reaction to proceed<ref>[http://www.ch.ic.ac.uk/rzepa/] Henry S. Rzepa, Imperial College, Organic Pericyclic Reactions: Categories</ref>.

==='''Regioselectivity'''===

The reaction can form two different products depending on the transition state. The exo product is formed when the approach of the maleic anhydride occurs with the oxygens pointing away from the double bond, and the endo product is formed when the oxygens point towards it.
In order to determine the energies of the transition states first the reactants were optimized separately and then together using the HF/3-21G level of theory. Then the endo and exo products were optimized the same way. The transition state was found using the QTS2 method, an Opt+Freq calculation under the HF/3-21G level of theory, this gave an optimized version of each transition state. This was used in order to utilise the reactants and products to find the transition state as the initial guess for the transition state was not a confident enough one.

The imaginary vibrations of these transition at -647.54 and -643.32cm-1 for the exo and endo transition states respectively. These vibrations show the coming together of the reactant molecules to form the product as is expected. These Transition state structures were then optimized under a higher level of theory in order to obtain a more accurate energy, and the HOMO and LUMO orbitals of each visualized.

[[File:BE_Exo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Exo Transition State]][[File:BE_Endo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Endo Transition State]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Energies of the Exo and Endo Transition States under the B3LYP/6-31G(d) Level of Theory
!
! Exo Transition State
! Endo Transition State
|-
|
|<jmol><jmolApplet><title>EXO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_Exo_TS_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>ENDO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_ENDO_TS_3_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|-
!Total Energy/Hartrees
| -612.66122
| -612.68340
|-
!Sum of Electronic and Zero-Point Energies/Hartrees
| -612.47669
| -612.50214
|-
!Sum of Electronic and Thermal Energies/Hartrees
| -612.46612
| -612.49179
|-
!Activation Energy at 0K/kcalmol-1
| 34.90
| 18.93
|-
!Activation Energy at 298.15K/kcalmol-1
| 33.58
| 17.47
|-
!Dihedral Angle/o
| -169.56
| -175.67
|-
!Intermolecular Bond Lengths/A
|1.5
|2.27
|}

Activation energies were calculated by comparing the energy of the transition state to the energy of the reactants (At 0K B3LYP/6-31G(d) = -612.53231, At 298.15K B3LYP/6-31G(d) = -612.51964). The endo product should be the kinetic product, this is shown by its much lower activation energy. The exo product has a tighter dihedral angle than the endo, this is due to the repulsion between the two reactants as seen in the HOMO of the exo transition state below<ref name="Steric Effects vs. Secondary Orbital Overlap in Diels-Alder Reactions. MNDO and AM1 Studies">, M. A. Fox, R. Cardons and N. J. Kiwiet, ''J. Org. Chem.'', '''1987''', ''52'',1469-1474</ref>.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The HOMO and LUMO Molecular Orbitals of the Exo and Endo Transition States
! colspan="2" | Exo Transition State
! colspan="2" | Endo Transition State
|-
!HOMO
!LUMO
!HOMO
!LUMO
|-
|[[File:BE_HOMO_exo.jpg|center|200px]]
|[[File:BE_LUMO_Exo.jpg|center|200px]]
|[[File:BE_HOMO_Endo.jpg|center|200px]]
|[[File:BE_LUMO_endo.jpg|center|200px]]
|}

The endo and exo products will be stabilised by secondary orbital overlap (SOO). This is the mixing of the HOMO orbital of one reactant with the LUMO orbital of the other<ref name="Orbital symmetries and endo-exo relationships in concerted cycloaddition reactions">, R. Hoffmann, R. B. Woodward, ''J. Am. Chem. Soc.'', '''1965''', ''87'', 4388-4389</ref>. This can be seen in both transition state HOMO orbitals above. In the case of the endo transition state the orbitals from the central oxygen shows a node around it, the orbitals on that oxygen are of the same symmetry and hence will interact with the orbitals from the double bond that is formed in the product above it. For the exo transition state the position of the oxygen and double bond are much separated and therefore there is no SOO. This SOO effect in the endo transition state, and hence the greater stabilisation, is a reason that the energy of the endo transition state is lower than that of the endo.

==Conclusion==

These reactions have shown the usefulness of molecular modelling in the visualization and analysis of transition states. A wide range of methods for optimizing the transition states have been considered and tested for their accuracy and ease of use. Despite initial difficulties the QTS2 method proved to be the most effective in determining the transition state of both the boat conformer in the Cope rearrangement and the endo and exo forms in the Diels-Alder reaction. It gave a good prediction for the geometries and energies of the transition states and allowed comparison between the reactants and products of the reactions. Further examples of Diels-Alder reactions could be looked at to determine the stereo and regio-selective implications of the mechanism and further prove the effectiveness of calculating activation energies by this method as compared to experimentally determined values. Also calculations from different basis sets could be run to determine the accuracy of the different levels of theory and to improve the calculated geometries and energies of the transition states studied in this experiment.

==References==

<references />

Rep:Mod:PhysBE411

2014-03-21T16:33:47Z

Be411: /* The Boat Transition State */

'''Physical Module: Transition States and Reactivity'''

Computational methods can be used to study reaction pathways and more specifically transition states. In this experiment the Cope rearrangement reaction and Diels-Alder cycloaddition reaction are used as examples of this. Using various methods the lowest energy conformation of 1,5-hexadiene can be found as well as the conformation that exists as a reactant in the Cope rearrangement reaction. The two possible transition states of this reaction can be studied and their activation energies calculated to gain some understanding of the mechanism of this reaction. The Diels-Alder cycloaddition will also be studied. The origin of the symmetry allowed reaction between ethene and cis-butadiene will be visualized and the reaction between maleic anhydride and hexa-1,5-diene studied in terms of the regioselectivity of its two conformational different transition states.

=='''The Cope Rearrangement Tutorial'''==

The Cope rearrangement is a specific [3,3] sigmatropic rearrangement in which a sigma bond moves and the pi system is redistributed across the molecule. In this case the Cope rearrangement of hexa-1,5-diene is considered with regard to its conformations and the boat and chair forms of its reaction's transition state.

==='''Optimizing the Reactants and Products'''===

===='''Conformation of 1,5-hexadiene'''====
[[File:BE_1,5-HEXADIENE_CONFORTMATIONS.jpg|thumb|The Conformations of 1,5-Hexadiene]]

1,5-hexadiene is one of the simplest precursors in the Cope rearrangement. The three rotating carbon-carbon bonds within the center of the molecule cause there to be 10 energetically different conformations, when taking into account enantiomers and symmetry.

Within these conformations you would expect that where two carbons are in the eclipsed form (F-J) the conformer will be of higher energy than conformers with CH-eclipsed bonds (A-E)<ref name="Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols">B. W. Gung, Z. Zhu and R. A. Fouch, ''J. Am. Chem. Soc'', '''1995''', ''117'', 1783-1788.</ref>. This is demonstrated in the case of butane where the difference in energy between the CC eclipsed form and the CH eclipsed form is 1.5kcal/mol<ref name="Website - Butane Conformational Analysis">[http://research.cm.utexas.edu/nbauld/teach/butane.html]N. Bauld, '''"Butane Conformational Analysis"''', ''University of Texas'', Retreived 20 March 2014.</ref>. The energies of these forms can be optimized by computational methods in order to find the most suitable candidates for the reactants and products in the Cope Rearrangement reaction.

The table below shows the total energy for all energetically distinct conformers of the molecule, it shows that in the case of this molecule the anti conformer is generally of higher energy. Relating these structures to the literature<ref name="Instructional Wiki">[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3]M. Bearpark, Imperial College, Physical Module: Transition states and reactivity</ref>, the energies found are all in line with the energies expected, as is also the case with the point groups.

The lowest energy conformation is the Gauche Conformer D (-231.69266Hartrees) this maybe due to an attractive interaction between the pi orbital of the double bond and the neighboring proton, whose orbital would interact with that of the pi bond<ref name="An Analysis of the Conformers of 1,5-hexadiene">B. G. Rocque, J. M. Gonzales, H. F. Schaeffer III, ''Mol. Phys'', '''2002''', ''100'', 441-446.</ref>. This conjugation will stabilize the molecule making it lower in energy.
The highest energy conformer of the CH eclipsed forms is C (-231.69153Hartrees) this maybe explained by interactions between the two alkene protons which are 2.51A apart. This proximity will have a repulsive effect increasing the overall energy of the molecule.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimized Energies for the Anti and Gauche Conformations of 1,5-Hexadiene (Relating to the Instructional Wiki<ref name="Instructional Wiki"/>)
!
! align="center" | '''<jmolFile text="Conformer A - Gauche2">BE_A.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer B - Anti1 ">BE_B.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer C - Gauche4">BE_C.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer D - Gauche3">BE_D.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer E - Anti2">BE_E.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer F - Gauche6">BE_F.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer G - Anti4">BE_G.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer H - Gauche5">BE_H.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer I - Gauche1">BE_I.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer J - Anti3">BE_J.mol</jmolFile>'''
|-
|'''Total Energy/Hartrees'''
| -231.69167
| -231.69260
| -231.69153
| -231.69266
| -231.69254
| -231.68916
| -231.69097
| -231.68962
| -231.68772
| -231.68907
|-
|'''Point Group'''
|C2
|C2
|C2
|C1
|Ci
|C1
|C1
|C1
|C2
|C2h
|-
|'''Dihedral Angle (C2-C3-C4-C5)'''
|64.17
| -176.91
| -63.65
|67.70
| -179.996
|70.23
| -178.32
| -71.09
|75.76
| -179.98
|}

===='''Analysis of Conformer E'''====

Further Analysis of Conformer E was done at a higher basis set in order to get an even better approximation for its lowest energy conformation. An optimization calculation was run as before at the B3LYP/6-31G(d)level of theory.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimised Conformations of Conformer E using the HF/3-21G and B3LYP/6-31G(d) Basis Sets
!<jmol><jmolApplet><title>Anti 2 Conformation at HF level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_E.mol</uploadedFileContents> </jmolApplet></jmol>
!<jmol><jmolApplet><title>Anti 2 Conformation at B3LYP level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_DFT_optmisation.mol</uploadedFileContents> </jmolApplet></jmol>
|}

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Geometrical Features of Conformer E Optimized Under Different Basis Sets
! rowspan ="2" | '''Basis Set'''
! colspan ="3" | '''Bond Length'''
! colspan ="3" | '''Angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3
|C2-C3-C4
|C2-C3-C4-C5
|-
|HF/3-21G
|1.32
|1.51
|1.55
|179.996
|124.81
|111.35
|-
|B3LYP/6-31G(d)
|1.33
|1.50
|1.55
|180
|125.29
|112.68
|}

[[image:BE_IR_Spectrum_of_Anti2.jpg|thumb|200px|IR Spectrum of Conformer E]]

This shows that the geometries of the results of the two different approaches don't differ wildly, the bond lengths especially are very similar, however there is some deviation between their angles.

Following on from this a frequency calculation was run using the same higher level of theory (B3LYP/6-31G(d)). This calculation can be used to confirm that this conformation is of a minimum energy by the absence of an imaginary vibrational frequency. The infrared spectrum of the molecule can be generated from the molecular vibrations and compared to literature.

The thermochemical energies for this molecule can be found in the output file of the calculation:

Sum of electronic and zero-point Energies= -234.46921 Hartrees
Sum of electronic and thermal Energies= -234.46186 Hartrees
Sum of electronic and thermal Enthalpies= -234.46091 Hartrees
Sum of electronic and thermal Free Energies= -234.50082 Hartrees

The various information above can be used to consider the potential and free energy of the system. The sum of the electronic and zero-point energies (E= Eelec + ZPE) show the potential energy at 0K and the sum of electronic and thermal energies (E = E + Evib + Erot + Etrans) considers it at 298.15K and 1atm. The sum of electronic and thermal enthalpies (H = E + RT) shows the enthalpy of the system containing a correction for thermal energy and the sum of electronic and thermal free energies ( G= H - TS) suggests the entropic contribution to the free energy of the system.

This data shows that the energy at 0K is lower than that at room temperature, this is as expected because at 0K there will be no internal movement of the molecule and hence no contribution from vibrational, rotational or translational energies.

==='''Optimizing the "Chair" and "Boat" Transition Structures'''===

Conformer E was considered for this reaction and used to model both the chair and boat transition states for the Cope rearrangement reaction. A number of different ways of optimizing the geometry of the transition states were considered including using redundant coordinates and QTS2(TS)optimization and frequency calculation.

===='''Chair Transition State'''====

The chair transition state was constructed from two allyl fragments. Firstly the geometry of a single allyl fragment was optimized using the HF/3-21G level of theory giving an energy of -117.61330Hartrees. Two allyl fragments are placed 2.2A apart and then using this as a guess for the transition state, the molecule is then optimized in two different ways.

[[File:BE_B_vibration.gif|thumb|Cope Rearrangement Chair Transition State Vibration]]

Firstly it can be optimized using a opt + freq calculation to a TS using the Berny Method. By calculating the force constant once an imaginary vibration can be found. The imaginary vibration was found at 817.95cm-1 (expected value of 818cm-1<ref name="Instructional Wiki"/>) on animation it can be seen that this corresponds to the coming together of the two terminal carbons on each allyl molecule, the movement expected during this reaction. In this case the total energy is -231.61932Hartrees.

[[File:BE_D_ChairTS.jpg|thumb|left|Chair Transition State by Redundant Coordinate Method]]
Secondly the transition state was optimized by freezing the reaction coordinate in order to minimize the rest of the molecule first and then unfreezing it in order to find the minimum for the transition state. This means that the force constant Hessian matrix doesn't need to be calculated and therefore is a faster method of computing the transition state.

By looking at the atom distances between the bond that will be formed the suitability of both of these methods to find the true transition state can be studied. All the measured geometric parameters were identical in both versions of the transition state, namely the intranuclear bond lengths was 1.39A, the internuclear distance between the two allyl molecules is 2.02A and the angle within the each molecule is 120.05o. This suggests that both methods for determining the transition state are equally effect from the same guess input file. It also confirms that the guess structure was close to the exact structure as the first method was as successful in calculating the force constants as the second which doesn't need to calculate them.

===='''The Boat Transition State'''====

The boat transition state was constructed in much the same way as the chair using two pre-optimized allyl fragments, placing them together in the correct geometry and setting them to 2.2A apart. In the case of the boat transition state the QTS2 method was used to optimize the energy. By ensuring that the same labeling system is used in the reactants and products, an optimization and frequency calculation was run that could help define the geometry of the boat transition state. This calculation was run with a predicted transition state with a dihedral angle (C2-C3-C4-C5) of 0o and inside C2-C3-C4 and C3-C4-C5 angles of 100o.

[[File:BE_E_Vibration.gif|thumb|200px|Cope Rearrangement Boat Transition State Vibration]]

The result of this calculation shows the boat transition state with a single imaginary vibration of -840nm and energy of -231.60280Hartrees. Visualizing this vibration shows that it corresponds to the coming together of the two terminal carbons and the separation of the two central carbons. This is the expected vibration seen during the Cope rearrangement.
The QTS2 method is advantageous because it is fully automated however it required a good starting guess for the transition state and therefore a lot of permutations of structure, in the input file, were needed for a successful calculation.

{| class="wikitable" style="margin: auto;"
|+ Summary Table of the Energies of the Two Transition States
! rowspan="2" | Transition State
! colspan="3" | Energy
|-
|<jmolFile text="'''Berny TS'''">BE_ChairB.mol</jmolFile>
|<jmolFile text="'''Redunant Coordinates'''">BE_chairD.mol</jmolFile>
|<jmolFile text="'''QTS2 (TS)'''">BE_Boat_TS_Again2.mol</jmolFile>
|-
!chair
| -231.61932
| -231.61932
|
|-
!Boat
|
|
| -231.60280
|}

===='''The Reaction Pathway'''====

The reaction pathway can be studied through both the minimum energy position on its potential energy surface and the activation energies of the two transition states.

====='''IRC'''=====

[[File:BE_IRC_Plot.jpg|thumb|400px|IRC Energy Plot of the Chair Transition State]]
An intrinsic reaction coordinate (IRC) was run as it can be used to determine the minimum energy pathway from the transition state to its local minimum by using many points along the pathway to the point where the energy plateaus. The IRC in this case has been run in the forward direction only and the force constants calculated at every step.

The lowest energy conformation is achieved after 44 steps. The structure of the first and final conformations can be compared by considering their different geometry and energy. The IRC determines the minimum energy conformation and therefore the energy of step 44 (-231.69158Hartrees) is, as expected, lower than that of the first step (-231.61932Hartrees), this confirms that the calculation has found a minimum. The most noticeable different between the two conformations is the bond symmetry, in step 1 the bond lengths within each molecule were identical (1.39A) and the distance between the two molecules is the same (2.02A) for both sets of terminal carbons. However in step 44 the bond lengths within each molecule are different in preparation for the formation of two different types of bonds - single and double - this means one bond is shorter than the other (1.32 and 1.51A). As well as this difference in intranuclear distances there is also a difference in the internuclear distances as the one end approaches much closer (1.55 and 4.23A) almost within the van der Waals radius of the carbon atom as it prepares to form a new bond whereas the other distance is much larger as there is no bond formation at those carbons.

{| class="wikitable" style="margin: auto;"
|+ Structures of Steps 1 and 44 in the Determination of the IRC of the Chair Transition State
! [[File:BE_F1_Image.jpg|400px]]
! [[File:BE_F44_Image.jpg|400px]]
|}

====='''Activation Energies'''=====

In order to determine the activation energy of the reaction via the two different transition states a higher level of theory needed to be used. The B3LYP/6-31G(d) was used to optimize the two transition states to a minimum. Comparing this to the activation energies obtained at the lower level of theory and literature values<ref name="Instructional Wiki"/> shows the large difference in the energies obtained at each level of theory. The Energies at 0 and 298.15K were then found by running a frequency calculation, using the same basis set, on the optimised structure and checking the sum of the electronic and zero-point (0K) and sum of electronic and thermal energies (298.15K).

{| class="wikitable" style="margin: auto;"
|+ Geometrical Features of the Reactant, Boat and Chair Transition States
!
! colspan="2" | Reactants
! colspan="2" | Chair Transition State
! colspan="2" | Boat Transition State
|-
|
|<jmol><jmolApplet><title>Reactants Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Reactants Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Transition State Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Transition State Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Transition State Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again2.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Transition State Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again3_-_EA.mol</uploadedFileContents> </jmolApplet></jmol>
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
!Intramolecular Bond Lengths/A
|1.32, 1.51
|1.33, 1.50
|1.39
|1.41
|1.38
|1.39
|-
!Intermolecular Bond Lengths/A
|1.55
|1.55
|2.02
|1.97
|2.14
|2.21
|-
!Dihedral Angle/o
|179.996
|180
|54.96
|54
|0
|0
|}

In the reactants the higher level of theory shows a tightening of the dihedral angle to exactly 180o as well as a shortening of the single bonds and lengthening of the double bonds within the molecule. This acts to bring the bond lengths within the molecule closer to the expected bond lengths of 1.47 and 1.54A<ref name="Wiki Book">M. A. Fox, J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen.'', '''1995''', Springer.</ref>. Within chair transition state the same effect can be seen, increasing the dihedral angle and double bond length and decreasing the single and intermolecular bond distances.

{| class="wikitable" style="margin: auto;"
|+ Electronic Energies and Energies of the Boat and Chair Transition States at 0K
!
! colspan="2" | '''Electronic Energy/Hartrees'''
! colspan="2" | '''Sum of Electronic and Zero-Point Energies'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.60280
| -234.54309
| -231.45093
| -234.40234
| 0.08861
| 0.06687
| 55.60
| 41.96
| 44.7 ± 2.0
|-
|'''Chair Transition State'''
| -231.69167
| -234.55698
| -231.46669
| -234.41488
| 0.07286
| 0.05433
| 45.72
| 34.09
| 33.5 ± 0.5
|}

The activation energies were calculated with reference to the conformer E energies as the reactant ( At 0K HF/3-21G = -231.53954 and B2LYP/6-31G(d) = -234.46921Hartrees, At 298.15K HF/3-21G = -231.53257 and B2LYP/6-31G(d) = -234.46186Hartrees ).

{| class="wikitable" style="margin: auto;"
|+ Energies of the Boat and Chair Transition States at 298.15K
!
! colspan="2" | '''Sum of Electronic and Thermal Energies'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.44530
| -234.39599
| 0.08727
| 0.06587
| 54.76
| 41.33
| 41.32
|-
|'''Chair Transition State'''
| -231.46133
| -234.40895
| 0.07124
| 0.05291
| 44.70
| 33.20
| 33.17
|}

From this it is clear why a higher level of theory is needed. The energies calculated at the B3LYP/6-31G(d) level yield much more accurate activation energies that those at HF/3-21G. In both cases this increase in accuracy has brought the activation energies significantly closer to the expected values.

The activation energy of the Chair transition state is a good match for that in the literature and is very close to the boundaries of error given. When comparing this to the energy obtained at the lower level of theory there is a much greater difference and therefore illustrates the ability of the higher level of energy to calculate a much more accurate transition state.

=='''Diels-Alder Cycloaddition Reaction'''==

The Diels-Alder cycloaddition reaction is a pericyclic reaction that occurs between two pi systems, a diene and a dienophile, in order to form a cyclohexene. It offers good regio- and stereo-selective control that can be extended to other conjugated systems<ref name="The Diels-Alder Reaction in Total Synthesis">K. C. Nicolaou, S. A. Snyder, T. Montagnon and G. Vassilikogiannakis ''Angewandte Chemie International Edition'','''2002''', ''41'', 1668-1698.</ref>.

In this experiment the reaction between cis butadiene and ethene will be studied in order to determine the symmetry of both the reactant and product molecular orbitals, as well as the geometry of the reaction's transition state. The regioselectivity of the reaction will also be looked at into order to determine the preferred product of the reaction, endo or exo.

==='''Cis-butadiene'''===

Cis-butadiene will be used as the diene in this studied reaction with ethene. Its geometry was first optimized using the HF/3-21G level of theory. The final energy of the product was -154.05514Hartrees. The molecular orbitals of the optimized structure can be modeled and used further to look at the electronic contribution of the diene to the final molecule.

{| class="wikitable" style="margin: auto;"
|+ Images of the Molecular Orbitals of Cis-Butadiene
! [[File:BE_I_HOMO.jpg|200px]]
! rowspan="3" | [[File:BE_Diene.jpg|150px]]
! [[File:BE_I_LUMO.jpg|200px]]
|-
![[File:BE_I_HOMO2.jpg|200px]]
![[File:BE_I_LUMO2.jpg|200px]]
|-
| align="center" | The HOMO orbital of the molecule is asymmetric. The nodes in the molecule fall between the red and green lobes of the HOMO. This shows the two separate pi systems within the molecule, but the proximity of the two systems suggests that the molecule could be fully conjugated.
| align="center" | The LUMO orbital of the molecule is symmetric. It shows the pi* orbitals of the two pi systems and the conjugation of these across the central single bond. The nodes fall within the plane of the molecule and between the double bonds of the molecule suggesting that for this orbital there is no conjugation over the double bonds.
|}

==='''The Transition State'''===

A number of methods, that had previously been discussed above for the Cope rearrangement reaction, were considered to optimize the transition state of this reaction. However it was decided to use a method previously used in optimizing the chair transition state of 1,5-hexadiene due to it's simplicity and that a good guess for the transition state could be made initially.

Firstly the reactants were optimized separately under the HF/3-21G level of theory. Then the two molecules were combined in an envelope like geometry and the internuclear distances between the two molecules set to 2.2A in order to give a best guess for the transition state. An optimization and frequency calculation was then run, optimizing to a transition state using the Berny method, calculating the force constants only once.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Structure and Molecular Orbitals of the Diels-Alder Transition State
![[File:BE_TSHOMO.jpg|center|200px]][[File:BE_TSHOMO2.jpg|center|200px]]
!<jmol><jmolApplet><title>1,5-hexdiene formation transition state</title><color>white</color> <size>300</size><uploadedFileContents>BE_Reactants.mol</uploadedFileContents> </jmolApplet></jmol>
|[[File:BE_TSLUMO.jpg|center|200px]][[File:BE_TSLUMO2.jpg|center|200px]]
|-
! align="center" | HOMO
| align="center" | The transition state for the Diels-Alder reaction between ethene and cis-butadiene has an energy of -231.60321Hartrees
! align="center" | LUMO
|}

[[File:BE_TSVibration.gif|thumb|left|Imaginary Vibration of the Diels-Alder Transition State]][[File:BE_Positive_Vibration.gif|thumb|right|Lowest Energy Positive Vibration of the Diels-Alder Transition State]]

This gave a good approximation for the transition state with the imaginary vibration, of -817.71cm-1, showing the two molecules approaching each other in a synchronous motion. Comparatively the lowest positive vibration at 166.54cm-1 is an asynchronous rocking motion that does not contribute to the coming together of the two molecules.

[[File:BE_TSDIELSALDERIR_Spectrum.png|thumb|center|300px|Predicted IR Spectrum of the Diels-Alder Transition State]]

The justification of calling this the transition state is also shown in the final bond lengths between the two molecules, the length of approach is 2.21A. Typically an sp3-sp3 carbon bond length is 1.54A and for a sp2-sp2 carbon it is 1.47A<ref name="Wiki Book"/>this suggests that a bond has not been fully formed in the transition state. However it is within the van der Waals radius of the carbon atom, which is 1.7A<ref name="Van der Waals Volumes and Radii">A. Bondi, ''J. Phys. Chem.'','''1964''', ''68 '', 441-51.</ref> so a partial bond may have started to form at this point in the reaction.

Looking at the HOMO and LUMO orbitals for the reactants there is a clear contribution from both to the molecular orbitals of the transition state, this is especially the case for the ethene molecule. The visualization of the HOMO orbital throws some doubt onto the final structure obtained however considering the imaginary vibration is as expected, it can be assumed that this is a form of the transition state, if not the most minimal energy one. Further calculations were run in order to find the transition state with the correct HOMO orbital however it could not be found with the correct geometry and imaginary vibration.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Molecular Orbitals of Ethene and Cis-Butadiene
! colspan="4" | Ethene
! colspan="4" | cis-butadiene
|-
|[[File:BE_Ethene_HOMO.jpg|center|100px]]
|[[File:BE_Ethene_HOMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO.jpg|center|100px]]
|[[File:BE_I_HOMO.jpg|center|100px]]
|[[File:BE_I_HOMO2.jpg|center|100px]]
|[[File:BE_I_LUMO.jpg|center|100px]]
|[[File:BE_I_LUMO2.jpg|center|100px]]
|-
! colspan="2" | HOMO - Symmetric
! colspan="2" | LUMO - Asymmetric
! colspan="2" | HOMO - Asymmetric
! colspan="2" | LUMO - Symmetric
|}

The reactants orbitals above display frontier molecular orbital theory. The HOMO of cis-butadiene and the LUMO of ethene are of correct symmetry to interact to form a bonding molecular orbital in the product. This means that the reaction is symmetry allowed and therefore no other conditions are needed, such as light, in order for the reaction to proceed<ref>[http://www.ch.ic.ac.uk/rzepa/] Henry S. Rzepa, Imperial College, Organic Pericyclic Reactions: Categories</ref>.

==='''Regioselectivity'''===

The reaction can form two different products depending on the transition state. The exo product is formed when the approach of the maleic anhydride occurs with the oxygens pointing away from the double bond, and the endo product is formed when the oxygens point towards it.
In order to determine the energies of the transition states first the reactants were optimized separately and then together using the HF/3-21G level of theory. Then the endo and exo products were optimized the same way. The transition state was found using the QTS2 method, an Opt+Freq calculation under the HF/3-21G level of theory, this gave an optimized version of each transition state. This was used in order to utilise the reactants and products to find the transition state as the initial guess for the transition state was not a confident enough one.

The imaginary vibrations of these transition at -647.54 and -643.32cm-1 for the exo and endo transition states respectively. These vibrations show the coming together of the reactant molecules to form the product as is expected. These Transition state structures were then optimized under a higher level of theory in order to obtain a more accurate energy, and the HOMO and LUMO orbitals of each visualized.

[[File:BE_Exo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Exo Transition State]][[File:BE_Endo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Endo Transition State]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Energies of the Exo and Endo Transition States under the B3LYP/6-31G(d) Level of Theory
!
! Exo Transition State
! Endo Transition State
|-
|
|<jmol><jmolApplet><title>EXO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_Exo_TS_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>ENDO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_ENDO_TS_3_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|-
!Total Energy/Hartrees
| -612.66122
| -612.68340
|-
!Sum of Electronic and Zero-Point Energies/Hartrees
| -612.47669
| -612.50214
|-
!Sum of Electronic and Thermal Energies/Hartrees
| -612.46612
| -612.49179
|-
!Activation Energy at 0K/kcalmol-1
| 34.90
| 18.93
|-
!Activation Energy at 298.15K/kcalmol-1
| 33.58
| 17.47
|-
!Dihedral Angle/o
| -169.56
| -175.67
|-
!Intermolecular Bond Lengths/A
|1.5
|2.27
|}

Activation energies were calculated by comparing the energy of the transition state to the energy of the reactants (At 0K B3LYP/6-31G(d) = -612.53231, At 298.15K B3LYP/6-31G(d) = -612.51964). The endo product should be the kinetic product, this is shown by its much lower activation energy. The exo product has a tighter dihedral angle than the endo, this is due to the repulsion between the two reactants as seen in the HOMO of the exo transition state below<ref name="Steric Effects vs. Secondary Orbital Overlap in Diels-Alder Reactions. MNDO and AM1 Studies">, M. A. Fox, R. Cardons and N. J. Kiwiet, ''J. Org. Chem.'', '''1987''', ''52'',1469-1474</ref>.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The HOMO and LUMO Molecular Orbitals of the Exo and Endo Transition States
! colspan="2" | Exo Transition State
! colspan="2" | Endo Transition State
|-
!HOMO
!LUMO
!HOMO
!LUMO
|-
|[[File:BE_HOMO_exo.jpg|center|200px]]
|[[File:BE_LUMO_Exo.jpg|center|200px]]
|[[File:BE_HOMO_Endo.jpg|center|200px]]
|[[File:BE_LUMO_endo.jpg|center|200px]]
|}

The endo and exo products will be stabilised by secondary orbital overlap (SOO). This is the mixing of the HOMO orbital of one reactant with the LUMO orbital of the other<ref name="Orbital symmetries and endo-exo relationships in concerted cycloaddition reactions">, R. Hoffmann, R. B. Woodward, ''J. Am. Chem. Soc.'', '''1965''', ''87'', 4388-4389</ref>. This can be seen in both transition state HOMO orbitals above. In the case of the endo transition state the orbitals from the central oxygen shows a node around it, the orbitals on that oxygen are of the same symmetry and hence will interact with the orbitals from the double bond that is formed in the product above it. For the exo transition state the position of the oxygen and double bond are much separated and therefore there is no SOO. This SOO effect in the endo transition state, and hence the greater stabilisation, is a reason that the energy of the endo transition state is lower than that of the endo.

==Conclusion==

These reactions have shown the usefulness of molecular modelling in the visualization and analysis of transition states. A wide range of methods for optimizing the transition states have been considered and tested for their accuracy and ease of use. Despite initial difficulties the QTS2 method proved to be the most effective in determining the transition state of both the boat conformer in the Cope rearrangement and the endo and exo forms in the Diels-Alder reaction. It gave a good prediction for the geometries and energies of the transition states and allowed comparison between the reactants and products of the reactions. Further examples of Diels-Alder reactions could be looked at to determine the stereo and regio-selective implications of the mechanism and further prove the effectiveness of calculating activation energies by this method as compared to experimentally determined values. Also calculations from different basis sets could be run to determine the accuracy of the different levels of theory and to improve the calculated geometries and energies of the transition states studied in this experiment.

==References==

<references />

File:BE IR Spectrum of Anti2.jpg

2014-03-21T16:32:58Z

Be411: uploaded a new version of "File:BE IR Spectrum of Anti2.jpg"

Rep:Mod:PhysBE411

2014-03-21T16:31:43Z

Be411: /* Analysis of Conformer E */

'''Physical Module: Transition States and Reactivity'''

Computational methods can be used to study reaction pathways and more specifically transition states. In this experiment the Cope rearrangement reaction and Diels-Alder cycloaddition reaction are used as examples of this. Using various methods the lowest energy conformation of 1,5-hexadiene can be found as well as the conformation that exists as a reactant in the Cope rearrangement reaction. The two possible transition states of this reaction can be studied and their activation energies calculated to gain some understanding of the mechanism of this reaction. The Diels-Alder cycloaddition will also be studied. The origin of the symmetry allowed reaction between ethene and cis-butadiene will be visualized and the reaction between maleic anhydride and hexa-1,5-diene studied in terms of the regioselectivity of its two conformational different transition states.

=='''The Cope Rearrangement Tutorial'''==

The Cope rearrangement is a specific [3,3] sigmatropic rearrangement in which a sigma bond moves and the pi system is redistributed across the molecule. In this case the Cope rearrangement of hexa-1,5-diene is considered with regard to its conformations and the boat and chair forms of its reaction's transition state.

==='''Optimizing the Reactants and Products'''===

===='''Conformation of 1,5-hexadiene'''====
[[File:BE_1,5-HEXADIENE_CONFORTMATIONS.jpg|thumb|The Conformations of 1,5-Hexadiene]]

1,5-hexadiene is one of the simplest precursors in the Cope rearrangement. The three rotating carbon-carbon bonds within the center of the molecule cause there to be 10 energetically different conformations, when taking into account enantiomers and symmetry.

Within these conformations you would expect that where two carbons are in the eclipsed form (F-J) the conformer will be of higher energy than conformers with CH-eclipsed bonds (A-E)<ref name="Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols">B. W. Gung, Z. Zhu and R. A. Fouch, ''J. Am. Chem. Soc'', '''1995''', ''117'', 1783-1788.</ref>. This is demonstrated in the case of butane where the difference in energy between the CC eclipsed form and the CH eclipsed form is 1.5kcal/mol<ref name="Website - Butane Conformational Analysis">[http://research.cm.utexas.edu/nbauld/teach/butane.html]N. Bauld, '''"Butane Conformational Analysis"''', ''University of Texas'', Retreived 20 March 2014.</ref>. The energies of these forms can be optimized by computational methods in order to find the most suitable candidates for the reactants and products in the Cope Rearrangement reaction.

The table below shows the total energy for all energetically distinct conformers of the molecule, it shows that in the case of this molecule the anti conformer is generally of higher energy. Relating these structures to the literature<ref name="Instructional Wiki">[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3]M. Bearpark, Imperial College, Physical Module: Transition states and reactivity</ref>, the energies found are all in line with the energies expected, as is also the case with the point groups.

The lowest energy conformation is the Gauche Conformer D (-231.69266Hartrees) this maybe due to an attractive interaction between the pi orbital of the double bond and the neighboring proton, whose orbital would interact with that of the pi bond<ref name="An Analysis of the Conformers of 1,5-hexadiene">B. G. Rocque, J. M. Gonzales, H. F. Schaeffer III, ''Mol. Phys'', '''2002''', ''100'', 441-446.</ref>. This conjugation will stabilize the molecule making it lower in energy.
The highest energy conformer of the CH eclipsed forms is C (-231.69153Hartrees) this maybe explained by interactions between the two alkene protons which are 2.51A apart. This proximity will have a repulsive effect increasing the overall energy of the molecule.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimized Energies for the Anti and Gauche Conformations of 1,5-Hexadiene (Relating to the Instructional Wiki<ref name="Instructional Wiki"/>)
!
! align="center" | '''<jmolFile text="Conformer A - Gauche2">BE_A.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer B - Anti1 ">BE_B.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer C - Gauche4">BE_C.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer D - Gauche3">BE_D.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer E - Anti2">BE_E.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer F - Gauche6">BE_F.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer G - Anti4">BE_G.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer H - Gauche5">BE_H.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer I - Gauche1">BE_I.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer J - Anti3">BE_J.mol</jmolFile>'''
|-
|'''Total Energy/Hartrees'''
| -231.69167
| -231.69260
| -231.69153
| -231.69266
| -231.69254
| -231.68916
| -231.69097
| -231.68962
| -231.68772
| -231.68907
|-
|'''Point Group'''
|C2
|C2
|C2
|C1
|Ci
|C1
|C1
|C1
|C2
|C2h
|-
|'''Dihedral Angle (C2-C3-C4-C5)'''
|64.17
| -176.91
| -63.65
|67.70
| -179.996
|70.23
| -178.32
| -71.09
|75.76
| -179.98
|}

===='''Analysis of Conformer E'''====

Further Analysis of Conformer E was done at a higher basis set in order to get an even better approximation for its lowest energy conformation. An optimization calculation was run as before at the B3LYP/6-31G(d)level of theory.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimised Conformations of Conformer E using the HF/3-21G and B3LYP/6-31G(d) Basis Sets
!<jmol><jmolApplet><title>Anti 2 Conformation at HF level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_E.mol</uploadedFileContents> </jmolApplet></jmol>
!<jmol><jmolApplet><title>Anti 2 Conformation at B3LYP level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_DFT_optmisation.mol</uploadedFileContents> </jmolApplet></jmol>
|}

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Geometrical Features of Conformer E Optimized Under Different Basis Sets
! rowspan ="2" | '''Basis Set'''
! colspan ="3" | '''Bond Length'''
! colspan ="3" | '''Angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3
|C2-C3-C4
|C2-C3-C4-C5
|-
|HF/3-21G
|1.32
|1.51
|1.55
|179.996
|124.81
|111.35
|-
|B3LYP/6-31G(d)
|1.33
|1.50
|1.55
|180
|125.29
|112.68
|}

[[image:BE_IR_Spectrum_of_Anti2.jpg|thumb|200px|IR Spectrum of Conformer E]]

This shows that the geometries of the results of the two different approaches don't differ wildly, the bond lengths especially are very similar, however there is some deviation between their angles.

Following on from this a frequency calculation was run using the same higher level of theory (B3LYP/6-31G(d)). This calculation can be used to confirm that this conformation is of a minimum energy by the absence of an imaginary vibrational frequency. The infrared spectrum of the molecule can be generated from the molecular vibrations and compared to literature.

The thermochemical energies for this molecule can be found in the output file of the calculation:

Sum of electronic and zero-point Energies= -234.46921 Hartrees
Sum of electronic and thermal Energies= -234.46186 Hartrees
Sum of electronic and thermal Enthalpies= -234.46091 Hartrees
Sum of electronic and thermal Free Energies= -234.50082 Hartrees

The various information above can be used to consider the potential and free energy of the system. The sum of the electronic and zero-point energies (E= Eelec + ZPE) show the potential energy at 0K and the sum of electronic and thermal energies (E = E + Evib + Erot + Etrans) considers it at 298.15K and 1atm. The sum of electronic and thermal enthalpies (H = E + RT) shows the enthalpy of the system containing a correction for thermal energy and the sum of electronic and thermal free energies ( G= H - TS) suggests the entropic contribution to the free energy of the system.

This data shows that the energy at 0K is lower than that at room temperature, this is as expected because at 0K there will be no internal movement of the molecule and hence no contribution from vibrational, rotational or translational energies.

==='''Optimizing the "Chair" and "Boat" Transition Structures'''===

Conformer E was considered for this reaction and used to model both the chair and boat transition states for the Cope rearrangement reaction. A number of different ways of optimizing the geometry of the transition states were considered including using redundant coordinates and QTS2(TS)optimization and frequency calculation.

===='''Chair Transition State'''====

The chair transition state was constructed from two allyl fragments. Firstly the geometry of a single allyl fragment was optimized using the HF/3-21G level of theory giving an energy of -117.61330Hartrees. Two allyl fragments are placed 2.2A apart and then using this as a guess for the transition state, the molecule is then optimized in two different ways.

[[File:BE_B_vibration.gif|thumb|Cope Rearrangement Chair Transition State Vibration]]

Firstly it can be optimized using a opt + freq calculation to a TS using the Berny Method. By calculating the force constant once an imaginary vibration can be found. The imaginary vibration was found at 817.95cm-1 (expected value of 818cm-1<ref name="Instructional Wiki"/>) on animation it can be seen that this corresponds to the coming together of the two terminal carbons on each allyl molecule, the movement expected during this reaction. In this case the total energy is -231.61932Hartrees.

[[File:BE_D_ChairTS.jpg|thumb|left|Chair Transition State by Redundant Coordinate Method]]
Secondly the transition state was optimized by freezing the reaction coordinate in order to minimize the rest of the molecule first and then unfreezing it in order to find the minimum for the transition state. This means that the force constant Hessian matrix doesn't need to be calculated and therefore is a faster method of computing the transition state.

By looking at the atom distances between the bond that will be formed the suitability of both of these methods to find the true transition state can be studied. All the measured geometric parameters were identical in both versions of the transition state, namely the intranuclear bond lengths was 1.39A, the internuclear distance between the two allyl molecules is 2.02A and the angle within the each molecule is 120.05o. This suggests that both methods for determining the transition state are equally effect from the same guess input file. It also confirms that the guess structure was close to the exact structure as the first method was as successful in calculating the force constants as the second which doesn't need to calculate them.

===='''The Boat Transition State'''====

The boat transition state was constructed in much the same way as the chair using two pre-optimized allyl fragments, placing them together in the correct geometry and setting them to 2.2A apart. In the case of the boat transition state the QTS2 method was used to optimize the energy. By ensuring that the same labeling system is used in the reactants and products, an optimization and frequency calculation was run that could help define the geometry of the boat transition state. This calculation was run with a predicted transition state with a dihedral angle (C2-C3-C4-C5) of 0o and inside C2-C3-C4 and C3-C4-C5 angles of 100o.

[[File:BE_E_Vibration.gif|thumb|200px|Cope Rearrangement Boat Transition State Vibration]]

The result of this calculation shows the boat transition state with a single imaginary vibration of -840nm and energy of -231.60280Hartrees. Visualizing this vibration shows that it corresponds to the coming together of the two terminal carbons and the separation of the two central carbons. This is the expected vibration seen during the Cope rearrangement.
The QTS2 method is advantageous because it is fully automated however it required a good starting guess for the transition state and therefore a lot of permutations of structure, in the input file, were needed for a successful calculation.

{| class="wikitable" style="margin: auto;"
|+ Summary Table of the Energies of the Two Transition States
! rowspan="2" | Transition State
! colspan="3" | Energy
|-
|<jmolFile text="Berny TS">BE_ChairB.mol</jmolFile>
|<jmolFile text="Redunant Coordinates">BE_chairD.mol</jmolFile>
|<jmolFile text="QTS2 (TS)">BE_Boat_TS_Again2.mol</jmolFile>
|-
|chair
| -231.61932
| -231.61932
|
|-
|Boat
|
|
| -231.60280
|}

===='''The Reaction Pathway'''====

The reaction pathway can be studied through both the minimum energy position on its potential energy surface and the activation energies of the two transition states.

====='''IRC'''=====

[[File:BE_IRC_Plot.jpg|thumb|400px|IRC Energy Plot of the Chair Transition State]]
An intrinsic reaction coordinate (IRC) was run as it can be used to determine the minimum energy pathway from the transition state to its local minimum by using many points along the pathway to the point where the energy plateaus. The IRC in this case has been run in the forward direction only and the force constants calculated at every step.

The lowest energy conformation is achieved after 44 steps. The structure of the first and final conformations can be compared by considering their different geometry and energy. The IRC determines the minimum energy conformation and therefore the energy of step 44 (-231.69158Hartrees) is, as expected, lower than that of the first step (-231.61932Hartrees), this confirms that the calculation has found a minimum. The most noticeable different between the two conformations is the bond symmetry, in step 1 the bond lengths within each molecule were identical (1.39A) and the distance between the two molecules is the same (2.02A) for both sets of terminal carbons. However in step 44 the bond lengths within each molecule are different in preparation for the formation of two different types of bonds - single and double - this means one bond is shorter than the other (1.32 and 1.51A). As well as this difference in intranuclear distances there is also a difference in the internuclear distances as the one end approaches much closer (1.55 and 4.23A) almost within the van der Waals radius of the carbon atom as it prepares to form a new bond whereas the other distance is much larger as there is no bond formation at those carbons.

{| class="wikitable" style="margin: auto;"
|+ Structures of Steps 1 and 44 in the Determination of the IRC of the Chair Transition State
! [[File:BE_F1_Image.jpg|400px]]
! [[File:BE_F44_Image.jpg|400px]]
|}

====='''Activation Energies'''=====

In order to determine the activation energy of the reaction via the two different transition states a higher level of theory needed to be used. The B3LYP/6-31G(d) was used to optimize the two transition states to a minimum. Comparing this to the activation energies obtained at the lower level of theory and literature values<ref name="Instructional Wiki"/> shows the large difference in the energies obtained at each level of theory. The Energies at 0 and 298.15K were then found by running a frequency calculation, using the same basis set, on the optimised structure and checking the sum of the electronic and zero-point (0K) and sum of electronic and thermal energies (298.15K).

{| class="wikitable" style="margin: auto;"
|+ Geometrical Features of the Reactant, Boat and Chair Transition States
!
! colspan="2" | Reactants
! colspan="2" | Chair Transition State
! colspan="2" | Boat Transition State
|-
|
|<jmol><jmolApplet><title>Reactants Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Reactants Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Transition State Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Transition State Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Transition State Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again2.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Transition State Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again3_-_EA.mol</uploadedFileContents> </jmolApplet></jmol>
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
!Intramolecular Bond Lengths/A
|1.32, 1.51
|1.33, 1.50
|1.39
|1.41
|1.38
|1.39
|-
!Intermolecular Bond Lengths/A
|1.55
|1.55
|2.02
|1.97
|2.14
|2.21
|-
!Dihedral Angle/o
|179.996
|180
|54.96
|54
|0
|0
|}

In the reactants the higher level of theory shows a tightening of the dihedral angle to exactly 180o as well as a shortening of the single bonds and lengthening of the double bonds within the molecule. This acts to bring the bond lengths within the molecule closer to the expected bond lengths of 1.47 and 1.54A<ref name="Wiki Book">M. A. Fox, J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen.'', '''1995''', Springer.</ref>. Within chair transition state the same effect can be seen, increasing the dihedral angle and double bond length and decreasing the single and intermolecular bond distances.

{| class="wikitable" style="margin: auto;"
|+ Electronic Energies and Energies of the Boat and Chair Transition States at 0K
!
! colspan="2" | '''Electronic Energy/Hartrees'''
! colspan="2" | '''Sum of Electronic and Zero-Point Energies'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.60280
| -234.54309
| -231.45093
| -234.40234
| 0.08861
| 0.06687
| 55.60
| 41.96
| 44.7 ± 2.0
|-
|'''Chair Transition State'''
| -231.69167
| -234.55698
| -231.46669
| -234.41488
| 0.07286
| 0.05433
| 45.72
| 34.09
| 33.5 ± 0.5
|}

The activation energies were calculated with reference to the conformer E energies as the reactant ( At 0K HF/3-21G = -231.53954 and B2LYP/6-31G(d) = -234.46921Hartrees, At 298.15K HF/3-21G = -231.53257 and B2LYP/6-31G(d) = -234.46186Hartrees ).

{| class="wikitable" style="margin: auto;"
|+ Energies of the Boat and Chair Transition States at 298.15K
!
! colspan="2" | '''Sum of Electronic and Thermal Energies'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.44530
| -234.39599
| 0.08727
| 0.06587
| 54.76
| 41.33
| 41.32
|-
|'''Chair Transition State'''
| -231.46133
| -234.40895
| 0.07124
| 0.05291
| 44.70
| 33.20
| 33.17
|}

From this it is clear why a higher level of theory is needed. The energies calculated at the B3LYP/6-31G(d) level yield much more accurate activation energies that those at HF/3-21G. In both cases this increase in accuracy has brought the activation energies significantly closer to the expected values.

The activation energy of the Chair transition state is a good match for that in the literature and is very close to the boundaries of error given. When comparing this to the energy obtained at the lower level of theory there is a much greater difference and therefore illustrates the ability of the higher level of energy to calculate a much more accurate transition state.

=='''Diels-Alder Cycloaddition Reaction'''==

The Diels-Alder cycloaddition reaction is a pericyclic reaction that occurs between two pi systems, a diene and a dienophile, in order to form a cyclohexene. It offers good regio- and stereo-selective control that can be extended to other conjugated systems<ref name="The Diels-Alder Reaction in Total Synthesis">K. C. Nicolaou, S. A. Snyder, T. Montagnon and G. Vassilikogiannakis ''Angewandte Chemie International Edition'','''2002''', ''41'', 1668-1698.</ref>.

In this experiment the reaction between cis butadiene and ethene will be studied in order to determine the symmetry of both the reactant and product molecular orbitals, as well as the geometry of the reaction's transition state. The regioselectivity of the reaction will also be looked at into order to determine the preferred product of the reaction, endo or exo.

==='''Cis-butadiene'''===

Cis-butadiene will be used as the diene in this studied reaction with ethene. Its geometry was first optimized using the HF/3-21G level of theory. The final energy of the product was -154.05514Hartrees. The molecular orbitals of the optimized structure can be modeled and used further to look at the electronic contribution of the diene to the final molecule.

{| class="wikitable" style="margin: auto;"
|+ Images of the Molecular Orbitals of Cis-Butadiene
! [[File:BE_I_HOMO.jpg|200px]]
! rowspan="3" | [[File:BE_Diene.jpg|150px]]
! [[File:BE_I_LUMO.jpg|200px]]
|-
![[File:BE_I_HOMO2.jpg|200px]]
![[File:BE_I_LUMO2.jpg|200px]]
|-
| align="center" | The HOMO orbital of the molecule is asymmetric. The nodes in the molecule fall between the red and green lobes of the HOMO. This shows the two separate pi systems within the molecule, but the proximity of the two systems suggests that the molecule could be fully conjugated.
| align="center" | The LUMO orbital of the molecule is symmetric. It shows the pi* orbitals of the two pi systems and the conjugation of these across the central single bond. The nodes fall within the plane of the molecule and between the double bonds of the molecule suggesting that for this orbital there is no conjugation over the double bonds.
|}

==='''The Transition State'''===

A number of methods, that had previously been discussed above for the Cope rearrangement reaction, were considered to optimize the transition state of this reaction. However it was decided to use a method previously used in optimizing the chair transition state of 1,5-hexadiene due to it's simplicity and that a good guess for the transition state could be made initially.

Firstly the reactants were optimized separately under the HF/3-21G level of theory. Then the two molecules were combined in an envelope like geometry and the internuclear distances between the two molecules set to 2.2A in order to give a best guess for the transition state. An optimization and frequency calculation was then run, optimizing to a transition state using the Berny method, calculating the force constants only once.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Structure and Molecular Orbitals of the Diels-Alder Transition State
![[File:BE_TSHOMO.jpg|center|200px]][[File:BE_TSHOMO2.jpg|center|200px]]
!<jmol><jmolApplet><title>1,5-hexdiene formation transition state</title><color>white</color> <size>300</size><uploadedFileContents>BE_Reactants.mol</uploadedFileContents> </jmolApplet></jmol>
|[[File:BE_TSLUMO.jpg|center|200px]][[File:BE_TSLUMO2.jpg|center|200px]]
|-
! align="center" | HOMO
| align="center" | The transition state for the Diels-Alder reaction between ethene and cis-butadiene has an energy of -231.60321Hartrees
! align="center" | LUMO
|}

[[File:BE_TSVibration.gif|thumb|left|Imaginary Vibration of the Diels-Alder Transition State]][[File:BE_Positive_Vibration.gif|thumb|right|Lowest Energy Positive Vibration of the Diels-Alder Transition State]]

This gave a good approximation for the transition state with the imaginary vibration, of -817.71cm-1, showing the two molecules approaching each other in a synchronous motion. Comparatively the lowest positive vibration at 166.54cm-1 is an asynchronous rocking motion that does not contribute to the coming together of the two molecules.

[[File:BE_TSDIELSALDERIR_Spectrum.png|thumb|center|300px|Predicted IR Spectrum of the Diels-Alder Transition State]]

The justification of calling this the transition state is also shown in the final bond lengths between the two molecules, the length of approach is 2.21A. Typically an sp3-sp3 carbon bond length is 1.54A and for a sp2-sp2 carbon it is 1.47A<ref name="Wiki Book"/>this suggests that a bond has not been fully formed in the transition state. However it is within the van der Waals radius of the carbon atom, which is 1.7A<ref name="Van der Waals Volumes and Radii">A. Bondi, ''J. Phys. Chem.'','''1964''', ''68 '', 441-51.</ref> so a partial bond may have started to form at this point in the reaction.

Looking at the HOMO and LUMO orbitals for the reactants there is a clear contribution from both to the molecular orbitals of the transition state, this is especially the case for the ethene molecule. The visualization of the HOMO orbital throws some doubt onto the final structure obtained however considering the imaginary vibration is as expected, it can be assumed that this is a form of the transition state, if not the most minimal energy one. Further calculations were run in order to find the transition state with the correct HOMO orbital however it could not be found with the correct geometry and imaginary vibration.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Molecular Orbitals of Ethene and Cis-Butadiene
! colspan="4" | Ethene
! colspan="4" | cis-butadiene
|-
|[[File:BE_Ethene_HOMO.jpg|center|100px]]
|[[File:BE_Ethene_HOMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO.jpg|center|100px]]
|[[File:BE_I_HOMO.jpg|center|100px]]
|[[File:BE_I_HOMO2.jpg|center|100px]]
|[[File:BE_I_LUMO.jpg|center|100px]]
|[[File:BE_I_LUMO2.jpg|center|100px]]
|-
! colspan="2" | HOMO - Symmetric
! colspan="2" | LUMO - Asymmetric
! colspan="2" | HOMO - Asymmetric
! colspan="2" | LUMO - Symmetric
|}

The reactants orbitals above display frontier molecular orbital theory. The HOMO of cis-butadiene and the LUMO of ethene are of correct symmetry to interact to form a bonding molecular orbital in the product. This means that the reaction is symmetry allowed and therefore no other conditions are needed, such as light, in order for the reaction to proceed<ref>[http://www.ch.ic.ac.uk/rzepa/] Henry S. Rzepa, Imperial College, Organic Pericyclic Reactions: Categories</ref>.

==='''Regioselectivity'''===

The reaction can form two different products depending on the transition state. The exo product is formed when the approach of the maleic anhydride occurs with the oxygens pointing away from the double bond, and the endo product is formed when the oxygens point towards it.
In order to determine the energies of the transition states first the reactants were optimized separately and then together using the HF/3-21G level of theory. Then the endo and exo products were optimized the same way. The transition state was found using the QTS2 method, an Opt+Freq calculation under the HF/3-21G level of theory, this gave an optimized version of each transition state. This was used in order to utilise the reactants and products to find the transition state as the initial guess for the transition state was not a confident enough one.

The imaginary vibrations of these transition at -647.54 and -643.32cm-1 for the exo and endo transition states respectively. These vibrations show the coming together of the reactant molecules to form the product as is expected. These Transition state structures were then optimized under a higher level of theory in order to obtain a more accurate energy, and the HOMO and LUMO orbitals of each visualized.

[[File:BE_Exo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Exo Transition State]][[File:BE_Endo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Endo Transition State]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Energies of the Exo and Endo Transition States under the B3LYP/6-31G(d) Level of Theory
!
! Exo Transition State
! Endo Transition State
|-
|
|<jmol><jmolApplet><title>EXO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_Exo_TS_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>ENDO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_ENDO_TS_3_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|-
!Total Energy/Hartrees
| -612.66122
| -612.68340
|-
!Sum of Electronic and Zero-Point Energies/Hartrees
| -612.47669
| -612.50214
|-
!Sum of Electronic and Thermal Energies/Hartrees
| -612.46612
| -612.49179
|-
!Activation Energy at 0K/kcalmol-1
| 34.90
| 18.93
|-
!Activation Energy at 298.15K/kcalmol-1
| 33.58
| 17.47
|-
!Dihedral Angle/o
| -169.56
| -175.67
|-
!Intermolecular Bond Lengths/A
|1.5
|2.27
|}

Activation energies were calculated by comparing the energy of the transition state to the energy of the reactants (At 0K B3LYP/6-31G(d) = -612.53231, At 298.15K B3LYP/6-31G(d) = -612.51964). The endo product should be the kinetic product, this is shown by its much lower activation energy. The exo product has a tighter dihedral angle than the endo, this is due to the repulsion between the two reactants as seen in the HOMO of the exo transition state below<ref name="Steric Effects vs. Secondary Orbital Overlap in Diels-Alder Reactions. MNDO and AM1 Studies">, M. A. Fox, R. Cardons and N. J. Kiwiet, ''J. Org. Chem.'', '''1987''', ''52'',1469-1474</ref>.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The HOMO and LUMO Molecular Orbitals of the Exo and Endo Transition States
! colspan="2" | Exo Transition State
! colspan="2" | Endo Transition State
|-
!HOMO
!LUMO
!HOMO
!LUMO
|-
|[[File:BE_HOMO_exo.jpg|center|200px]]
|[[File:BE_LUMO_Exo.jpg|center|200px]]
|[[File:BE_HOMO_Endo.jpg|center|200px]]
|[[File:BE_LUMO_endo.jpg|center|200px]]
|}

The endo and exo products will be stabilised by secondary orbital overlap (SOO). This is the mixing of the HOMO orbital of one reactant with the LUMO orbital of the other<ref name="Orbital symmetries and endo-exo relationships in concerted cycloaddition reactions">, R. Hoffmann, R. B. Woodward, ''J. Am. Chem. Soc.'', '''1965''', ''87'', 4388-4389</ref>. This can be seen in both transition state HOMO orbitals above. In the case of the endo transition state the orbitals from the central oxygen shows a node around it, the orbitals on that oxygen are of the same symmetry and hence will interact with the orbitals from the double bond that is formed in the product above it. For the exo transition state the position of the oxygen and double bond are much separated and therefore there is no SOO. This SOO effect in the endo transition state, and hence the greater stabilisation, is a reason that the energy of the endo transition state is lower than that of the endo.

==Conclusion==

These reactions have shown the usefulness of molecular modelling in the visualization and analysis of transition states. A wide range of methods for optimizing the transition states have been considered and tested for their accuracy and ease of use. Despite initial difficulties the QTS2 method proved to be the most effective in determining the transition state of both the boat conformer in the Cope rearrangement and the endo and exo forms in the Diels-Alder reaction. It gave a good prediction for the geometries and energies of the transition states and allowed comparison between the reactants and products of the reactions. Further examples of Diels-Alder reactions could be looked at to determine the stereo and regio-selective implications of the mechanism and further prove the effectiveness of calculating activation energies by this method as compared to experimentally determined values. Also calculations from different basis sets could be run to determine the accuracy of the different levels of theory and to improve the calculated geometries and energies of the transition states studied in this experiment.

==References==

<references />

Rep:Mod:PhysBE411

2014-03-21T16:31:23Z

Be411: /* Analysis of Conformer E */

'''Physical Module: Transition States and Reactivity'''

Computational methods can be used to study reaction pathways and more specifically transition states. In this experiment the Cope rearrangement reaction and Diels-Alder cycloaddition reaction are used as examples of this. Using various methods the lowest energy conformation of 1,5-hexadiene can be found as well as the conformation that exists as a reactant in the Cope rearrangement reaction. The two possible transition states of this reaction can be studied and their activation energies calculated to gain some understanding of the mechanism of this reaction. The Diels-Alder cycloaddition will also be studied. The origin of the symmetry allowed reaction between ethene and cis-butadiene will be visualized and the reaction between maleic anhydride and hexa-1,5-diene studied in terms of the regioselectivity of its two conformational different transition states.

=='''The Cope Rearrangement Tutorial'''==

The Cope rearrangement is a specific [3,3] sigmatropic rearrangement in which a sigma bond moves and the pi system is redistributed across the molecule. In this case the Cope rearrangement of hexa-1,5-diene is considered with regard to its conformations and the boat and chair forms of its reaction's transition state.

==='''Optimizing the Reactants and Products'''===

===='''Conformation of 1,5-hexadiene'''====
[[File:BE_1,5-HEXADIENE_CONFORTMATIONS.jpg|thumb|The Conformations of 1,5-Hexadiene]]

1,5-hexadiene is one of the simplest precursors in the Cope rearrangement. The three rotating carbon-carbon bonds within the center of the molecule cause there to be 10 energetically different conformations, when taking into account enantiomers and symmetry.

Within these conformations you would expect that where two carbons are in the eclipsed form (F-J) the conformer will be of higher energy than conformers with CH-eclipsed bonds (A-E)<ref name="Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols">B. W. Gung, Z. Zhu and R. A. Fouch, ''J. Am. Chem. Soc'', '''1995''', ''117'', 1783-1788.</ref>. This is demonstrated in the case of butane where the difference in energy between the CC eclipsed form and the CH eclipsed form is 1.5kcal/mol<ref name="Website - Butane Conformational Analysis">[http://research.cm.utexas.edu/nbauld/teach/butane.html]N. Bauld, '''"Butane Conformational Analysis"''', ''University of Texas'', Retreived 20 March 2014.</ref>. The energies of these forms can be optimized by computational methods in order to find the most suitable candidates for the reactants and products in the Cope Rearrangement reaction.

The table below shows the total energy for all energetically distinct conformers of the molecule, it shows that in the case of this molecule the anti conformer is generally of higher energy. Relating these structures to the literature<ref name="Instructional Wiki">[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3]M. Bearpark, Imperial College, Physical Module: Transition states and reactivity</ref>, the energies found are all in line with the energies expected, as is also the case with the point groups.

The lowest energy conformation is the Gauche Conformer D (-231.69266Hartrees) this maybe due to an attractive interaction between the pi orbital of the double bond and the neighboring proton, whose orbital would interact with that of the pi bond<ref name="An Analysis of the Conformers of 1,5-hexadiene">B. G. Rocque, J. M. Gonzales, H. F. Schaeffer III, ''Mol. Phys'', '''2002''', ''100'', 441-446.</ref>. This conjugation will stabilize the molecule making it lower in energy.
The highest energy conformer of the CH eclipsed forms is C (-231.69153Hartrees) this maybe explained by interactions between the two alkene protons which are 2.51A apart. This proximity will have a repulsive effect increasing the overall energy of the molecule.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimized Energies for the Anti and Gauche Conformations of 1,5-Hexadiene (Relating to the Instructional Wiki<ref name="Instructional Wiki"/>)
!
! align="center" | '''<jmolFile text="Conformer A - Gauche2">BE_A.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer B - Anti1 ">BE_B.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer C - Gauche4">BE_C.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer D - Gauche3">BE_D.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer E - Anti2">BE_E.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer F - Gauche6">BE_F.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer G - Anti4">BE_G.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer H - Gauche5">BE_H.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer I - Gauche1">BE_I.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer J - Anti3">BE_J.mol</jmolFile>'''
|-
|'''Total Energy/Hartrees'''
| -231.69167
| -231.69260
| -231.69153
| -231.69266
| -231.69254
| -231.68916
| -231.69097
| -231.68962
| -231.68772
| -231.68907
|-
|'''Point Group'''
|C2
|C2
|C2
|C1
|Ci
|C1
|C1
|C1
|C2
|C2h
|-
|'''Dihedral Angle (C2-C3-C4-C5)'''
|64.17
| -176.91
| -63.65
|67.70
| -179.996
|70.23
| -178.32
| -71.09
|75.76
| -179.98
|}

===='''Analysis of Conformer E'''====

Further Analysis of Conformer E was done at a higher basis set in order to get an even better approximation for its lowest energy conformation. An optimization calculation was run as before at the B3LYP/6-31G(d)level of theory.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimised Conformations of Conformer E using the HF/3-21G and B3LYP/6-31G(d) Basis Sets
!<jmol><jmolApplet><title>Anti 2 Conformation at HF level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_E.mol</uploadedFileContents> </jmolApplet></jmol>
!<jmol><jmolApplet><title>Anti 2 Conformation at B3LYP level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_DFT_optmisation.mol</uploadedFileContents> </jmolApplet></jmol>
|}

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Geometrical Features of the Anti 2 Conformer Optimized Under Different Basis Sets
! rowspan ="2" | '''Basis Set'''
! colspan ="3" | '''Bond Length'''
! colspan ="3" | '''Angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3
|C2-C3-C4
|C2-C3-C4-C5
|-
|HF/3-21G
|1.32
|1.51
|1.55
|179.996
|124.81
|111.35
|-
|B3LYP/6-31G(d)
|1.33
|1.50
|1.55
|180
|125.29
|112.68
|}

[[image:BE_IR_Spectrum_of_Anti2.jpg|thumb|200px|IR Spectrum of Conformer E]]

This shows that the geometries of the results of the two different approaches don't differ wildly, the bond lengths especially are very similar, however there is some deviation between their angles.

Following on from this a frequency calculation was run using the same higher level of theory (B3LYP/6-31G(d)). This calculation can be used to confirm that this conformation is of a minimum energy by the absence of an imaginary vibrational frequency. The infrared spectrum of the molecule can be generated from the molecular vibrations and compared to literature.

The thermochemical energies for this molecule can be found in the output file of the calculation:

Sum of electronic and zero-point Energies= -234.46921 Hartrees
Sum of electronic and thermal Energies= -234.46186 Hartrees
Sum of electronic and thermal Enthalpies= -234.46091 Hartrees
Sum of electronic and thermal Free Energies= -234.50082 Hartrees

The various information above can be used to consider the potential and free energy of the system. The sum of the electronic and zero-point energies (E= Eelec + ZPE) show the potential energy at 0K and the sum of electronic and thermal energies (E = E + Evib + Erot + Etrans) considers it at 298.15K and 1atm. The sum of electronic and thermal enthalpies (H = E + RT) shows the enthalpy of the system containing a correction for thermal energy and the sum of electronic and thermal free energies ( G= H - TS) suggests the entropic contribution to the free energy of the system.

This data shows that the energy at 0K is lower than that at room temperature, this is as expected because at 0K there will be no internal movement of the molecule and hence no contribution from vibrational, rotational or translational energies.

==='''Optimizing the "Chair" and "Boat" Transition Structures'''===

Conformer E was considered for this reaction and used to model both the chair and boat transition states for the Cope rearrangement reaction. A number of different ways of optimizing the geometry of the transition states were considered including using redundant coordinates and QTS2(TS)optimization and frequency calculation.

===='''Chair Transition State'''====

The chair transition state was constructed from two allyl fragments. Firstly the geometry of a single allyl fragment was optimized using the HF/3-21G level of theory giving an energy of -117.61330Hartrees. Two allyl fragments are placed 2.2A apart and then using this as a guess for the transition state, the molecule is then optimized in two different ways.

[[File:BE_B_vibration.gif|thumb|Cope Rearrangement Chair Transition State Vibration]]

Firstly it can be optimized using a opt + freq calculation to a TS using the Berny Method. By calculating the force constant once an imaginary vibration can be found. The imaginary vibration was found at 817.95cm-1 (expected value of 818cm-1<ref name="Instructional Wiki"/>) on animation it can be seen that this corresponds to the coming together of the two terminal carbons on each allyl molecule, the movement expected during this reaction. In this case the total energy is -231.61932Hartrees.

[[File:BE_D_ChairTS.jpg|thumb|left|Chair Transition State by Redundant Coordinate Method]]
Secondly the transition state was optimized by freezing the reaction coordinate in order to minimize the rest of the molecule first and then unfreezing it in order to find the minimum for the transition state. This means that the force constant Hessian matrix doesn't need to be calculated and therefore is a faster method of computing the transition state.

By looking at the atom distances between the bond that will be formed the suitability of both of these methods to find the true transition state can be studied. All the measured geometric parameters were identical in both versions of the transition state, namely the intranuclear bond lengths was 1.39A, the internuclear distance between the two allyl molecules is 2.02A and the angle within the each molecule is 120.05o. This suggests that both methods for determining the transition state are equally effect from the same guess input file. It also confirms that the guess structure was close to the exact structure as the first method was as successful in calculating the force constants as the second which doesn't need to calculate them.

===='''The Boat Transition State'''====

The boat transition state was constructed in much the same way as the chair using two pre-optimized allyl fragments, placing them together in the correct geometry and setting them to 2.2A apart. In the case of the boat transition state the QTS2 method was used to optimize the energy. By ensuring that the same labeling system is used in the reactants and products, an optimization and frequency calculation was run that could help define the geometry of the boat transition state. This calculation was run with a predicted transition state with a dihedral angle (C2-C3-C4-C5) of 0o and inside C2-C3-C4 and C3-C4-C5 angles of 100o.

[[File:BE_E_Vibration.gif|thumb|200px|Cope Rearrangement Boat Transition State Vibration]]

The result of this calculation shows the boat transition state with a single imaginary vibration of -840nm and energy of -231.60280Hartrees. Visualizing this vibration shows that it corresponds to the coming together of the two terminal carbons and the separation of the two central carbons. This is the expected vibration seen during the Cope rearrangement.
The QTS2 method is advantageous because it is fully automated however it required a good starting guess for the transition state and therefore a lot of permutations of structure, in the input file, were needed for a successful calculation.

{| class="wikitable" style="margin: auto;"
|+ Summary Table of the Energies of the Two Transition States
! rowspan="2" | Transition State
! colspan="3" | Energy
|-
|<jmolFile text="Berny TS">BE_ChairB.mol</jmolFile>
|<jmolFile text="Redunant Coordinates">BE_chairD.mol</jmolFile>
|<jmolFile text="QTS2 (TS)">BE_Boat_TS_Again2.mol</jmolFile>
|-
|chair
| -231.61932
| -231.61932
|
|-
|Boat
|
|
| -231.60280
|}

===='''The Reaction Pathway'''====

The reaction pathway can be studied through both the minimum energy position on its potential energy surface and the activation energies of the two transition states.

====='''IRC'''=====

[[File:BE_IRC_Plot.jpg|thumb|400px|IRC Energy Plot of the Chair Transition State]]
An intrinsic reaction coordinate (IRC) was run as it can be used to determine the minimum energy pathway from the transition state to its local minimum by using many points along the pathway to the point where the energy plateaus. The IRC in this case has been run in the forward direction only and the force constants calculated at every step.

The lowest energy conformation is achieved after 44 steps. The structure of the first and final conformations can be compared by considering their different geometry and energy. The IRC determines the minimum energy conformation and therefore the energy of step 44 (-231.69158Hartrees) is, as expected, lower than that of the first step (-231.61932Hartrees), this confirms that the calculation has found a minimum. The most noticeable different between the two conformations is the bond symmetry, in step 1 the bond lengths within each molecule were identical (1.39A) and the distance between the two molecules is the same (2.02A) for both sets of terminal carbons. However in step 44 the bond lengths within each molecule are different in preparation for the formation of two different types of bonds - single and double - this means one bond is shorter than the other (1.32 and 1.51A). As well as this difference in intranuclear distances there is also a difference in the internuclear distances as the one end approaches much closer (1.55 and 4.23A) almost within the van der Waals radius of the carbon atom as it prepares to form a new bond whereas the other distance is much larger as there is no bond formation at those carbons.

{| class="wikitable" style="margin: auto;"
|+ Structures of Steps 1 and 44 in the Determination of the IRC of the Chair Transition State
! [[File:BE_F1_Image.jpg|400px]]
! [[File:BE_F44_Image.jpg|400px]]
|}

====='''Activation Energies'''=====

In order to determine the activation energy of the reaction via the two different transition states a higher level of theory needed to be used. The B3LYP/6-31G(d) was used to optimize the two transition states to a minimum. Comparing this to the activation energies obtained at the lower level of theory and literature values<ref name="Instructional Wiki"/> shows the large difference in the energies obtained at each level of theory. The Energies at 0 and 298.15K were then found by running a frequency calculation, using the same basis set, on the optimised structure and checking the sum of the electronic and zero-point (0K) and sum of electronic and thermal energies (298.15K).

{| class="wikitable" style="margin: auto;"
|+ Geometrical Features of the Reactant, Boat and Chair Transition States
!
! colspan="2" | Reactants
! colspan="2" | Chair Transition State
! colspan="2" | Boat Transition State
|-
|
|<jmol><jmolApplet><title>Reactants Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Reactants Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Transition State Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Transition State Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Transition State Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again2.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Transition State Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again3_-_EA.mol</uploadedFileContents> </jmolApplet></jmol>
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
!Intramolecular Bond Lengths/A
|1.32, 1.51
|1.33, 1.50
|1.39
|1.41
|1.38
|1.39
|-
!Intermolecular Bond Lengths/A
|1.55
|1.55
|2.02
|1.97
|2.14
|2.21
|-
!Dihedral Angle/o
|179.996
|180
|54.96
|54
|0
|0
|}

In the reactants the higher level of theory shows a tightening of the dihedral angle to exactly 180o as well as a shortening of the single bonds and lengthening of the double bonds within the molecule. This acts to bring the bond lengths within the molecule closer to the expected bond lengths of 1.47 and 1.54A<ref name="Wiki Book">M. A. Fox, J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen.'', '''1995''', Springer.</ref>. Within chair transition state the same effect can be seen, increasing the dihedral angle and double bond length and decreasing the single and intermolecular bond distances.

{| class="wikitable" style="margin: auto;"
|+ Electronic Energies and Energies of the Boat and Chair Transition States at 0K
!
! colspan="2" | '''Electronic Energy/Hartrees'''
! colspan="2" | '''Sum of Electronic and Zero-Point Energies'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.60280
| -234.54309
| -231.45093
| -234.40234
| 0.08861
| 0.06687
| 55.60
| 41.96
| 44.7 ± 2.0
|-
|'''Chair Transition State'''
| -231.69167
| -234.55698
| -231.46669
| -234.41488
| 0.07286
| 0.05433
| 45.72
| 34.09
| 33.5 ± 0.5
|}

The activation energies were calculated with reference to the conformer E energies as the reactant ( At 0K HF/3-21G = -231.53954 and B2LYP/6-31G(d) = -234.46921Hartrees, At 298.15K HF/3-21G = -231.53257 and B2LYP/6-31G(d) = -234.46186Hartrees ).

{| class="wikitable" style="margin: auto;"
|+ Energies of the Boat and Chair Transition States at 298.15K
!
! colspan="2" | '''Sum of Electronic and Thermal Energies'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.44530
| -234.39599
| 0.08727
| 0.06587
| 54.76
| 41.33
| 41.32
|-
|'''Chair Transition State'''
| -231.46133
| -234.40895
| 0.07124
| 0.05291
| 44.70
| 33.20
| 33.17
|}

From this it is clear why a higher level of theory is needed. The energies calculated at the B3LYP/6-31G(d) level yield much more accurate activation energies that those at HF/3-21G. In both cases this increase in accuracy has brought the activation energies significantly closer to the expected values.

The activation energy of the Chair transition state is a good match for that in the literature and is very close to the boundaries of error given. When comparing this to the energy obtained at the lower level of theory there is a much greater difference and therefore illustrates the ability of the higher level of energy to calculate a much more accurate transition state.

=='''Diels-Alder Cycloaddition Reaction'''==

The Diels-Alder cycloaddition reaction is a pericyclic reaction that occurs between two pi systems, a diene and a dienophile, in order to form a cyclohexene. It offers good regio- and stereo-selective control that can be extended to other conjugated systems<ref name="The Diels-Alder Reaction in Total Synthesis">K. C. Nicolaou, S. A. Snyder, T. Montagnon and G. Vassilikogiannakis ''Angewandte Chemie International Edition'','''2002''', ''41'', 1668-1698.</ref>.

In this experiment the reaction between cis butadiene and ethene will be studied in order to determine the symmetry of both the reactant and product molecular orbitals, as well as the geometry of the reaction's transition state. The regioselectivity of the reaction will also be looked at into order to determine the preferred product of the reaction, endo or exo.

==='''Cis-butadiene'''===

Cis-butadiene will be used as the diene in this studied reaction with ethene. Its geometry was first optimized using the HF/3-21G level of theory. The final energy of the product was -154.05514Hartrees. The molecular orbitals of the optimized structure can be modeled and used further to look at the electronic contribution of the diene to the final molecule.

{| class="wikitable" style="margin: auto;"
|+ Images of the Molecular Orbitals of Cis-Butadiene
! [[File:BE_I_HOMO.jpg|200px]]
! rowspan="3" | [[File:BE_Diene.jpg|150px]]
! [[File:BE_I_LUMO.jpg|200px]]
|-
![[File:BE_I_HOMO2.jpg|200px]]
![[File:BE_I_LUMO2.jpg|200px]]
|-
| align="center" | The HOMO orbital of the molecule is asymmetric. The nodes in the molecule fall between the red and green lobes of the HOMO. This shows the two separate pi systems within the molecule, but the proximity of the two systems suggests that the molecule could be fully conjugated.
| align="center" | The LUMO orbital of the molecule is symmetric. It shows the pi* orbitals of the two pi systems and the conjugation of these across the central single bond. The nodes fall within the plane of the molecule and between the double bonds of the molecule suggesting that for this orbital there is no conjugation over the double bonds.
|}

==='''The Transition State'''===

A number of methods, that had previously been discussed above for the Cope rearrangement reaction, were considered to optimize the transition state of this reaction. However it was decided to use a method previously used in optimizing the chair transition state of 1,5-hexadiene due to it's simplicity and that a good guess for the transition state could be made initially.

Firstly the reactants were optimized separately under the HF/3-21G level of theory. Then the two molecules were combined in an envelope like geometry and the internuclear distances between the two molecules set to 2.2A in order to give a best guess for the transition state. An optimization and frequency calculation was then run, optimizing to a transition state using the Berny method, calculating the force constants only once.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Structure and Molecular Orbitals of the Diels-Alder Transition State
![[File:BE_TSHOMO.jpg|center|200px]][[File:BE_TSHOMO2.jpg|center|200px]]
!<jmol><jmolApplet><title>1,5-hexdiene formation transition state</title><color>white</color> <size>300</size><uploadedFileContents>BE_Reactants.mol</uploadedFileContents> </jmolApplet></jmol>
|[[File:BE_TSLUMO.jpg|center|200px]][[File:BE_TSLUMO2.jpg|center|200px]]
|-
! align="center" | HOMO
| align="center" | The transition state for the Diels-Alder reaction between ethene and cis-butadiene has an energy of -231.60321Hartrees
! align="center" | LUMO
|}

[[File:BE_TSVibration.gif|thumb|left|Imaginary Vibration of the Diels-Alder Transition State]][[File:BE_Positive_Vibration.gif|thumb|right|Lowest Energy Positive Vibration of the Diels-Alder Transition State]]

This gave a good approximation for the transition state with the imaginary vibration, of -817.71cm-1, showing the two molecules approaching each other in a synchronous motion. Comparatively the lowest positive vibration at 166.54cm-1 is an asynchronous rocking motion that does not contribute to the coming together of the two molecules.

[[File:BE_TSDIELSALDERIR_Spectrum.png|thumb|center|300px|Predicted IR Spectrum of the Diels-Alder Transition State]]

The justification of calling this the transition state is also shown in the final bond lengths between the two molecules, the length of approach is 2.21A. Typically an sp3-sp3 carbon bond length is 1.54A and for a sp2-sp2 carbon it is 1.47A<ref name="Wiki Book"/>this suggests that a bond has not been fully formed in the transition state. However it is within the van der Waals radius of the carbon atom, which is 1.7A<ref name="Van der Waals Volumes and Radii">A. Bondi, ''J. Phys. Chem.'','''1964''', ''68 '', 441-51.</ref> so a partial bond may have started to form at this point in the reaction.

Looking at the HOMO and LUMO orbitals for the reactants there is a clear contribution from both to the molecular orbitals of the transition state, this is especially the case for the ethene molecule. The visualization of the HOMO orbital throws some doubt onto the final structure obtained however considering the imaginary vibration is as expected, it can be assumed that this is a form of the transition state, if not the most minimal energy one. Further calculations were run in order to find the transition state with the correct HOMO orbital however it could not be found with the correct geometry and imaginary vibration.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Molecular Orbitals of Ethene and Cis-Butadiene
! colspan="4" | Ethene
! colspan="4" | cis-butadiene
|-
|[[File:BE_Ethene_HOMO.jpg|center|100px]]
|[[File:BE_Ethene_HOMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO.jpg|center|100px]]
|[[File:BE_I_HOMO.jpg|center|100px]]
|[[File:BE_I_HOMO2.jpg|center|100px]]
|[[File:BE_I_LUMO.jpg|center|100px]]
|[[File:BE_I_LUMO2.jpg|center|100px]]
|-
! colspan="2" | HOMO - Symmetric
! colspan="2" | LUMO - Asymmetric
! colspan="2" | HOMO - Asymmetric
! colspan="2" | LUMO - Symmetric
|}

The reactants orbitals above display frontier molecular orbital theory. The HOMO of cis-butadiene and the LUMO of ethene are of correct symmetry to interact to form a bonding molecular orbital in the product. This means that the reaction is symmetry allowed and therefore no other conditions are needed, such as light, in order for the reaction to proceed<ref>[http://www.ch.ic.ac.uk/rzepa/] Henry S. Rzepa, Imperial College, Organic Pericyclic Reactions: Categories</ref>.

==='''Regioselectivity'''===

The reaction can form two different products depending on the transition state. The exo product is formed when the approach of the maleic anhydride occurs with the oxygens pointing away from the double bond, and the endo product is formed when the oxygens point towards it.
In order to determine the energies of the transition states first the reactants were optimized separately and then together using the HF/3-21G level of theory. Then the endo and exo products were optimized the same way. The transition state was found using the QTS2 method, an Opt+Freq calculation under the HF/3-21G level of theory, this gave an optimized version of each transition state. This was used in order to utilise the reactants and products to find the transition state as the initial guess for the transition state was not a confident enough one.

The imaginary vibrations of these transition at -647.54 and -643.32cm-1 for the exo and endo transition states respectively. These vibrations show the coming together of the reactant molecules to form the product as is expected. These Transition state structures were then optimized under a higher level of theory in order to obtain a more accurate energy, and the HOMO and LUMO orbitals of each visualized.

[[File:BE_Exo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Exo Transition State]][[File:BE_Endo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Endo Transition State]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Energies of the Exo and Endo Transition States under the B3LYP/6-31G(d) Level of Theory
!
! Exo Transition State
! Endo Transition State
|-
|
|<jmol><jmolApplet><title>EXO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_Exo_TS_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>ENDO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_ENDO_TS_3_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|-
!Total Energy/Hartrees
| -612.66122
| -612.68340
|-
!Sum of Electronic and Zero-Point Energies/Hartrees
| -612.47669
| -612.50214
|-
!Sum of Electronic and Thermal Energies/Hartrees
| -612.46612
| -612.49179
|-
!Activation Energy at 0K/kcalmol-1
| 34.90
| 18.93
|-
!Activation Energy at 298.15K/kcalmol-1
| 33.58
| 17.47
|-
!Dihedral Angle/o
| -169.56
| -175.67
|-
!Intermolecular Bond Lengths/A
|1.5
|2.27
|}

Activation energies were calculated by comparing the energy of the transition state to the energy of the reactants (At 0K B3LYP/6-31G(d) = -612.53231, At 298.15K B3LYP/6-31G(d) = -612.51964). The endo product should be the kinetic product, this is shown by its much lower activation energy. The exo product has a tighter dihedral angle than the endo, this is due to the repulsion between the two reactants as seen in the HOMO of the exo transition state below<ref name="Steric Effects vs. Secondary Orbital Overlap in Diels-Alder Reactions. MNDO and AM1 Studies">, M. A. Fox, R. Cardons and N. J. Kiwiet, ''J. Org. Chem.'', '''1987''', ''52'',1469-1474</ref>.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The HOMO and LUMO Molecular Orbitals of the Exo and Endo Transition States
! colspan="2" | Exo Transition State
! colspan="2" | Endo Transition State
|-
!HOMO
!LUMO
!HOMO
!LUMO
|-
|[[File:BE_HOMO_exo.jpg|center|200px]]
|[[File:BE_LUMO_Exo.jpg|center|200px]]
|[[File:BE_HOMO_Endo.jpg|center|200px]]
|[[File:BE_LUMO_endo.jpg|center|200px]]
|}

The endo and exo products will be stabilised by secondary orbital overlap (SOO). This is the mixing of the HOMO orbital of one reactant with the LUMO orbital of the other<ref name="Orbital symmetries and endo-exo relationships in concerted cycloaddition reactions">, R. Hoffmann, R. B. Woodward, ''J. Am. Chem. Soc.'', '''1965''', ''87'', 4388-4389</ref>. This can be seen in both transition state HOMO orbitals above. In the case of the endo transition state the orbitals from the central oxygen shows a node around it, the orbitals on that oxygen are of the same symmetry and hence will interact with the orbitals from the double bond that is formed in the product above it. For the exo transition state the position of the oxygen and double bond are much separated and therefore there is no SOO. This SOO effect in the endo transition state, and hence the greater stabilisation, is a reason that the energy of the endo transition state is lower than that of the endo.

==Conclusion==

These reactions have shown the usefulness of molecular modelling in the visualization and analysis of transition states. A wide range of methods for optimizing the transition states have been considered and tested for their accuracy and ease of use. Despite initial difficulties the QTS2 method proved to be the most effective in determining the transition state of both the boat conformer in the Cope rearrangement and the endo and exo forms in the Diels-Alder reaction. It gave a good prediction for the geometries and energies of the transition states and allowed comparison between the reactants and products of the reactions. Further examples of Diels-Alder reactions could be looked at to determine the stereo and regio-selective implications of the mechanism and further prove the effectiveness of calculating activation energies by this method as compared to experimentally determined values. Also calculations from different basis sets could be run to determine the accuracy of the different levels of theory and to improve the calculated geometries and energies of the transition states studied in this experiment.

==References==

<references />

Rep:Mod:PhysBE411

2014-03-21T16:30:40Z

Be411:

'''Physical Module: Transition States and Reactivity'''

Computational methods can be used to study reaction pathways and more specifically transition states. In this experiment the Cope rearrangement reaction and Diels-Alder cycloaddition reaction are used as examples of this. Using various methods the lowest energy conformation of 1,5-hexadiene can be found as well as the conformation that exists as a reactant in the Cope rearrangement reaction. The two possible transition states of this reaction can be studied and their activation energies calculated to gain some understanding of the mechanism of this reaction. The Diels-Alder cycloaddition will also be studied. The origin of the symmetry allowed reaction between ethene and cis-butadiene will be visualized and the reaction between maleic anhydride and hexa-1,5-diene studied in terms of the regioselectivity of its two conformational different transition states.

=='''The Cope Rearrangement Tutorial'''==

The Cope rearrangement is a specific [3,3] sigmatropic rearrangement in which a sigma bond moves and the pi system is redistributed across the molecule. In this case the Cope rearrangement of hexa-1,5-diene is considered with regard to its conformations and the boat and chair forms of its reaction's transition state.

==='''Optimizing the Reactants and Products'''===

===='''Conformation of 1,5-hexadiene'''====
[[File:BE_1,5-HEXADIENE_CONFORTMATIONS.jpg|thumb|The Conformations of 1,5-Hexadiene]]

1,5-hexadiene is one of the simplest precursors in the Cope rearrangement. The three rotating carbon-carbon bonds within the center of the molecule cause there to be 10 energetically different conformations, when taking into account enantiomers and symmetry.

Within these conformations you would expect that where two carbons are in the eclipsed form (F-J) the conformer will be of higher energy than conformers with CH-eclipsed bonds (A-E)<ref name="Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols">B. W. Gung, Z. Zhu and R. A. Fouch, ''J. Am. Chem. Soc'', '''1995''', ''117'', 1783-1788.</ref>. This is demonstrated in the case of butane where the difference in energy between the CC eclipsed form and the CH eclipsed form is 1.5kcal/mol<ref name="Website - Butane Conformational Analysis">[http://research.cm.utexas.edu/nbauld/teach/butane.html]N. Bauld, '''"Butane Conformational Analysis"''', ''University of Texas'', Retreived 20 March 2014.</ref>. The energies of these forms can be optimized by computational methods in order to find the most suitable candidates for the reactants and products in the Cope Rearrangement reaction.

The table below shows the total energy for all energetically distinct conformers of the molecule, it shows that in the case of this molecule the anti conformer is generally of higher energy. Relating these structures to the literature<ref name="Instructional Wiki">[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3]M. Bearpark, Imperial College, Physical Module: Transition states and reactivity</ref>, the energies found are all in line with the energies expected, as is also the case with the point groups.

The lowest energy conformation is the Gauche Conformer D (-231.69266Hartrees) this maybe due to an attractive interaction between the pi orbital of the double bond and the neighboring proton, whose orbital would interact with that of the pi bond<ref name="An Analysis of the Conformers of 1,5-hexadiene">B. G. Rocque, J. M. Gonzales, H. F. Schaeffer III, ''Mol. Phys'', '''2002''', ''100'', 441-446.</ref>. This conjugation will stabilize the molecule making it lower in energy.
The highest energy conformer of the CH eclipsed forms is C (-231.69153Hartrees) this maybe explained by interactions between the two alkene protons which are 2.51A apart. This proximity will have a repulsive effect increasing the overall energy of the molecule.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimized Energies for the Anti and Gauche Conformations of 1,5-Hexadiene (Relating to the Instructional Wiki<ref name="Instructional Wiki"/>)
!
! align="center" | '''<jmolFile text="Conformer A - Gauche2">BE_A.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer B - Anti1 ">BE_B.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer C - Gauche4">BE_C.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer D - Gauche3">BE_D.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer E - Anti2">BE_E.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer F - Gauche6">BE_F.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer G - Anti4">BE_G.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer H - Gauche5">BE_H.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer I - Gauche1">BE_I.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer J - Anti3">BE_J.mol</jmolFile>'''
|-
|'''Total Energy/Hartrees'''
| -231.69167
| -231.69260
| -231.69153
| -231.69266
| -231.69254
| -231.68916
| -231.69097
| -231.68962
| -231.68772
| -231.68907
|-
|'''Point Group'''
|C2
|C2
|C2
|C1
|Ci
|C1
|C1
|C1
|C2
|C2h
|-
|'''Dihedral Angle (C2-C3-C4-C5)'''
|64.17
| -176.91
| -63.65
|67.70
| -179.996
|70.23
| -178.32
| -71.09
|75.76
| -179.98
|}

===='''Analysis of Conformer E'''====

Further Analysis of Conformer E was done at a higher basis set in order to get an even better approximation for its lowest energy conformation. An optimization calculation was run as before at the B3LYP/6-31G(d)level of theory.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimised Conformations for the Anti2 Conformer using the HF/3-21G and B3LYP/6-31G(d) Basis Sets
!<jmol><jmolApplet><title>Anti 2 Conformation at HF level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_E.mol</uploadedFileContents> </jmolApplet></jmol>
!<jmol><jmolApplet><title>Anti 2 Conformation at B3LYP level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_DFT_optmisation.mol</uploadedFileContents> </jmolApplet></jmol>
|}

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Geometrical Features of the Anti 2 Conformer Optimized Under Different Basis Sets
! rowspan ="2" | '''Basis Set'''
! colspan ="3" | '''Bond Length'''
! colspan ="3" | '''Angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3
|C2-C3-C4
|C2-C3-C4-C5
|-
|HF/3-21G
|1.32
|1.51
|1.55
|179.996
|124.81
|111.35
|-
|B3LYP/6-31G(d)
|1.33
|1.50
|1.55
|180
|125.29
|112.68
|}

[[image:BE_IR_Spectrum_of_Anti2.jpg|thumb|200px|IR Spectrum of Conformer E]]

This shows that the geometries of the results of the two different approaches don't differ wildly, the bond lengths especially are very similar, however there is some deviation between their angles.

Following on from this a frequency calculation was run using the same higher level of theory (B3LYP/6-31G(d)). This calculation can be used to confirm that this conformation is of a minimum energy by the absence of an imaginary vibrational frequency. The infrared spectrum of the molecule can be generated from the molecular vibrations and compared to literature.

The thermochemical energies for this molecule can be found in the output file of the calculation:

Sum of electronic and zero-point Energies= -234.46921 Hartrees
Sum of electronic and thermal Energies= -234.46186 Hartrees
Sum of electronic and thermal Enthalpies= -234.46091 Hartrees
Sum of electronic and thermal Free Energies= -234.50082 Hartrees

The various information above can be used to consider the potential and free energy of the system. The sum of the electronic and zero-point energies (E= Eelec + ZPE) show the potential energy at 0K and the sum of electronic and thermal energies (E = E + Evib + Erot + Etrans) considers it at 298.15K and 1atm. The sum of electronic and thermal enthalpies (H = E + RT) shows the enthalpy of the system containing a correction for thermal energy and the sum of electronic and thermal free energies ( G= H - TS) suggests the entropic contribution to the free energy of the system.

This data shows that the energy at 0K is lower than that at room temperature, this is as expected because at 0K there will be no internal movement of the molecule and hence no contribution from vibrational, rotational or translational energies.

==='''Optimizing the "Chair" and "Boat" Transition Structures'''===

Conformer E was considered for this reaction and used to model both the chair and boat transition states for the Cope rearrangement reaction. A number of different ways of optimizing the geometry of the transition states were considered including using redundant coordinates and QTS2(TS)optimization and frequency calculation.

===='''Chair Transition State'''====

The chair transition state was constructed from two allyl fragments. Firstly the geometry of a single allyl fragment was optimized using the HF/3-21G level of theory giving an energy of -117.61330Hartrees. Two allyl fragments are placed 2.2A apart and then using this as a guess for the transition state, the molecule is then optimized in two different ways.

[[File:BE_B_vibration.gif|thumb|Cope Rearrangement Chair Transition State Vibration]]

Firstly it can be optimized using a opt + freq calculation to a TS using the Berny Method. By calculating the force constant once an imaginary vibration can be found. The imaginary vibration was found at 817.95cm-1 (expected value of 818cm-1<ref name="Instructional Wiki"/>) on animation it can be seen that this corresponds to the coming together of the two terminal carbons on each allyl molecule, the movement expected during this reaction. In this case the total energy is -231.61932Hartrees.

[[File:BE_D_ChairTS.jpg|thumb|left|Chair Transition State by Redundant Coordinate Method]]
Secondly the transition state was optimized by freezing the reaction coordinate in order to minimize the rest of the molecule first and then unfreezing it in order to find the minimum for the transition state. This means that the force constant Hessian matrix doesn't need to be calculated and therefore is a faster method of computing the transition state.

By looking at the atom distances between the bond that will be formed the suitability of both of these methods to find the true transition state can be studied. All the measured geometric parameters were identical in both versions of the transition state, namely the intranuclear bond lengths was 1.39A, the internuclear distance between the two allyl molecules is 2.02A and the angle within the each molecule is 120.05o. This suggests that both methods for determining the transition state are equally effect from the same guess input file. It also confirms that the guess structure was close to the exact structure as the first method was as successful in calculating the force constants as the second which doesn't need to calculate them.

===='''The Boat Transition State'''====

The boat transition state was constructed in much the same way as the chair using two pre-optimized allyl fragments, placing them together in the correct geometry and setting them to 2.2A apart. In the case of the boat transition state the QTS2 method was used to optimize the energy. By ensuring that the same labeling system is used in the reactants and products, an optimization and frequency calculation was run that could help define the geometry of the boat transition state. This calculation was run with a predicted transition state with a dihedral angle (C2-C3-C4-C5) of 0o and inside C2-C3-C4 and C3-C4-C5 angles of 100o.

[[File:BE_E_Vibration.gif|thumb|200px|Cope Rearrangement Boat Transition State Vibration]]

The result of this calculation shows the boat transition state with a single imaginary vibration of -840nm and energy of -231.60280Hartrees. Visualizing this vibration shows that it corresponds to the coming together of the two terminal carbons and the separation of the two central carbons. This is the expected vibration seen during the Cope rearrangement.
The QTS2 method is advantageous because it is fully automated however it required a good starting guess for the transition state and therefore a lot of permutations of structure, in the input file, were needed for a successful calculation.

{| class="wikitable" style="margin: auto;"
|+ Summary Table of the Energies of the Two Transition States
! rowspan="2" | Transition State
! colspan="3" | Energy
|-
|<jmolFile text="Berny TS">BE_ChairB.mol</jmolFile>
|<jmolFile text="Redunant Coordinates">BE_chairD.mol</jmolFile>
|<jmolFile text="QTS2 (TS)">BE_Boat_TS_Again2.mol</jmolFile>
|-
|chair
| -231.61932
| -231.61932
|
|-
|Boat
|
|
| -231.60280
|}

===='''The Reaction Pathway'''====

The reaction pathway can be studied through both the minimum energy position on its potential energy surface and the activation energies of the two transition states.

====='''IRC'''=====

[[File:BE_IRC_Plot.jpg|thumb|400px|IRC Energy Plot of the Chair Transition State]]
An intrinsic reaction coordinate (IRC) was run as it can be used to determine the minimum energy pathway from the transition state to its local minimum by using many points along the pathway to the point where the energy plateaus. The IRC in this case has been run in the forward direction only and the force constants calculated at every step.

The lowest energy conformation is achieved after 44 steps. The structure of the first and final conformations can be compared by considering their different geometry and energy. The IRC determines the minimum energy conformation and therefore the energy of step 44 (-231.69158Hartrees) is, as expected, lower than that of the first step (-231.61932Hartrees), this confirms that the calculation has found a minimum. The most noticeable different between the two conformations is the bond symmetry, in step 1 the bond lengths within each molecule were identical (1.39A) and the distance between the two molecules is the same (2.02A) for both sets of terminal carbons. However in step 44 the bond lengths within each molecule are different in preparation for the formation of two different types of bonds - single and double - this means one bond is shorter than the other (1.32 and 1.51A). As well as this difference in intranuclear distances there is also a difference in the internuclear distances as the one end approaches much closer (1.55 and 4.23A) almost within the van der Waals radius of the carbon atom as it prepares to form a new bond whereas the other distance is much larger as there is no bond formation at those carbons.

{| class="wikitable" style="margin: auto;"
|+ Structures of Steps 1 and 44 in the Determination of the IRC of the Chair Transition State
! [[File:BE_F1_Image.jpg|400px]]
! [[File:BE_F44_Image.jpg|400px]]
|}

====='''Activation Energies'''=====

In order to determine the activation energy of the reaction via the two different transition states a higher level of theory needed to be used. The B3LYP/6-31G(d) was used to optimize the two transition states to a minimum. Comparing this to the activation energies obtained at the lower level of theory and literature values<ref name="Instructional Wiki"/> shows the large difference in the energies obtained at each level of theory. The Energies at 0 and 298.15K were then found by running a frequency calculation, using the same basis set, on the optimised structure and checking the sum of the electronic and zero-point (0K) and sum of electronic and thermal energies (298.15K).

{| class="wikitable" style="margin: auto;"
|+ Geometrical Features of the Reactant, Boat and Chair Transition States
!
! colspan="2" | Reactants
! colspan="2" | Chair Transition State
! colspan="2" | Boat Transition State
|-
|
|<jmol><jmolApplet><title>Reactants Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Reactants Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Transition State Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Transition State Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Transition State Optimized at HF/3-21G Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again2.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Transition State Optimized at B3LYP/6-31G(d) Theory</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again3_-_EA.mol</uploadedFileContents> </jmolApplet></jmol>
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
!Intramolecular Bond Lengths/A
|1.32, 1.51
|1.33, 1.50
|1.39
|1.41
|1.38
|1.39
|-
!Intermolecular Bond Lengths/A
|1.55
|1.55
|2.02
|1.97
|2.14
|2.21
|-
!Dihedral Angle/o
|179.996
|180
|54.96
|54
|0
|0
|}

In the reactants the higher level of theory shows a tightening of the dihedral angle to exactly 180o as well as a shortening of the single bonds and lengthening of the double bonds within the molecule. This acts to bring the bond lengths within the molecule closer to the expected bond lengths of 1.47 and 1.54A<ref name="Wiki Book">M. A. Fox, J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen.'', '''1995''', Springer.</ref>. Within chair transition state the same effect can be seen, increasing the dihedral angle and double bond length and decreasing the single and intermolecular bond distances.

{| class="wikitable" style="margin: auto;"
|+ Electronic Energies and Energies of the Boat and Chair Transition States at 0K
!
! colspan="2" | '''Electronic Energy/Hartrees'''
! colspan="2" | '''Sum of Electronic and Zero-Point Energies'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.60280
| -234.54309
| -231.45093
| -234.40234
| 0.08861
| 0.06687
| 55.60
| 41.96
| 44.7 ± 2.0
|-
|'''Chair Transition State'''
| -231.69167
| -234.55698
| -231.46669
| -234.41488
| 0.07286
| 0.05433
| 45.72
| 34.09
| 33.5 ± 0.5
|}

The activation energies were calculated with reference to the conformer E energies as the reactant ( At 0K HF/3-21G = -231.53954 and B2LYP/6-31G(d) = -234.46921Hartrees, At 298.15K HF/3-21G = -231.53257 and B2LYP/6-31G(d) = -234.46186Hartrees ).

{| class="wikitable" style="margin: auto;"
|+ Energies of the Boat and Chair Transition States at 298.15K
!
! colspan="2" | '''Sum of Electronic and Thermal Energies'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.44530
| -234.39599
| 0.08727
| 0.06587
| 54.76
| 41.33
| 41.32
|-
|'''Chair Transition State'''
| -231.46133
| -234.40895
| 0.07124
| 0.05291
| 44.70
| 33.20
| 33.17
|}

From this it is clear why a higher level of theory is needed. The energies calculated at the B3LYP/6-31G(d) level yield much more accurate activation energies that those at HF/3-21G. In both cases this increase in accuracy has brought the activation energies significantly closer to the expected values.

The activation energy of the Chair transition state is a good match for that in the literature and is very close to the boundaries of error given. When comparing this to the energy obtained at the lower level of theory there is a much greater difference and therefore illustrates the ability of the higher level of energy to calculate a much more accurate transition state.

=='''Diels-Alder Cycloaddition Reaction'''==

The Diels-Alder cycloaddition reaction is a pericyclic reaction that occurs between two pi systems, a diene and a dienophile, in order to form a cyclohexene. It offers good regio- and stereo-selective control that can be extended to other conjugated systems<ref name="The Diels-Alder Reaction in Total Synthesis">K. C. Nicolaou, S. A. Snyder, T. Montagnon and G. Vassilikogiannakis ''Angewandte Chemie International Edition'','''2002''', ''41'', 1668-1698.</ref>.

In this experiment the reaction between cis butadiene and ethene will be studied in order to determine the symmetry of both the reactant and product molecular orbitals, as well as the geometry of the reaction's transition state. The regioselectivity of the reaction will also be looked at into order to determine the preferred product of the reaction, endo or exo.

==='''Cis-butadiene'''===

Cis-butadiene will be used as the diene in this studied reaction with ethene. Its geometry was first optimized using the HF/3-21G level of theory. The final energy of the product was -154.05514Hartrees. The molecular orbitals of the optimized structure can be modeled and used further to look at the electronic contribution of the diene to the final molecule.

{| class="wikitable" style="margin: auto;"
|+ Images of the Molecular Orbitals of Cis-Butadiene
! [[File:BE_I_HOMO.jpg|200px]]
! rowspan="3" | [[File:BE_Diene.jpg|150px]]
! [[File:BE_I_LUMO.jpg|200px]]
|-
![[File:BE_I_HOMO2.jpg|200px]]
![[File:BE_I_LUMO2.jpg|200px]]
|-
| align="center" | The HOMO orbital of the molecule is asymmetric. The nodes in the molecule fall between the red and green lobes of the HOMO. This shows the two separate pi systems within the molecule, but the proximity of the two systems suggests that the molecule could be fully conjugated.
| align="center" | The LUMO orbital of the molecule is symmetric. It shows the pi* orbitals of the two pi systems and the conjugation of these across the central single bond. The nodes fall within the plane of the molecule and between the double bonds of the molecule suggesting that for this orbital there is no conjugation over the double bonds.
|}

==='''The Transition State'''===

A number of methods, that had previously been discussed above for the Cope rearrangement reaction, were considered to optimize the transition state of this reaction. However it was decided to use a method previously used in optimizing the chair transition state of 1,5-hexadiene due to it's simplicity and that a good guess for the transition state could be made initially.

Firstly the reactants were optimized separately under the HF/3-21G level of theory. Then the two molecules were combined in an envelope like geometry and the internuclear distances between the two molecules set to 2.2A in order to give a best guess for the transition state. An optimization and frequency calculation was then run, optimizing to a transition state using the Berny method, calculating the force constants only once.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Structure and Molecular Orbitals of the Diels-Alder Transition State
![[File:BE_TSHOMO.jpg|center|200px]][[File:BE_TSHOMO2.jpg|center|200px]]
!<jmol><jmolApplet><title>1,5-hexdiene formation transition state</title><color>white</color> <size>300</size><uploadedFileContents>BE_Reactants.mol</uploadedFileContents> </jmolApplet></jmol>
|[[File:BE_TSLUMO.jpg|center|200px]][[File:BE_TSLUMO2.jpg|center|200px]]
|-
! align="center" | HOMO
| align="center" | The transition state for the Diels-Alder reaction between ethene and cis-butadiene has an energy of -231.60321Hartrees
! align="center" | LUMO
|}

[[File:BE_TSVibration.gif|thumb|left|Imaginary Vibration of the Diels-Alder Transition State]][[File:BE_Positive_Vibration.gif|thumb|right|Lowest Energy Positive Vibration of the Diels-Alder Transition State]]

This gave a good approximation for the transition state with the imaginary vibration, of -817.71cm-1, showing the two molecules approaching each other in a synchronous motion. Comparatively the lowest positive vibration at 166.54cm-1 is an asynchronous rocking motion that does not contribute to the coming together of the two molecules.

[[File:BE_TSDIELSALDERIR_Spectrum.png|thumb|center|300px|Predicted IR Spectrum of the Diels-Alder Transition State]]

The justification of calling this the transition state is also shown in the final bond lengths between the two molecules, the length of approach is 2.21A. Typically an sp3-sp3 carbon bond length is 1.54A and for a sp2-sp2 carbon it is 1.47A<ref name="Wiki Book"/>this suggests that a bond has not been fully formed in the transition state. However it is within the van der Waals radius of the carbon atom, which is 1.7A<ref name="Van der Waals Volumes and Radii">A. Bondi, ''J. Phys. Chem.'','''1964''', ''68 '', 441-51.</ref> so a partial bond may have started to form at this point in the reaction.

Looking at the HOMO and LUMO orbitals for the reactants there is a clear contribution from both to the molecular orbitals of the transition state, this is especially the case for the ethene molecule. The visualization of the HOMO orbital throws some doubt onto the final structure obtained however considering the imaginary vibration is as expected, it can be assumed that this is a form of the transition state, if not the most minimal energy one. Further calculations were run in order to find the transition state with the correct HOMO orbital however it could not be found with the correct geometry and imaginary vibration.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Molecular Orbitals of Ethene and Cis-Butadiene
! colspan="4" | Ethene
! colspan="4" | cis-butadiene
|-
|[[File:BE_Ethene_HOMO.jpg|center|100px]]
|[[File:BE_Ethene_HOMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO.jpg|center|100px]]
|[[File:BE_I_HOMO.jpg|center|100px]]
|[[File:BE_I_HOMO2.jpg|center|100px]]
|[[File:BE_I_LUMO.jpg|center|100px]]
|[[File:BE_I_LUMO2.jpg|center|100px]]
|-
! colspan="2" | HOMO - Symmetric
! colspan="2" | LUMO - Asymmetric
! colspan="2" | HOMO - Asymmetric
! colspan="2" | LUMO - Symmetric
|}

The reactants orbitals above display frontier molecular orbital theory. The HOMO of cis-butadiene and the LUMO of ethene are of correct symmetry to interact to form a bonding molecular orbital in the product. This means that the reaction is symmetry allowed and therefore no other conditions are needed, such as light, in order for the reaction to proceed<ref>[http://www.ch.ic.ac.uk/rzepa/] Henry S. Rzepa, Imperial College, Organic Pericyclic Reactions: Categories</ref>.

==='''Regioselectivity'''===

The reaction can form two different products depending on the transition state. The exo product is formed when the approach of the maleic anhydride occurs with the oxygens pointing away from the double bond, and the endo product is formed when the oxygens point towards it.
In order to determine the energies of the transition states first the reactants were optimized separately and then together using the HF/3-21G level of theory. Then the endo and exo products were optimized the same way. The transition state was found using the QTS2 method, an Opt+Freq calculation under the HF/3-21G level of theory, this gave an optimized version of each transition state. This was used in order to utilise the reactants and products to find the transition state as the initial guess for the transition state was not a confident enough one.

The imaginary vibrations of these transition at -647.54 and -643.32cm-1 for the exo and endo transition states respectively. These vibrations show the coming together of the reactant molecules to form the product as is expected. These Transition state structures were then optimized under a higher level of theory in order to obtain a more accurate energy, and the HOMO and LUMO orbitals of each visualized.

[[File:BE_Exo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Exo Transition State]][[File:BE_Endo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Endo Transition State]]
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Energies of the Exo and Endo Transition States under the B3LYP/6-31G(d) Level of Theory
!
! Exo Transition State
! Endo Transition State
|-
|
|<jmol><jmolApplet><title>EXO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_Exo_TS_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>ENDO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_ENDO_TS_3_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|-
!Total Energy/Hartrees
| -612.66122
| -612.68340
|-
!Sum of Electronic and Zero-Point Energies/Hartrees
| -612.47669
| -612.50214
|-
!Sum of Electronic and Thermal Energies/Hartrees
| -612.46612
| -612.49179
|-
!Activation Energy at 0K/kcalmol-1
| 34.90
| 18.93
|-
!Activation Energy at 298.15K/kcalmol-1
| 33.58
| 17.47
|-
!Dihedral Angle/o
| -169.56
| -175.67
|-
!Intermolecular Bond Lengths/A
|1.5
|2.27
|}

Activation energies were calculated by comparing the energy of the transition state to the energy of the reactants (At 0K B3LYP/6-31G(d) = -612.53231, At 298.15K B3LYP/6-31G(d) = -612.51964). The endo product should be the kinetic product, this is shown by its much lower activation energy. The exo product has a tighter dihedral angle than the endo, this is due to the repulsion between the two reactants as seen in the HOMO of the exo transition state below<ref name="Steric Effects vs. Secondary Orbital Overlap in Diels-Alder Reactions. MNDO and AM1 Studies">, M. A. Fox, R. Cardons and N. J. Kiwiet, ''J. Org. Chem.'', '''1987''', ''52'',1469-1474</ref>.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The HOMO and LUMO Molecular Orbitals of the Exo and Endo Transition States
! colspan="2" | Exo Transition State
! colspan="2" | Endo Transition State
|-
!HOMO
!LUMO
!HOMO
!LUMO
|-
|[[File:BE_HOMO_exo.jpg|center|200px]]
|[[File:BE_LUMO_Exo.jpg|center|200px]]
|[[File:BE_HOMO_Endo.jpg|center|200px]]
|[[File:BE_LUMO_endo.jpg|center|200px]]
|}

The endo and exo products will be stabilised by secondary orbital overlap (SOO). This is the mixing of the HOMO orbital of one reactant with the LUMO orbital of the other<ref name="Orbital symmetries and endo-exo relationships in concerted cycloaddition reactions">, R. Hoffmann, R. B. Woodward, ''J. Am. Chem. Soc.'', '''1965''', ''87'', 4388-4389</ref>. This can be seen in both transition state HOMO orbitals above. In the case of the endo transition state the orbitals from the central oxygen shows a node around it, the orbitals on that oxygen are of the same symmetry and hence will interact with the orbitals from the double bond that is formed in the product above it. For the exo transition state the position of the oxygen and double bond are much separated and therefore there is no SOO. This SOO effect in the endo transition state, and hence the greater stabilisation, is a reason that the energy of the endo transition state is lower than that of the endo.

==Conclusion==

These reactions have shown the usefulness of molecular modelling in the visualization and analysis of transition states. A wide range of methods for optimizing the transition states have been considered and tested for their accuracy and ease of use. Despite initial difficulties the QTS2 method proved to be the most effective in determining the transition state of both the boat conformer in the Cope rearrangement and the endo and exo forms in the Diels-Alder reaction. It gave a good prediction for the geometries and energies of the transition states and allowed comparison between the reactants and products of the reactions. Further examples of Diels-Alder reactions could be looked at to determine the stereo and regio-selective implications of the mechanism and further prove the effectiveness of calculating activation energies by this method as compared to experimentally determined values. Also calculations from different basis sets could be run to determine the accuracy of the different levels of theory and to improve the calculated geometries and energies of the transition states studied in this experiment.

==References==

<references />

File:BE F44 Image.jpg

2014-03-21T16:28:39Z

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File:BE F1 Image.jpg

2014-03-21T16:28:21Z

Be411: uploaded a new version of "File:BE F1 Image.jpg"

2014-03-21T11:56:28Z

Be411: /* Regioselectivity */

'''Physical Module: Transition States and Reactivity'''

Computational methods can be used to study reaction pathways from reactants to products. In this experiment the Cope rearrangement reaction and Diels-Alder cycloaddition reaction are used as examples of this. Using various methods the lowest energy conformation of 1,5-hexadiene can be found as well as the conformation that exists as a reactant in the Cope rearrangement reaction. The two possible transition states of this reaction can be studied and their activation energies calculated to gain some understanding of the mechanism of this reaction. The Diels-Alder cycloaddition will also be studied, in terms of it's transition state and the regioselectivity of its products.

=='''The Cope Rearrangement Tutorial'''==

==='''Optimizing the Reactants and Products'''===

===='''Conformation of 1,5-hexadiene'''====
[[File:BE_1,5-HEXADIENE_CONFORTMATIONS.jpg|thumb|The Conformations of 1,5-Hexadiene]]

1,5-hexadiene is one of the simplest precursors in the Cope rearrangement. The three rotating carbon-carbon bonds within the center of the molecule cause there to be 10 energetically different conformations, when taking into account enantiomers and symmetry.

You would expect that where two carbons are in the eclipsed form (F-J) the conformer will be of higher energy than conformers with CH-eclipsed bonds (A-E)<ref name="Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols">B. W. Gung, Z. Zhu and R. A. Fouch, ''J. Am. Chem. Soc'', '''1995''', ''117'', 1783-1788.</ref>. This is demonstrated in the case of butane where the difference in energy between the CC eclipsed form and the CH eclipsed form is 1.5kcal/mol<ref name="Website - Butane Conformational Analysis">[http://research.cm.utexas.edu/nbauld/teach/butane.html]N. Bauld, '''"Butane Conformational Analysis"''', ''University of Texas'', Retreived 20 March 2014.</ref>. The energies of these forms can be optimized by computational methods in order to find the most suitable candidates for the reactants and products in the Cope Rearrangement reaction.

The table below shows the total energy for all energetically distinct conformers of the molecule, it shows that in the case of this molecule the anti conformer is generally of higher energy. Relating these structures to the literature<ref name="Instructional Wiki">[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3]M. Bearpark, Imperial College, Physical Module: Transition states and reactivity</ref>. The energies found are all in line with the energies expected, as is also the case with the point groups.

The lowest energy conformation is the Gauche Conformer D (-231.69266Hartrees) this maybe due to an attractive interaction between the why? -an attractive interaction may be present between the n orbital and the vinyl proton

The highest energy conformer of the CH eclipsed forms is C (-231.69153Hartrees) this maybe due to interactions between the two alkene protons which are 2.51A apart. This proximity will have a repulsive effect increasing the overall energy of the molecule.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimised energies for the anti and gauche conformations of 1,5-hexadiene (relating to the instructional wiki<ref name="Instructional Wiki"/>)
!
! align="center" | '''<jmolFile text="Conformer A - Gauche2">BE_A.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer B - Anti1 ">BE_B.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer C - Gauche4">BE_C.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer D - Gauche3">BE_D.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer E - Anti2">BE_E.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer F - Gauche6">BE_F.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer G - Anti4">BE_G.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer H - Gauche5">BE_H.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer I - Gauche1">BE_I.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer J - Anti3">BE_J.mol</jmolFile>'''
|-
|'''Total Energy/Hartrees'''
| -231.69167
| -231.69260
| -231.69153
| -231.69266
| -231.69254
| -231.68916
| -231.69097
| -231.68962
| -231.68772
| -231.68907
|-
|'''Point Group'''
|C2
|C2
|C2
|C1
|Ci
|C1
|C1
|C1
|C2
|C2h
|-
|'''Dihedral Angle (C2-C3-C4-C5)'''
|64.17
| -176.91
| -63.65
|67.70
| -179.996
|70.23
| -178.32
| -71.09
|75.76
| -179.98
|}

===='''Analysis of Conformer E'''====

Further Analysis of Conformer E was done at a higher basis set in order to get an even better approximation for its lowest energy conformation. An optimisation calculation was run as before at the B3LYP/6-31G(d)level of theory.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimised Conformations for the Anti2 Conformer using the HF/3-21G and B3LYP/6-31G(d) Basis Sets
!<jmol><jmolApplet><title>Anti 2 Conformation at HF level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_E.mol</uploadedFileContents> </jmolApplet></jmol>
!<jmol><jmolApplet><title>Anti 2 Conformation at B3LYP level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_DFT_optmisation.mol</uploadedFileContents> </jmolApplet></jmol>
|}

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Geometrical Features of the Anti 2 Conformer Optimized Under Different Basis Sets
! rowspan ="2" | '''Basis Set'''
! colspan ="3" | '''Bond Length'''
! colspan ="3" | '''Angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3
|C2-C3-C4
|C2-C3-C4-C5
|-
|HF/3-21G
|1.32
|1.51
|1.55
|179.996
|124.81
|111.35
|-
|B3LYP/6-31G(d)
|1.33
|1.50
|1.55
|180
|125.29
|112.68
|}

[[image:BE_IR_Spectrum_of_Anti2.jpg|right|200px|Thumb|IR Spectrum of the Anti2 Conformer]]

This shows that the geometries of the results of the two different approaches don't differ wildly, the bond lengths especially being very similar. However there is some deviation between the two in the angles.

Following on from this a frequency calculation was run using the same higher level of theory (B3LYP/6-31G(d)). This calculation can be used to confirm that this conformation is of a minimum energy by the absence of an imaginary vibrational frequency. The infrared spectrum of the molecule can be generated from the molecular vibrations and compared to literature.

The thermochemical energies for this molecule can be found in the ouput file of the calculation:

Sum of electronic and zero-point Energies= -234.46921
Sum of electronic and thermal Energies= -234.46186
Sum of electronic and thermal Enthalpies= -234.46091
Sum of electronic and thermal Free Energies= -234.50082

The various information above can be used to consider the potential and free energy of the system. The sum of the electronic and zero-point energies (E= Eelec + ZPE) show the potential energy at 0K and the sum of electronic and thermal energies (E = E + Evib + Erot + Etrans) considers it at 298.15K and 1atm. The sum of electronic and thermal enthalpies (H = E + RT) shows the enthalpy of the system containing a correction for thermal energy and the sum of electronic and thermal free energies ( G= H - TS) suggests the entropic contribution to the free energy of the system.

==='''Optimizing the "Chair" and "Boat" Transition Structures'''===

The Anti 2 molecule was considered for this reaction and used to model both the chair and boat transition states for the Cope rearrangement reaction. A number of different ways of optimizing the geometry of the transition states were considered including using redundant coordinates and QTS2(TS)optimization and frequency calculations.

===='''Chair Transition State'''====

The chair transition state was constructed from two allyl fragments. Firstly the geometry of a single allyl fragment was optimized using the HF/3-21G level of theory giving an energy of -117.61330Hartrees. Two allyl fragments are placed 2.2A apart and then using this as a guess for the transition state, the molecule is then optimized in two different ways.

[[File:BE_B_vibration.gif|thumb|Cope Rearrangement Chair Transition State Vibration]]

Firstly it can be optimized using a optimization and frequency calculation to a TS - Berny Method. By calculating the force constant once an imaginary vibration can be found. The imaginary vibration was found at 817.95cm-1 (expected value of 818cm-1<ref name="Instructional Wiki"/>) on animation it can be seen that this corresponds to the coming together of the two terminal carbons on each allyl molecule, the movement expected during this reaction. In this case the total energy is -231.61932Hartrees.

[[File:BE_D_ChairTS.jpg|thumb|left|Chair Transition State by Redundant Coordinate Method]]
Secondly the transition state was optimized by freezing the reaction coordinate in order to minimize the rest of the molecule first and then unfreezing it in order to find the minimum for the transition state. This means that the force constant Hessian matrix doesn't need to be calculated and therefore is a faster method of computing the transition state.

By looking at the atom distances between the bond that will be formed the suitability of both of these methods to find the true transition state can be studied. All the measured geometric parameters were identical in both versions of the transition state, namely the intranuclear bond lengths was 1.39A, the internuclear distance between the two allyl molecules is 2.02A and the angle within the each molecule is 120.05o. This suggests that both methods for determining the transition state are equally effect from the same guess input file. It also confirms that the guess structure was close to the exact structure as the first method was at successful in calculating the force constants as the second which doesn't need to calculate them.

===='''The Boat Transition State'''====

The boat transition state was constructed in much the same way as the chair using two pre-optimized allyl fragments, placing them together in the correct geometry and setting them to 2.2A apart. In the case of the boat transition state the QST2 method was used to optimize the energy. By ensuring that the same labeling system is used in the reactants and products an optimization and frequency calculation was run that could help define the geometry of the boat transition state. This calculation was run with a predicted transition state with a dihedral angle (C2-C3-C4-C5) of 0o and inside C2-C3-C4 and C3-C4-C5 angles of 100o.

[[File:BE_E_Vibration.gif|thumb|200px|Cope Rearrangement Boat Transition State Vibration]]

The result of this calculation shows the boat transition state with a single imaginary vibration of -840nm and energy of -231.60Hartrees. Visualizing this vibration shows that it corresponds to the coming together of the two terminal carbons and the separation of the two central carbons. This is the expected vibration seen during the Cope rearrangement.
The QTS2 method is advantageous because it is fully automated however it required a good starting guess for the transition state and therefore a lot of permutations of structure were needed for a successful calculation.

{| class="wikitable" style="margin: auto;"
|+ Summary Table of the Energies of the Two Transition States
! rowspan="2" | Transition State
! colspan="3" | Energy
|-
|<jmolFile text="Berny TS">BE_ChairB.mol</jmolFile>
|<jmolFile text="Redunant Coordinates">BE_chairD.mol</jmolFile>
|<jmolFile text="OTS2 (TS)">BE_Boat_TS_Again2.mol</jmolFile>
|-
|chair
| -231.61932
| -231.61932
|
|-
|Boat
|
|
| -231.60280
|}

===='''The Reaction Pathway'''====

The reaction pathway can be studied through both the minimum energy position on its potential energy surface and the activation energies of the two transition states.

====='''IRC'''=====

[[File:BE_IRC_Plot.jpg|thumb|400px|IRC Energy Plot ofr the Chair Transition State]]
An IRC was run as it can be used to determine the minimum energy pathway from the transition state to its local minimum by using many point along the pathway to the point where the energy plateaus.The IRC in this case has been run in the forward direction only and the force constants calculated at every step.

The lowest energy conformation is achieved after 44 steps. The structure of the first and final conformations can be compared by considering their different geometry and energy. The IRC determines the minimum energy conformation and therefore the energy of step 44 (-231.69158Hartrees) is, as expected, lower than that of the first step (-231.61932Hartrees), this confirms that the calculation has found a minimum. The most noticeable different between the two conformations is the symmetry, in step 1 the bond lengths within each molecule were identical (1.39A) and the distance between the two molecules is the same (2.02A) for both sets of terminal carbons. However in step 44 the bond lengths within each molecule are different in preparation for the formation of two different types of bonds - single and double - this means one bond is shorter than the other (1.32 and 1.51A). As well as this difference in intranuclear distances there is also a difference in the internuclear distances as the one end approaches much closer (1.55 and 4.23A) almost within the van der Waals radius of the carbon atom as it prepares to form a new bond whereas the other distance is much larger as there is no bond formation at those carbons.

{| class="wikitable" style="margin: auto;"
|+ Images of Steps 1 and 44 in the Determination of the IRC of the Chair Transition State
! [[File:BE_F1_Image.jpg|400px]]
! [[File:BE_F44_Image.jpg|400px]]
|}

====='''Activation Energies'''=====

In order to determine the activation energy of the reaction via the two different transition states a higher level of theory needed to be used. The B3LYP/6-31G* was used to optimize the two transition states to a minimum. Comparing this to the activation energies obtained at the lower level of theory and literature values<ref name="Instructional Wiki"/> shows the large difference in the energies obtained at each level of theory.

{| class="wikitable" style="margin: auto;"
!
| colspan="2" | Reactants
| colspan="2" | Chair Transition State
| colspan="2" | Boat Transition State
|-
|
|<jmol><jmolApplet><title>Reactants Optimized at HF</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Reactants Optimized at B2LYP</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Optimized at HF</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Optimized at B2LYP</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Optimized at HF</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again2.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Optimized at B3LYP</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again3_-_EA.mol</uploadedFileContents> </jmolApplet></jmol>
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
!Intramolecular Bond Lengths/A
|1.32, 1.51
|1.33, 1.50
|1.39
|1.41
|1.38
|1.39
|-
!Intermolecular Bond Lengths/A
|1.55
|1.55
|2.02
|1.97
|2.14
|2.21
|-
!Dihedral Angle/o
| -179.996
|180
|54.96
|54
|0
|0
|}

In the reactants the higher level of theory shows a tightening of the dihedral angle to exactly 180o as well as a shortening of the single bonds and lengthening of the double bonds within the molecule. This acts to bring the bond lengths within the molecule closer to the expected bond lengths '''quote values mentioned later'''. Within chair transition state the same effect can be seen, increasing the dihedral angle and double bond length and decreasing the single and intermolecular bond distances.

*compare the geometries of these and lower theory TSs

The activation energies with calculated with reference to the anti2 conformer energies as the reactant (HF/3-21G = -231.69253529 and B2LYP/6-31G(d) = -234.61171Hartrees).

{| class="wikitable" style="margin: auto;"
!
! colspan="2" | '''Energy/Hartrees'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.60280
| -234.54309
| 0.0897
| 0.0686
| 56.31
| 43.06
| 44.7 ± 2.0
|-
|'''Chair Transition State'''
| -231.69167
| -234.55698
| 0.0732
| 0.0547
| 45.94
| 34.34
| 33.5 ± 0.5
|}

*compare the energy differences between chair and boat at different theories
*compare the activation energies (energies at 0K- Sum of electronic and zero-point energies) to the literature (chair - 33.5 ± 0.5 kcal/mol, boat - 44.7 ± 2.0 kcal/mol)
*Consider the effect at 298.15K (Sum of electronic and thermal energies)

The activation energy of the Chair transition state is a good match for that in the literature and is very close to the boundaries of error given. When comparing this to the energy obtained at the lower level of theory there is a much greater difference and therefore illustrates the ability of the higher level of energy to calculate a much more accurate transition state.

=='''Diels-Alder Cycloaddition Reaction'''==

The Diels-Alder cycloaddition reaction is a pericyclic reaction that occurs between two pi systems, a diene and a dienophile, in order to form a cyclohexene. It offers good regio- and stereo-selective control that can be extended to other conjugated systems<ref name="The Diels-Alder Reaction in Total Synthesis">K. C. Nicolaou, S. A. Snyder, T. Montagnon and G. Vassilikogiannakis ''Angewandte Chemie International Edition'','''2002''', ''41'', 1668-1698.</ref>.

In this experiment the reaction between cis butadiene and ethene will be studied in order to determine the symmetry of both the reactant and product molecular orbitals, as well as the geometry of the reaction's transition state. The regioselectivity of the reaction will also be looked at into order to determine the preferred product of the reaction, endo or exo.

==='''Cis-butadiene'''===

Cis-butadiene will be used as the diene in this studied reaction. Its geometry was first optimized using the HF/3-21G level of theory. The final energy of the product was -154.05514Hartrees. The molecular orbitals of the optimized structure can be modeled and used further to look at the electronic contribution of the diene to the final molecule.

{| class="wikitable" style="margin: auto;"
|+ Images of the Molecule orbitals of cis-butadiene
! [[File:BE_I_HOMO.jpg|200px]]
! rowspan="3" | [[File:BE_Diene.jpg|150px]]
! [[File:BE_I_LUMO.jpg|200px]]
|-
![[File:BE_I_HOMO2.jpg|200px]]
![[File:BE_I_LUMO2.jpg|200px]]
|-
| align="center" | The HOMO orbital of the molecule is asymmetric. The nodes in the molecule fall between the red and green lobes of the HOMO. This shows the two seperate pi systems within the molecule, but the proximity of the two systems suggests that the molecule could be fully conjugated.
| align="center" | The LUMO orbital of the molecule is symmetric. It shows the pi* orbitals of the two pi systems and the conjugation of these across the central single bond. The nodes fall within the plane of the molecule and between the double bonds of the molecule suggesting that for this orbital there is no conjugation over the double bonds.
|}

==='''The Transition State'''===

A number of methods, that had previously been discussed above for the Cope rearrangement reaction, were considered to optimize the transition state of this reaction. However it was decided to use a method previously used in optimizing the chair transition state of 1,5-hexadiene because...

Firstly the reactants were optimized separately under the HF/3-21G level of theory. Then the two molecules were combined in an envelope like geometry and the internuclear distances between the two molecules set to 2.2A in order to give a best guess for the transition state. An optimization and frequency calculation was then run, optimizing to a transition state using the Berny method, calculating the force constants only once.

{| class="wikitable" style="margin: 1em auto 1em auto;"
![[File:BE_TSHOMO.jpg|center|200px]][[File:BE_TSHOMO2.jpg|center|200px]]
!<jmol><jmolApplet><title>1,5-hexdiene formation transition state</title><color>white</color> <size>300</size><uploadedFileContents>BE_Reactants.mol</uploadedFileContents> </jmolApplet></jmol>
|[[File:BE_TSLUMO.jpg|center|200px]][[File:BE_TSLUMO2.jpg|center|200px]]
|-
| align="center" | HOMO
| align="center" | The transition state for the Diels-Alder reaction between ethene and cis-butadiene has an energy of -231.60321Hartrees
| align="center" | LUMO
|}

[[File:BE_TSVibration.gif|thumb|left|Imaginary Vibration of the Diels-Alder Transition State]][[File:BE_Positive_Vibration.gif|thumb|right|Lowest Energy Positive Vibration of the Diels-Alder Transition State]]

This gave a good approximation for the transition state with the imaginary vibration, of -817.71nm, showing the two molecules approaching each other in a synchronous motion. Comparatively the lowest positive vibration at 166.54nm is an asynchronous rocking motion that does not contribute to the coming together of the two molecules.

[[File:BE_TSDIELSALDERIR_Spectrum.png|thumb|center|300px|Predicted IR Spectrum of the Diels-Alder Transition State]]

The justification of calling this the transition state is also shown in the final bond lengths between the two molecules, the length of approach is 2.21A. Typically an sp3-sp3 carbon bond length is 1.54A and for a sp2-sp2 carbon it is 1.47A<ref name="Wiki Book">M. A. Fox, J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen.'', '''1995''', Springer.</ref> this suggests that a bond has not been fully formed in the transition state. However it is within the van der Waals radius of the carbon atom, which is 1.7A<ref name="Van der Waals Volumes and Radii">A. Bondi, ''J. Phys. Chem.'','''1964''', ''68 (3)'', 441-51.</ref> so a partial bond may have started to form at this point in the reaction.

The HOMO and LUMO orbitals of the transition state are both symmetric. This is not what you would expect, you would expect the HOMO orbital to be asymmetric, due to the contributions from the two reactants' molecular orbitals. Looking at the HOMO and LUMO orbitals for the reactants there is a clear contribution from both to the molecular orbitals of the transition state, this is especially the case for the ethene molecule. The visualization of the HOMO orbital throws some doubt onto the final structure obtained however considering the imagianry vibration is as expected, it can be assumed that this is a form of the transition state, if not the most minimal energy one. Further calculations were run in order to find the transition state with the correct HOMO orbital however it could not be found with the correct geometry and imaginary vibration.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Molecular Orbitals of ethene and cis-butadiene
! colspan="4" | Ethene
! colspan="4" | cis-butadiene
|-
|[[File:BE_Ethene_HOMO.jpg|center|100px]]
|[[File:BE_Ethene_HOMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO.jpg|center|100px]]
|[[File:BE_I_HOMO.jpg|center|100px]]
|[[File:BE_I_HOMO2.jpg|center|100px]]
|[[File:BE_I_LUMO.jpg|center|100px]]
|[[File:BE_I_LUMO2.jpg|center|100px]]
|-
|colspan="2" | HOMO - Symmetric
|colspan="2" | LUMO - Asymmetric
|colspan="2" | HOMO - Asymmetric
|colspan="2" | LUMO - Symmetric
|}

Explain why the reaction is allowed.

==='''Regioselectivity'''===

iii)For the cyclohexa-1,3-diene reaction with maleic anhydride: Give the relative energies of the exo and endo transition structures. Comment on the structural difference between the endo and exo form. Why do you think that the exo form could be more strained?

The reaction can form two different products depending on the transition state. The exo product is formed when .... and the endo product is formed when....
In order to determine the energies of the transition states first the reactants were optimized separately and then together using the HF/3-21G level of theory. Then the endo and exo products were optimized the same way. The transition state was found using the QTS2 method, an Opt+Freq calculation under the HF/3-21G level of theory, this gave an optimized version of each transition state. The imaginary vibrations of these transition at -647.54 and -643.32nm for the exo and endo transition states respectively. These vibrations show the coming together of the reactant molecule to form the product as is expected. These Transition state structures were then optimized under a higher level of theory in order to obtain a more accurate energy, and the HOMO and LUMO orbitals of each visualized.

[[File:BE_Exo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Exo Transition State]][[File:BE_Endo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Endo Transition State]]

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Energies of the Exo and Endo Transition States
!
! Exo Transition State
! Endo Transition State
|-
|
|<jmol><jmolApplet><title>EXO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_Exo_TS_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>ENDO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_ENDO_TS_3_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|-
!Total Energy/Hartrees
| -612.66122
| -612.68340
|-
!Activation Energy/kcalmol-1
| 31.67265
| 17.75488
|-
!Dihedral Angle/o
| -169.56
| 54.77
|-
!Intermolecular Bond Lengths/A
|1.5
|2.27
|}

Activation energies were calculated by comparing the energy of the transition state to the energy of the reactants (for the HF/3-21G = -605.65223 and B3LYP/6-31G(d) = -612.71169. The endo product should be the kinetic product, this is shown by its much lower activation energy. The exo product appears to be more strained

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The HOMO and LUMO molecular orbitals of the exo and endo transition states
! colspan="2" | Exo Transition State
! colspan="2" | Endo Transition State
|-
|HOMO - Symmetric
|LUMO - Asymmetric
|HOMO - Symmetric
|LUMO - Asymmetric
|-
|[[File:BE_HOMO_exo.jpg|center|200px]]
|[[File:BE_LUMO_Exo.jpg|center|200px]]
|[[File:BE_HOMO_Endo.jpg|center|200px]]
|[[File:BE_LUMO_endo.jpg|center|200px]]
|}

The endo and exo products will be stabilised by secondary orbital overlap (COO). This is the mixing of the HOMO orbital of one reactant with the LUMO orbital of the other<ref name="Orbital symmetries and endo-exo relationships in concerted cycloaddition reactions">, R. Hoffmann, R. B. Woodward, ''J. Am. Chem. Soc.'', '''1965''', 87, 4388-4389</ref> .'''. This can be seen in both transition state HOMO orbitals above. In the case of the endo transition state the orbitals from the anomeric oxygen shows a node around it, the orbitals on that oxygen are of the same symmetry and hence will interact with the orbitals from the double bond that is formed in the product above it. For the exo transition state ...
This greater COO effect in the endo transition state, and hence the greater stabilisation, maybe a reason that the energy of the endo transition state is lower than that of the endo.

==References==

<references />

Rep:Mod:PhysBE411

2014-03-21T11:55:34Z

Rep:Mod:PhysBE411

2014-03-21T11:07:26Z

Be411: /* Diels-Alder Cycloaddition Reaction */

'''Physical Module: Transition States and Reactivity'''

Computational methods can be used to study reaction pathways from reactants to products. In this experiment the Cope rearrangement reaction and Diels-Alder cycloaddition reaction are used as examples of this. Using various methods the lowest energy conformation of 1,5-hexadiene can be found as well as the conformation that exists as a reactant in the Cope rearrangement reaction. The two possible transition states of this reaction can be studied and their activation energies calculated to gain some understanding of the mechanism of this reaction. The Diels-Alder cycloaddition will also be studied, in terms of it's transition state and the regioselectivity of its products.

=='''The Cope Rearrangement Tutorial'''==

==='''Optimizing the Reactants and Products'''===

===='''Conformation of 1,5-hexadiene'''====
[[File:BE_1,5-HEXADIENE_CONFORTMATIONS.jpg|thumb|The Conformations of 1,5-Hexadiene]]

1,5-hexadiene is one of the simplest precursors in the Cope rearrangement. The three rotating carbon-carbon bonds within the center of the molecule cause there to be 10 energetically different conformations, when taking into account enantiomers and symmetry.

You would expect that where two carbons are in the eclipsed form (F-J) the conformer will be of higher energy than conformers with CH-eclipsed bonds (A-E)<ref name="Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols">B. W. Gung, Z. Zhu and R. A. Fouch, ''J. Am. Chem. Soc'', '''1995''', ''117'', 1783-1788.</ref>. This is demonstrated in the case of butane where the difference in energy between the CC eclipsed form and the CH eclipsed form is 1.5kcal/mol<ref name="Website - Butane Conformational Analysis">[http://research.cm.utexas.edu/nbauld/teach/butane.html]N. Bauld, '''"Butane Conformational Analysis"''', ''University of Texas'', Retreived 20 March 2014.</ref>. The energies of these forms can be optimized by computational methods in order to find the most suitable candidates for the reactants and products in the Cope Rearrangement reaction.

The table below shows the total energy for all energetically distinct conformers of the molecule, it shows that in the case of this molecule the anti conformer is generally of higher energy. Relating these structures to the literature<ref name="Instructional Wiki">[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3]M. Bearpark, Imperial College, Physical Module: Transition states and reactivity</ref>. The energies found are all in line with the energies expected, as is also the case with the point groups.

The lowest energy conformation is the Gauche Conformer D (-231.69266Hartrees) this maybe due to an attractive interaction between the why? -an attractive interaction may be present between the n orbital and the vinyl proton

The highest energy conformer of the CH eclipsed forms is C (-231.69153Hartrees) this maybe due to interactions between the two alkene protons which are 2.51A apart. This proximity will have a repulsive effect increasing the overall energy of the molecule.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimised energies for the anti and gauche conformations of 1,5-hexadiene (relating to the instructional wiki<ref name="Instructional Wiki"/>)
!
! align="center" | '''<jmolFile text="Conformer A - Gauche2">BE_A.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer B - Anti1 ">BE_B.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer C - Gauche4">BE_C.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer D - Gauche3">BE_D.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer E - Anti2">BE_E.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer F - Gauche6">BE_F.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer G - Anti4">BE_G.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer H - Gauche5">BE_H.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer I - Gauche1">BE_I.mol</jmolFile>'''
! align="center" | '''<jmolFile text="Conformer J - Anti3">BE_J.mol</jmolFile>'''
|-
|'''Total Energy/Hartrees'''
| -231.69167
| -231.69260
| -231.69153
| -231.69266
| -231.69254
| -231.68916
| -231.69097
| -231.68962
| -231.68772
| -231.68907
|-
|'''Point Group'''
|C2
|C2
|C2
|C1
|Ci
|C1
|C1
|C1
|C2
|C2h
|-
|'''Dihedral Angle (C2-C3-C4-C5)'''
|64.17
| -176.91
| -63.65
|67.70
| -179.996
|70.23
| -178.32
| -71.09
|75.76
| -179.98
|}

===='''Analysis of Conformer E'''====

Further Analysis of Conformer E was done at a higher basis set in order to get an even better approximation for its lowest energy conformation. An optimisation calculation was run as before at the B3LYP/6-31G(d)level of theory.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Optimised Conformations for the Anti2 Conformer using the HF/3-21G and B3LYP/6-31G(d) Basis Sets
!<jmol><jmolApplet><title>Anti 2 Conformation at HF level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_E.mol</uploadedFileContents> </jmolApplet></jmol>
!<jmol><jmolApplet><title>Anti 2 Conformation at B3LYP level of Theory</title><color>white</color> <size>200</size><uploadedFileContents>BE_DFT_optmisation.mol</uploadedFileContents> </jmolApplet></jmol>
|}

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Geometrical Features of the Anti 2 Conformer Optimized Under Different Basis Sets
! rowspan ="2" | '''Basis Set'''
! colspan ="3" | '''Bond Length'''
! colspan ="3" | '''Angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3
|C2-C3-C4
|C2-C3-C4-C5
|-
|HF/3-21G
|1.32
|1.51
|1.55
|179.996
|124.81
|111.35
|-
|B3LYP/6-31G(d)
|1.33
|1.50
|1.55
|180
|125.29
|112.68
|}

[[image:BE_IR_Spectrum_of_Anti2.jpg|right|200px|Thumb|IR Spectrum of the Anti2 Conformer]]

This shows that the geometries of the results of the two different approaches don't differ wildly, the bond lengths especially being very similar. However there is some deviation between the two in the angles.

Following on from this a frequency calculation was run using the same higher level of theory (B3LYP/6-31G(d)). This calculation can be used to confirm that this conformation is of a minimum energy by the absence of an imaginary vibrational frequency. The infrared spectrum of the molecule can be generated from the molecular vibrations and compared to literature.

The thermochemical energies for this molecule can be found in the ouput file of the calculation:

Sum of electronic and zero-point Energies= -234.46921
Sum of electronic and thermal Energies= -234.46186
Sum of electronic and thermal Enthalpies= -234.46091
Sum of electronic and thermal Free Energies= -234.50082

The various information above can be used to consider the potential and free energy of the system. The sum of the electronic and zero-point energies (E= Eelec + ZPE) show the potential energy at 0K and the sum of electronic and thermal energies (E = E + Evib + Erot + Etrans) considers it at 298.15K and 1atm. The sum of electronic and thermal enthalpies (H = E + RT) shows the enthalpy of the system containing a correction for thermal energy and the sum of electronic and thermal free energies ( G= H - TS) suggests the entropic contribution to the free energy of the system.

==='''Optimizing the "Chair" and "Boat" Transition Structures'''===

The Anti 2 molecule was considered for this reaction and used to model both the chair and boat transition states for the Cope rearrangement reaction. A number of different ways of optimizing the geometry of the transition states were considered including using redundant coordinates and QTS2(TS)optimization and frequency calculations.

===='''Chair Transition State'''====

The chair transition state was constructed from two allyl fragments. Firstly the geometry of a single allyl fragment was optimized using the HF/3-21G level of theory giving an energy of -117.61330Hartrees. Two allyl fragments are placed 2.2A apart and then using this as a guess for the transition state, the molecule is then optimized in two different ways.

[[File:BE_B_vibration.gif|thumb|Cope Rearrangement Chair Transition State Vibration]]

Firstly it can be optimized using a optimization and frequency calculation to a TS - Berny Method. By calculating the force constant once an imaginary vibration can be found. The imaginary vibration was found at 817.95cm-1 (expected value of 818cm-1<ref name="Instructional Wiki"/>) on animation it can be seen that this corresponds to the coming together of the two terminal carbons on each allyl molecule, the movement expected during this reaction. In this case the total energy is -231.61932Hartrees.

[[File:BE_D_ChairTS.jpg|thumb|left|Chair Transition State by Redundant Coordinate Method]]
Secondly the transition state was optimized by freezing the reaction coordinate in order to minimize the rest of the molecule first and then unfreezing it in order to find the minimum for the transition state. This means that the force constant Hessian matrix doesn't need to be calculated and therefore is a faster method of computing the transition state.

By looking at the atom distances between the bond that will be formed the suitability of both of these methods to find the true transition state can be studied. All the measured geometric parameters were identical in both versions of the transition state, namely the intranuclear bond lengths was 1.39A, the internuclear distance between the two allyl molecules is 2.02A and the angle within the each molecule is 120.05o. This suggests that both methods for determining the transition state are equally effect from the same guess input file. It also confirms that the guess structure was close to the exact structure as the first method was at successful in calculating the force constants as the second which doesn't need to calculate them.

===='''The Boat Transition State'''====

The boat transition state was constructed in much the same way as the chair using two pre-optimized allyl fragments, placing them together in the correct geometry and setting them to 2.2A apart. In the case of the boat transition state the QST2 method was used to optimize the energy. By ensuring that the same labeling system is used in the reactants and products an optimization and frequency calculation was run that could help define the geometry of the boat transition state. This calculation was run with a predicted transition state with a dihedral angle (C2-C3-C4-C5) of 0o and inside C2-C3-C4 and C3-C4-C5 angles of 100o.

[[File:BE_E_Vibration.gif|thumb|200px|Cope Rearrangement Boat Transition State Vibration]]

The result of this calculation shows the boat transition state with a single imaginary vibration of -840nm and energy of -231.60Hartrees. Visualizing this vibration shows that it corresponds to the coming together of the two terminal carbons and the separation of the two central carbons. This is the expected vibration seen during the Cope rearrangement.
The QTS2 method is advantageous because it is fully automated however it required a good starting guess for the transition state and therefore a lot of permutations of structure were needed for a successful calculation.

{| class="wikitable" style="margin: auto;"
|+ Summary Table of the Energies of the Two Transition States
! rowspan="2" | Transition State
! colspan="3" | Energy
|-
|<jmolFile text="Berny TS">BE_ChairB.mol</jmolFile>
|<jmolFile text="Redunant Coordinates">BE_chairD.mol</jmolFile>
|<jmolFile text="OTS2 (TS)">BE_Boat_TS_Again2.mol</jmolFile>
|-
|chair
| -231.61932
| -231.61932
|
|-
|Boat
|
|
| -231.60280
|}

===='''The Reaction Pathway'''====

The reaction pathway can be studied through both the minimum energy position on its potential energy surface and the activation energies of the two transition states.

====='''IRC'''=====

[[File:BE_IRC_Plot.jpg|thumb|400px|IRC Energy Plot ofr the Chair Transition State]]
An IRC was run as it can be used to determine the minimum energy pathway from the transition state to its local minimum by using many point along the pathway to the point where the energy plateaus.The IRC in this case has been run in the forward direction only and the force constants calculated at every step.

The lowest energy conformation is achieved after 44 steps. The structure of the first and final conformations can be compared by considering their different geometry and energy. The IRC determines the minimum energy conformation and therefore the energy of step 44 (-231.69158Hartrees) is, as expected, lower than that of the first step (-231.61932Hartrees), this confirms that the calculation has found a minimum. The most noticeable different between the two conformations is the symmetry, in step 1 the bond lengths within each molecule were identical (1.39A) and the distance between the two molecules is the same (2.02A) for both sets of terminal carbons. However in step 44 the bond lengths within each molecule are different in preparation for the formation of two different types of bonds - single and double - this means one bond is shorter than the other (1.32 and 1.51A). As well as this difference in intranuclear distances there is also a difference in the internuclear distances as the one end approaches much closer (1.55 and 4.23A) almost within the van der Waals radius of the carbon atom as it prepares to form a new bond whereas the other distance is much larger as there is no bond formation at those carbons.

{| class="wikitable" style="margin: auto;"
|+ Images of Steps 1 and 44 in the Determination of the IRC of the Chair Transition State
! [[File:BE_F1_Image.jpg|400px]]
! [[File:BE_F44_Image.jpg|400px]]
|}

====='''Activation Energies'''=====

In order to determine the activation energy of the reaction via the two different transition states a higher level of theory needed to be used. The B3LYP/6-31G* was used to optimize the two transition states to a minimum. Comparing this to the activation energies obtained at the lower level of theory and literature values<ref name="Instructional Wiki"/> shows the large difference in the energies obtained at each level of theory.

{| class="wikitable" style="margin: auto;"
!
| colspan="2" | Reactants
| colspan="2" | Chair Transition State
| colspan="2" | Boat Transition State
|-
|
|<jmol><jmolApplet><title>Reactants Optimized at HF</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Reactants Optimized at B2LYP</title><color>white</color> <size>100</size><uploadedFileContents>BE_Product-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Optimized at HF</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair_HF.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Chair Optimized at B2LYP</title><color>white</color> <size>100</size><uploadedFileContents>BE_G_chair.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Optimized at HF</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again2.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>Boat Optimized at B3LYP</title><color>white</color> <size>100</size><uploadedFileContents>BE_Boat_TS_Again3_-_EA.mol</uploadedFileContents> </jmolApplet></jmol>
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
!Intramolecular Bond Lengths/A
|1.32, 1.51
|1.33, 1.50
|1.39
|1.41
|1.38
|1.39
|-
!Intermolecular Bond Lengths/A
|1.55
|1.55
|2.02
|1.97
|2.14
|2.21
|-
!Dihedral Angle/o
| -179.996
|180
|54.96
|54
|0
|0
|}

In the reactants the higher level of theory shows a tightening of the dihedral angle to exactly 180o as well as a shortening of the single bonds and lengthening of the double bonds within the molecule. This acts to bring the bond lengths within the molecule closer to the expected bond lengths '''quote values mentioned later'''. Within chair transition state the same effect can be seen, increasing the dihedral angle and double bond length and decreasing the single and intermolecular bond distances.

*compare the geometries of these and lower theory TSs

The activation energies with calculated with reference to the anti2 conformer energies as the reactant (HF/3-21G = -231.69253529 and B2LYP/6-31G(d) = -234.61171Hartrees).

{| class="wikitable" style="margin: auto;"
!
! colspan="2" | '''Energy/Hartrees'''
! colspan="2" | ''' Activation Energy/Hartrees'''
! colspan="2" | '''Activation Energy/kcalmol-1'''
! rowspan="2" | '''Expected Activation Energy/kcalmol-1'''
|-
! Basis Set
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
! HF/3-21G
! B3LYP/6-31G(d)
|-
|'''Boat Transition State'''
| -231.60280
| -234.54309
| 0.0897
| 0.0686
| 56.31
| 43.06
| 44.7 ± 2.0
|-
|'''Chair Transition State'''
| -231.69167
| -234.55698
| 0.0732
| 0.0547
| 45.94
| 34.34
| 33.5 ± 0.5
|}

*compare the energy differences between chair and boat at different theories
*compare the activation energies (energies at 0K- Sum of electronic and zero-point energies) to the literature (chair - 33.5 ± 0.5 kcal/mol, boat - 44.7 ± 2.0 kcal/mol)
*Consider the effect at 298.15K (Sum of electronic and thermal energies)

The activation energy of the Chair transition state is a good match for that in the literature and is very close to the boundaries of error given. When comparing this to the energy obtained at the lower level of theory there is a much greater difference and therefore illustrates the ability of the higher level of energy to calculate a much more accurate transition state.

=='''Diels-Alder Cycloaddition Reaction'''==

The Diels-Alder cycloaddition reaction is a pericyclic reaction that occurs between two pi systems, a diene and a dienophile, in order to form a cyclohexene. It offers good regio- and stereo-selective control that can be extended to other conjugated systems<ref name="The Diels-Alder Reaction in Total Synthesis">K. C. Nicolaou, S. A. Snyder, T. Montagnon and G. Vassilikogiannakis ''Angewandte Chemie International Edition'','''2002''', ''41'', 1668-1698.</ref>.

In this experiment the reaction between cis butadiene and ethene will be studied in order to determine the symmetry of both the reactant and product molecular orbitals, as well as the geometry of the reaction's transition state. The regioselectivity of the reaction will also be looked at into order to determine the preferred product of the reaction, endo or exo.

==='''Cis-butadiene'''===

Cis-butadiene will be used as the diene in this studied reaction. Its geometry was first optimized using the HF/3-21G level of theory. The final energy of the product was -154.05514Hartrees. The molecular orbitals of the optimized structure can be modeled and used further to look at the electronic contribution of the diene to the final molecule.

{| class="wikitable" style="margin: auto;"
|+ Images of the Molecule orbitals of cis-butadiene
! [[File:BE_I_HOMO.jpg|200px]][[File:BE_I_HOMO2.jpg|200px]]
! rowspan="2" | [[File:BE_Diene.jpg|150px]]
! [[File:BE_I_LUMO.jpg|200px]][[File:BE_I_LUMO2.jpg|200px]]
|-
|The HOMO orbital of the molecule is asymmetric. The nodes in the molecule fall between the red and green lobes of the HOMO. This shows the two seperate pi systems within the molecule, but the proximity of the two systems suggests that the molecule could be fully conjugated.
|The LUMO orbital of the molecule is symmetric. It shows the pi* orbitals of the two pi systems and the conjugation of these across the central single bond. The nodes fall within the plane of the molecule and between the double bonds of the molecule suggesting that for this orbital there is no conjugation over the double bonds.
|}

==='''The Transition State'''===

A number of methods, that had previously been discussed above for the Cope rearrangement reaction, were considered to optimize the transition state of this reaction. However it was decided to use a method previously used in optimizing the chair transition state of 1,5-hexadiene because...

Firstly the reactants were optimized separately under the HF/3-21G level of theory. Then the two molecules were combined in an envelope like geometry and the internuclear distances between the two molecules set to 2.2A in order to give a best guess for the transition state. An optimization and frequency calculation was then run, optimizing to a transition state using the Berny method, calculating the force constants only once.

{| class="wikitable" style="margin: 1em auto 1em auto;"
![[File:BE_TSHOMO.jpg|center|200px]][[File:BE_TSHOMO2.jpg|center|200px]]
!<jmol><jmolApplet><title>1,5-hexdiene formation transition state</title><color>white</color> <size>300</size><uploadedFileContents>BE_Reactants.mol</uploadedFileContents> </jmolApplet></jmol>
|[[File:BE_TSLUMO.jpg|center|200px]][[File:BE_TSLUMO2.jpg|center|200px]]
|-
| align="center" | HOMO
| align="center" | The transition state for the Diels-Alder reaction between ethene and cis-butadiene has an energy of -231.60321Hartrees
| align="center" | LUMO
|}

[[File:BE_TSVibration.gif|thumb|left|Imaginary Vibration of the Diels-Alder Transition State]][[File:BE_Positive_Vibration.gif|thumb|right|Lowest Energy Positive Vibration of the Diels-Alder Transition State]]

This gave a good approximation for the transition state with the imaginary vibration, of -817.71nm, showing the two molecules approaching each other in a synchronous motion. Comparatively the lowest positive vibration at 166.54nm is an asynchronous rocking motion that does not contribute to the coming together of the two molecules.

[[File:BE_TSDIELSALDERIR_Spectrum.png|thumb|center|300px|Predicted IR Spectrum of the Diels-Alder Transition State]]

The justification of calling this the transition state is also shown in the final bond lengths between the two molecules, the length of approach is 2.21A. Typically an sp3-sp3 carbon bond length is 1.54A and for a sp2-sp2 carbon it is 1.47A<ref name="Wiki Book">M. A. Fox, J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen.'', '''1995''', Springer.</ref> this suggests that a bond has not been fully formed in the transition state. However it is within the van der Waals radius of the carbon atom, which is 1.7A<ref name="Van der Waals Volumes and Radii">A. Bondi, ''J. Phys. Chem.'','''1964''', ''68 (3)'', 441-51.</ref> so a partial bond may have started to form at this point in the reaction.

The HOMO and LUMO orbitals of the transition state are both symmetric. This is not what you would expect, you would expect the HOMO orbital to be asymmetric, due to the contributions from the two reactants' molecular orbitals. Looking at the HOMO and LUMO orbitals for the reactants there is a clear contribution from both to the molecular orbitals of the transition state, this is especially the case for the ethene molecule. The visualization of the HOMO orbital throws some doubt onto the final structure obtained however considering the imagianry vibration is as expected, it can be assumed that this is a form of the transition state, if not the most minimal energy one. Further calculations were run in order to find the transition state with the correct HOMO orbital however it could not be found with the correct geometry and imaginary vibration.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Molecular Orbitals of ethene and cis-butadiene
! colspan="4" | Ethene
! colspan="4" | cis-butadiene
|-
|[[File:BE_Ethene_HOMO.jpg|center|100px]]
|[[File:BE_Ethene_HOMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO2.jpg|center|100px]]
|[[File:BE_Ethene_LUMO.jpg|center|100px]]
|[[File:BE_I_HOMO.jpg|center|100px]]
|[[File:BE_I_HOMO2.jpg|center|100px]]
|[[File:BE_I_LUMO.jpg|center|100px]]
|[[File:BE_I_LUMO2.jpg|center|100px]]
|-
|colspan="2" | HOMO - Symmetric
|colspan="2" | LUMO - Asymmetric
|colspan="2" | HOMO - Asymmetric
|colspan="2" | LUMO - Symmetric
|}

Explain why the reaction is allowed.

==='''Regioselectivity'''===

iii)For the cyclohexa-1,3-diene reaction with maleic anhydride: Give the relative energies of the exo and endo transition structures. Comment on the structural difference between the endo and exo form. Why do you think that the exo form could be more strained? Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').

The reaction can form two different products depending on the transition state. The exo product is formed when .... and the endo product is formed when....
In order to determine the energies of the transition states first the reactants were optimized separately and then together using the HF/3-21G level of theory. Then the endo and exo products were optimized the same way. The transition state was found using the QTS2 method, an Opt+Freq calculation under the HF/3-21G level of theory, this gave an optimized version of each transition state. The imaginary vibrations of these transition at -647.54 and -643.32nm for the exo and endo transition states respectively. These vibrations show the coming together of the reactant molecule to form the product as is expected. These Transition state structures were then optimized under a higher level of theory in order to obtain a more accurate energy, and the HOMO and LUMO orbitals of each visualized.

[[File:BE_Exo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Exo Transition State]][[File:BE_Endo_TS_Vibration.gif|thumb|left|Imaginary Vibration of the Endo Transition State]]

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The Energies of the Exo and Endo Transition States
!
! Exo Transition State
! Endo Transition State
|-
|
|<jmol><jmolApplet><title>EXO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_Exo_TS_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|<jmol><jmolApplet><title>ENDO Transition state at the B3LYP/6-31G(d) level of theory</title><color>white</color> <size>150</size><uploadedFileContents>BE_ENDO_TS_3_-_B3LYP.mol</uploadedFileContents> </jmolApplet></jmol>
|-
!Total Energy/Hartrees
| -612.66122
| -612.68340
|-
!Activation Energy/kcalmol-1
| 31.67265
| 17.75488
|-
!Dihedral Angle/o
| -169.56
| 54.77
|-
!Intermolecular Bond Lengths/A
|1.5
|2.27
|}

Activation energies were calculated by comparing the energy of the transition state to the energy of the reactants (for the HF/3-21G = -605.65223 and B3LYP/6-31G(d) = -612.71169. The endo product should be the kinetic product, this is shown by its much lower activation energy. The exo product appears to be more strained

==References==

<references />

File:BE Ethene LUMO2.jpg

2014-03-21T11:07:04Z

Be411: uploaded a new version of "File:BE Ethene LUMO2.jpg"

File:BE Ethene LUMO.jpg

2014-03-21T11:05:50Z

Be411: uploaded a new version of "File:BE Ethene LUMO.jpg"

File:BE Ethene HOMO2.jpg

2014-03-21T11:05:32Z

Be411: uploaded a new version of "File:BE Ethene HOMO2.jpg"